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Thin film dielectric properties characterization by scanning near-field microwave microscopy

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THIN FILM DIELECTRIC PROPERTIES CHARACTERIZATION
BY SCANNING NEAR-FIELD MICROWAVE MICROSCOPY
by Shuogang Huang
Master of Philosophy, May 2008, The George Washington University
Bachelor of Science, July 1996, Shanghai JiaoTong University
A Dissertation submitted to
The Faculty of
Columbian College of Arts and Sciences
of The George Washington University
in partial fulfillment of the requirements
for the degree of Doctor of Philosophy
May 17th, 2009
Dissertation directed by
Mark E. Reeves
Professor of Physics
UMI Number: 3349661
Copyright 2009 by
Huang, Shuogang
All rights reserved
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THIN FILM DIELECTRIC PROPERTIES CHARACTERIZATION BY
SCANNING NEAR-FIELD MICROWAVE MICROSCOPY
SHUOGANG HUANG
Dissertation Research Committee:
Mark Edwin Reeves, Professor of Physics, Dissertation Director
William Carleton Parke, Professor of Physics, Committee Member
Jasper Nijdam, Research Assistant Professor of Physics, Committee
Member
ii
c Copyright 2009 by Shuogang Huang
All rights reserved
iii
Dedication
To my family for their love, support, and encouragement.
iv
Acknowledgements
First, I would like to express my deep gratitude to my thesis adviser, Professor Mark
E. Reeves, for making this research possible through his guidance, and for his many
suggestions and constant support over the past years. I would also like to thank
Mr. Yiheli Tesfu for his help during my first year working in lab. Thanks are also
given to the committee members for their time on my dissertation. I would also like
to acknowledge Professor David P. Norton from Department of Materials Science &
Engineering of University of Florida and Dr. Hans M. Christen from Materials Science and Technology Division of Oak Ridge National Laboratory, for providing me
the samples and helpful discussions as well as many scientific and technical interactions. I would like to thank the machine shop personnel, William Rutkowski, for his
instructions on the use of the machines, and the construction of the experimental
apparatus. I also want to thank our IT coordinator Peter Kovac, who helped me a
lot in building the Linux environment for the finite element calculation. Finally, I
am greatly indebted to my parents for all the things they have done for me. Without
their patience and love, this work would never have come into existence.
v
Abstract
Thin Film Dielectric Properties Characterization By Scanning Near-Field
Microwave Microscopy
We present a method for extracting high-spatial resolution dielectric constant data
at microwave frequencies. A scanning near field microwave microscope probes samples with the data being acquired in the form of the frequency and quality factor
shifts of a resonant cavity coupled to the sample. We find this technique is extremely
sensitive to the behavior of the system in a small volume at the end of the scanning tip. In our experiments, the near field is confined to a 1 µm region below the
sample surface, Thus, We are able to observe the dielectric response with a spatial
resolution as fine as 200 nm even with an 18 cm radiation wavelength. The theoretical part of the approach addressed here is to determine material parameters (in this
case, the complex dielectric constant) from the measured shifts. Our approach is to
apply cavity perturbation theory, which connects the shift in resonant frequency, to
computations of the energy stored in the electromagnetic fields of the cavity and its
surroundings in order to characterize the real part of the permittivity, and by applying a high frequency heat dissipation model to characterize the imaginary part of the
permittivity. The electromagnetic fields are simulated with the finite element method
(FEM) both in the quasi-static and high frequency modes. In this thesis, the details
of the experiment and FEM calculation will be given. Examples of determining the
dielectric properties in reduced dimensional thin-film materials will be presented.
vi
Table of Contents
Dedication
iv
Acknowledgements
v
Abstract
vi
Table of Contents
vii
List of Figures
ix
List of Tables
xiii
1 Introduction
1.1 Dielectric constant measurement at different length scales . . . . . . .
1.2 What is ‘Near Field’ and how it is applied in microscopy . . . . . . .
1.3 Near Field Microwave Microscopy in thin film dielectric measurements
1.4 Previous work on near-field microwave microscopy . . . . . . . . . . .
1.5 Our work on near-field microwave microscopy . . . . . . . . . . . . .
1
1
5
8
10
13
2 Theory and Numerical Calculation
2.1 Real part of Dielectric Constant – Quasi Static with Cavity Perturbation
2.1.1 Bulk samples– Calibration of scaling factor . . . . . . . . . . .
2.1.2 Thin Films– Ratio of energy integrals . . . . . . . . . . . . . .
2.2 Imaginary Part of Dielectric Constant – Dielectric Power Dissipation
2.3 Spatial Resolution and Sensitivity . . . . . . . . . . . . . . . . . . . .
17
19
21
27
30
39
3 Experiment
3.1 The Principles of the SNMM Design . . . . . . . . . . . . . . . . . .
3.1.1 Resonator construction . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Tip preparation . . . . . . . . . . . . . . . . . . . . . . . . . .
43
43
44
47
vii
3.2
3.3
3.1.3 Tip-sample distance control . . . . . . . . . . . . . . . . . . .
Our SNMM Experimental Setup with temperature control . . . . . .
The Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
52
58
4 Crystallization temperature and Ferroelectric phase transition
4.1 Relationship between crystallization and dielectric response of epitaxial
rare-earth scandate films . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Ferroelectric phase transition study in KTN Perovskite thin films . .
4.2.1 Ferroelectricity in Perovskite Oxides . . . . . . . . . . . . . .
4.2.2 Ferroelectric Phase transition . . . . . . . . . . . . . . . . . .
4.2.3 Ferroelectric phase transition in a coupled KTN superlattice .
4.2.4 Temperature dependent dielectric response of KTN solid solution
63
64
71
71
74
77
97
5 Conclusion
108
Bibliography
112
Appendix A Manual for SNMM operation and Femlab simulation
122
A.1 Procedures of experiment and numerical calculation . . . . . . . . . . 122
A.2 Source code of simulatation of a 200nm thin film reference curve on
LaO (=26) substrate . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
viii
List of Figures
2.1
Schematic mesh for tip - dielectric sample at contact mode. . . . . . .
22
2.2
The SEM image of the tungsten tip used in bulk T iO2 measurements.
23
2.3
The the near-field electric potential at the tip end. . . . . . . . . . . .
25
2.4
Calibration of scaling factor α with bulk T iO2 . . . . . . . . . . . . . .
26
2.5
The fitting curves of thin-film samples on LaAlO3 substrate with varying thickness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
2.6
Geometry definition in high frequency mode. . . . . . . . . . . . . . .
33
2.7
Mesh domains in high frequency mode. . . . . . . . . . . . . . . . . .
35
2.8
Solved EM field in high frequency mode. . . . . . . . . . . . . . . . .
36
2.9
The electric field polarization on tip end. . . . . . . . . . . . . . . . .
40
2.10 Sensitivity analysis of tips with different flatness. . . . . . . . . . . .
42
3.1
The System Diagram of Scanning Near-field Microwave Microscope. .
45
3.2
The picture of cavity of Scanning Near-field Microwave Microscope. .
46
3.3
AC-prepared wire-tip.
. . . . . . . . . . . . . . . . . . . . . . . . . .
48
3.4
AC-etched wire-tip after nitric acid dipped. . . . . . . . . . . . . . . .
48
3.5
Experimental setup of for tungsten wire etching. . . . . . . . . . . . .
49
3.6
Scanning near field microwave measurement on heating and cooling stage 55
3.7
The comparison measurements of df /dz vs. tip-sample distance ‘z’
before and after redesign the temperature stage.The sample measured
here is bulk LaAlO3 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
56
3.8
The frequency f, quality factor Q and df/dz vs. z of a thin film sample
by motor approach. Motor stopped at df/dz = 0.08. . . . . . . . . . .
3.9
59
The scanning frequency of a thin film sample in room temperature.
The value of df/dz at the contact point is 7. . . . . . . . . . . . . . .
60
3.10 The scanning Q factor of a thin film sample in room temperature. . .
62
4.1
Dielectric constant of rare-earth scan dates deposited at 800◦ C. . . .
66
4.2
X-ray diffraction in determining the crystal orientation. . . . . . . . .
67
4.3
Dielectric constant of NdScO3 films deposited at different temperature. 69
4.4
Symmetry of perovskite crystal structure ABO3 . . . . . . . . . . . . .
4.5
Temperature dependence of dielectric constant of KH2 PO3 at 9.2GHz.
