Copyright

by

Lu Zheng

2020

The Dissertation Committee for Lu Zhengcertifies that this is the approved version of the following dissertation:

Imaging Electromechanical Phenomena with Microwave

Impedance Microscopy

Committee:

Keji Lai, Supervisor

Gregory A. Fiete

Xiaoqin (Elaine) Li

Maxim Tsoi

Edward T. Yu

Imaging Electromechanical Phenomena with Microwave

Impedance Microscopy

by

Lu Zheng

DISSERTATION

Presented to the Faculty of the Graduate School of

The University of Texas at Austin

in Partial Fulfillment

of the Requirements

for the Degree of

DOCTOR OF PHILOSOPHY

THE UNIVERSITY OF TEXAS AT AUSTIN

May 2020

Dedicated to my beloved parents.

Acknowledgments

First and foremost, I would like to express my deep and sincere grati-

tude to my supervisor, Prof. Keji Lai. I am really fortunate to be his student.

He is one of the smartest people I know. He has guided me through every phase

of scientific research and inspired me to become an independent researcher. It

would be impossible to count all the ways that he has helped me over my six-

year graduate study. He was and remains my best role model for a scientist,

mentor, and teacher.

My sincere thanks must also go to the other members in my dissertation

committee: Prof. Gregory Fiete, Prof. Xiaoqin (Elaine) Li, Prof. Maxim

Tsoi, and Prof. Edward Yu. They generously offered their expertise, time,

and insightful comments to improve my work.

There is no way to express how much it meant to me to have been a

member of our lab. Many thanks to all my colleagues. Dr. Yuan Ren and

Dr. Di Wu are experienced physicists and offered a great amount of assis-

tance for my life and research. Dr. Xiaoyu Wu and Dr. Yen-Lin Huang are

fantastic mentors who taught me so many useful skills and guided me to be

a professional researcher. Thanks to Zhaodong Chu and Zhanzhi Jiang for

all the constructive discussions and suggestions. I also had the great pleasure

of working with Ashish Gangshettiwar, David Wannlund, Daehun Lee, Xue-

v

jian Ma, Zifan Xu, and Jia Yu. Thanks for their generous help and valuable

comments. Thanks should also go to all the visiting students and scholars

Dr. Qing He, Dr. Zhenqi Hao, Dr. Shinichi Nishihaya, Ryo Noguchi, Zhiran

Zhang, Weixiong Wu, and He Liu for bringing in their expertise.

I am extremely grateful to the collaborators for lending me their exper-

tise and intuition. They always inspired me over my six-year graduate research.

Prof. Ying-Hao Chu and Prof. Ramamoorthy Ramesh keep providing us with

high-quality ferroelectric samples. Prof. Sergey Artyukhin and his postdoc

Dr.Peng Chen and Prof. Long-Qing Chen and his student Dr. Xiaoxing Cheng

offered theoretical models to explain the giant microwave conductivity in ferro-

electric domain walls. Dr.Hui Dong and his advisor Prof. Zheng Wang helped

us simulate the energy transduction in LiNbO3. Prof.Marko Loncar and his

students Dr. Linbo Shao and Smarak Maity provide us with many acoustic

devices and helpful suggestions for the application of our T-MIM technique.

Prof A.T. Charlie Johnson and his postdoc Dr.Qicheng Zhang provide us with

surface acoustic wave phononic crystals and many insightful designs.

Special thanks must go to all my friends. Thank my high school class-

mate Yihan Sun and her genius husband Yan Gu (now Prof. Sun and Prof.

Gu at the University of California, Riverside) for organizing our annual travel

group, including Fan Gao, Weibing Pan, Wanjian Tang, Yanzhe Yin, Tao Yu,

and Kairui Zhang. We have explored many beautiful places, and the won-

derful memories will never fade. I would also like to thank my college room-

mates and friends in the states Xiaoqiang Li, Yize Jin, Tianzong Wang, and

vi

Yicheng Wang, for all the good times, dreams, and encouragement we have

shared. Many thanks to my close friends Yifan Chen, Wei Guo, Xingxiao Xu,

Mengzhen Zhang, and Yongxing Zhang for continually inspiring me to push

my limit and to explore more.

I dedicate this dissertation to my parents, Hong Zheng and Suhong Ji,

for their constant love. They selflessly encouraged me to chase my dream and

seek my own destiny. This journey would not have been possible without their

unlimited support. I owe them everything.

vii

Imaging Electromechanical Phenomena with Microwave

Impedance Microscopy

Abstract

Lu Zheng, Ph.D.

The University of Texas at Austin, 2020

Supervisor: Keji Lai

Electromechanics combines processes from electrical and mechanical

systems and focuses on their interactions, which lead to wide applications in

modern electronics. The observation of microscale and nanoscale electrome-

chanical phenomena is critical for understanding the underlying physics and in-

spiring scientific and technological innovations. Near-field scanning microscopy

is a promising tool that utilizes evanescent waves to detect local physical prop-

erties at a length scale much smaller than the far-field resolution limit. This

dissertation demonstrates the discoveries of novel electromechanical phenom-

ena revealed by microwave impedance microscopy (MIM) and shows the inven-

tion and application of new scanning microwave microscopy technique inspired

by the discoveries.

In this dissertation, I first introduce the development of near field scan-

ning probe microscopy. In Chapter 2, I begin by reviewing the basic compo-

viii

nents and the system design of microwave impedance microscopy (MIM), fol-

lowed by a description of data analysis and its main application - local conduc-

tivity mapping. I then elaborate its other applications in the following chap-

ters, categorized by the different properties probed by the technique. Chapter

3 demonstrates the discovery of a unique phonon mode existing in the ferro-

electric domain walls. Chapter 4 shows a novel electromechanical transduction

phenomenon in the ferroelectric domains of LiNbO3. In Chapter 5, I present

the invention of a new microwave microscopy technique, transmission-mode

MIM (T-MIM), which can be used to visualize the microwave field directly

and has achieved great success on mapping surface acoustic wave (SAW). I

conclude the thesis in Chapter 6 with a summary of the published discoveries

and an outlook of the ongoing projects and future plans in this area.

ix

Table of Contents

Acknowledgments v

Abstract viii

List of Tables xii

List of Figures xiii

Chapter 1. Introduction 1

1.1 Near Field Microwave Microscopy . . . . . . . . . . . . . . . . 1

1.2 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . 3

Chapter 2. Microwave Impedance Microscopy 5

2.1 Basic Principles . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 System Components . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Probe Design . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.2 Tip-Sample Interaction . . . . . . . . . . . . . . . . . . 10

2.2.3 Impedance Matching . . . . . . . . . . . . . . . . . . . . 13

2.2.4 Microwave Electronics . . . . . . . . . . . . . . . . . . . 15

2.3 Application: Conductivity Imaging . . . . . . . . . . . . . . . 17

2.4 Beyond Conductivity Imaging . . . . . . . . . . . . . . . . . . 19

Chapter 3. Microwave Absorption at Ferroelectric DomainWalls 20

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.1.1 Ferroelectrics . . . . . . . . . . . . . . . . . . . . . . . . 21

3.1.2 Domain and Domain Walls . . . . . . . . . . . . . . . . 21

3.1.3 DC conduction and AC conduction . . . . . . . . . . . . 22

3.2 Nanoscale Image of BiFeO3 . . . . . . . . . . . . . . . . . . . . 23

3.2.1 71 Domain Walls . . . . . . . . . . . . . . . . . . . . . 25

x

3.2.2 Control Experiments on Charged Domain Walls . . . . . 31

3.3 Theoretical Analysis . . . . . . . . . . . . . . . . . . . . . . . . 35

3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Chapter 4. Electromechanical Power Transduction in Ferroelec-tric Domains 41

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2 Image of LiNbO3 . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.2.1 Experiment Setup . . . . . . . . . . . . . . . . . . . . . 43

4.2.2 MIM Images around a Single LiNbO3 Domain Wall . . . 44

4.3 Theoretical Analysis . . . . . . . . . . . . . . . . . . . . . . . . 52

4.4 Further Experiments . . . . . . . . . . . . . . . . . . . . . . . 60

4.4.1 Double Domain Walls . . . . . . . . . . . . . . . . . . . 60

4.4.2 Two Dimensional Patterns . . . . . . . . . . . . . . . . 63

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Chapter 5. Imaging Acoustic Wave 67

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.1.1 Surface Acoustic Wave (SAW) . . . . . . . . . . . . . . 68

5.1.2 Visualization Tools . . . . . . . . . . . . . . . . . . . . . 70

5.2 Transmission-mode MIM . . . . . . . . . . . . . . . . . . . . . 72

5.3 Imaging SAW . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.3.1 Traveling Wave . . . . . . . . . . . . . . . . . . . . . . . 75

5.3.2 Standing Wave . . . . . . . . . . . . . . . . . . . . . . . 79

5.3.3 Modelling and Analysis . . . . . . . . . . . . . . . . . . 81

5.4 Ferroelectric Domain and SAW . . . . . . . . . . . . . . . . . . 83

5.5 More Applications . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.5.1 SAW Resonator . . . . . . . . . . . . . . . . . . . . . . 84

5.5.2 Other Devices . . . . . . . . . . . . . . . . . . . . . . . 88

5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Chapter 6. Summary and Outlook 90

Bibliography 93

xi

List of Tables

4.1 Velocities and properties of the acoustic waves in LiNbO3 . . . 49

xii

List of Figures

2.1 Schematic setup of MIM . . . . . . . . . . . . . . . . . . . . . 7

2.2 MIM Probes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Equivalent circuit of the tip-sample interaction . . . . . . . . . 11

2.4 Impedance-match sections at different frequencies . . . . . . . 14

2.5 S11 measurement after impedance match . . . . . . . . . . . . 15

2.6 MIM imaging on MoS2 field-effect transistors . . . . . . . . . . 18

3.1 Microwave imaging on BiFeO3 domain walls . . . . . . . . . . 27

3.2 Quantitative analysis of domain wall conductivity . . . . . . . 28

3.3 Tip-sample geometry for the FEA modeling in (a) Sample Aand (b) Sample B . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.4 C-AFM data of Sample A . . . . . . . . . . . . . . . . . . . . 30

3.5 Frequency dependence of DW signals in Sample A . . . . . . . 31

3.6 MIM result on Sample B . . . . . . . . . . . . . . . . . . . . . 32

3.7 C-AFM data of Sample B . . . . . . . . . . . . . . . . . . . . 33

3.8 Detailed analysis of MIM and PFM images in Sample B . . . 34

3.9 Theoretical analysis of DW oscillation in Sample A . . . . . . 36

3.10 Stress and polarization fields in both samples . . . . . . . . . 37

3.11 Configuration for the dynamical phase-field simulation of Sam-ple A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.1 Schematics of the MIM setup . . . . . . . . . . . . . . . . . . 44

4.2 Microwave imaging around a single LiNbO3 domain wall . . . 46

4.3 MIM-Re image analysis . . . . . . . . . . . . . . . . . . . . . . 47

4.4 Full set of MIM data of the single DW . . . . . . . . . . . . . 48

4.5 DW reflection of SAW and P-SAW . . . . . . . . . . . . . . . 50

4.6 Analytically calculated mechanical force density . . . . . . . . 54

4.7 Numerical simulation of the power transduction near a singledomain wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

xiii

4.8 Details of the FEA modeling . . . . . . . . . . . . . . . . . . . 56

4.9 P-SAW versus SAW . . . . . . . . . . . . . . . . . . . . . . . . 59

4.10 Imaging and simulation of the double-DW sample . . . . . . . 62

4.11 Complete set of data of the enclosed domains . . . . . . . . . 63

4.12 Interference of piezoelectric transduction in corral domains . . 66

5.1 Visualization tools of SAW . . . . . . . . . . . . . . . . . . . . 70

5.2 Schematics of R-MIM and T-MIM . . . . . . . . . . . . . . . . 74

5.3 T-MIM Microwave Circuit Analysis . . . . . . . . . . . . . . . 76

5.4 Characterization of a SAW device . . . . . . . . . . . . . . . . 79

5.5 T-MIM image of traveling SAW . . . . . . . . . . . . . . . . . 81

5.6 T-MIM image of traveling SAW . . . . . . . . . . . . . . . . . 82

5.7 Finite-element Modeling . . . . . . . . . . . . . . . . . . . . . 83

5.8 SAW passing through domains . . . . . . . . . . . . . . . . . . 86

5.9 SAW resonator . . . . . . . . . . . . . . . . . . . . . . . . . . 88

xiv

Chapter 1

Introduction

Understanding the microscopic electromechanical properties of a mate-

rial at different frequencies is of fundamental importance for material science,

condensed matter physics, and device engineering. Microscopy, the technical

field of using microscopes to view tiny objects, is critical for exploring the mi-

croscale and nanoscale electromechanical interactions. In this dissertation, I

will present several microscopic electromechanical phenomena revealed by im-

plementing microwave microscopy on ferroelectric materials and acoustic-wave

systems.

1.1 Near Field Microwave Microscopy

For the microscopy working in the far-field regime, the result of a mea-

surement is always averaged over a length scale defined by the electromagnetic

(EM) wavelength [1], which limits the spatial resolution by the Abbe diffrac-

tion limit

∆x =λ

2NA(1.1)

where λ is the wavelength of the illuminating EM wave and NA is the numer-

ical aperture. As a result, a nanoscale resolution requires a nanometer-size

1

wavelength, which is much shorter than the wavelength of visible light and

microwave.

Going beyond this diffraction limit would demand the use of a sharp

near-field probe in close proximity with the specimen, usually implemented

on a scanning probe microscopy (SPM) platform. By utilizing the evanescent

wave, the spatial resolution is determined by the size of the aperture, which

can be much smaller the wavelength. In the infrared and visible range of the

EM spectrum, such near-field scanning optical microscopy (NSOM) has greatly

expanded our knowledge on the mesoscale optical properties of functional ma-

terials [2–4]. The analogue of NSOM in the microwave regime, or near-field

scanning microwave microscopy (NSMM), has also gained tremendous mo-

mentum [4–7]. In most cases, the interaction between microwave radiation

and materials is much more classical than the optical counterpart. The rela-

tive simplicity in the design of NSMM and straightforward data interpretation

have led to new types of spatially resolved experiments that are otherwise not

feasible.

Near Field Microwave Microscopy (NSMM) was first envisioned in 1928

[8], followed by a few early prototypes [9–12]. With the invention of modern

SPMs, most notably scanning tunneling microscopy (STM) [13] and atomic-

force microscopy (AFM) [14], various designs of NSMM were demonstrated in

the 1990’s [15–20] and the focus was mainly on the understanding of contrast

mechanism, the modeling of tip-sample interaction, and the system calibra-

tion using standard dielectrics and semiconductors. Excellent reviews of these

2

efforts can be found in Ref. [4–6]. In the past decade, much progress was

made to integrate microwave probes with state-of-the-art scanning platforms

[21–26] and low-temperature/high-magnetic-field chambers [27]. Propelled by

the rise of quantum materials that are inherently heterogeneous, NSMM has

evolved into an important SPM mode for cutting-edge scientific research.

1.2 Dissertation Outline

With the history of near field microwave microscopy briefly reviewed,

the rest of this dissertation will be organized as follows.

Chapter 2 covers the fundamentals of microwave impedance microscopy

(MIM) [28], including the working mechanism, individual components, overall

microwave circuits, signal interpretation, and data analysis. The first and

foremost application of MIM, local conductivity mapping, will be introduced.

Chapter 3-5 elaborate multiple projects explored during my graduate

research. The results are based on several previous publications [29–31]. They

include the discoveries of many novel electromechanical phenomena investi-

gated by MIM, which are different from the general mapping of permittivity

and conductivity. These studies broadly expand our knowledge on electrome-

chanics at the microwave regime and pave the way for many possible practical

applications.

Chapter 3 reviews the discovery of a giant microwave conductivity in

BiFeO3 ferroelectric domain walls. I will show that it originates from the

3

microwave absorption of a nominally silent phonon mode excited by the out-

of-plane electric field on the probe.

Chapter 4 displays a novel electromechanical transduction phenomenon

found in the ferroelectric domains of LiNbO3. This work reveals certain in-

ternal degrees of freedom in piezoelectric and elastic tensors, which are not

accessible by measurements of the acoustic displacement fields.

Chapter 5 begins with the invention of a new microwave microscopy

technique: transmission-mode MIM (T-MIM). After we modified the config-

uration of traditional reflection mode MIM to the transmission mode, the

visualization of the microwave field on the sample surface becomes straight-

forward.

Chapter 6 concludes the dissertation and gives the outlook for the fu-

ture.

4

Chapter 2

Microwave Impedance Microscopy

Microwave is commonly defined as the electromagnetic (EM) radiation

with free-space wavelength l ranging from 1 m (frequency f = 300 MHz) to

1 mm (f = 300 GHz). The interaction between matter and EM wave in this

regime is well understood and widely exploited in day-to-day technology. An-

tennas in cellphones and televisions receive and broadcast microwaves, which

propagate through the air and other dielectrics. An airplane made of metal, on

the other hand, reflects microwave and appears on the radar screen. Moreover,

food can be heated in a microwave oven due to the absorption of the electro-

magnetic energy in water. While sharing the same characteristics, NSMM

explores these phenomena in a tiny region determined by the typical tip size,

which is about 6 orders of magnitude smaller than the wavelength. By raster

scanning the tip on the sample surface based on the SPM technique, NSMM

provides a near-field image with the local electrodynamic response informa-

tion.

As one of the latest advances in near-field scanning microwave mi-

croscopy, microwave impedance microscopy (MIM) measures the admittance

(inversion of impedance) between a sharp conductive probe and the sample un-

5

derneath at radio and microwave frequency (∼10 MHz to 10 GHz) [7, 32, 33].

In this chapter, I will first walk through the working principles of MIM,

then scrutinize every important section of the instrument. Finally, a typical

example of a MIM experiment will be introduced. More details about MIM

technique can be found in Ref. [28].

2.1 Basic Principles

Figure 2.1 illustrates the schematic setup of MIM. Microwave is gen-

erated via the source and delivered to the tip through a directional coupler.

Since the impedance of an electrically open tip is very different from 50 Ω in

typical transmission lines, an impedance-match (Z-match) section is required

to ensure efficient power transfer to the probe. The reflected signal is am-

plified and demodulated by an in-phase quadrature mixer, similar to the S11

measurement in a vector network analyzer (VNA) [24, 25, 32]. The two or-

thogonal outputs: MIM-Im and MIM-Re, after proper calibration on standard

sample Aluminum dot (Al dots on an insulating sapphire substrate), should be

proportional to the change of the imaginary and real parts of tip-sample ad-

mittance respectively. The local electronic properties of materials are revealed

because the admittance is correlated with the local complex permittivity ε and

conductivity σ.

6

Figure 2.1: Schematic setup of MIM

2.2 System Components

The actual system consists of the probe, detection electronics, and scan-

ning platform. In this dissertation, all the MIM measurements are performed

on a commercial AFM platform (XE-70, Park Systems). However, it is ob-

vious that the scanning platform will not affect the implementation of MIM,

and the measurements can be performed on any other commercial platforms

or home-built systems.

2.2.1 Probe Design

The successful design of a MIM probe is the key part of microwave

microscopy. Some general principles are listed here. First of all, MIM probes

need to be AFM-compatible so that we can take full advantage of the scanning

capability offered by modern AFM platforms. Second, although metalized tips

with nanoscale apex are readily available for conductive AFM (C-AFM), they

are unsuitable for MIM due to the strong stray-field contribution from the

cantilever body. Therefore, specific measures are required to mitigate this

issue. Moreover, minimized series resistance and background capacitance of

7

the probe itself are critical to ensure high sensitivity to the minute change of

tip-sample admittance when implemented in a resonant circuit. Last but not

least, in order to get consistent results in an economic manner, batch-processed

fabrication of durable probes is highly desired.

Figure 2.2: MIM Probes. (a) Top: Layer structure of the shielded cantileverprobe Bottom: Scanning electron micrograph (SEM) of the cantilever andpyramidal tip. The dashed line indicates the buried center conductor. Theinset shows the apex of the Au/Ti/W tip. (b) Top: Schematic of the solidmetal probe with a contact pad. Bottom: Picture of the cantilever probe witha tall shank of the metal tip. The inset shows the apex of the etched Pt tip. (c)Top: Sketch of the tuning-fork-based probe. Bottom: Picture of the etched Wwire (SEM image in the inset) glued to a tuning fork. Panel (a) adapted withpermission from Reference [34], copyright 2012, IOP Publishing. Panel (c)adapted with permission from Reference [35], copyright 2016, AIP Publishing.

In practice, the first and most widely used design is the batch fabri-

cation of shielded cantilever probes, which are commercially available from

PrimeNano Inc. This type of design has evolved through several generations

[23, 34, 36] and Figure 2.2(a) shows the most accepted one, which is commonly

referred as “Gen 5” because it is the 5th generation design. As shown in

8

the upper panel of Figure 2.2(a), different from common conductive tip, the

Au/Ti/W center conductor is sandwiched between SiNx layers and shielded

by metal layers. This symmetric structure of cantilever is designed to bal-

ance the stress and thermal expansion on both sides. The sub-100nm tip apex

is obtained by oxidation sharpening of the sacrificial Si pit and the subse-

quent metal refill. Thanks to advanced MEMS technology, it can be routinely

achieved for mass-produced probes at the Si wafer scale.

Another design to mitigate strong stray-field contribution, as shown in

Figure 2.2(b), is to increase the sample-cantilever distance by a tall (∼ 80 µm)

shank of metal tip [24, 25], which is commercially available from Rocky Moun-

tain Nanotechnology, LLC. Specially, this type of probe is also suitable for a

contact-mode scan without a feedback loop because of its relatively flexible

structure.

However, a drawback for the above two probes is the inevitable wearing

during the contact-mode scans, which strongly affects the MIM signal level.

In this regard, quartz tuning-fork (TF) probes with etched metal tips (Figure

2.2(c)) are desirable not only for the preservation of tip apex but also for their

self-sensing capability in cryogenic environment [35, 37]. The lack of shielding

is again circumvented by the high-aspect ratio probe. The distance modulation

in a TF-MIM also rejects the background drift and provides absolute sample

information such as the dielectric constant [37]. The disadvantage of TF probes

is the much harder operation compared with cantilevers and the difficulty to

track rough surfaces due to the small oscillation amplitude (usually ≤ 10 nm).

9

The batch-processed fabrication technique of TF probes is not developed yet.

Gen 5 tip is the main type of MIM probe used for the measurements

demonstrated in this dissertation. It should be noticed that it is also suit-

able for other SPM measurements, such as C-AFM and piezoresponse force

microscopy (PFM).

2.2.2 Tip-Sample Interaction

A thorough analysis of tip-sample interaction is crucial to understand

how the electrical properties of the sample affect the admittance probed by

the tip, which further leads to a change of output MIM signal.

Regardless of specific probe designs, the tip-sample interaction can al-

ways be modeled as an interconnection of lumped elements, such as capacitors,

resistors, and inductors, due to the much smaller size of the probes compared

with the wavelength of the microwave. The values and equivalent circuit are

determined by local physical properties and detailed probe geometry. There-

fore, in this section, I will first discuss the equivalent circuit of the MIM probe

and the tip-sample interaction. Then, the tip-sample admittance will be nu-

merically calculated using the method of finite element analysis (FEA).

Figure 2.3 displays the equivalent circuit of the MIM probe with the

general form of tip-sample interaction marked in the dashed rectangle. The

probe itself can be regarded as an inductor, a resistor, and a capacitor in

series. The exact values of these lumped elements are determined by the

material properties and the geometry of the probe. For example, the typical

10

Figure 2.3: Equivalent circuit of the tip-sample interaction.

values for a ”Gen 5” tip are 1 nH, 4 Ω, and 1 pF, which shows the impedance

of tip at 1 GHz is dominant by its capacitance. These values will also be used

for the impedance matching later.

The tip-sample interaction can be modeled as the capacitance between

the tip apex and the sample (Ctip-sample), in series with the impedance of sam-

ple. The impedance of sample usually can be represented by a simple parallel

circuit with both resistive and capacitive components. Note that the existence

of tip-sample capacitance is very common due to separation induced by surface

contamination, “dead” layer of material, or non-ohmic contact. Only in rare

cases, such as a metal surface free of oxide, the tip-sample capacitance can be

absent. Finally, the overall tip-sample interaction should be in parallel with

the tip-shield capacitance. Since the tip-sample interaction is limited to the

tip apex region, the induced tip-sample admittance is rather small compared

with the original tip admittance. Therefore, we treat this induced tip-sample

admittance as a perturbation to the admittance of a tip in the air without any

surrounding samples. The imaginary (MIM-Im) and real (MIM-Re) parts of

the output signals are proportional to the imaginary (susceptance) and real

11

(conductance) parts of the effective tip-sample admittance, respectively. A

rigorous derivation of the MIM signals can be found in Ref. [38].

In order to quantitatively analyze the tip-sample admittance, we nu-

merically model the tip-sample interaction with a commercial FEA software,

COMSOL, to solve the Maxwell’s equations at such complicated conditions.

No matter what version of COMSOL we use, the general procedures

are similar. First of all, choose the dimension of the model. Although a 3D

model usually works, reducing the dimension to 2D or 2D-axisymmetric can

greatly reduce the computing time and requirement. Secondly, choose a phys-

ical module depending on the question. For example, for most cases of MIM

application, the electric current mode in AC/DC module is enough for simu-

lation related to the sample’s permittivity ε and conductivity σ. However, if

the piezoelectric effect is involved, modules like piezoelectric devices should be

considered. Thirdly, draw the geometry of the model, such as tip and sample.

Usually, the built-in plot tools are enough to draw the geometry, but import-

ing complex structures from other CAD tools is also allowed. Fourthly, set the

material and values of the material for each region, and set the boundary con-

ditions, which is the trickiest step. Fifthly, generate mesh automatically with

some adjustable settings. Finer mesh gives more accurate results but requires

larger memory and longer computing time. Lastly, specify the frequency (usu-

ally 1 GHz or 3 GHz for MIM) and compute the solution. COMSOL provides

rich functions and tools for us to obtain and display the physical properties.

Details about these simulation steps can be found in Ref. [39]. The official

12

documents and forums of COMSOL are also very good resources.

2.2.3 Impedance Matching

As we have discussed in the previous sections, in order to ensure efficient

power transfer to the probe, the load impedance of the probe should be 50 Ω,

which is also the typical characteristic impedance Z0 of a uniform transmission

line. However, not every microwave component shows an impedance with a

purely real part of 50 Ω. Therefore, if such load impedance ZL is connected to

a transmission line, the propagating EM wave will be partially reflected and

the reflection coefficient or return loss is defined as

Γ =ZL − Z0

ZL + Z0

. (2.1)

For example, ”Gen 5” tip has around −i150 Ω impedance at 1GHz. Most of

the microwave would be reflected back, and the power would not be delivered

to the probe. The solution to this problem is called impedance matching,

which is a crucial topic in microwave engineering. The interested readers can

fill in the required details from the famous textbook written by Prof. David

Pozar [40].

Figure 2.4 displays all sets of impedance matching sections used in our

lab to obtain the best sensitivity around different frequencies. The dashed box

marks the primary setup we will discuss in this dissertation. At 1 GHz, in order

to match the tip impedance to 50 Ω prior to the coupler, a flexible quarter-

wave cable (AstroLab, Astro-Boa-Flex III, ∼ 4.2 cm) is connected to the MIM

13

Figure 2.4: Impedance-match sections at different frequencies.

tip. The connection is made by wire bonding the tip bonding pad to the inner

conductor of the cable. An open-end tuning stub (Micro-Coax, UT-085C-TP,

∼ 5 cm) is needed to route the tip impedance to the 50 Ω transmission line

[32]. A three-cable junction is formed after connecting the impedance match

section to electronics via a rigid cable (Micro-Coax, UT-085C-TP).

In practice, after the length of the quarter-wave cable (QWC) is de-

termined, we can tune the length of the open stub while monitoring S11 on a

vector network analyzer (VNA). For each value of the open stub length, the

VNA displays S11 as a function of frequency f . By trimming a relatively long

stub, we can observe a deep trough caused by the resonator formed by the

14

Figure 2.5: S11 measurement after impedance match.

tip and impedance match section. As shown in Figure 2.5, in this particu-

lar example, the center frequency of the trough/valley is 957 MHz, which is

determined by the length of the QWC. The shorter the QWC is, the higher

the frequency is. Since our MIM tip is very sensitive to a small change of

admittance at the resonance frequency, we choose this center frequency as the

operating frequency.

2.2.4 Microwave Electronics

In principle, the MIM electronics (as shown in Figure 2.1) measures the

reflection from the matching network mentioned in the last section. The signal

from a microwave source is first split into two lines. One line is for the reference

signal required by the quadrature mixer. The other line is further split into two

parts. The first part is delivered to a directional coupler and to the tip. Tip

receives the reflected signal, which contains a large background or common-

mode signal. The common-mode signal is the background signal coming from

15

the internal and external reflection in the electronics and environment. It could

saturate the amplifiers and randomly drift over a long term. The second part

signal is used as a cancellation signal to sensitively pick up the small change

from the large background. It cancels the common-mode signal by tuning

its amplitude to be equal to and its phase to be opposite to the background.

Because the background varies from tip to tip and the reflection is unknown,

an attenuator and a phase shifter are added on the cancellation signal before

the second coupler. The contrast signal is then amplified by multiple RF

amplifiers (usually 3, with the first one having the best noise performance),

demodulated by an IQ mixer, and amplified again in the DC stage. The total

gain of the system, including RF and DC, can be calibrated by feeding an RF

signal of small power and measuring the output voltages. The instrument here

is calibrated to have a gain of 106 dB (2×105) at 1 kHz bandwidth. Therefore,

assuming -20 dBm (10 µ W) input power, a 1 aF capacitance change at the

tip produces ∼ 30mV in the output:

∆Vout = Vin ·∆S11 ·G = 22mV · 2π × 10−6 · 2× 105 ≈ 30mV

where ∆S11 per capacitance change of 1 aF is from transmission line simulation

[40]. Johnson noise of the system can be calculated as

VJ =√

4kBTBR =√

4kB(300K)(1kHz)(50Ω) ≈ 30nV,

and shot noise is

VS =√

2eV BR =√

2e(0.02V )(1kHz)(50Ω) ≈ 20nV.