72
The curve of a is for field along the a-axis, the curve of c is for field
along the c-axis.[1] . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6
Temperature dependence of dielectric constant of KNbO3 in different
symmetries and of ‘incipient ferroelctric’ KTaO3 .[2] . . . . . . . . . .
4.7
76
79
Atomic resolution Z-contrast scanning transmission electron micrograph of KTN superlattice.[3] . . . . . . . . . . . . . . . . . . . . . .
81
4.8
The contact frequency vs. temperature of KTN 1x1 superlattice. . . .
82
4.9
The temperature dependent dielectric constant of KTN 1x1 superlattice. The peak at 195o C coincides with XRD structural transition, but
the peak at 137o C does not. . . . . . . . . . . . . . . . . . . . . . . .
83
4.10 X-ray study of KTN superlattice phase transition by H. M. Christen.(a)
is the plot of lattice parameters of both in-plane and out-of-plane (normal)directions. (b) is the plot of paraelectric to antiferroelectric transition temperatures at different layer thicknesses.[4] . . . . . . . . . .
86
4.11 Domains and depolarization field in coupled ferroelectric layers in a
ferroelectric/paraelectric superlattice close to Tc . Plot (a) shows the
weak coupling for thicker layers and plot (b) is the strong coupling for
thinner layers (below critical thickness).[5] . . . . . . . . . . . . . . .
x
87
4.12 Measurement of KTN 8x8 superlattice. Plot of the temperature dependence of the dielectric constant. . . . . . . . . . . . . . . . . . . .
89
4.13 Measurement of KTN 16x16 superlattice. Plot of the temperature
dependence of the dielectric constant. . . . . . . . . . . . . . . . . . .
90
4.14 Measurement of KTN 1x7 superlattice. Plot of the temperature dependence of the dielectric constant. . . . . . . . . . . . . . . . . . . .
91
4.15 Measurement of KTN 2x16 superlattice. Plot of the temperature dependence of the dielectric constant. . . . . . . . . . . . . . . . . . . .
92
4.16 Tc of KTN with different periods. The red and pink dots of superlattice samples represent the temperature of structural transition, the
green dots represent the temperature of antiferroelectric to ferroelectric
transition. For comparison, the data of KT a0.5 N b0.5 O3 solid solution
is also included. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
4.17 Temperature dependence of dielectric constant of KT a0.63 N b0.37 O3 thin
film on M gAl2 O4 substrate. . . . . . . . . . . . . . . . . . . . . . . .
99
4.18 The phase diagram of bulk KT a0.64 N b0.36 O3 vs. pressure. At atmosphere pressure 1 bar, the crystal undergoes 3 phase transitions.[6] . . 100
4.19 X-ray diffraction pattern of the solid solution thin film at the temperature both above and below the transition. . . . . . . . . . . . . . . . 101
4.20 The off-center of the Nb cation in KTN solid solution. Eight diagnose
< 111 > directions are shown.[7] . . . . . . . . . . . . . . . . . . . . . 103
4.21 Temperature dependence of the dielectric constant of KTN solid solution under different pressure. The concentration is x=0.023. The
transition vanishes at 9.2kbar.[6] . . . . . . . . . . . . . . . . . . . . . 106
A.1 A SEM picture of a tip with 22◦ cone angle. . . . . . . . . . . . . . . 123
A.2 A motor approach of a thin film sample, recorded by f oQV − z motor.vi.125
A.3 A Piezocrystal approach of a thin film sample, recorded by contactbypiezo.vi. The piezocrystal voltage step is set to be -0.1V, corresponding
to about 30nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
xi
A.4 Define tip and sample geometry in femlab GUI. . . . . . . . . . . . . 127
A.5 Define boundary conditions of tip and sample in femlab GUI. . . . . . 128
A.6 Define subdomain properties of sample in femlab GUI. The figure shows
the thin film dielectric constant is defined as 26, same as substrate. In
this case, the sample is equivalent to a bulk material. . . . . . . . . . 129
A.7 Generate mesh in femlab GUI. Notice in this figure, the mesh area
around the contact point has been refined for several times. Make sure
the actual mesh element size there is less than 30mn. That is critical
for an accurate solution. . . . . . . . . . . . . . . . . . . . . . . . . . 130
A.8 Export the solution (electric field) to matlab.
. . . . . . . . . . . . . 131
A.9 Add another application through femlab multiphysics menu. . . . . . 132
A.10 Define the second mode as ‘Electrostatics’, define its dependent variable as ‘W’, in order to consistent with matlab program. . . . . . . . 133
xii
List of Tables
2.1
The integrations of energy loss by Femlab. . . . . . . . . . . . . . . .
39
3.1
Table of dielectric constant measurement for bulk samples. . . . . . .
62
4.1
The measurement of rare-earth scandate thin films on LaAlO3 substrates. By using the fitting curves from the Femlab simulation, the
films’ permittivities with errors are shown on the last column. . . . .
xiii
70
Chapter 1
Introduction
1.1
Dielectric constant measurement at different
length scales
Much of our understanding of insulating materials comes from the study of the interaction of electromagnetic fields with matter. Different insulators have various
electrical characteristics that are dependent on their dielectric properties. A material is classified as ‘dielectric’ if it has the ability to store energy when an external
electric field is applied. As in a parallel plate capacitor, more charge is stored when
a dielectric material is between the plates than when vacuum is between the plates.
Accurate measurements of dielectric properties can provide scientists and engineers
with critical design parameters to better select materials for an intended application.
For example, the transmission loss in a cable’s insulator, the impedance of a substrate, and the center frequency of a dielectric resonator are all related to dielectric
properties. This knowledge is also beneficial for improving ferrited absorber materials, and packaging designs in the area of industrial microwave processing of food,
rubber, plastic and ceramics.
In most cases, the dielectric properties are expressed in the form of a complex
1
value, the dielectric constant or permittivity, which is a physical quantity that describes how an electric field affects and is affected by a dielectric medium, and is
determined by the extent to which a material polarizes in response to an applied electric field, and thereby reduces the field inside the material. The complex dielectric
constant consists of a real part 0 which is a measure of how much energy from
an external electric field is stored in a material and an imaginary part 00 which is a
measure of how dissipative or lossy a material is to an external electric field. The
imaginary part of dielectric constant 00 is also called the loss factor and is always
greater than zero and is usually much smaller than 0 . The loss factor includes the
effects of both dielectric loss and conductivity. The loss tangent or tanδ is defined
as the ratio of the imaginary part of the dielectric constant to the real part. It is
important to note that the permittivity is not a constant; it can change with frequency, temperature, orientation, mixture, pressure, and molecular structure of the
material. The reason is that at the microscopic level, a material may have several
dielectric mechanisms or polarization effects that contribute to its overall permittivity. Among them, dipole orientation and ionic conduction contribute strongly at
microwave frequencies; others such as atomic and electronic mechanisms are relatively
weak and are usually constant over the microwave region. Each dielectric mechanism
has a characteristic cutoff frequency. As the frequency of the applied field increases,
the slower mechanisms drop out in turn, leaving the others to contribute to 0 . The
loss factor 00 correspondingly peaks at each critical frequency. The magnitude and
cutoff frequency of each mechanism is unique for different materials. Water has a
strong dipolar effect at low frequencies, but its dielectric constant rolls off dramatically around 22 GHz. Teflon, on the other hand, has no dipolar mechanisms and its
2
permittivity is remarkably constant well up to 300 GHz. [8][9]
There are a number of techniques that have been traditionally used to measure
the dielectric constant, including the following[8]:
• The parallel plate capacitor method: A thin sheet of dielectric material is
sandwiched between two conducting plates. The capacitance is measured by an
LCR meter and then plotted versus the reciprocal of the sheet thickness to find the
dielectric constant.
• Free-space method: A non-contacting reflectivity measurement of the
electromagnetic wave from the sample surface, where an antenna focuses microwave
energy at slab of the material. This method can be applied to test samples at high
temperatures and in hostile environments.
• Resonant cavity method: The frequency shift is measured after introducing a
sample into a high-Q resonant cavity. For dielectric measurements the sample
should be placed at a spatial maximum of the electric field, which causes the cavity
to shift to a lower frequency with a broader resonance curve, corresponding to a
lower quality factor, Q. A vector network analyzer is often used and the sample’s
cross-section dimensions must be known precisely. Normally, this technique is
extremely accurate but the results can only be acquired at the discrete resonant
frequencies of the cavity.
• Coaxial probe method: An open-ended coaxial probe is formed from a cut off
section of transmission line. The material under study is placed at the end of the
probe and pressed into close contact during the measurements. The reflected signal
(S11) is measured and then related to . For the simplest data reduction, the
sample must be thick enough to appear ‘infinite’ to the probe.