16

Therefore, total output noise is

Vn = (VJ + VS) ·G = 10mV,

which shows the sensitivity of our MIM setup at room temperature.

2.3 Application: Conductivity Imaging

The imaging of local conductivity is arguably the most successful ap-

plication of microwave microscopy. In particular, the ability to visualize sub-

surface conduction is of great interest across many research disciplines. Here, I

will demonstrate a material research of novel low dimensional semiconductors.

Semiconductor devices, in which the electrical conductivity is controlled

by doping, gating, temperature, or illumination, are the backbone of modern

information technology. For the same reason, doped Si and GaAs devices have

been the preferred samples for NSMM calibration [25, 32, 41–44].

Semiconducting transition metal dichalcogenides (TMDs) such as MoS2

and WSe2 are in the limelight of current material research [45]. These layered

materials can be exfoliated into atomically thin 2D sheets with unique electri-

cal and optical properties, which are attractive for nanoelectronics and opto-

electronics. Figure 2.6(a) shows the transfer curve of a few-layer MoS2 flake in

the field-effect transistor (FET) configuration [46]. The device was covered by

15-nm-thick Al2O3 to avoid direct contact between the metallic tip and the 2D

sheet. Figure 2.6(b) displays selected MIM images within the channel region

as a function of the back-gate voltage VBG. As VBG increased, the conductance

17

Figure 2.6: MIM imaging on MoS2 field-effect transistors. (a) Transfer char-acteristics of the back-gated MoS2 transistor. The inset shows a picture of thedevice. (b) Selected MIM images of the sample in the dashed box in (a). (c)Averaged MIM signals as a function of the source-drain conductance for the4ML and 3ML regions in dashed boxes in (b).

signal emerged initially at the edges and then in the interior, with appreciable

spatial non-uniformity [46]. For quantitative analysis, average MIM-Im/Re

signals within the white dashed boxes on the four-monolayer (4ML) and 3ML

segments are plotted in Figure 2.6(c), which are consistent with the simulated

response curves. The results suggest that the contribution of defect-induced

edge states to the total conductance is significant in the subthreshold regime

but negligible once the bulk becomes conductive. The observation of conduc-

tance inhomogeneity also provides a guideline for future improvement of the

device performance. Similar MIM works have been carried out on other 2D

18

materials [47–56], photovoltaics [57–59], and device engineering [60].

2.4 Beyond Conductivity Imaging

In this section, we review the basic components and data interpretation

of microwave impedance microscopy. In addition to the general-purpose map-

ping of permittivity and conductivity, MIM is now exploited to perform quan-

titative measurements on semiconductor devices, photosensitive materials, fer-

roelectric domains and domain walls, and acoustic-wave systems. Implemen-

tation of the technique in low-temperature and high-magnetic-field chambers

has also led to major discoveries in quantum materials with strong correlation

and topological order.

While the majority of MIM applications to date are the local conduc-

tivity mapping, during my graduate research, I mainly explored the material

properties beyond the conductivity. In the next three chapters, I will intro-

duce the imaging on three different properties: dielectric loss in ferroelectric

domain walls, electromechanical power transduction in ferroelectric domains,

and microwave fields in electroacoustic devices, individually.

19

Chapter 3

Microwave Absorption at Ferroelectric

Domain Walls

The properties of ferroelectric domain walls are of fundamental impor-

tance for their device applications. In particular, it is desirable to understand

the microwave conductivity of individual domain walls, which is largely un-

known to date, for practical high-speed electronics. In this chapter, a giant

giga-Hertz (GHz) conductivity, which is ∼ 100,000 times greater than the

carrier-induced dc conductivity, is observed in certain BiFeO3 domain walls

by microwave microscopy. Ginzburg-Landau analysis and phase-field model-

ing suggest that the imbalanced polarization across the wall is responsible for

the dielectric loss under the microwave fields1.

1The results in this chapter are primarily based on the previous publication: Y.-L. Huang,L. Zheng, P. Chen, X. Cheng, S.-L. Hsu, T. Yang, X. Wu, L. Ponet, R. Ramesh, L.-Q.Chen, S. Artyukhin, Y.-H. Chu, and K. Lai, Unexpected giant microwave conductivity in anominally silent bifeo3 domain wall,” Advanced Materials, vol. 32, no. 9, p. 1905132, 2020.As one of the first co-authors, I performed the MIM experiment and numerical analysis withY.-L. Huang, participated in the discussion of results, and edited the manuscript.

20

3.1 Introduction

3.1.1 Ferroelectrics

As indicated by the prefix ‘ferro’, ferroelectrics, in analogy to ferro-

magnets, are certain materials that have a spontaneous electric polarization

in the absence of an applied electric field. This spontaneous polarization can

be reversed by the application of an external electric field, and the switching

displays history-dependent behavior or hysteresis loop (P-E loop) [61].

Due to the existence of remnant polarization, ferroelectric materials

have been widely used in various applications, such as non-volatile memory,

capacitor, etc. Utilizing the polarization to manipulate material properties is

also an exciting topic in many research fields.

3.1.2 Domain and Domain Walls

Similar to ferromagnets, ferroelectrics normally present separate regions

called domains. Each domain is an assembly of unit cells with the same electric

dipole, and therefore has a spontaneous polarization. Adjacent domains that

have different spontaneous polarization directions are separated by domain

walls (DWs) which usually have a much smaller length scale of a few unit cells

comparing to ferromagnetic DWs.

Both domain and domain walls can work as functional elements in

nanoscale electronic devices. By simply utilizing the bipolar domain orien-

tations to store “0” and “1” states, scientists developed ferroelectric non-

volatile random access memory (FeRAM) with many advantages, including

21

high-density storage, low-energy consumption, and unlimited cycle endurance

[62, 63]. Moreover, DWs may exhibit novel functionalities that are different

from the surrounding domains. Due to their extremely small width, the con-

cept of using DWs as active parts has given rise to the rapid development of

prototypical DW-based nano-devices, including diodes [64], nonvolatile mem-

ory [65], and tunnel junctions [66], among others.

3.1.3 DC conduction and AC conduction

The ground-breaking discovery that DWs in BiFeO3 is more conduc-

tive than surrounding domains [67] has led to many significant works that

demonstrate the potential of these conducting interfaces for promising device

applications [65, 68–71]. Although most of the applications are inspired by the

abnormal dc conductivity due to the presence of free carriers, some of these

devices can function at the low mega-Hertz (MHz) range [70, 71]. Practical

electronics, however, usually demand much higher operation frequencies in the

giga-Hertz (GHz) regime, where the effect of dipolar oscillation becomes im-

portant. Specifically, it is well-known that when a multi-domain ferroelectric is

placed in microwave fields, the coupling between the spontaneous polarization

and the alternating E-field may lead to the periodic oscillation of the DWs

[72–74]. Because of the inertia of dipole moments and frictional effects in the

material, such DW motion gives rise to dielectric loss at the microwave fre-

quency (f) , which limits the use of bulk ferroelectrics for RF tunable devices

[75]. The situation could be different, however, if DWs can be individually

22

addressed and their dielectric response fully understood. For instance, one

may take advantage of the ac conductivity of DWs due to the dipolar loss

σac1 = ωε′′, where ω = 2πf and ε

′′is the imaginary part of the permittivity, to

function as device interconnects or high-f waveguides. At microwave frequen-

cies, the free-carrier contribution to the ac conductivity σac2 is indistinguishable

from σac1 at the circuit level [75]. It is therefore imperative to obtain a uni-

fied picture on both mobile-carrier conduction and bound-charge oscillation in

ferroelectrics DWs, which is crucial for their applications in nanoelectronics.

As we have introduced in the previous chapter, our MIM technique

is able to visualize the local admittance of the sample, which contains the

information of local electronic conduction. According to the definition:

ε = ε′+ i(

σ

ω+ ε

′′),

at a fixed frequency, the dipolar loss ε′′

is indistinguishable from the electronic

conduction σ at the circuit level. Therefore, MIM would be the only technique

to map out these GHz dynamics in the nanoscale.

Here, we report the observation of a giant ac conductivity of 103 S/m

at 1 GHz in certain BiFeO3 DWs [31], which is 100,000 times greater than the

carrier-induced dc conductivity of the same walls.

3.2 Nanoscale Image of BiFeO3

The giga-Hertz (GHz) response of ferroelectric DWs has been spa-

tially resolved by microwave impedance microscopy (MIM) [7, 32]. For weakly

23

charged walls in KNbO3 crystals [76] and transiently formed charged walls in

lead zirconate (PZT) thin films [77], it was shown that the DW ac conductiv-

ity is dominated by the charge carriers, i.e., σac = σac2 . In contrast, the MIM

result on hexagonal manganites (h-RMnO3) [78] and ferrites (h-RFeO3) [79]

depends strongly on the DW orientation. On the (001) surface with uncharged

180 DWs, σacDW is 4 ∼ 6 orders of magnitude higher than σdcDW , suggesting

that the ac E-field from the MIM tip can effectively drive the DW oscillation

[78]. For charged walls on the (110) surface, however, the MIM E-field does not

favor the polarization on either side of the wall. As a result, no enhancement

of σacDW over the dc value was observed [78].

In this section, I will report the nanoscale microwave conductivity imag-

ing on BiFeO3 domain walls that nominally do not vibrate under out-of-plane

ac E-fields. Surprisingly, while σac ≈ σdc is indeed observed in the control

sample with vertical 71 walls, the effective GHz conductivity of the inclined

71 walls is 105 times higher than its σdc, which appears to violate the selec-

tion rule of DW oscillations. Using a simplified Ginzburg-Landau theory [80],

we find that the inclination of the wall leads to an asymmetric profile of the

out-of-plane polarization, which is responsible for its vibration under the ac E-

field. The dielectric loss due to displacement current at the tilted DWs is also

confirmed by the phase-field modeling. Our results highlight the importance

of local symmetry in the structural dynamics of ferroelectric DWs, which may

be utilized for radio-frequency nanoelectronics.

24

3.2.1 71 Domain Walls

The main sample in this work – 150 nm BiFeO3 (BFO) on 3 nm con-

ductive SrRuO3 (SRO) thin film, hereafter referred to as Sample A – was epi-

taxially grown on DyScO3 (DSO) substrates by pulse laser deposition [81, 82].

The BFO layer was deposited on SRO electrodes at 700C with an O2 pressure

of 100 mTorr with a laser fluence of ∼ 1 J/cm2 and a repetition rate of 5 Hz.

BFO is one of the most promising multiferroic materials [83] and its DW prop-

erties are technologically important. The coherent growth is facilitated by the

close match between the DSO lattice on the (110)O orthorhombic surface and

the BFO lattice on the (001)C pseudocubic surface.

As illustrated in Figure 3.1(a), the BFO film displays an array of stripe

domains oriented along the [010]C direction [81, 82]. The 71 DWs, categorized

by the angle between the polarization vector of two neighboring domains (inset

of Figure 3.1(a)), are uncharged since the polar discontinuity is parallel to the

wall plane. Moreover, the change of polarization vector across the wall lies

in the plane of film surface. Consequently, an out-of-plane oscillating E-field,

which is most relevant for thin-film devices with bottom electrodes, should not

couple to the DWs here.

In order to study the GHz dielectric response of Sample A, we per-

formed the MIM experiment [7, 32], where the microwave signal is delivered to

the center conductor of a shielded cantilever probe [34]. By amplifying and de-

modulating the reflected signal, the imaginary (MIM-Im) and real (MIM-Re)

parts of the tip-sample admittance can be spatially resolved. Figure 3.1(b)

25

26

shows the atomic-force microscopy (AFM), in-plane piezoresponse force mi-

croscopy (PFM), and MIM (f = 1 GHz) images of Sample A. With virtually

no crosstalk to the surface topography, the stripe domains are clearly visual-

ized in the PFM image. The microwave images, on the other hand, exhibit

strong signals at the walls.

The MIM contrast between DWs and domains in Sample A can be

vividly seen from the line profiles in Figure 3.2(a). It shows that the DW

signals are ∼ 10 times greater in the MIM-Im channel than that in MIM-Re.

To quantify the result, we carried out finite-element analysis (FEA,

Figure 3.3) by finite-element modeling using COMSOL 4.4 to compute the

complex tip-sample admittance and convert them to the MIM signals [32].

The program can directly compute the admittance between the tip and the

ground based on the tip-sample geometry in Figure 3.3. Due to the lack of

axisymmetry when a DW is involved, 3D modeling is needed for this work,

which requires large memory and long computing time. Dimensions of the

sample structure are labeled in Figure 3.3. The DWs are modeled as a thin

Figure 3.1 (preceding page): Microwave imaging on BiFeO3 domain walls. (a)Schematics of the MIM setup and domain structures in Sample A. The mi-crowave signal at 1 GHz is sent to the center conductor of a shielded cantileverprobe and the reflected signal is demodulated to form the MIM-Im/Re images.The polarization vectors (green arrows) of the BFO domains are labeled onboth the surface and cross-section of the film. The inset shows a close-up viewof the polarization direction on both sides of the DW (yellow slab). (b) Fromleft to right: AFM, phase image of in-plane PFM, and MIM-Im/Re images onSample A. All images (1 µm× 1 µm) were taken at the same location.

27

Figure 3.2: Quantitative analysis of domain wall conductivity. (a) TypicalMIM line profiles in Sample A, showing that the DW signals are ∼ 10 timesgreater in the MIM-Im channel than that in MIM-Re. Positions of the DWsare marked by dotted lines. (b) Simulated MIM signals as a function of the ef-fective DW ac conductivity in Sample A. The corresponding σacDW that matchesthe measured signals is marked by the dashed arrow. The DW dc conductivityis also indicated for comparison. The inset illustrates the tip-sample geometry,as well as the directions of the MIM E-fields and polarization vectors.

slab with a thickness of 2 nm. Parameters for the FEA are as follows: tip

radius r = 50 nm; tip height h = 500 nm; half-cone angle of the pyramidal

tip θ = 22; tip-sample distance (in order to avoid a divergent signal when

σ is very large) t = 1 nm; dielectric constant of bulk BFO εr = 30. Note

that for Sample A, DWs near the surface bend towards the normal direction

(see theory part). However, the effect on the FEA result is small and not

considered here.

By comparing the simulated and measured signals (Figure 3.2(b)), it

can be shown that the effective DW ac conductivity σacDW,A is ∼ 103 S/m, the

highest value reported in BFO to date [67, 84].

28

Figure 3.3: Tip-sample geometry for the FEA modeling in (a) Sample A and(b) Sample B.

In contrast, the dc conductivity of DWs in Sample A can be estimated

from the conductive AFM (C-AFM) data (Figure 3.4). Figure 3.4 shows the

typical I-V curve of Sample A when the tip is on top of the DW, from which

the slope of ∼ 0.6 pA/V at high tip biases can be extracted. Assuming that

the tip-DW contact area is 5 nm × 2 nm and the cross-sectional length of the

DW is ∼ 200 nm (150 nm film thickness with 45 tilt), it is easy to calculate

that the dc conductivity of inclined DWs in Sample A is on the order of 10−2

S/m.

It should be noted that for the similar behaviors were also observed at

other frequencies ranging from 20 MHz to 5 GHz, as shown in Figure 3.5.

Since the data were taken using different sets of MIM electronics, it is

not possible to compare the absolute signal strengths at different frequencies.

It is clear that the DWs exhibit strong MIM-Im signals from 20 MHz to 5 GHz,

although we only focus on the 1 GHz data with good signal-to-noise ratios in

29

Figure 3.4: C-AFM data of Sample A. (a) AFM and C-AFM images in SampleA. (b) I-V curve on the DW and domain in Sample A. The slope is ∼ 0.6 pA/V.

both MIM channels. Because of the very weak MIM-Re signals, it is difficult

to determine the effective DW ac conductivity in a quantitative manner using

the response curves in Figure 3.1(b) for 1 GHz data. In the table of Figure

3.5(b), we provide rough estimates of σacDW based on the FEA curve. Note that

the overall characteristics are similar to that observed in h-RMnO3 [78].

In conclusion, the drastic difference between ac and dc conductivity

(103 S/m vs. 10−2 S/m) implies that the microwave response of the inclined

71 DWs is dominated by the dipolar loss rather than the Ohmic loss. As

discussed before, the vibrational motion of this DW, if exists at all, is in

30

Figure 3.5: Frequency dependence of DW signals in Sample A. (a) MIM-Imand MIM-Re images taken with 5 different operation frequencies. All scalebars are 200 nm. (b) DW ac conductivity at various frequencies estimatedfrom the ratio of MIM-Re/Im contrast.

principle decoupled from the out-of-plane microwave fields (inset of Figure

3.2(b)). The excitation of such a nominally silent mode in our MIM experiment

is thus unexpected and highly nontrivial.

3.2.2 Control Experiments on Charged Domain Walls

Before further investigating the DW vibration in Sample A, it is in-

structive to evaluate the result of a control sample, 50 nm BFO / 30 nm SRO

thin film on (110)C cubic SrTiO3 substrate [7], hereafter referred to as Sample

31

B. As depicted in Figure 3.6, the (110)C-oriented BFO film exhibits an irreg-

ular domain pattern. DWs in this sample, also categorized as 71 walls, are

perpendicular to the film surface. The two polarization vectors in neighbor-

ing domains span from the ‘head-to-head’ (HtH), neutral, to the ‘tail-to-tail’

(TtT) configurations.

Figure 3.6: MIM result on Sample B. (a) Schematics of the layer and domainstructures of Sample B. The inset shows the polarization vectors on bothsides of the DW (yellow slab) in the ‘head-to-head’ (HtH) configuration. (b)From left to right: phase image of in-plane PFM and MIM-Im/Re images onSample B. Green dotted lines in the MIM images mark the contour of domainsdetermined from the PFM data. (c) Simulated MIM signals as a function of theeffective DW ac conductivity in Sample B. Solid and dashed arrows indicatedthe dc and ac conductivity of the HtH DWs in this sample, respectively. Theinset illustrates the tip-sample geometry, as well as the directions of the MIME-fields and polarization vectors.

In contrast with C-AFM result of sample A (Figure 3.4), only segments

of the charged DWs in Sample B exhibit a much larger current, as shown in

Figure 3.7(a). The combined C-AFM and PFM images indicate that these

32

high-current regions correspond to the ‘head-to-head’ polarization configura-

tion. Following the same steps for the calculation of the conduction of Sample

A, as estimated from the I-V curve in Figure 3.7(b), the maximum DW dc

conductivity due to free-carrier conduction is around 3 S/m in the HtH con-

figuration, consistent with previous reports [84, 85].

Figure 3.7: C-AFM data of Sample B. (a) AFM, C-AFM, and phase/amplitudeof in-plane PFM images in Sample B. Dashed lines in the C-AFM image showthe contour of domains from the PFM data. Scale bars in (a) are 200 nm. (b)I-V curve on a head-to-head segment of DW and domain in Sample B. Theslope is ∼ 0.7 nA/V.

In Figure 3.6, the contours of ferroelectric domains obtained from the

in-plane PFM are overlaid on the MIM images. Neither domain nor DW

contrast appears in the MIM-Im channel, whereas broken sections of DWs

are seen in MIM-Re. Detailed analysis (Figure 3.8) shows that these isolated

segments coincide with the HtH sections of DWs.

33

Figure 3.8: Detailed analysis of MIM and PFM images in Sample B. (a) In-plane PFM phase image in Sample B. The in-plane polarization directionsof the domains are denoted by green arrows. (b) Shaded image of (a), withneutral, HtH, and TtT segments overlaid in the plot. (c) MIM-Re image ofthe same area. The scale bar is 100 nm.

The domain structures are resolved by the in-pane PFM image in Figure

3.8(a), from which the neutral, head-to-head (HtH), and tail-to-tail (TtT) seg-

ments of the DWs can be determined (Figure 3.8(b)). A comparison with the

MIM-Re image in the same area (Figure 3.8(c)) shows that only the HtH sec-

tions with free-carrier accumulation display appreciable microwave response.

Based on the FEA simulation in Figure 3.6, σdcDW and σacDW are essen-

tially the same for the HtH walls in Sample B. In other words, the microwave

response of vertical 71 DWs can be fully accounted for by the mobile-carrier

contribution.

34

3.3 Theoretical Analysis

The major difference between the two BFO samples is the orientation

of DWs with respect to the surface. In an infinite bulk sample with one DW,

the diagonal mirror plane indicated by the dashed line in Figure 3.9(a) is a

symmetry operation. As a result, the nominal orientation of DWs in Sam-

ple A is tilted 45 away from the surface normal. In thin films with a finite

thickness, however, the strain near the surface can be relaxed and violation of

the compatibility constraint [86] does not lead to infinite elastic energy. The

stress near the surface is therefore imbalanced on two sides of the wall. Using a

simplified Ginzburg-Landau model [80], one can show that the DW bends to-

wards the normal direction near the surface [87]. We note that the bending of

71 DWs near the surface of BFO thin films has been experimentally observed

by transmission electron microscopy (TEM) and PFM studies [88]. Figure

3.9(a) shows the spatial distribution of Pz (out-of-plane component of the po-

larization) in the cross-section of the film, as simulated by electromechanical

finite-element method [87]. Because of the large spontaneous polarization in

BFO [83], the imbalance of Pz across the wall is comparable to polarization

change across the 180 walls in h-RMnO3 [78] and h-RFeO3 [79]. The asym-

metric Pz profile can now couple to the out-of-plane MIM E-fields and induce

DW vibration in Sample A.

In contrast, since DWs in Sample B are perpendicular to the film sur-

face, no asymmetry in the stress/strain or polarization is induced around the

wall, thus the absence of DW dielectric loss.

35

Figure 3.9: Theoretical analysis of DW oscillation in Sample A. (a) Simulatedout-of-plane polarization Pz in Sample A, showing the imbalanced Pz on twosides of the wall. Arrows in the dotted boxes represent the strength of stressalong different directions. The net effect causes the DW to bend towards thesurface normal. (b) Simulated ∂Pz/∂t at t = T/4 when a uniform ac E-fieldis applied between the top surface and the substrate. The DW sliding underoscillating electric fields is evident from the bipolar line profile across the wall.(c) Spatial distribution of the time-averaged dielectric loss density when a tip-like potential is placed (top) on top of and (bottom) away from the DW. Thetip position is indicated by the orange and black arrows. The DW is depictedby the white dash-dotted line in the bottom image. Gray arrows represent theE-field from the tip. (d) Simulated ac conductivity as a function of the tipposition, showing a sharp peak when the tip scans across the DW.

Figure 3.10 shows the details of our Ginzburg-Landau analysis on both

samples [87]. For Sample A, the asymmetric stress (dashed boxes in Figure

3.10(a)) near the film surface leads to deviation of the DW from its orientation

in the bulk. The resultant asymmetric strain induces imbalanced Pz on two

sides of the wall via electrostriction, as seen in Figure 3.10(b). For Sample B,

however, the DW is perpendicular to the film surface, therefore no asymmetry

36

in the stress/strain (Figure 3.10(c)) or polarization (Figure 3.10(d)) is induced

around the wall.

Figure 3.10: Stress and polarization fields in both samples. (a) Illustrationof strain field near the inclined DW in Sample A. The dashed line indicatesthe mirror plane perpendicular to the wall. The polarization components areshown on both sides of the DW. Arrows in the dotted boxes represent thestrength of stress along different directions. The net effect causes the DWto bend towards the surface normal. (b) Simulated out-of-plane polarizationPz in Sample A, showing the imbalanced Pz on two sides of the wall. (c)Illustration of strain field near the neutral section of the vertical DW in SampleB. The stress is balanced on two sides of the wall. (d) Simulated out-of-planepolarization in Sample B, showing the same Pz as that in the bulk. The dashedline shows the unperturbed wall.

The excitation of DW vibration in Sample A can be validated by our

dynamical phase-field model [89], where the time-dependent response of the

polarization vector is computed using the polarization dynamics equation [90,

91].

A phase-field model taking into account the polarization dynamics [89]

37

Figure 3.11: Configuration for the dynamical phase-field simulation of SampleA.

was used to simulate the domain and domain wall response in the BFO thin

films under an ac electric field. The dynamic response of the local polarization

field P (r), where r is the position vector, is described by a modified time-

dependent Ginzburg-Landau equation with an additional term of second-order

time-derivative of P accounting for its intrinsic oscillation, i.e.,

µ∂2P

∂t2+ γ

∂P

∂t+δF

δP= 0

where µ and γ are kinetic coefficients related to the domain wall mobility.

The equation was numerically solved using a semi-implicit Fourier spectral

method. F = Flandau + Fgradient + Felectric + Felastic is the total free energy of

the ferroelectric BFO. Flandau, Fgradient, and Felectric are the ferroelectric landau

free energy, ferroelectric gradient energy, and electrostatic energy, respectively

[90, 91]. The elastic energy Felastic is expressed as follows.

Felastic =

∫1

2cijkl(εij − ε0ij)(εkl − ε0kl) dr3

where c is the elastic stiffness tensor and ε0ij = QijklPkPl is the stress-free

strain related to the local ferroelectric order (Q denoting the electrostrictive

38

coefficient tensor). The time step in the simulation is 0.01 ps. The polarization,

permittivity, stiffness, and electrostrictive parameters of BFO were taken from

Ref. [92]. The kinetic coefficients µ = 10−16J ·m/A2 and γ = 10−4J ·m/(A2·s)

were used in the simulation. In particular, the effective damping coefficient

γ corresponds to a bulk dielectric loss of tan δ ∼ 0.01 at 1 GHz, which is

consistent with the literature. Figure 3.11 shows the geometry of our 2D

simulation, with 512 grids in the x-direction and 64 grids in the z-direction.

Each grid here represents 0.4 nm. The BFO thin film, 50 grids in height, is

terminated by 4 grids of vacuum on the top surface and 10 grids of conductive

substrate on the bottom surface. The periodic boundary condition is applied

in the x-direction. Two types of external potential were used in the simulation.

For Figure 3.9(b), a uniform ac electric field E = E0 sin(2πt/T ), where E0 =

105 V/m and T = 1 ns, is applied between the top surface and the substrate.

For Figure 3.9(c), a Lorentz distribution of tip-like electric field (maximum field

105 V/m) is scanned across the top surface. Finally, the displacement current

density jP = ∂P /∂t is simulated by the dynamical phase-field modeling and

the time-averaged dielectric loss density ∂P /∂t ·E is plotted in Figure 3.9(d).

As a result, Figure 3.9(b) shows a snapshot of ∂Pz/∂t at t = T/4 when

a sinusoidal E-field (∝ sin 2πt/T ) is uniformly applied between the top surface

and the conductive substrate. The bipolar line shape around the wall clearly

manifests the DW sliding [78] under the out-of-plane ac E-field. To under-

stand the influence of DW vibration on electrical energy loss, we performed

a series of simulations by moving a tip-induced potential profile across the

39

sample surface. Figure 3.9(c) shows the spatial distribution of power density

∂P /∂t ·E integrated over one period when the tip is on top of DW and away

from the wall. The effective ac conductivity, estimated from the spatial sum-

mation of the time-averaged value (∂P /∂t · E)/E2, is plotted as a function

of the tip location in Figure 3.9(d). The appearance of strong σac1 at the DW

demonstrates that our simple 2D dynamical phase-field simulations capture

the essential physics behind the experimental observation in Sample A.

3.4 Conclusion

In summary, we discover the excitation of a nominally silent mode

in BiFeO3 domain walls by the out-of-plane electric field from a microwave

probe. The effective ac conductivity of such inclined 71 DWs is about 105

times greater than the dc value, signifying the predominance of bound-charge

oscillation over mobile-carrier conduction in the sample. Our analysis based

on the electrostriction effect shows that the out-of-plane polarization is imbal-

anced around the wall. Phase-field simulation further indicates that such an

asymmetric polarization profile can couple to the ac electric field from the tip,

resulting in strong power dissipation at the DW. We emphasize that, while the

physical origin is different, the ac conductivity due to dipolar loss is equivalent

to that from the electron conduction at the circuit level. In that sense, this

work represents an important step towards implementing DW nanoelectronics

for radio-frequency applications.

40

Chapter 4

Electromechanical Power Transduction in

Ferroelectric Domains

The electrical generation and detection of elastic waves are the foun-

dation for acousto-electronic and acousto-optic systems. For surface-acoustic-

wave devices, micro-/nano-electromechanical systems (MEMS/NEMS), and

phononic crystals, tailoring the spatial variation of material properties such as

piezoelectric and elastic tensors may bring significant improvements to the sys-

tem performance. Due to the much smaller speed of sound than speed of light

in solids, it is desirable to study various electroacoustic behaviors at the meso-

scopic length scale. In this section, we demonstrate the interferometric imag-

ing of electromechanical power transduction in ferroelectric lithium niobate

domain structures by typical MIM setup. In sharp contrast to the traditional

standing-wave patterns caused by the superposition of counter-propagating

waves, the constructive and destructive fringes in microwave dissipation images

exhibit an intriguing one-wavelength periodicity. We show that such unusual

interference patterns, which are fundamentally different from the acoustic dis-

placement fields, stem from the nonlocal interaction between electric fields and

elastic waves. The results are corroborated by numerical simulations taking

into account the sign reversal of piezoelectric tensor in oppositely polarized

41

domains. 1

4.1 Introduction

The hallmark of wave interference, a ubiquitous phenomenon in na-

ture, is the appearance of time-independent spatially varying patterns of the

oscillation amplitude [93]. In the famous Young’s double-slit experiment, al-

ternating bright and dark bands on the detector screen vividly demonstrate

the wave nature of light, where the periodicity of the interference pattern is

proportional to the wavelength. Two counter-propagating waves, one usually

generated by boundary-induced reflection of the other, can also interfere with

each other to form a standing-wave pattern with a half-wavelength periodicity.