3
• Transmission line method: Transmission line methods rely on the re-distribution
of the TEM wave in an enclosed coaxial line partially filled with the lossy material
under study. The dielectric constant is computed from the reflected signal (S11) and
transmitted signal (S21).
The methods list above all work in the far-field of the radiation, which requires
the sample to be large, smooth with flat faces, pure, and homogeneous on the scale
of the free-space radiation’s wavelength to obtain reasonable measurements. Thus,
the electromagnetic properties of the sample are actually a macroscopic average over
a large sampling volume. For some material, the dielectric properties may be varying
on a much shorter length, typically in the sub-micron range, as in artificially layered
structures of dielectric material that are achieved via doping or in microstructures
fabricated at micrometer or even nanometer length scales. In other cases, such as
of thin-film samples with sub-micron thickness, the substrate generally changes the
crystal structure in film, due to size or strain effects. This causes the thin-film material
to have different dielectric properties than in the bulk. Traditional techniques are less
sensitive to changes of dielectric constant at small length scales, either because the
samples in this small dimension do not cause a measurable shift in the signal or
because of the impossibility of deconvolving the contribution of the film from that of
the substrate.
In order to probe the localized electrodynamic properties of small-dimensional
samples, a new microscopy technique based on near-field measurements and 3D finite
element analysis has been developed to map dielectric properties with very high spatial resolution. By utilizing the strongly confined near field, the dielectric constants
of samples as thin as a few hundred nanometers have been measured at microwave
4
frequencies. The details of this work will be presented in the following chapters.
1.2
What is ‘Near Field’ and how it is applied in
microscopy
Generally speaking, near field concerns phenomena of confined electromagnetic fields
concentrated near the surface of materials and of evanescent waves within the materials. In the view of diffraction and antenna design, the near field is that part of the
radiated field nearest to the antenna, where the radiation pattern depends strongly
on the geometry of the antenna. If we consider the space surrounding the antenna,
it generally can be divided into three zones by comparing with the size of antenna D
with the wavelength of the radiation λ: 1) near field (static), 2) intermediate field,
and 3) far field (radiation). The far field, where radius vector r D λ, is the radiating field with an angular distribution essentially independent of distance from the
source, the fields are in-phase, transverse and decay as 1/r. The far-field simplifications are very useful in engineering calculations and the propagating wave is governed
by the wavenumber k = ω/c (where ω is angular frequency and c is speed of light)
and wave impedance Z = E/H (where E and H are electric and magnetic filed, respectively). In the near-field zone, the electromagnetic fields are strongly determined
by the antenna’s geometry and by the electrodynamic properties of its surroundings.
The range of the near field is D ≤r λ. Because kr 1, the exponential eikr ' 1
and the fields can be treated as quasi-stationary with a harmonic oscillating factor
eiωt .[10] The fields are not transverse and decay with increasing distance from the
antenna as 1/r2 or faster, far more rapidly than do the classical radiated EM far-field
waves. Typically, near-field effects are not important farther than a few wavelengths
5
away from the antenna.
More generally, the concept of near and far fields is one of mathematical convenience and engineering simplicity than of actual separate kinds of fields in space. The
solutions to Maxwell’s equations for the electric and magnetic fields for a localized oscillating source can be written as a multipole expansion. The terms in this expansion
are spherical harmonics, which give the angular dependence, multiplied by spherical
Bessel functions, which give the radial dependence. For large r, the spherical Bessel
functions decay as 1/r, giving the radiated far field. As one gets closer to the source,
approaching the near field, the induction term proportional to 1/r2 becomes significant. One can envision that the energy stored in the near field is returned to the
antenna every half-cycle. For even smaller r, the electrostatic field term proportional
to 1/r3 become significant, which can be thought of as stemming from the oscillating electrical charge in the antenna element. Very close to the source, the multipole
expansion requires even more terms for an accurate description of the fields. Modern
computer techniques are needed for such calculations.
For microscopy, there has been a long history of using the far field to produce images based on transmitted or reflected radiation. However, the resolution power has
a limit of about one-half of the optical wavelength. Let’s consider an electromagnetic
wave beam, which is incident on to a flat object and which is diffracted by the object’s
features. According to the diffraction theory, the smaller the detail, the higher number of diffraction orders will be observed. Hence, there is generally information lost
because of the interference of waves and only structures with lateral dimensions larger
than λ/2 can be imaged accurately. This is the smallest resolvable spacing, known as
Abbe’s limit or the far-field diffraction limit.[11] However, it is possible to overcome
6
this limit when near field interactions are taken into account. The near field interacts with the sample by the stored reactive energy, electric or magnetic, in the near
zone of the probe. When a dielectric or magnetic sample is brought in close vicinity
to the probe, this energy changes and the response can be detected. Unfortunately,
the importance of such waves has been ignored for a long time in materials science
and surface physics until the emergence of scanning near-field microwave microscopy
(SNMM). In this technique, a probe approaches close to the surface of a sample and
scans the sample in order to investigate local variations in the electrodynamic properties of the sample on nanometer-length scales. This resolution can be practically
realized to about one millionth of the wavelength.
Scanning near-field microscopy based on this principle was first mentioned by E.
Synge in 1928[12], in a paper that became the basis of modern near-field microscopy,
super-resolution, and evanescent wave detection. Synge even exchanged correspondence with Einstein about the ways to increase the spatial resolution[13]. A few years
later, he proposed an idea for using a piezo-electric actuator to moving the detector, as
we see in today’s scanning probe microscopes. But it was not until 1962, that Soohoo
successfully demonstrated this effect at microwave frequencies[14]. In 1972, E. Ash
and G. Nicholls again demonstrated a near field microscope at microwave frequencies
with sub-wavelength resolution, λ/60, by using a small pinhole to scan the object
surface.[15] In the early 1980’s, G. Binning and H. Rohrer described a new instrument for measuring the tunneling current between a conductor and a sharp metallic
tip, the STM. Since the tip was necessarily to detect current over a distance of about
1 nanometer, they introduced a feedback loop for maintaining a constant current and
thereby allowing the tip to float a few angstroms above the sample surface.[16] At
7
visible wavelengths, near-field scanning optical microscopy (NSOM) was also demonstrated in eighties by several different groups: D. W. Pohl,[17] G. A. Massey,[18] and
A. Lewis.[19] They demonstrated an instrument with about 30∼50 nanometers resolution at visible wavelengths. Since then, the notion of near field detection, involving
new theoretical and experimental models has developed rapidly.
1.3
Near Field Microwave Microscopy in thin film
dielectric measurements
Thin film ferroelectrics and dielectrics offer several advantages over bulk forms for
applications as filters and other passive components in high-frequency circuits. The
ability to control the dielectric properties in a simple way allows thin film based devices to be easily optimized for maximum tunability and minimum loss at the desired
frequency and operating temperature. The main reason is that the modern electronic
industry needs new techniques to accurately measure thin film materials and structures, and to more fully characterize their properties and performance. Researchers
are developing new measurement approaches to resolve or analyze component structures, material layers or active regions that measure hundreds of nanometers or less.
These approaches can be used in the development, fabrication and analysis of the
next generation electronic processing and data storage devices.
For a complete understanding of the physics of these device, it is necessary to
develop local probes for the study of a film’s dielectric response to electromagnetic
fields. To this end, the development of the near-field microwave microscope gives
researchers the capability for precise quantitative measurements of the permittivity
8
of thin dielectric films. The technique is non-destructive, requires no sample preparation, and can resolve a one-micron sampling spot. Combined with data reduction
strategies, this approach allows one to study material composition and performance
at the nanoscale.
For the research addressed in this thesis, we will present quantitative measurements on a variety of dielectric thin films by our scanning near-field microwave microscope. The first investigation is so-called high-κ dielectric films, which are of great
interest to the semiconductor industry as replacements for the SiO2 insulating layers
of on-chip wiring in high-density, high-speed integrated circuits. From these measurements, we obtain information relating the films’ crystallization to their dielectric
response. These are the most complete set of data for epitaxial scandate films crystallization temperatures and high dielectric constants. The results indicate the potential
of these materials as high-k dielectrics for application in field-effect transistors.
A second investigation is of perovskite thin film materials, which are an important
class of ferroelectrics. Their relatively simple chemical and crystallographic structures
have contributed significantly to our understanding of ferroelectricity. They are one
of the most technologically important ferroelectrics, finding a broad range of applications as transducers, actuators, positioners, and memory devices[2, 20]. We have
investigated the temperature dependence of the dielectric properties of the thin films,
both in the form of superlattice and of solid solution, obtained information about
ferroelectric phase transitions, and revealed the antiferroelectric mechanism in the
heterostructures. We also compared our results with x-ray diffraction and capacitance measurements.
9
1.4
Previous work on near-field microwave microscopy
Recently, several groups have developed versions of the near-field microwave microscope with individual approaches for converting the frequency shift to the material
permittivity. These are reviewed below.
C. Gao and X.D. Xiang at Lawrence Berkeley National Laboratory describe a
near-field microwave microscope based on a λ/4 coaxial resonator with a scanning
tunneling microscope (STM)-like tip protruding from the end cap. A dielectric sample placed near the tip causes a shift in the cavity’s resonant frequency and its quality
factor. To demonstrate their system’s sensitivity, which is about ∆/ ≈ 1E − 5,
they approximated the electric field inside the cavity as an equivalent lumped series
resonant circuit with effective capacitance, inductance and resistance; the tip is represented as a small capacitor, whose value depends on the tip-sample interaction. The
relative resonant frequency shift is then determined by the charge and field redistribution within this small capacitor. To actually calculate the value of permittivity,
they model the interaction by representing an image charge located in the sample
and the redistribution is represented by another image charge inside the tip sphere.
This process is iterative and forms an infinite series of image point charges to meet
the boundary conditions at both the tip and the dielectric sample surfaces. For samples whose thicknesses are much greater than the tip radius, the shifts in f and Q,
which are directly related to the sample dielectric constant, are calculated to first
order from cavity perturbation theory (see below) and the dipole-dipole interaction
model.[21, 22]
Steven M. Anlage’s group at the University of Maryland built an open-ended
coaxial probe with a sharp, protruding center conductor. The probe is connected to a
10
coaxial transmission line resonator, which is powered by a microwave source through
a capacitive coupler. The probe tip is held fixed, while the sample is supported by a
spring-loaded cantilever. In addition, a local dc electric field is applied to the sample
by means of a bias tee in the resonator for measuring the local dielectric tunability
of thin films. They also developed a new method, called distance-following (DF), to
operate this microscope with an improved spatial resolution of 1 µm and enhanced
material property sensitivity; the probe-sample separation is modulated at fixed amplitude by adding an ac voltage to the voltage from the feedback circuit applied to
the piezoactuator. The height modulated signal is proportional, to first approximation, to the slope of the frequency shift versus distance curve at a given sample-probe
separation. These slopes, at a given resonance frequency, are very sensitive to local
materials properties and can approach values of 1kHz/nm near the surface. Therefore,
if the lateral dimensions of topographic features are much larger or much smaller than
the probe diameter, the height modulation images of the local electrical properties
are free from topographic artifacts. In addition, the height modulation approach is
less susceptible to vibrational noise. Later, they tried to keep the probe at a constant
height of about 1nm above the sample with the help of scanning tunneling microscope
(STM) feedback, which makes their microwave microscope sensitive to sample sheet
resistance; they also develop a quantitative transmission line model for treating the
tip to sample interaction as a series combination of capacitance and sheet resistance
in the sample.[23, 24]
J. H. Lee, S. Hyun, and K. Char at Seoul National University also set up a
scanning microwave microscope using a λ/4 coaxial resonator with their experimental
data for bulk dielectrics modeled by the Gao and Xiang model. However, in the case
11
of dielectric thin films, the Gao and Xiang model fails, as it is independent of the
thickness of the film. A careful numerical analysis based on this monopole model
shows a discrepancy between the simulation and the approximate formula for thin
films. Because fields emanating from the side of the tip are significant in size, the
shifts of the resonant frequency are enhanced more than expected for a tip monopole
source. For this reason, the Korean group uses a numerical calculation (finite element
method) for computing of the frequency shifts for bulk and thin film dielectrics. The
model treats the tip as a perfectly conducting sphere (which is not always realized
experimentally). By normalizing all length scales to the tip radius R and all potentials
to the probing tip voltage V, fitting functions for the dielectric constant of a thin film
are derived as dimensionless quantities. These fitting functions for films are quite
complicated and not unique. It is also found that fitting the global range of dielectric
constants from extremely low to extremely high value with a single set of parameters
is not easy.[25]
Another Korean group lead by Kiejin Lee from Sogang University, Korea developed an improved near-field scanning microwave microscope system with a tunable
resonance cavity instead of a fixed λ/4 coaxial resonator. The tunable resonance
cavity probe consists of two parts. The main part is a movable λ/4 coaxial resonator
for tuning to achieve the highest possible sensitivity and improved spatial resolution.
The second part is a fixed center conductor with a stainless steel tip that protrudes
out from the aperture. A stabilized RF signal is input to the resonant cavity and
the cavity output is connected to a power meter and a spectrum analyzer. As the
tip approaches the sample, its resonance is shifted due to the interaction between the
tip and the sample, and the cavity’s impedance changes. By adjusting the length of
12
the resonant cavity, it is possible to optimize the impedance at the new resonance
frequency to maximize the coupling between the SNMM and the frequency source
and increase the sensitivity by obtaining an improvement of the signal-to-noise ratio. In order to understand the function of the probe, they fabricate three different
tips using a conventional chemical etching technique; only a hybrid tip combining
a reduced length of the tapered part with a small apex demonstrated an improved,
high-contrast SNMM image. They also implemented a tuning-fork, shear-force feedback method to control the distance between the tip and the sample. This distance
control method is independent of local microwave characteristics. The probe tip for
the SNMM is attached to one prong of a quartz tuning fork and directly coupled
to the resonator. The amplitude of the tuning fork vibration is used as a distance
control parameter. Based on these techniques, they demonstrate SNMM in a liquid
environment; the bottom of the probe tip is immersed in water, and the top remains
above the liquid, attached to one prong of a quartz tuning fork and directly coupled to
a high quality factor dielectric resonator. The amplitude of the tuning fork vibration
is used to set the distance control between the tip and the sample. By this method,
they successfully obtain topographic and SNMM images of samples, including a DNA
film in buffer solution. [26, 27, 28]
1.5
Our work on near-field microwave microscopy
In this thesis, we will describe our contributions to the understanding of near field
generated by the scanning microwave microscope, especially in the sub-micron range.
In fact, one must be careful about tip-field interaction in order to avoid artifacts. We
will present a few measurements and simulations in order to answer this fundamental
13
question and how we apply this technique in dielectric material research.
This thesis is structured in three main chapters. Chapter 2 introduces the theoretical basis and numerical calculation of our near-filed microwave microscope. It is the
fundamental part of this work. We simulate the tip-sample response by a 3D finite
element calculation, both in static mode and high frequency mode, which reveals interesting phenomena in the near-field region. We model the tip as a cylinder capped
by a cone with a flat end, all held at a constant potential, then we obtain the field
polarization from this geometry. The change of resonant frequency is converted to the
real part of the permittivity by using perturbation theory, and the imaginary part of
the permittivity is obtained from the change of the quality factor via a high frequency
power dissipation model. Some theoretical noise aspects are also introduced in order
to know the spatial resolution and sensitivity of this method.
In chapter 3, we present a detailed description of the complete scanning near-field
microwave microscope. We show what the basic concepts have been used in designing
this coaxial cavity based scanning probe microscope in order to measure temperature
dependent dielectric materials. As a first test, we present the measurements of a
number of bulk materials in room temperature.
Chapter 4 is the main interest of this work. We apply this integrated technique to
two practical applications, the studies of semiconductor crystallization and ferroelectric phase transitions. The first project, we study the crystallization temperatures
and dielectric constant of epitaxial rare-earth scandate films. The scandates thin films
are deposited by a pulsed laser deposition onto LaAlO3 substrates at 800◦ C. We use
x-ray diffraction to determine the films’ onset crystallization temperatures and crystalline orientations. For the films whose crystallization temperatures are greater than
14
800◦ C, we observe how dielectric constants consistent with those of polycrystalline
materials. For the films that are well crystallized by 800◦ C, we find that their thin-film
values were significantly larger than those in the polycrystalline bulk form. Even
for the films just starting to crystalize at the deposition temperature, we find their values to be almost the same as the single-crystal bulk values along the c-axis, 33 .
In combination with the large observed band gaps, these results indicate the potential for application of these materials as high-κ dielectrics for field-effect transistors.