In both cases, the interference fringes reveal the hidden phase information of

the wave, which enables measurements with superior sensitivity, information

capacity [94], and resolution [95] far beyond the wavelength limit. As a result,

interferometry has become the basis for nearly all ultra-precision metrology

in science and technology, ranging from astronomy [96] and quantum physics

[97] to radar [98] and medical imaging [99].

Wave interference is generally caused by the superposition of local os-

1The results in this chapter are primarily based on the previous publication: L. Zheng,H. Dong, X. Wu, Y.-L. Huang, W. Wang, W. Wu, Z. Wang, and K. Lai, ”Interferometricimaging of nonlocal electromechanical power transduction in ferroelectric domains,” Pro-ceedings of the National Academy of Sciences, vol. 115, no. 21, pp. 5338-5342, 2018. As thefirst author, I performed the experiment with X. Wu and Y.-L. Huang and performed thenumerical simulations with H. Dong. I also participated in data analysis and manuscriptrevision.

42

cillating fields of individual waves at each point in the space. In this work

[30], we demonstrate a special type of interference from the superposition of

nonlocal interaction between electric fields and elastic waves in ferroelectric

domain structures. Because of the sign reversal of piezoelectric tensor in op-

positely polarized domains, the fringe patterns in microwave impedance maps

are fundamentally different from that of the underlying acoustic fields. Our re-

sults are corroborated by first-principle numerical simulations. Microscopy on

piezoelectric energy transduction is highly desirable for the design and char-

acterization of novel surface acoustic wave (SAW) devices [100], microwave

micro/nano-electromechanical systems [101], and phonon-polariton systems

[102]. In this context, our work may open a new research frontier to explore

various nanoscale elastic phenomena in these systems by near-field electromag-

netic imaging.

4.2 Image of LiNbO3

4.2.1 Experiment Setup

The experimental technique and setup in this study is the same MIM

technique we discussed in previous chapters [7], as schematically illustrated in

Figure 4.1.

Again, the excitation signal V = V0ei2πft (voltage V0 around 0.1 V and

frequency f from 100 MHz to 10 GHz) is delivered to the center conductor

of a shielded cantilever probe [34]. The tip can be viewed as a point voltage

source since its diameter at the apex (∼ 100nm) is much smaller than the

43

Figure 4.1: Schematics of the MIM setup. Schematics of the probe, electron-ics, and the z-cut LiNbO3 sample with a single domain wall. The microwavesignal is delivered to the cantilever tip by a directional coupler, and the re-flected signal are amplified and mixed with the reference signal to form theMIM-Im and MIM-Re images. The top-left inset shows the scanning electronmicroscopy image of a typical tip apex.

acoustic wavelength at these frequencies. The MIM electronics detect the real

and imaginary components of the tip-sample admittance Y = G + iB (G:

conductance, B: susceptance), which are displayed as MIM-Re and MIM-Im

images, respectively [32]. Instead of connecting the measured admittance to

the local permittivity [103] and conductivity [46] as we discussed before, we

will show that it can also reveal information on the electroacoustic power

transduction in piezoelectric materials.

4.2.2 MIM Images around a Single LiNbO3 Domain Wall

Our sample is single-crystalline lithium niobate (LiNbO3), which is

technologically important because of its high piezoelectric constants [104], low

acoustic attenuation [105, 106], and strong 2nd-order nonlinear optical coeffi-

cients [107, 108]. LiNbO3 has a trigonal (class 3m) crystal structure with a

44

mirror yz-plane and a direct triad z-axis along the polar direction [104]. The

polarization can be switched by electrical poling [107, 109], allowing artificial

domain patterns at micrometer sizes to be created for microwave signal pro-

cessing [105, 106] and nonlinear optics [107, 108].

We start with the simplest scenario around a straight domain wall

(DW) on a z-cut LiNbO3 sample (Figure 4.1). At a first glance, the system

is akin to the electron-wave interference near an atomic step edge imaged by

scanning tunneling microscopy [110]. Since the domain inversion flips the sign

of odd-rank tensors (the 1st rank polarization P and the 3rd rank piezoelectric

tensor e) [104], the two oppositely polarized domains can be visualized by

piezoresponse force microscopy (PFM) in Figure 4.2. The MIM data at f =

967 MHz in the same area are also displayed in Figure 4.2.

The MIM-Im image, which represents the non-dissipative dielectric re-

sponse, only shows weak contrast possibly due to the static surface charge. The

MIM-Re image, on the other hand, exhibits clear interference fringes around

the DW. Since the electrical conductance of LiNbO3 due to free carriers is neg-

ligible, the MIM-Re contrast indicates that the microwave energy is dissipated

through the piezoelectric transduction rather than the Ohmic loss.

Figure 4.3(a) shows the averaged MIM-Re line profile across the DW.

Neglecting a small spike on the wall due to the dielectric loss associated with

DW vibrations [78], the main features include a prominent dip at the DW

and damped oscillations with a periodicity of λ away from the wall. Here λ is

found to be 4.55 µm from the Fourier transform of the ripples (inset of Figure

45

Figure 4.2: Microwave imaging around a single LiNbO3 domain wall. From topto bottom: AFM, PFM amplitude (PFM-amp) and phase (PFM-ph) images,and MIM-Im/Re (f = 967 MHz) images of the sample. All scale bars are 10µm.

4.3(b)) and the oscillation amplitude decays quadratically with the distance to

the wall. We notice that the first pair of crests only develop as weak shoulders

and the first pair of troughs are separated by 1.8λ rather than 2λ. Similar

MIM results are observed from 285 MHz to 6 GHz (Figure 4.4).

As shown in Figure 4.3(b), the measured 1/λ scales linearly with f

and the slope corresponds to an apparent phase velocity of 4.4 ± 0.2 km/s.

Comparing it with the velocities of x-propagating acoustic waves [106, 111, 112]

on z-cut LiNbO3 (Table 4.1), it is clear that the results have the closest match

to the pseudo surface acoustic wave (P-SAW) [113], whose dispersion lies in

46

Figure 4.3: MIM-Re image analysis. (a) MIM-Re line profile, in which thepeaks and valleys are marked by red and blue dashed lines, respectively. Notethat the first pair of troughs (labeled by pink dashed lines) are separated by1.8λ. The oscillation amplitude at each peak (red circles) and valley (bluesquares) is the difference between its signal and the average signal of the twoadjacent valleys and peaks, respectively. The black dashed lines are fits to theinverse square of the distance to the wall. (b) Linear relation between λ−1 andthe frequency. The slope corresponds to a wave velocity of 4.4±0.2 km/s. Theinset shows the Fourier transform of the data in a with a spatial frequency of0.22 (µm)−1.

the continuum of bulk waves.

Unlike the Rayleigh SAW (hereafter denoted as SAW) that exists on

the surface of all solids [113], this electroacoustic Bleustein-Gulyaev [114, 115]

SAW only exists on the surface of piezoelectric materials. The displacement

fields of the P-SAW are primarily polarized in the y-direction [116], although

the wave is not purely transverse-horizontal due to the lack of an even-order

symmetry axis in LiNbO3.

It is tempting to interpret the MIM-Re fringes as the standing-wave

47

Figure 4.4: Full set of MIM data of the single DW. MIM-Re images and thecorresponding line profiles (averaged over 100 repeated line scans) at 6 differentfrequencies. The first two dips nearby the DW are labeled by pink dashed lines.Peaks and other valleys are labeled by red and blue dashed lines, respectively.The oscillation amplitude of each peak (valley) is determined as the differencebetween its signal and the average signal of nearby valleys (peaks). The blackdashed lines are fits to the inverse square of the distance to the DW. All scalebars are 5 µm.

patterns of the acoustic displacement fields underneath the tip, similar to

those measured by scanning laser vibrometry [117, 118], scanning electron mi-

croscopy [119], and scanning probe microscopy [120, 121]. However, should

the data represent a standing-wave pattern due to strong reflection off a DW,

as suggested by an earlier MIM work [122], the measured periodicity would

indicate the dominance of a guided wave with an extraordinarily large phase

velocity of 8.8 km/s. In fact, since the acoustic impedance is the same for

both domains, the reflection of displacement fields from the yz-DW is rather

48

Annotation v (km/s) Properties Polarization

Rayleigh surface wave 3.82Surface guided,

non-leakyx- and

z-dominantSlow transverse bulk wave 4.08 Bulk continuum y-dominant

Bleustein-Gulyaevpseudo surface wave

4.55Surface guided,

leakyy-dominant

Fast transverse bulk wave 4.79 Bulk continuum z-dominantLongitudinal bulk wave 6.55 Bulk continuum x-dominant

Table 4.1: Velocities and properties of the acoustic waves in LiNbO3. Velocitiesand properties of the acoustic waves propagating along the x-axis on z-cutLiNbO3 surface. The Rayleigh SAW is confined to the surface (non-leaky).The Bleustein-Gulyaev pseudo-SAW has a velocity greater than that of theslow transverse bulk wave. Its energy therefore leaks to the bulk during thepropagation.

weak for both SAW and P-SAW, whereas the associated electric fields change

sign across the DW due to the opposite piezoelectric coefficients (Figure 4.5).

The reflection of the displacement fields of SAW and P-SAW from the

DW located in the yz-plane can be analyzed by FEA modeling. Assuming a

+x-propagating wave (ω: frequency, k: wave vector, α: attenuation coefficient)

is reflected at x = x0 with a reflection coefficient of r and a phase slip of ϕ,

the total displacement within x ∈ (0, x0) can be expressed as follows (Figure

4.5(a)).

u = ei(ωt−kx) · e−αx + rei(ωt+kx+ϕ) · e−α(x0−x)

The attenuation and reflection coefficients can be evaluated by taking the

product of u and its complex conjugate. The four situations based on the

49

Figure 4.5: DW reflection of SAW and P-SAW. (a) Schematic of the one-dimensional incident, reflected, and transmitted waves. (b) Simulated y-component of the displacement map without (top) and with the DW (bottom),where the spacing of the interdigital transducer (IDT) matches the wavelengthof the P-SAW. (c) Normalized uy ·u∗y near the surface as a function of the dis-tance from the DW. (d) Simulated z-component of the displacement map with-out (top) and with the DW (bottom), where the spacing of the IDT matchesthat of the Rayleigh-like SAW. (e) Normalized uz · u∗z near the surface as afunction of the distance from the DW.

values of r and α are listed below.

u · u∗ =

1, [α = 0, r = 0]1 + r2 + 2r cos (2kx+ ϕ), [α = 0, r 6= 0]e−2αx, [α 6= 0, r = 0]e−2αx + r2e−2α(x0−x) + 2re−2αx0 cos (2kx+ ϕ), [α 6= 0, r 6= 0]

(4.1)

Figure 4.5(b) shows the simulated y-component of the displacement

fields, where the source is an interdigital transducer (IDT) ∼ 100 µm away

50

from the DW. The spacing in the IDT is set to excite the P-SAW only. From

the wave amplitude near the surface, it is obvious that the energy of the P-SAW

continuously leaks into the bulk (slow transverse bulk wave) as it propagates

away from the source. In Figure 4.5(c), we plot the simulated uy · u∗y at ∼ 100

nm below the surface. By fitting to an exponential decay, we can obtain a wave

attenuation of ∼ 0.4 dB/wavelength, consistent with the literature [111]. The

simulated results with a DW are shown in Figure 4.5(b) and (c). The small

modulation in uy · u∗y is due to the DW reflection. From Equation 4.1, one

can show that 4re−αx0 = 0.07 and the reflection coefficient r ≈ 2.6%. Similar

results are also obtained for the Rayleigh-type SAW in 4.5(d) and (e), which

does not attenuate during the propagation. Here uz is analyzed since the SAW

is primarily polarized along the z-direction. A reflection coefficient of r ∼ 2%

is calculated for the SAW. The voltage standing wave ratio (VSWR) can be

calculated as follows.

V SWR =1 + r

1− r(4.2)

Due to the small r, the VSWR ≈ 1.05 is very close to 1 for both SAW

and P-SAW, indicative of the weak DW reflection for these surface waves. We

emphasize that since the acoustic impedance is the same for up and down

domains, a vanishing r is indeed expected for the displacement fields. The

electric fields associated with the propagating waves, however, will experience

a sign reversal across the DW due to the opposite piezoelectric coupling coef-

ficient. A careful analysis of the tip-sample interaction is therefore necessary

to understand the intriguing interference pattern in Figure 4.2.

51

4.3 Theoretical Analysis

In LiNbO3, the local mechanical strain and electric field are coupled by

the piezoelectric effect. As a result, the tip displacement under an AC bias

in the PFM measurement can be quantitatively analyzed by 3D finite-element

modeling [123, 124]. The MIM, on the other hand, measures the total power

dissipation and numerical simulations have to take into account the energy

transduction in the entire sample rather than the local displacement under-

neath the tip. Here, energy conservation dictates that the loss in electrical

power ~J · ~E∗(j: current density; E: electric field) is equal to the mechanical

power ~F · ~u∗ (F : electromechanical force density; u: displacement field; u:

time derivative of u), which excites various acoustic waves in solids [113]. The

electromechanical force can be derived from the divergence of the stress field

as ~F = divT .

The MIM tip can be viewed as a point voltage source in the sagittal

xz-plane since its diameter is much smaller than the acoustic wavelength at

GHz frequencies. The three components of the electric field from such a point

charge are as follows.

Ex =q

4πε0· xr3, Ey =

q

4πε0· yr3, Ez =

q

4πε0· zr3. (4.3)

The electromechanical force induced by the piezoelectric coupling can be calcu-

lated as the divergence of the stress tensor T = cS−eTE [113]. For simplicity,

we only consider the second term associated with the external electric fields.

~F = divText = ∇ · (−eTE) (4.4)

52

We can then plug the piezoelectric tensor of LiNbO3 [104, 125] into the

equation as follows.

−

e11 e21 e31e12 e22 e32e13 e23 e33e14 e24 e34e15 e25 e35e16 e26 e36

ExEyEz

= −

0 −2.5 0.230 2.5 0.230 0 1.30 3.7 0

3.7 0 0−2.5 0 0

ExEyEz

(4.5)

Using vector calculus, the three components of the force density as the diver-

gence of the external stress tensor are:

Fx = 2.5∂xEy − 0.23∂xEz + 2.5∂yEx − 3.7∂zEx

Fy = 2.5∂xEx − 2.5∂yEy − 0.23∂yEz − 3.7∂zEy

Fz = −3.7∂xEx − 3.7∂yEy − 1.3∂zEz

(4.6)

The analytically calculated force densities are shown in Figure 4.6.

When the tip is on top of a single domain (Figure 4.6(a)), Fx is an odd func-

tion (antisymmetric) with respect to x = xtip, whereas Fy and Fz are even

functions (symmetric) with respect to the tip. On the other hand, due to the

sign flip of the piezoelectric tensor in opposite domains, the parity of all three

components of the force density is reversed when the tip is on top of the DW

(Figure 4.6(b)).