The second project is the study of ferroelectric phase transitions in thin film KTN
superlattices and solid solutions. For superlattice thin films, our measurements show
that the antiferroelectric to ferroelectric transition occurs in every sample within a
fairly narrow temperature range. But the paraelectric to antiferroelectric transition
temperatures is revealed to be determined by inter layer coupling. Measurements
of symmetric superlattice are well matched to previously reported data from x-ray
diffraction and capacitance measurements, and the 2 asymmetric superlattice were
measured for the first time and reported in this thesis. For the solid solution thin
film, our measurement shows an abnormal dielectric transition compared to the bulk
case, with the x-ray diffraction data show that this dielectric transition is distinct
from any structural transition. Two possible models, the order-disorder transition
and the strain driven relaxor behavior. are applied to explain this observation.
This thesis also covers some work did by our cooperators,like Prof. David P. Norton and Dr. J. Sigman grew the KTN thin film samples and made the capacitance
measurements, Dr. Hans M. Christen grew the rare-earth scandate thin film samples and made the X-ray diffraction measurements. Based on their contribution, I
15
make the measurements on dielectric and phase-transition properties with our scanning near-field microwave microscope, build numerical simulations with finite element
method and compare my data with theirs. Some results, like paraelectric to antiferroelectric transitions in superlattice, are highly consistent with previous capacitance
and x-ray measurements, and some results, like ferroelectric to antiferroelectric transition in superlattice and abnormal dielectric transition in solid solution, are first time
observed. I also bring some discussions for possible theoretical explanations of these
newly found phenomena.
16
Chapter 2
Theory and Numerical Calculation
Most theories researchers have today to describe near-field techniques are focused on
describing the interaction between the incident wave and sample in the macroscopic
range. For the microscopic range, such as a few hundred nanometers, our knowledge
remains limited, and this normally requires a detailed solution of Maxwell’s Equations.
For solving Maxwell’s Equations, a numerical method is needed, especially in complex
geometries and dielectric environments.
One of the most important facets in this research has been to convert the frequency shifts measured in the lab into values of the sample’s permittivity. Typically,
this computation proceeds by solving the Maxwell’s Equations under real boundary
conditions, which is difficult because the equations can not be solved for our experimental geometry in a closed form. Thus, numerical computation must be used for
this calculation. We will describe in this chapter our numerical simulations and develop a new technique for estimating the complex permittivity of a dielectric sample
at microwave frequencies based on 3-D finite element analysis.
The finite element method (FEM) is a numerical technique for finding solutions
to partial differential equations (PDE) in a complex space, even when the geometry
17
changes or when the desired precision varies over the entire space. It is perhaps the
most practical method for solving boundary values problems and is widely used to
solve electromagnetic problems. Mathematically, the finite element method models a
field, then transform the solution into energy-related functionals, and finally iterates
the solution by minimizing these functionals. The origin of the finite element method
can be traced back to the work by Alexander Hrennikoff (1941) and Richard Courant
(1942), who used it to solve complex structural problems in civil engineering and
aircraft analysis. Finite element modeling in electrical engineering began in the late
1960s, mainly for the analysis of waveguides and cavities. Later developments, including the addition of scattering boundary conditions and hybridization with boundary
integral methods, led to the widespread application of the finite element method to
EM scattering, microwave circuits, and antennas. In 1973, with the contributions
from William Gilbert Strang and George J. Fix at MIT, the method was grounded
with a rigorous mathematical foundation and has since been generalized to be a useful
tool for the numerical modeling of physical systems in a wide variety of engineering
disciplines. The main advantage of the finite element method is its adaptability, which
enables treatment of complex geometries and inhomogeneous materials. Moreover, it
is versatile, does not need to associate the boundary conditions with the trial functions, has low memory requirements for storing sparse matrices, and yields stable and
accurate solutions. [29][30]
The steps involved in modeling and solving a PDE by FEM are as follows:
Define the boundary conditions of the PDE.
Obtain a variational formulation of the PDE in terms of energy-related
functionals or weighted residual expressions.
18
Discretize the solution volume into small enough elements, meshes, and
high-order shape functions.
Choose a trial solution in terms of nodal values of the elements.
Choose a solver to minimize the functional.
Convert the potential field data to the parameters of interest.
In this work, Femlab, a powerful interactive commercial software package for modeling and solving partial differential equations (PDEs), is used for the computational
evaluation of the electric field distribution in the near-field zone of an SNMM probe.
By exporting Femlab structures into a Matlab program based on cavity perturbation
theory (where the cross integration of the perturbation energy is converted to the
frequency shift), the real part of the permittivity can be calculated. The loss tangent of the thin-film is mainly calculated from the shift of the cavity quality factor,
in a similar Femlab geometry, but much larger in size, to find the losses associated
with the tip and sample. We compare numerical simulation results with experimental
values for various tip shape and film thickness, and find that this approach yields a
good approximation for calculating dielectric properties with a sub-micron spatial
resolution.
2.1
Real part of Dielectric Constant – Quasi Static
with Cavity Perturbation
In many applications, cavity resonators are modified by making small changes in
their shape, or by introducing small pieces of dielectric material. By measuring the
shift in the resonant frequency, the material’s dielectric constant can be determined.
19
In some cases, one can exactly calculate the change of the cavity’s response, but
more often approximations must be used. The cavity perturbation method provides
a formalism for this by first assuming that the eigen modes of the fields in the cavity
are unchanged from those of the unperturbed cavity. This is similar to first order
perturbation theory in quantum mechanics.
For our geometry, the sample is placed in the near-field range of the tip, where the
tip radius and effective field distribution range are much smaller than the wavelength
of the electromagnetic field. Thus, a quasi-static approximation can be used, and
the problem reduces to finding the electrostatic potential from Poisson’s equation.
The major concern here, in using the quasi-static model, is the ‘electrical size’ of the
structure, a dimensionless ratio of the size of the region under study to the wave length
of the electromagnetic field. The quasi-static approach is suitable for simulations of
structures with an electrical size up to 1/10. In our set-up, the typical tip length
is about 3 mm, but more importantly, its end dimension is of micron scale. The
wavelength of electromagnetic wave in the cavity is 17 cm, yielding a ratio well below
0.1. The actual physical assumption behind the ‘electrical size’ is that the currents
and charges generating electromagnetic fields at every point inside the structure are
practically the same at every instant as if they had been generated simultaneously by
stationary sources. Field retardation effects can be ignored.
Furthermore, since the majority of the microwave energy is concentrated in the
cavity, the field distribution inside the cavity will not be disturbed significantly by
the tip-sample interaction. Thus many groups, including ours, find that cavity perturbation theory is a good approach for calculating the frequency shift caused by
changes in the dielectric environment. With this theory, we can relate the resonant
20
frequency shift of the cavity to a change in the permittivity ∆, or permeability ∆µ
in the volume occupied by sample, as shown in Eq. 2.1.1.
RRR
~ ·E
~ ∗ + ∆µH
~ ·H
~ ∗ )dv
(∆E
−
f − f0
0
0
V0
=
RRR
∗
∗
~
~
~
~
f
(E · E0 + µH · H0 )dv
(2.1.1)
V0
Here E, H, and f refer to the electric and magnetic fields and the resonant frequency, the subscript 0 represents the unperturbed fields (when sample is not present
or is equivalently replaced by vacuum).[31] Normally, this equation is not in a very
useful form since we generally do not know the exact unperturbed fields in the cavity. In order to get these values, we use a commercial software package Femlab3.0 to
help determine the fields and then evaluate the integral numerically. If we assume
that the volume involving the change of permittivity or the change of permeability is
small, then the perturbed fields vary little from the original fields. Thus the equation
predicts that for any sample with ∆ > 1 or ∆µ > 1, the stored electric and magnetic
energy will increase, resulting in a decrease of the resonant frequency.
2.1.1
Bulk samples– Calibration of scaling factor
The numerator on the right side of Eq. 2.1.1 is the actual contribution from the
perturbed field of the sample. Because the change in permittivity only happens within
the volume occupied by the sample, we can concentrate our computational resources
on this sub-domain with a dense mesh. In our setup, a piezoelectric actuator is used
to control the sample’s up and down motion in small step sizes, as small as 30 nm
per step. This sets the lower limit for the triangular elements of the refined mesh to
get an accurate simulation. Assuming the sample is nonmagnetic, which is true for
all materials here, ∆µ = 0, and the actual perturbed energy can be simplified as:
21
Figure 2.1: Schematic mesh for tip - dielectric sample at contact mode.
Z Z Z
~ ·E
~ 0∗ dv.
∆E
Epert = −
(2.1.2)
V0
The right side denominator of Eq. 2.1.1 is the total energy of the cavity and
surrounding space, which can be expressed as a constant α.
Epert
∆f
=
f
α
(2.1.3)
Based on this simplification, We simulate the Femlab geometry shown in Fig. 2.1.
Since the tip is made of tungsten, a very good conductor, its skin depth at microwave frequencies is a few hundred nanometers. One can assume that the high
frequency current only exists on the surface and that there is no EM field inside the
tip. Thus, the tip is represented by an equal-potential, φ = 1V . It is also found that
after a series of measurements, the end of the tungsten tip is inevitably flattened, and
it can not be simply treated as a sphere. In fact, we will show that the calculation of
22
Figure 2.2: The SEM image of the tungsten tip used in bulk T iO2 measurements.
the electric field shows a considerable qualitative difference between a perfectly spherical tip and a flattened tip. Thus in this analysis, the exact tip shape is determined by
scanning electron microscopy (SEM) after the experiment. Furthermore, since this is
an axially symmetric problem, it can be solved in polar coordinates to save memory
and computation time. The actual tip used here in Fig. 2.1 has a flatness of 5.02 µm
and its SEM image is shown as Fig. 2.2.
Placed beneath the tip is the dielectric material under study. In Fig. 2.1 and
Fig. 2.3 are depicted the ‘soft contact’ moment when the tip barely touches the
sample. Here We draw a sample which is a 300 nm thick film, but the sample could
also be a bulk material to represent the substrate on which the film is deposited. To
be consistent with experiments, a metal plane representing a ground plane is added
below and above the sample. After drawing the geometry with the exact dimension
23
of the experimental setup, a mesh is generated. In this model, as in many others
dealing with electromagnetic phenomena, the effects of fields near the interface of
different mediums are of special interest. In order to obtain accurate results, the
mesh near the tip end is refined several times, to make the size of the elements less
than 30 nm. The program allows us to calculate over 500,000 field points within a 10
µm*10 µm area. Fig. 2.3 is the near-field electric potential near the tip end found
by solving the Poisson’s Equation by FEM in this geometry. The inset is a magnified
view of the tip-film-substrate geometry. The dielectric constants in all domains are
set to 1, so this is the case without the sample and the tip is just surrounded by air
with the grounded stage placed at a distant equal to the sample thickness. The color
bar shows the strength of the near-field electric potential. Clearly, a strong electric
field is confined to a small region (approximately, 1 µm*1 µm) at the tip-end. The
gradient of the potential gives the solution for the the unperturbed field E0 at these
500,000 field points, the values of which are exported to Matlab in the form of a FEM
structure; ‘fem1’.
By setting the same dielectric constant to both thin film and substrate domains
and redoing the simulation without changing the mesh, one can calculate the perturbed field for the substrate. The values of this solution over the entire geometry
are exported as another FEM structure; ‘fem2’. At this point, a Matlab script calculates the integrals of the inner product of these 2 vectors over the entire sample
volume and returns the scalar quantity, Epert . See Eq. 2.1.2.
It is difficult to calculate the α numerically, as this would require a precision
greater than 1ppm when integrated over the entire space (sample + cavity), and the
limited computational resources had been focused on the near field region. Thus the
24
Figure 2.3: The the near-field electric potential at the tip end.
25
Figure 2.4: Calibration of scaling factor α with bulk T iO2 .
approach is to treat α as a parameter by fitting the perturbed energy to experimental
data, as shown in Fig. 2.4.
The sample shown in Fig. 2.4 is bulk TiO2 . The graph is a Double-ordinate plot
with the abscissa being the separation between the tip and sample. The circle points
are the experimental data and the left ordinate is the calculated ∆f /f . During the
experiment, the step size by which the sample approaches the tip is approximately
30 nm. When the tip and sample are far apart, >1 µm, the shift of the resonant
frequency is so small that the data points look like a continuous curve. However, as
the distance decreases, the strong dielectric interaction causes a larger frequency shift.
The triangle dots are the calculated perturbed energies from the FEM simulation,
found by varying the gap between the tip and sample. The data obtained are at
10 µm, 5 µm, 2 µm, 1 µm, 500 nm, 300 nm, 200 nm, 180 nm, 150 nm, 120 nm,
26
90 nm, 60 nm, 30 nm and 0 nm(contact). Alter scaling these points to fit to the
blue experimental curve, one can see the agreement between the simulation and the
experimental data is good, and the calculated scaling factor is 1.6E − 12. From Eq.
2.1.3, this is the α factor. Of course, it is not the exact value of the total energy
over the entire space, because in most cases the arbitrarily defined equal-potential at
the tip surface is set to be 1V. But a change in the potential at the tip would enter
both the numerator and the denominator of Eq. 2.1.3, so this assumption is not a
problem. We measured several samples , MgO ( = 9.5) and LaAlO3 ( = 24), to
verify the α factor. These data have the identical scaling factors so one can conclude
that the α factor is robustly determined only by the cavity and the tip, not by the
sample beneath.
2.1.2
Thin Films– Ratio of energy integrals
It is expected that thin films will be the most common samples for SNMM, so We
focus on a geometry similar to Fig. 2.1 for determining the thin film permittivity.
We started from Eq. 2.1.3 for ∆f /f and assume that we will not only measure the
thin film sample, but also its bulk substrate. The measurements of these 2 samples
will give 2 sets of data:
∆f
Epert−thinf ilm
∆f
≡
=
f thinf ilm+substrate
f thinf ilm
α
(2.1.4)
∆f
Epert−substrate
=
.
f substrate
α
(2.1.5)
If we take the ratio of these two expressions, then
∆f
=
f ratio
∆f
f thinf ilm
∆f
f substrate
27
=
Epert−thinf ilm
.
Epert−substrate
(2.1.6)
In this construction, the α factor is canceled and we can determine the film’s
dielectric constant without fitting parameters. Practically, the thin films are normally
no more than few hundred nanometers thick, compared to the substrate which is 500
micron thick. This yields a very small change of the frequency shift compared with
the bulk case, especially when the thin film’s permittivity is close to substrate’s.
This procedure leads to the cancelation of systematic error such as an inaccurate
representation of the experimental geometry.
The following is an example of determining the dielectric constant of a batch of
rare earth metal oxide thin films. The films were grown on LaAlO3 substrates (=24)
with different film thicknesses. Both the thin-film samples and the bulk substrate
materials were measured separately and the experimental data yielded a ratio of
∆f /f that was very sensitive to the film thickness. In the simulation, by varying the
thin film permittivity from 1 to 300 in the FEM geometry, and then calculating a
set of reference curves for the perturbed energy with different film thickness and tip
flatness, the experimental frequency shift can be converted to the thin-film dielectric
constant, as shown in Fig. 2.5.
Where, ∆f /fratio vs. film permittivity are plotted.
Each curve in Fig. 2.5 is calculated for a fixed film thickness by Femlab’s parametric linear solver. The perturbed energy is represented by the frequency ratio. Since
the film dielectric constants are close to those of the substrate, one can increase the
density of points in calculation at the low end ( < 50). The results show that the
thinner films have a smaller ratio as they cause less perturbation, which is consistent
with our intuition. All curves pass through a common point where the ratio equals
to 1; this is the case where the film’s dielectric constant is the same value as the
28
Figure 2.5: The fitting curves of thin-film samples on LaAlO3 substrate with varying
thickness.
29
substrate’s dielectric constant: the bulk case. By matching the experimental data,
one can find the thin film’s dielectric constant from the graph. For example, if the
measurement of a 1000 nm thin film gives the ratio of 1.2, one can find the corresponding is 70, as shown in Fig. 2.5. To test the effect of the substrate thickness, we
used 500 µm, 1 mm and 2 mm substrates in the same FEMLAB model, and observed
no measurable difference in the results. Thus the substrates can be treated as being
infinitely thick.
By this method, we also find from the fitting curves that with the same measurement uncertainty, the error will be smaller when thin film permittivity is lower than
the substrate’s, because of the steep slope of the fitting curve. When calculating the
dielectric constant of a thin-film, special attention must be paid to error propagation because the signal change is small: for some samples it is only 1/10000th of the
background frequency.