Since P-SAW is dominated by the y-component of its displacement

fields, the power transduction is predominantly determined by the overlap

between Fy and uy. The vector calculus result also shows that Fy is symmetric

around the tip on a single domain (Figure 4.6). To satisfy the continuity

condition, uy is also an even function with respect to x = xtip. Because of the

53

Figure 4.6: Analytically calculated mechanical force density. Three compo-nents of the mechanical force density (force per unit volume) analytically cal-culated from the piezoelectric coupling when (a) the tip is on top of one singledomain and (b) the tip is on the DW.

sign flip of e across the DW, the overlap integral of Re(∫Fy · u∗y dx) in the two

shaded areas in Figure 4.7(a) cancels each other, leading to a drop in power

transduction when the tip is close to the wall.

As the tip moves away from the DW, the truncated overlap integral

within x ∈ (0, 2xtip) oscillates with the same periodicity as uy and shows the

power-law decrease of amplitude. Here the overlap integral of nonlocal acous-

tic wave sources are analogous to that in phase-array antennas [98], where

distributed electromagnetic source configuration and position result in corre-

sponding variations of the antenna impedance. Since the fringe patterns are a

54

Figure 4.7: Numerical simulation of the power transduction near a single do-main wall. (a) (top) Schematic view of the tip-sample configuration in thexz-plane. (middle) Sketches of the y-components of the mechanical force and(bottom) time derivative of the displacement near the surface. The sign of Fyis flipped in opposite domains due to the sign reversal of piezoelectric coeffi-cients. The overlap integral of Fy · uy in the two shaded areas cancels eachother. (b) (top to bottom) Snap shots of the x-, y-, and z-components of thesimulated time derivative of displacement fields in the xz-plane. The P-SAWwith a wavelength of 4.55 µm at 1 GHz is seen in the y-component of thevelocity fields. (c) Numerical simulation of the dissipated electrical power asa function of the tip position. The strength of the oscillation amplitude scaleswith 1/x (dashed lines) instead of 1/x2 from the wall due to the use of a linesource in the modeling.

result of the interference of nonlocal power transduction rather than the local

displacement fields of counter-propagating waves, oscillations in the microwave

dissipation can be observed even in the absence of DW reflection. In reality,

partial reflection of the SAW and P-SAW displacements always exist at the

DW. The fact that only one-wavelength oscillation is visible in Figure 4.3,

however, indicates that DW reflection is insignificant and does not change the

overall picture here.

The qualitative picture above is confirmed by numerical simulations

55

using finite-element analysis (FEA). Due to the prohibitive computational cost

of a full 3D modeling for the sample volume exceeding 1000 λ3, we truncate

the material in the y-direction using periodic boundary condition, effectively

treating the tip as an infinitely long line source (1 V at 1 GHz) along the y-axis

(Figure 4.8).

Figure 4.8: Details of the FEA modeling. Snap shots of (a) x-, (b) y-, and (c)z-components of the simulated velocity (time derivative of the displacement)fields. The inner half-circle is LiNbO3 and the outer half-ring is the perfectlymatched layer (PML). The DW (dashed line) lies in the middle of the sample.The slow transverse and longitudinal bulk waves propagating along the z-direction can be seen from the uy and uz plots, respectively.

Basically, we simulate a thin plate (1 µm in thickness) with periodic

56

boundary condition along the y-direction. The finite thickness is to enable

the motion in the y-axis. Except for the fact that the E-field decays as 1/r

from the tip rather than 1/r2 in the actual case, this approximation captures

the essential physics in our experiment, especially the phase velocities of the

acoustic waves. The DW separating the two oppositely polarized domains is

positioned in the yz-plane. The LiNbO3 region, whose permittivity, piezoelec-

tric coefficient, and elasticity tensor are taken from the literature [104, 125], is

bounded by the perfectly matched layer (PML) to avoid wave reflection from

the boundary. The tip on the sample surface is modeled as a line source along

the y-axis with an oscillating voltage of V = V0ei2πft, where V0 = 1 V and

f = 1 GHz. With these input conditions, the FEA software (COMSOL4.3,

Structural Mechanics module, linear solver) can compute the time-dependent

displacement (~u) fields, from which the velocity (~u), electric field ( ~E), current

density ( ~J), and other physical parameters can be derived and analyzed.

Figure 4.8 shows snap shots (t = 0) of ~u fields when the tip is 15 µm

away from the DW. A video clip showing the time evolution of the propagating

waves is also included in the Supplementary Information Movie S1 of Reference

[30]. As is evident from the plots, the tip excites several acoustic modes due to

the piezoelectric coupling. Near the surface, the pseudo surface acoustic wave

(P-SAW) is clearly seen in the uy map since the P-SAW is mostly transverse-

horizontal. The Rayleigh-type SAW, on the other hand, is predominately

polarized along the z-axis. Consequently, the beat between SAW (λ = 3.82

µm at 1 GHz) and P-SAW (λ = 4.55 µm at 1 GHz) appears in the uz map.

57

The leaking of energy from the P-SAW to the slow transverse bulk wave is

obvious in all three plots. Finally, for the bulk waves propagating along the

z-axis, the slow transverse wave (λ = 3.6 µm at 1 GHz) and the longitudinal

wave (λ = 7.3 µm at 1 GHz) can be seen from the uy and uz plots, respectively.

Note that these bulk velocities [112] are different from the values in Table 4.1,

which are along the x-axis of LiNbO3.

Among the various acoustic waves excited by the tip (Figure 4.7(b)),

bulk waves do not contribute to periodic oscillations as the tip moves away

from the DW. Further analysis (Figure 4.9) shows that the power of the P-

SAW excited by the tip is ∼ 5 times higher than that of the SAW, supporting

our qualitative description above.

The tip voltage at microwave frequencies is capable of exciting both

the SAW and P-SAW. In order to analyze the relative strength between the

two waves, we first use the linear solver to compute the displacement fields

when the tip is on top of a single domain, as shown in Figure 4.9(a). The beat

between P-SAW and SAW can be seen near the surface in the uz plot. Next,

we solve the eigen-modes of P-SAW ~uP−SAW (Figure 4.9(b)) and SAW ~uSAW

(Figure 4.9(c)) by FEA. The leaky nature of the P-SAW (displacement not

confined to the surface) is evident in the depth profile in Figure 4.9(d). Note

that the P-SAW is primarily polarized in the y-direction on the surface, rem-

iniscent of the electroacoustic Bleustein-Gulyaev SAW [114, 115]. In contrast,

the non-leaky SAW is confined to the surface and its primary polarization is

along the z-axis.

58

Figure 4.9: P-SAW versus SAW. (a) (Top to bottom) Snap shots of x-, y-, and z-components of the simulated displacement fields when the tip is ontop of a single domain. The PML region is not shown. (b) Eigen modesof the P-SAW and (c) SAW displacement fields. Note that the two waveshave different wavelengths at 1 GHz due to the different velocities. (d) Depthprofiles along the dashed lines in (b) and (c) of the normalized P-SAW and (e)SAW displacement fields. (f) Overlap integral between the linear-solver resultsin (a) and eigen-solver results in (b) and (c). The relative strength betweenP-SAW and SAW excited by the tip voltage can be extracted.

The tip-induced displacement ~u in Figure 4.9(a) contains both P-SAW

and SAW on the surface. To analyze their relative strength, we take the

overlap integral of∫~u~uP−SAW dx and

∫~u~uSAW dx within the dashed box in

Figure 4.9(a), where the phase of ~u varies continuously from 0 to 2π. The result

in Figure 4.9(f) shows that the amplitude of P-SAW is ∼ 2.2 times of the SAW.

In other words, the transduction from electrical energy into mechanical energy

is ∼ 5 times more effective for the P-SAW than that for the SAW, which

59

explains the dominance of P-SAW in the experiment. Note that since there

are only a few oscillations in the MIM-Re data (Figure 4.3), it is difficult to

separate SAW (λ = 3.82µm at 1 GHz) and P-SAW (λ = 4.55µm at 1 GHz)

from the Fourier transform.

As seen in Figure 4.7(c), the simulated power dissipation Re(∫~J∗ ·

~E d3r) reproduces major features in the MIM-Re data, including the prominent

dip at the DW and the damped oscillations with one-λ periodicity. The use of

a line source in the modeling is responsible for the 1/x rather than 1/x2 decay

of the oscillation amplitude. The reduced separation between the first pair of

troughs, 1.8 λ rather than 2 λ, is also seen in the FEA result. We speculate

that the deviation is due to DW reflection being no longer negligible when the

tip-wall distance is less than one wavelength, although further work is needed

to understand this behavior.

4.4 Further Experiments

4.4.1 Double Domain Walls

We now move on to the LiNbO3 sample with two parallel DWs (PFM

image in Figure 4.10(a)).

The corresponding MIM-Re images at 5 different frequencies are shown

in Figure 4.10(b). The strength of the fringes increases as f goes from 850 MHz

to 967 MHz and decreases as f further increases towards 1115 MHz. Figure

4.10(c) and (d) display the simulated acoustic fields and microwave power loss

at f = 967 MHz. As summarized in Figure 4.10(e), when the DW spacing d

60

61

covers the distance between the first pair of troughs (∼ 1.8λ) and an integer

multiple the wavelength, e.g., d ≈ 1.8λ + 2λ at f = 967 MHz, the two sets

of ripples centered around the DWs reinforce each other, resulting in stronger

peak-to-valley contrast. On the contrary, when d−1.8λ ≈ 2±0.5λ at 850 MHz

and 1115 MHz, the two sets of ripples are opposite in phase by 180, resulting in

suppressed oscillation strengths. More importantly, the ripples remain strong

outside the two DWs at f = 967 MHz, which is a direct evidence of the non-

local power transduction nature of the MIM-Re images. If the data represent

the standing-wave amplitude of the acoustic displacement fields [122], this on-

resonance-like image should be free of oscillations outside the double-DW, since

a Fabry-Perot interferometer does not reflect on resonance [93]. In addition,

the fringes in Figure 4.10(b) exhibits periodicity identical to the wavelength of

the underlying acoustic wave, whereas the standing-wave patterns in a Fabry-

Perot interferometer has a periodicity of half-λ.

Figure 4.10 (preceding page): Imaging and simulation of the double-DW sam-ple. (a) Out-of-plane PFM phase image of the double-DW sample. (b) MIM-Re images and the corresponding line profiles at 5 different frequencies. Allscale bars are 10 µm. (c) Tip-sample configuration (top) and simulated uyfield at 967 MHz (bottom) of the double-DW sample. (d) Simulated electricalpower dissipation as a function of the tip position. The two dashed lines indi-cate the locations of the two walls. (e) (d− 1.8λ)/λ as a function of f . The 5frequency points, which are color coded as that in (b), are selected such that(d− 1.8λ) roughly equals to 1.5λ,1.75λ, 2.0λ, 2.25λ, and 2.5λ, respectively.

62

4.4.2 Two Dimensional Patterns

Finally, we present the MIM results on several closed domain structures.

Figure 4.11 show a complete set (AFM, PFM, and MIM) of data of four corral

domains shaped in an equilateral triangle, a hexagon, a circle, and a square,

respectively.

Figure 4.11: Complete set of data of the enclosed domains. All scale bars are10 µm.

The sample surface is very smooth after the electrical poling. As ex-

pected, the opposite LiNbO3 domains display DW contrast in the PFM ampli-

tude image and 180 contrast in the PFM phase image. The MIM-Im images

63

exhibit weak contrast since the tip-sample susceptance is dominated by the

dielectric response.

In order to analyze the interference patterns in the MIM-Re data caused

by piezoelectric transduction , we extract the PFM and MIM-Re images, which

are shown in Figure 4.12.

Because of the crystal symmetry [104], DWs on the z-cut LiNbO3 sur-

face can only form straight lines along the three y-equivalent axes and become

curved in other directions. Consequently, the domain designed to be a circle

(Figure 4.12(c)) appears as a rounded hexagon after electrical poling, and the

domain designed to be a square (Figure 4.12(d)) appears as a distorted rectan-

gle. Beautiful interference patterns due to the superposition of ripples around

each DW are observed in the MIM-Re images. For instance, the rectangular-

lattice-like pattern in Figure 4.12(d) can be viewed as an overlay of two sets of

oscillations parallel to the x-axis and y-axis. The different velocities of P-SAW

along the two directions [126], as calculated from the FEA (Figure 4.12(e)),

manifest in the different oscillation periods in the line profiles (Figure 4.12(f)).

Similar to the double-DW results, these features are different from the stand-

ing wave patterns in quantum corrals [127] in that the adjacent nodes are not

spaced by half-λ. And the existence of such patterns does not indicate the pres-

ence of acoustic resonance [122]. In other words, the bright/dark regions in the

MIM-Re images are not directly associated with the constructive/destructive

interference of acoustic waves underneath the tip. Instead, they mark the tip

locations around which the piezoelectric transduction over a distributed region

64

65

of tens of microns, much wider than the closed domains themselves, is highly

effective/ineffective.

4.5 Conclusion

Putting our findings in perspective, we have introduced a special type

of interferometry by spatial mapping and numerical modeling of the electroa-

coustic power conversion in ferroelectric materials. The images of microwave

dissipation reveal large internal degrees of freedom in piezoelectric and elastic

tensors, which are not accessible by measurements of the acoustic displace-

ment fields. For SAW devices, MEMS/NEMS, and phononic crystals, the

spatial variation of piezoelectric effect can substantially influence the system

performance. The submicron spatial resolution is also desirable to explore the

effect of wave scattering, diffraction, and localization on the energy transduc-

tion. In all, microwave imaging may emerge as a powerful tool to probe the

intimate coupling of electric and strain/stress fields in these systems.

Figure 4.12 (preceding page): Interference of piezoelectric transduction in cor-ral domains.(a-d) PFM-ph (top) and MIM-Re images at f = 955 MHz (bot-tom) of four closed LiNbO3 domains. Clear interference patterns due to thesuperposition of ripples around each DW are seen in the MIM-Re data. Allscale bars are 10 µm. (e) Velocities of P-SAW and SAW calculated from theCOMSOL eigen-solver as a function of the angle between propagation vectork and the y-axis on the z-cut LiNbO3 surface. (f) Line profiles in (d), showingdifferent oscillation periods along x-direction (red) and y-direction (blue). Theresult is consistent with the higher P-SAW velocity along the x-axis than thatalong the y-axis.

66

Chapter 5

Imaging Acoustic Wave

The working principle of microwave impedance microscopy (MIM) is

introduced in Chapter 2. It should be noted that most microwave microscopy

works, although with different acronyms, are based on the same underlying

physics. Inspired by the interference pattern discussed in the previous chap-

ter, we noticed that our MIM setup is very sensitive to electromechanical

transduction measurement because the microwave power on the tip is rela-

tively low. Therefore, if we use the tip as a detector only, the detection of

waves generated by external sources might be feasible, which is particularly

interesting in surface acoustic wave (SAW) devices field. In this section, we re-

port the visualization of SAWs on ferroelectric samples by transmission-mode

microwave impedance microscopy (T-MIM). The SAW potential launched by

the interdigital transducer is detected by the tip and demodulated by the mi-

crowave electronics as time-independent spatial patterns. Wave phenomena

such as interference and diffraction are imaged and the results are in excellent

agreement with the theoretical analysis. Our work opens up a new avenue to

study various electromechanical systems in a spatially resolved manner1.