2.2
Imaginary Part of Dielectric Constant – Dielectric Power Dissipation
By implementing a loss tangent calculation model into FEMLAB analysis, we can
determine the imaginary part of the permittivity at a sub-micron length scale, a
subject on which there are no reliable reports so far. Unlike the simulation of real
part of permittivity, this model is built on a high-frequency EM model of FEMLAB,
not a quasi-static one for the following reasons: first, the real part of the permittivity
can be obtained by solving the equations in a very small area, but the imaginary part
needs account for the radiation loss over the surrounding space, which increases the
30
‘electrical size’ over the critical limit of 0.1; second, the medium loss is caused by the
EM fields flipping dipoles up and down, the media could be nonlinear, inhomogeneous
or anisotropic and these properties are independent of the structure size and in most
cases are functions of frequency as well as of position.
The implementation is straightforward. A geometry similar to that used in the
quasi-static model is used except for its dimensions. It consists of a coaxial cavity
with a small ring shaped window on the bottom, through which the tip contacts the
inner conductor of cavity. The tip acts as an antenna operated at 1.74GHz. The
model is designed in 2D with cylindrical coordinates to take the advantage of the
symmetry of the experiment. The interior of the metallic conductors are modeled
by their boundaries having a tangential component of the electric field equal to zero.
Here we solve the full wave equation in the region surrounding the tip and sample for
the electric and magnetic fields both inside and outside the cavity, and accounting
for radiation losses and heat absorption in the thin film and substrate.
In this way, the energy loss in the medium is obtained from the value of its
dielectric absorption from EM field as a component of the total loss: resistive heat
loss, cavity loss, and tip radiation loss. Here the cavity loss is the only factor which
is almost impossible to precisely calculate from the model, and so it must be known
before measuring any sample. This is determined from a measurement of a standard
sample.
The EM wave in the coaxial cavity is characterized by TEM fields. Assuming time
harmonic fields with complex amplitudes containing the phase information we have:
~ = e~r C ej(ωt−kz)
E
r
~ = e~ϕ C ej(ωt−kz)
H
rZ
31
(2.2.1)
(2.2.2)
1~
C 2 router
ln
,
S~av = E
× H~ ∗ = e~z π
2
Z
rinner
(2.2.3)
where z is the direction of propagation and r,ϕ and z are the cylindrical coordinates
of the coaxial cavity. Z is the wave impedance, Sav is the time average power flow,
rmin and rmax are the inner and outer radii of the cavity, respectively. The angular
frequency is denoted by ω. The propagation constant is k.
In this model, the electric field inside the cavity has a finite axial component
whereas the magnetic field is purely in the azimuthal. Thus we use an axisymmetric
transverse magnetic (TM) formulation, and so the wave equation becomes scalar in
Hϕ :
∇ × ((r −
jσ −1
) ∇ × Hϕ ) − µr k20 Hϕ = 0.
ω0
(2.2.4)
The boundary conditions for the metallic surfaces are:
~ = 0.
~n × E
(2.2.5)
The feed point, which is the space between sapphire disk and tip edge in the plane
of cavity cap, connects the EM field inside the cavity to the excitation fields outside
the cavity. The microwave energy couples out of the cavity through this point. Part
of this field travels along the tip surface down to its end to form the near-field there,
and the rest dissipates into the free space between the sample and cavity cap as the
radiation. So from the view outside the cavity, it appears that the EM waves are fed
by the cavity, as shown in Fig. 2.6.
The feed point is modeled by a classical first order low-reflecting boundary condition combined with an excitation field Hϕ0 :
32
Figure 2.6: Geometry definition in high frequency mode.
~n ×
√
~−
E
√ ~
√
µHϕ = −2 µH~ϕ0 ,
(2.2.6)
where
Hϕ0 =
Constant
.
r
(2.2.7)
The constant here is deduced from the time-average power flow which can be
preset by the vector network analyzer.
The tip, like any antenna, radiates into the free space and into the sample where
a damped wave propagates. As we can only discretize a finite region, we truncate
the geometry 10 cm from the tip using a similar absorbing boundary condition as
the feed point but without excitation, and we also apply this boundary condition to
33
all external boundaries. Finally, we apply a symmetry boundary condition for the
boundaries at r=0:
E~r = 0,
(2.2.8)
∂ E~z
= 0.
∂r
(2.2.9)
In this model, we consider the dielectric power dissipation to be resistive heat
generated by the electromagnetic field in interaction with the material. The measured
samples have complex permittivity, where the imaginary part of accounts for loss
(heat) due to damping of the vibrating dipole moments. The loss within the dielectric
material may also be considered to be equivalent to conductor loss caused by a current:
~
J~ = σ E,
(2.2.10)
~
where σ is the conductivity. Apply Maxwell’s curl equation for H:
~ +(ω” +σ)E
~ = jω(0 −j” −j σ )E.
~ = jω D+
~ J~ = jωE
~ +σ E
~ = jω0 E
~ (2.2.11)
∇× H
ω
Here it is seen that loss due to dielectric damping (ω” ) is indistinguishable from
conductivity loss (σ). The term ω” + σ can then be considered as a total effective
conductivity σef f . A related quantity of interest is the loss tangent, defined to be
ω” + σ
tan δ =
,
ω0
(2.2.12)
which is the ratio of the real to the imaginary part of the total displacement
current. For this reason, microwave materials are usually characterized by specifying
the real permittivity and the loss tangent at a certain frequency.[31]
34
Figure 2.7: Mesh domains in high frequency mode.
The absorption over the entire volume of the dielectric sample, in the form of
resistive heat, can be expressed as:
Eresis.heat
1
=
2
Z Z Z
~ · E~∗ ].
Re[(σef f − jω)E
(2.2.13)
v
The calculation proceeds by drawing the geometry, defining the boundary conditions, and generating the mesh. In areas where the field is likely to be intense, such
as the feed point, the tip surface, and the contact point with the sample, the mesh
is refined to reduce cells to be as small as 100 nm. This is crucial for an accurate
calculation. The mesh domain is shown in Fig. 2.7 with 53321 elements and 427991
degrees of freedom, these are the upper-limits that our computational environment
can handle. The solution obtained is shown in Fig. 2.8.
The right side of Fig. 2.8 is the full view of the simulated EM field, where we
35
Figure 2.8: Solved EM field in high frequency mode.
can see that the field intensity is concentrated near the feed point, and it attenuates
down along the tip surface. In the open space between the cavity cap and sample, the
radiation field is weak. This can be used to explain the effect of the tip length in our
set-up. From our experiments, we find that the longer the tip, the lower the quality
factor. This is consistent with the calculation when for a long tip, the electromagnetic
waves lose energy by attenuation and by radiation into open space. A shorter tip has
better coupling, but it is also influenced more by the far-field component propagating
from the feed point, which increases the back ground noise in scanning. An optimized
tip length is a complicated function of the power dissipation and the geometry of the
sapphire disk. In our case, the sapphire disk is 2.5 mm in diameter with a 0.4 mm
center hole, the input power from the vector network analyzer is 1 dbm, and a 3 mm
tip usually gives very satisfied Q values and scanning results.
36
The left side of Fig. 2.8 is an amplified view of the field distribution at the tip
end, where we can see that the field is concentrated at the contact point with the
thin film. From the scale bars, the strongest field is not at the feed point but at the
contact point, and the penetration depth of the field is only few microns into the
sample. Here we point out that although the accuracy of the near-field distribution
in this model is much less than in the quasi-static model, because of the number and
size of the mesh elements. The essential characteristics of the near-field are however,
the same as in the quasi-static model. This increases our confidence in the validity
of this approach.
After solving the wave functions, the relationship between the quality factor and
energy losses can be established. The general definition of the quality factor, Q, of a
resonant cavity is:
Q=ω×
average energy stored in cavity
,
energy loss per second
(2.2.14)
where ω is the angular frequency of the system. Physically speaking, the Q factor
compares the frequency at which a system oscillates to the rate at which it dissipates
its energy. A higher Q indicates a lower rate of energy dissipation relative to the
oscillation frequency. In this case, the energy losses come from 3 sources: cavity wall
loss Ecavloss which only occurs inside cavity, tip loss Etiploss , including tip radiation and
tip resistive heat, and sample resistive heating loss Eresis.heat . Their corresponding Q
factors are:
Etotal
Ecavloss
Etotal
=ω×
Etiploss
Qcav = ω ×
(2.2.15)
Qtip
(2.2.16)
37
Qresis.heat = ω ×
Etotal
Eresis.heat
(2.2.17)
and the total Q is given by:
1
Qtotal
=
1
1
1
+
+
.