1The results in this chapter are primarily based on the previous publication: L. Zheng,D. Wu, X. Wu, and K. Lai, Visualization of surface-acoustic-wave potential by transmission-

67

5.1 Introduction

5.1.1 Surface Acoustic Wave (SAW)

Atoms in all materials are constantly shaking, which is largely respon-

sible for the transfer of heat and sound. In a uniform crystalline solid, the

motion of the lattice can be decomposed into just a handful of normal modes

of vibration, or in the language of quantum mechanics, a set of quantized eigen-

modes of the elastic structure known as phonons [61]. In the frequency (f)

regime of interest for microwave engineers, there exist three (one longitudinal,

two shear) branches of vibrational modes with long wavelengths – ‘long’ when

compared with the atomic spacing. Their frequencies, which represent the

energy of each quantum, are linearly proportional to the inverse wavelengths,

which represent the momentum of each quantum. The ratio between the two

is typically several kilometers per second. In good crystals with few imper-

fections, these vibrations travel very long distance – ‘long’ when compared

with the wavelength – with little decay of the amplitude under the ambient

temperature and pressure. At audio frequencies, we can hear these sounds

propagating in solids. They are commonly known as acoustic waves.

The connection between acoustic waves and microwave electronics dated

back to the 1950s, when scientists in the Bell Labs demonstrated the genera-

tion and detection of few-GHz longitudinal and shear waves in quartz crystals

mode microwave impedance microscopy,” Physical Review Applied, vol. 9, no. 6, p. 061002,2018. As the first author, I designed the experiment with K. Lai and performed the exper-iment. I also participated in data analysis and manuscript revision.

68

[128]. It was soon recognized that, because the speed of sound in solids is 105

times smaller than the speed of light, microwave acoustic devices using piezo-

electric transducers [129] could afford substantial miniaturization in the system

dimension. In particular, compact resonators that take advantage of the ul-

tralow loss of bulk acoustic waves (BAWs) in quartz have found widespread

application in professional electronic equipment [130]. Starting from the late

1960s, much attention was directed to the guided waves along the free surface

of materials [131], which were first explained by Lord Rayleigh in his semi-

nal 1885 paper [132]. Named after the discoverer, Rayleigh waves or surface

acoustic waves (SAWs) have lower speed than BAWs and low propagation loss.

They are easily accessible from the surface and readily compatible with the

planar technology developed for integrated semiconductor devices. To date,

SAW devices such as delay lines, filters, oscillators, transformers, and sensors

are indispensable in most microwave systems [100, 133] and will continue to

thrive in the 5G era.

The characteristic dimension of SAW devices using interdigital trans-

ducers (IDTs) is set by the acoustic wavelength in piezoelectric solids, which is

typically a few micrometers at 1 GHz. In other words, the inherent length scale

to observe wave phenomena, such as interference, diffraction, and localization

of SAWs, is in the mesoscopic regime. High-resolution imaging of the acous-

tic displacement field or the accompanying electric field is therefore highly

desirable for designing, characterizing, and ultimately improving the SAW de-

vices [133]. Moreover, it has been shown that, in artificial structures with

69

periodic piezoelectric properties, the coupling between GHz electromagnetic

waves (photons) and acoustic waves (phonons) can lead to polaritons with dif-

ferent phononic bands [102, 109, 134–137]. Spatial mapping of the SAW fields,

in conjunction with numerical simulations of the phononic metamaterials, is

thus of great interest to advance this research field.

5.1.2 Visualization Tools

In the past few decades, microscopy techniques have evolved rapidly to

provide various means to visualize GHz acoustic waves.

Figure 5.1: Visualization tools of SAW. (a) Stroboscopic X-ray topographof travelling Rayleigh waves on LiNbO3 taken with the driving signal phaselocked to the synchrotron radiation emission [138]. The SAW is excited at 38MHz. (b) Surface phonon image on the (001) TeO2 surface taken by pump-probe scanning laser interferometry [117]. SAWs are thermoelastically excitedby the 415 nm pump laser with a repetition rate of 80 MHz. (c) Scanningelectron photographs of the traveling SAW on LiNbO3 [119]. The excitationfrequency is 25.127 MHz. (d) SAFM amplitude image on (001) GaAs [121].The circular electrode is surrounded by an open ring electrode connected to theRF ground. The SAW is excited at 890 MHz and a nearby IDT (not shown)sends a slightly detuned signal for the SAFM imaging. These four panelsadapted with permissions from Ref. [138], copyright 1982, Springer Nature;Ref. [117], copyright 2002, American Physical Society; Ref. [119], copyright1992, AIP Publishing; and Ref. [121], copyright 2009, IOP Publishing.

70

Figure 5.1(a) shows the first image of radio-frequency Rayleigh waves

by stroboscopic x-ray topography [138]. Here the ‘strobe’ is obtained by phase

lock between the x-ray source and the excitation of the SAW device, resulting

in a stationary image of the propagating surface wave [138–140]. A major

drawback of the technique is the need of coherent X-rays from a synchrotron

radiation light source, which limits its widespread use in the laboratory scale.

A somewhat related technique that also utilizes light to visualize acoustic

displacement is the scanning laser reflectometry [141, 142] or interferometry

[117, 118, 143]. The technique can resolve sub-picometer out-of-plane displace-

ment with sub-nanosecond temporal resolution, as exemplified by the beautiful

snapshot of the propagating wave on (001) TeO2 surface [117] in Figure 5.1(b).

The lateral spatial resolution, however, is diffraction-limited to ∼ 1 µm due to

the use of visible laser. In the early 1990s, it was also demonstrated that the

secondary electrons in a scanning electron microscope (SEM) could be modu-

lated by the SAW electric field, forming a stationary pattern of the propagating

wave [119, 144, 145] (Figure 5.1(c)). The applicability of this method, however,

is rather limited due to the strong charging effect in insulating piezoelectric

crystals, resulting in a moderate resolution of about 1 µm and an operation

frequency below 0.5 GHz.

The advent of scanning probe microscopy has brought in new impetus

to image SAWs with nanoscale spatial resolution. In a typical atomic-force

microscopy (AFM) setup, the mechanical resonance of the cantilever of several

hundred kHz is far below the microwave regime. As a result, the scanning

71

acoustic force microscopy (SAFM) works in a heterodyne scheme [120, 121,

146], which mixes the acoustic sample wave with a slightly detuned reference

wave from another source. This leads to cantilever oscillations at a difference

frequency being measured by the conventional photodiode detection (Figure

5.1(d)). A similar scheme can be applied to the scanning tunneling microscopy

(STM) imaging of SAWs [147]. The interested readers are referred to the

review article by Hesjedal [121] for a comprehensive coverage of the technique.

Except for SEM imaging, most of these tools are trying to capture the

tiny vibration of the acoustic waves. However, this pico-meter or even femto-

meter level vibration amplitude is super challenging to observe in individual

laboratories. Since elastic waves propagating in piezoelectric materials are

accompanied by a time-varying electric potential, which is of critical impor-

tance for acousto-electronic applications. It should be much easier if we could

perform the spatial mapping of such a potential instead of the vibration.

5.2 Transmission-mode MIM

As we have introduced in Chapter 2, the schematic of a typical MIM

is depicted in Figure 2.1 and Figure 5.2(a). In this mode of operation, the

microwave signal is delivered to the center conductor of the tip through an

impedance-match section. The reflected signal is amplified and demodulated

by an in-phase quadrature (IQ) mixer. As seen from the equivalent circuit

in Figure 2.3, the microwave electronics is detecting the small variation of

tip-sample impedance Zt−s during the raster scan. By adjusting the local os-

72

cillator (LO) phase φ, the real and imaginary components of the admittance

change ∆Yt−s = ∆(Z−1t−s) can be mapped as MIM-Re and MIM-Im images,

respectively. It should be noted that this MIM setup is similar to the S11 mea-

surement in a vector network analyzer (VNA). Since S11 represents the input

reflection coefficient and MIM tip is used for probing the reflected microwave

signal, we can rename this typical MIM as reflection-mode MIM (R-MIM).

It is straightforward to reconfigure R-MIM to the transmission-mode

(T-MIM) to perform direct imaging of SAW electric fields [29]. As illustrated

in Figure 5.2(b), the microwave signal is delivered to the IDT on a piezo-

electric sample and the tip acts as a movable receiver detecting the local RF

voltage Vs. The oscillating SAW field coupled to the tip is then demodulated

as two orthogonal output channels T-MIM-Ch1 and -Ch2. The equivalent

circuits of R-MIM and T-MIM are schematically shown in Figure 5.2(c) and

(d), respectively. The time-varying source potential Vs is picked up through a

tip-sample coupling impedance Z′t−s, followed by the same amplification and

demodulation in the microwave electronics.

At our operation frequency of ∼ 1 GHz, the cantilever probe can be

modeled as a lumped RLC element with Rtip = 4 Ω, Ltip = 2nH, and Ctip =

1 pF . At f = 1 GHz, the effective tip impedance |Ztip| ∼ 150 Ω is dominated

by the capacitive reactance. As shown in Figure 5.3(a), an impedance-match

network consisting of a quarter-wave cable (AstroLab, Astro-Boa-Flex III, ∼ 5

cm) and a tuning stub (Micro-Coax, UT-085C-TP, ∼ 5 cm) is needed to route

the tip impedance to the 50 Ω transmission line [32]. Figure 5.3(b) shows the

73

Z-match

Zt-s Ztip

MIM electronics

VS

Z-match

Zt-s

Ztip

Zin

MIM electronics

VS

Source

Re

Im

Coupler Amp.MixerR-MIM

RF LOI

Q

Z-match

Stub

Q.W.

Source

Ch1

Ch2

Coupler Amp.MixerT-MIM

RF LOI

Q

Z-match

Stub

Q.W.

50

(a) (c)

(b) (d)

Figure 5.2: Schematics of R-MIM and T-MIM. (a) Schematic of the R-MIM.The excitation signal is delivered to the tip and the reflected signal is amplifiedand demodulated by the IQ mixer to form the R-MIM-Re/Im images. (b)Schematic of the T-MIM. The excitation signal is delivered to the IDT onthe sample and the transmitted signal is amplified and demodulated by theIQ mixer to form the T-MIM-Ch1/Ch2 images. (c) Equivalent circuits of theR-MIM and (d) T-MIM.

calculated return loss |Γ| of the Z-match circuit, which agrees with the result

measured by a vector network analyzer. Using the standard transmission-

line analysis [40], one can then compute the effective impedance viewed from

the tip side, i.e., the input impedance Zin of the receiver. For the T-MIM

experiment, a large |Zin| is desirable for signal pick-up. As shown in Figure

5.3(c), |Zin| reaches a maximum of ∼ 1kΩ at the matching frequency, which

is used as the operation frequency for both R-MIM and T-MIM.

The tip-sample coupling impedance Z′t−s is estimated as follows. First,

the relation between V1 and V2 (Figure 5.3(a)) can be analyzed by considering

the forward and backward propagating waves in the quarter-wave cable. On

the other hand, these two voltages provide the link between the source signal

74

Vs and the MIM output signal VMIM−out.

V1 = Vs · Zin/(Zin + Z′

t−s) ≈ Vs · Zin/Z′

t−s

V2 = VMIM−out/GMIM

(5.1)

Here GMIM = 86 dB is calibrated for the T-MIM electronics. Based on our

experimental data, a peak-to-peak SAW potential of 0.2 V (see Figure 5.7)

corresponds to a peak-to-peak T-MIM output signal of ∼ 2 V (Figure 5.5).

The computed tip-sample coupling impedance is plotted in Figure 5.3(d). The

result shows that in this particular experiment, |Z ′t−s| is around 160 kΩ, or an

effective capacitance of 1 fF, at 1 GHz. Note that Z′t−s strongly depends on

the tip apex condition and the sample properties. A small Z′t−s is desirable for

efficient T-MIM detection, which, however, usually comes at a price of blunt

tip and reduced spatial resolution.

In the next section, the images both traveling wave and standing wave

will be presented. It is worth noting that similar transmission-type probes

have been used to map out the RF fields in microwave resonators [148, 149]

and metamaterials [150, 151]. In those systems [152, 153], however, the char-

acteristic length scale is determined by the electromagnetic wavelength (30 cm

at 1 GHz) and a mesoscopic spatial resolution is not necessary.

5.3 Imaging SAW

5.3.1 Traveling Wave

Figure 5.4(a) shows the SEM image of a pair of IDTs used in our

experiment. The device was designed to excite the x-propagating Rayleigh-

75

0

-10

-20

-30

0

1

0.5

0

200

100

50

Zt-s

Ztip

Zin

VS Stub

Q.W.

1 2

|Zin| (k

)|Z t-

s| (k

)|

| (d

B)

0.8 1.0 1.2

f (GHz)

(b)

(c)

(d)

(a)

Figure 5.3: T-MIM Microwave Circuit Analysis. (a) Equivalent circuit ofthe T-MIM tip and the Z-match network. (b – d) Simulated return loss ofthe Z-match circuit, input impedance of the tip as a receiver, and tip-samplecoupling impedance. The matching frequency with minimal |Γ| is used as theoperation frequency for both R-MIM and T-MIM modes.

type SAW on the z-cut LiNbO3 substrate, which was poled to be a single

ferroelectric domain prior to the device fabrication. LiNbO3 has a trigonal

crystal structure with a mirror yz-plane and a direct triad z-axis along the polar

direction [104]. Using finite-element modeling (see the following Modelling

and Analysis section Figure 5.7), one can show that the piezoelectric SAW

potential Vs is about 10% of the excitation voltage at the transmitting IDT

(±1 V). In addition, since the coupling impedance Z′t−s between the tip and

metal electrodes is much smaller than that between the tip and LiNbO3, the

signals on the IDTs are very strong and saturate the T-MIM output. The S-

76

parameters of the two IDTs measured by a vector network analyzer are plotted

in Figure 5.4(b). The passband of ∼ 50 MHz around 1 GHz is consistent with

the use of 20 pairs of interdigital fingers [133]. The dip of S12 in the middle of

the passband is likely due to the SAW reflection from the receiving IDT.

Figure 5.4(c) displays the simultaneously acquired AFM and T-MIM-

Ch1 images when the excitation IDT is powered by 10 dBm microwave at

f = 957 MHz. While only the interdigital fingers are seen in the surface

topography, the electrical potential on both the IDT and the LiNbO3 surface

can be clearly imaged by the T-MIM.

In Figure 5.5, we focus on the data taken in an area of 10 µm× 20 µm

between the two IDTs. The featureless R-MIM images in Figure 5.5(a) indicate

that there is no permittivity or conductivity variation, whereas the two T-MIM

images in Figure 5.5(b) exhibit sinusoidal patterns. As discussed before, the

tip is picking up an input signal that is proportional to the SAW potential.

Without loss of generality, the signals at the RF and LO ports of the mixer

can be represented as VRF ∝ Vs ∝ ei(ωt−kx) and VLO ∝ ei(ωt+ϕ) (ω: angular

frequency, k: acoustic wave vector, ϕ: mixer phase), respectively. Ignoring

the terms containing 2ωt, we obtain the output signals from the quadrature

mixer as follows.

VCh1 ∝ Re(VRFV∗LO) = cos(kx+ ϕ)

VCh2 ∝ Im(VRFV∗LO) = − sin(kx+ ϕ)

(5.2)

In other words, the electronics demodulate the time-varying SAW potential

into time-independent spatial patterns, which is in good agreement with the

77

T-MIM-Ch1 (f = 957 MHz)

AFM

0 nm

10 V

100 nm

-10 V0.8 0.9 1.0 1.1

-20

-30

-40

-1

-2

S11

(dB

)S

12

(dB

)

f (GHz)

957 MHz

V

20 m

(a)

(b) (c)

4 m

78

T-MIM data. The line profiles in Figure 5.5(c) show the same amplitude and

a phase difference of 90 between the two channels. We have also confirmed

that a change in the mixer phase ϕ introduces the same phase shift to both

channels. By fitting the periodicity of the sinusoidal curves, a phase velocity

of v = 3.8 km/s is obtained, which is consistent with that of the x-propagating

Rayleigh SAW [106].

5.3.2 Standing Wave

We now turn to the T-MIM imaging of a standing wave formed by

two counter-propagating SAWs. Figure 5.6(a) shows the schematic of the

experimental setup, where two balanced signals (0 dBm in amplitude) with a

phase offset of θ are fed into the pair of IDTs. This geometry is technologically

important in that it can create acoustic trapping potentials for electrons [154].

Following the same analysis above, the input signals to the mixer can be

written as VRF ∝ ei(ωt−kx) + ei(ωt+kx+θ) and VLO ∝ eiωt. The LO phase φ is

omitted since it contributes the same phase to both channels. The mixer then

Figure 5.4 (preceding page): Characterization of a SAW device. (a) SEMimage of the SAW device with a pair of IDTs. (b) Return loss S11 and insertionloss S12 of the SAW device measured by a vector network analyzer. The T-MIM frequency of 957 MHz is labeled in the plots. (c) AFM and T-MIMimages in the dashed rectangular region in (a). Wave-like features are seen inthe T-MIM data. The scale bars are 4 µm.

79

1 V

-1 V

AFM

R-MIM-Re

R-MIM-Im

T-M

IM s

ign

als

(V

)

-1

0

1

0 5 10 15 20

Position (m)

Ch1 Ch2 Ch1, peak

Ch1, valley

Ch2, peak

Ch2, valley

v = f = 3.8 km/s

Number1 2 3 4 5

Po

sitio

n (

m)

0

10

20

T-MIM-Ch1

T-MIM-Ch2

0 V

0.2 V

0 nm

4 nm

0 mV

40 mV

4 m

(a) (b)

(c) (d)

80

generates two output signals as follows.

VCh1 ∝ Re(VRFV∗LO) = cos kx+ cos(kx+ θ)

VCh2 ∝ Im(VRFV∗LO) = − sin kx+ sin(kx+ θ)

(5.3)

By tuning the phase difference θ between the two counter-propagating

SAWs, the signal levels of the two T-MIM channels can be varied. When

θ = 0, the sinusoidal spatial patterns are expected to appear only in Ch1

(VCh1 ∝ 2 cos kx, VCh2 ∝ 0). The pattern should then be same in both channels

when θ = 90 (VCh1 = VCh2 ∝ cos kx − sin kx) and completely move to Ch2

when θ = 180 (VCh1 ∝ 0, VCh2 ∝ −2 sin kx). As seen in Figure 5.6(b), the

predicted evolution is again in excellent agreement with the measured T-MIM

data. The results demonstrate that T-MIM can probe the acoustic standing

waves in piezoelectric materials.

5.3.3 Modelling and Analysis

The piezoelectric SAW potential can be numerically computed by the

Structural Mechanics Module in commercial finite-element analysis (FEA)

software COMSOL 4.4. Here we simulate a thin plate (1 µm in thickness)

with periodic boundary condition along the y-direction. The LiNbO3 region,

Figure 5.5 (preceding page): T-MIM image of traveling SAW. (a) AFM andR-MIM-Re/Im images in an area between the two IDTs. (b) T-MIM-Ch1/Ch2images in the same area as (a). All scale bars are 4 µm. (c) Line profiles ofthe two T-MIM channels. The dash-dotted lines show that the two sinusoidalcurves are offset by 90. (d) Positions of the peaks and valleys in the T-MIMdata. The linear fits to the data points correspond to the Rayleigh SAW speedof 3.8 km/s.

81

20 m

𝜃

Splitter

Phase shifter Attenuator

Ch1

Ch2

= 0 = 90 = 180

4 m

0.25 V

-0.25 V

(a)

(b)

Figure 5.6: T-MIM image of traveling SAW. (a) Schematic diagram for theimaging of counter-propagating waves. Two signals (0 dBm in amplitude andphase offset by θ) split from the same source are fed into the two IDTs. (b)T-MIM images at different θ’s, showing the transition of signal strength fromCh1 to Ch2

whose permittivity, piezoelectric coefficient, and elasticity tensor are taken

from Ref.[104], is bounded by the perfectly matched layer (PML) to avoid

wave reflection from the boundary. The IDT spacing is set to be 3.8 µm. Al-

ternating voltage (1 V in amplitude and 1 GHz in frequency) and ground (0

V) are applied on the IDT fingers to excite the x-propagating Rayleigh-type

SAW on the z-cut LiNbO3 surface. Figure 5.7(a) shows the simulated piezo-

electric potential distribution in the sample, where the SAW is clearly seen.

82

The surface potential from the simulation (Figure 5.7(b)) indicates that the

peak-to-peak SAW potential is 0.2 V, which is the source signal for the T-MIM

measurement. Using this information in Eq.5.1, we are able to evaluate the

tip-sample coupling impedance Z′t−s, which is crucial to understand the signal

level in our T-MIM experiment in a quantitative manner.

0 V

0.6 V

PositionSu

rfa

ce

po

ten

tia

l

LiNbO3

IDT

0.2 V

3.8 m

10 m

(a) (b)

𝑥

𝑧

𝑦

Figure 5.7: Finite-element Modeling. (a) Piezoelectric potential distributionsimulated by finite-element modeling. The voltage on the IDT fingers in thissnapshot is 1V/0V. (b) Simulated SAW potential as a function of the position.

5.4 Ferroelectric Domain and SAW

Finally, we briefly discuss the visualization of SAW diffraction due to

the presence of a small domain. LiNbO3 wafers poled to be a single ferroelectric

domain are energetically unstable. Over an extended period, small domains

with opposite polarization may spontaneously form to reduce the electrostatic

energy. The domain inversion flips the sign of odd-rank tensors (polarization,

1st rank; piezoelectric tensor, 3rd rank), while leaving the even-rank tensors

(permittivity, 2nd rank; elasticity tensor, 4th rank) unchanged [104]. As the

acoustic impedance is mostly dependent on the density and elasticity of the

83

material, the SAW displacement field is not strongly affected by the domain

structure. In contrast, the SAW electric field, which is the gradient of the SAW

potential, changes sign across a domain wall due to the piezoelectric coupling.

In our sample, a small domain with spontaneous polarization reversal

is found near the left IDT (AFM image in Figure 5.8(a)), as seen in the PFM

image in Figure 5.8(b). Since its dimension is comparable to the acoustic

wavelength, wave diffraction is expected around the domain. In Figure 5.8(c),

the T-MIM surface potential maps indeed display very strong distortion of the

wave front in this region. The line profiles in Figure 5.8(d) further verify that

the T-MIM signals from both channels switch sign when passing through the

small domain.

5.5 More Applications

5.5.1 SAW Resonator

The capability to resolve the spatial distribution of SAW potential is of

particular interest to integrated phononic systems. For instance, when design-

ing SAW resonators [142, 155, 156], which feature stronger electrical coupling

and better confinement than the micromechanical counterparts, it is desir-

able to directly compare the simulated and measured spatial distribution of

energy intensity of each acoustic cavity mode. In a recent report, a high-Q

and small-mode-size SAW resonator on LiNbO3 was demonstrated by engi-

neering phononic band structures using adiabatically tapered structures [157].

As shown in Figure 5.9(a), the resonator consists of tapered couplers to en-

84

T-MIM-Ch1 T-MIM-Ch2

AFM PFM-ph

5 m 1 m

(a)

(c)

0

180

1 V

-1 V

4 nm

0 nm

Ch1

Ch2

0 5 10 15 20

Position (m)

T-M

IM s

ignals

(V

)

-1

0

1(d)

(b)

85

hance coupling to the source, phononic crystal sections to confine the phononic

modes, and the resonant cavity. Compared to conventional Fabry-Perot res-

onators [142] that employ unperturbed free surfaces at the center and Bragg

mirrors on the sides, the tapered grooves adiabatically change the reflectivity,

resulting in a significantly reduced scattering loss of acoustic waves into the

bulk and better confinement of phonons.

The SAW resonator device was designed to have two high-Q modes,

which are indeed observed in the S21 spectra measured by a vector network

analyzer (Figure 5.9(b)). The period of the central cavity part is fine-tuned to

align the resonant frequency of the fundamental mode to the center of phononic

crystal bandgap. Figure 5.9(c) displays the simulated energy density profiles

of the fundamental (Mode 1) and second-order (Mode 2) modes of the SAW

resonator. As shown in Figure 5.9(d), the single antinode in the middle of

the resonator in Mode 1 and dual antinodes in Mode 2 are vividly seen in

the SAW potential map measured by the T-MIM [157]. The high quality of

these modes can be further confirmed by taking T-MIM data near the resonant

frequency, as depicted in Figure 5.9(e). The substantial fall-off of the T-MIM

Figure 5.8 (preceding page): SAW passing through domains. (a) AFM imagein an area with a spontaneously reversed domain. (b) Close-up view of thePFM phase image inside the dashed square of (a). The polarization reversalof the internal domain is evident from the 180-phase contrast. (c) T-MIM-Ch1/Ch2 images in the same area as (a). Strong distortion of the wave frontis seen around the small domain. The scale bars are 5 µm. (d) T-MIM lineprofiles in (c), showing the sign change of piezoelectric potential in the oppositedomain.

86

87

signal strength within 1 MHz around f = 1.0384 GHz is a direct evidence of

the high-Q factor (∼ 10,000) of the second-order mode. We emphasize that,

while the simulation and S21 measurement demonstrate the basic function, the

spatially resolved T-MIM images offer much more information on each section

of the device. In-depth analysis of the local SAW images may reveal the

effect of fabrication imperfections, crystal defects, and scattering from other

phonons, which will provide insights on further improvement of the device

performance.

5.5.2 Other Devices

Besides the SAW modes in the high-Q resonator, we have also per-

formed a similar experiment on Gaussian shape IDTs to demonstrate the fo-

cusing of the generated SAW [158]. Moreover, it should be noticed that our

T-MIM measures the microwave field, which is not limited by SAW mode. For

Figure 5.9 (preceding page): SAW resonator. (a) Illustration of the SAWresonator on LiNbO3. The inset shows the optical microscope image of thefabricated device. The dark region at the center is the etched grooves, and thebright regions on the sides are metal IDTs [157]. (b) Transmission spectrumof a SAW resonator, showing the two high-Q modes in the bandgap of thephononic crystal. Note that the data are slightly different from that in Ref.[157] due to the use of a different device. (c) Mode profiles of the fundamentaland second-order modes of the SAW resonator. The color scale indicates thetotal energy density of electromagnetic, kinetic, and elastic energy densities.(d) T-MIM images of the measured SAW potential distribution at both modes[157]. (e) T-MIM images at three different frequencies. The scanned area iscentered at the right antinode of Mode 2. Color coding of the borders herematches that of the dots in (b).

88

example, we have also observed the profile of the acoustic mode in the thin

film acoustic resonators [159].

5.6 Conclusion

To summarize, we demonstrate the visualization of piezoelectric SAW

potential on z-cut LiNbO3 surface by transmission-mode microwave imaging.

The traveling or standing SAW potential generated by IDTs is demodulated

by the microwave electronics and mapped as stationary spatial patterns. The

signals can be explained by the standard microwave analysis. The wave diffrac-

tion due to a spontaneously reversed domain is also seen in the T-MIM images.

We also demonstrate the visualization of different SAW modes in other devices,

such as high-Q resonator. Our work paves the way to probe nanoscale acousto-

electronic behaviors in SAW devices, quantum materials, and phononic crys-

tals.

89

Chapter 6

Summary and Outlook

In this dissertation, I briefly reviewed the history of near-field scan-

ning microwave microscopy (NSMM) and introduced the working principles of

microwave impedance microscope (MIM). I then elaborated my exploration

of multiple unexpected electromechanical phenomena. Beyond its general

local conductivity mapping, MIM was implemented to perform the imag-

ing of dielectric loss and electroacoustic transduction. The development of

transmission-mode MIM offers a powerful tool for the direct visualization of

the microwave field. This unique setup has already attracted much attention

and continues to produce exciting results. Therefore, I can safely conclude

that we have made excellent progress in the past six years.

Besides all of these impressive achievements, future electromechanical

projects are foreseeable. The transmission-mode MIM is clearly advantageous

for the imaging of the SAW electric field in various devices such as delay

lines, filters, and oscillators. While the demonstration in the previous chapter

was performed in LiNbO3, it is straightforward to expand the study to other

popular acoustic materials such as quartz, AlN, GaAs, and ZnO even these

are only piezoelectrics. More importantly, SAWs at the frequency of 1 – 10

90

GHz have attracted great interest in quantum science and engineering as an

efficient means to couple superconducting qubits [160–162], drive spin qubits

[163–165], manipulate photons [166–168], and control two-dimensional (2D)

electrons [154, 169, 170]. Direct visualization of the SAW potential in different

platforms will help with the understanding of quantum entanglement, quantum

hybrid networks, and deterministic teleportation of electrons.

Another potentially fruitful area for T-MIM is 2D piezoelectric phononic

crystals [102, 136], which are artificial structures formed through periodic spa-

tial modulation of the elastic impedance. Through phonon-polariton coupling

between electromagnetic waves and acoustic phonons, it is possible to form

a complete band gap that phonons cannot propagate in all directions [102].

The spatial distribution of acoustic fields inside the 2D structures, which is

only obtainable by numerical simulation at present, is crucial to advance this

branch of research. Moreover, through band structure engineering, one may

even realize topological phononic crystals with robust edge channels [171]. The

topological protection against acoustic defects will be vividly demonstrated by

T-MIM imaging. Interestingly, a major research direction of the R-MIM is to

investigate the electronic topological edges in quantum Hall [172] and quantum

spin Hall [173] systems. In that sense, MIM may be the technique of choice to

visualize topological boundary states in both real materials and metamaterials.

At the time of this writing, we are still actively working with many

collaborators on these frontier studies, including topological modes of SAW,

zero-index metamaterial for SAW, and optomechanics. Continuing efforts will

91

surely generate more exciting works and accelerate the progress in both mi-

crowave microscopy and electromechanics fields.

92

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