Qcav Qtip Qresis.heat
(2.2.18)
Generally, the average energy stored in cavity; Etotal , and cavity wall loss, Ecavloss ,
are almost impossible to measure. In order to overcome that difficulty, we choose
to measure Q factors with and without the sample, as Qair and Qsample . Since the
majority energy in the system is stored inside cavity and the sample is treated as a
small perturbation. One can conclude that Etotal is almost the same in both measurements. Hence, Ecavloss is also the same because this factor is only determined by
cavity itself. So if the ratio of Qair to Qsample is taken, the Etotal can be canceled,
leaving Ecavloss as the only unknown factor. A standard sample with a known loss
tangent can be measured to determine Ecavloss , and having determined this parameter other samples can be measured. We perform the calibration with bulk lanthanum
aluminum oxide, LaAlO3 , whose literature value of loss tangent is 2E-5. The total
effective conductivity is then:
σef f −LaAlO3 = tanδ × ω × 0 = 4.9E − 5.
(2.2.19)
Without a sample, we measured resonant frequency ωair =2π×1756MHz, Qair =878.
With a sample, LaAlO3 , beneath the tip, the resonance frequency shifts to ωLaAlO3 =
2π×1747 MHz, QLaAlO3 =790. From Eq. 2.2.15 to Eq. 2.2.18,
Etotal
Ecavloss + Etiploss−air
Etotal
.
×
Ecavloss + Etiploss−LaAlO3 + Eresis.heat−LaAlO3
Qair = ωair ×
QLaAlO3 = ωLaAlO3
38
(2.2.20)
(2.2.21)
Here, Ecavloss is considered unchanged. Etiploss , the power flow by radiation, can
be integrated along the tip boundary using Femlab’s postprocessing function, which
changes in the presence of the sample. Similarly, Eresis.heat is the sum of the Femlab
integration of the resistive heating over the entire thin film and substrate domains.
The results of these calculation are listed in the Table 2.1.
Table 2.1: The
Energy
Value
Etotal
Unknown
Etiploss−air
1.113E-9
Ecavloss
19.132E-9
Etiploss−LaAlO3
2.331E-9
Eresis.heat−LaAlO3 9.212E-10
integrations of energy loss by Femlab.
Source
Canceled by taking ratio
Calculated by Femlab
Calculated by solving ration equation
Calculated by Femlab
Calculated by Femlab
By replacing the exact values, the ratio is:
790
1747
1.113E − 9 + (Ecavloss )air
QLaAlO3
=
=
×
.
Qair
878
1756 2.331E − 9 + (Ecavloss )LaAlO3 + 9.212E − 10
(2.2.22)
It can be seen here, that when a sample is present, the tip radiation loss increases
from 1.113E-9 to 2.331E-9. Keeping the assumption that that cavity wall loss is independent of the sample (Ecavloss )air =(Ecavloss )LaAlO3 . We find that Ecavloss =19.132E-9,
which is almost 10 times greater than the sum of tip loss and resistive heating loss.
LaAlO3 is relatively loss free, but other samples have higher loss tangents, which can
now be calculated knowing the cavity’s calibration factor.
2.3
Spatial Resolution and Sensitivity
As part of the theoretical modeling, the intrinsic spatial resolution and the sensitivity
are also studied by simulation. Technically, there are two types of resolution. One is
39
Figure 2.9: The electric field polarization on tip end.
called the imaging resolution, or the smallest feature can be detected from a high contrast sample such as a set of periodic strips. The other is the metrological resolution,
which is the largest material volume to which the tip response is insensitive.[32] The
latter is only dependent on the tip’s geometry and its near-field distribution. Some
groups have achieved a few nanometer imaging resolution by using metal or high
permittivity strip samples, materials that strongly interact with the tip and causes a
redistribution field at the tip’s end.[33, 34, 35] Here, in this thesis, the focus will be
on metrological resolution because we seldom treat samples with dielectric constants
greater than 900.
Since the spatial resolution can be seen from the field pattern, we plot the near
field electric field, both r and z components, at the tip end in Fig. 2.9.
The tip shown in Fig. 2.9 has a 4.5 µm flattened end. The inset shows the polar
40
coordinate system used in the simulation. Clearly from the Figure, the z-component
is a few hundred times bigger than r-component. So the dominant component of the
electric field is polarized along Z-axis, which reaches its maximum value at the edge of
the tip and then decays rapidly. This is qualitatively different from the field obtained
from a spherical tip, as anticipated earlier in this chapter.
Ez forms a ring of 2.25 µm radius and 250 nm width, so the actual interaction
area is a ring with that thickness circling the flattened surface of the tip. This implies
that the spatial resolution is almost the same as the tip flatness (typically, from 200
nm to 6 µm). Thus, as the tip radius decreases, the spatial resolution increases, and
in order to obtain high resolution, a small, sharp tip is essential.
The sensitivity, on the other hand, is also determined by the tip flatness and
tip-cone angle, because these two parameters determine the effective area of tip and
thus the probe’s output power. The tip end is inevitably deformed during the measurements, so we calculate the ∆f /f ratio with different tip flatness, as shown in
Fig. 2.10. The sample in this calculation is LaAlO3 ( = 24), coated by a thin film
whose permittivity is allowed vary from 1 to 300. In the calculation, the bulk case
is recovered when the thin film permittivity equals that of the substrate, where the
perturbation energy ratio is 1. For a given thin film sample, the greater the deviation
of the perturbation energy ratio from 1, the easier it is to determine the value of its
permittivity.
As shown on Fig. 2.10, a flatter tip normally has better sensitivity (shown by a
larger change in perturbed energy), but when the flatness exceeds 3 µm, the sensitivity
increment becomes negligible. Then it is obvious that our goal here is to determine
the best working condition to enhance the sensitivity without sacrificing too much
41
Figure 2.10: Sensitivity analysis of tips with different flatness.
in spatial resolution. In other words, one has to balance the tip flatness for the
sensitivity and spatial resolution needed to achieve the best measurement results.
42
Chapter 3
Experiment
As discussed in the introduction (Chapter 1), localized measurements at microwave
frequencies provide useful quantitative images of important characteristic properties
of dielectrics, such as permittivity, polarization, conductivity and surface topography.
Unlike optical radiation which interacts with matter via plasmon excitations, quantum
effects etc., the microwave interaction is much simpler, but it is classically limited to a
long length scale: larger than the electromagnetic wavelength. The direct simplicity
of microwave measurements combined with the spatial resolution of the near field
microwave microscopy enables many new types of investigations in materials science.
3.1
The Principles of the SNMM Design
The design of the SNMM can be basically divided into four parts, resonator construction, tip preparation, sample positioning, and signal detection.
43
3.1.1
Resonator construction
The resonator construction is used to couple the microwave energy from the source,
which generates a time harmonic signal, either via capacitors or coupling loops. The
resonator can be a coaxial waveguide with the advantage of no cutoff wavelength
limit for a broadband measurement, or it can be a cavity, which is more sensitive but
works only in a narrow frequency range. The former approach was pioneered by Steve
Anlage’s group at the University of Maryland,[36, 24, 37, 38] and the later by Xiang’s
group at Lawrence Berkeley National Laboratory.[11, 39, 40, 41] Both approaches
allow localization of the microwave field. At a fixed frequency, the reflection or
transmission (through coefficient S11 or S21 ) of the cavity is measured. When the
sample’s properties change, the stored energy and dissipated power in the cavity
are altered, and the resonant frequency and quality factor of the cavity shift. The
system is very sensitive to different dielectric environments and is also very efficient
in the narrow frequency band for which it is designed. The signal-to-noise ratio in a
cavity structure increases with cavity quality factor Q. Because the cavity resonator
produces two independent data streams, f and Q, it is advantageous to apply this
design to simultaneously measuring the dielectric response and loss tangent. Designs
coupling the cavity’s field to a sample through an aperture have been used for imaging
of ferroelectric materials.[21] However, when a simple aperture is used, its resolution
is limited to the millimeter range, when using frequencies around 1 GHz. If a sharp,
tapered STM-like tip is used to replace the aperture, this feature acts like a point
emitter. The cavity is closed all around except for a small hole, concentric with the
probe tip, which extends beyond the shielding. See Fig. 3.1.
44
Figure 3.1: The System Diagram of Scanning Near-field Microwave Microscope.
The resolution can be improved by reducing the size of the aperture and by sharpening the probe tip. A metallized sapphire ring is inserted into the aperture to shield
the far-field propagating components and reduce losses resulting from the proximity
of the shielding wall. This geometry maintains the highest possible Q factor, and
has been chosen by researchers to quantitatively map dielectric-constant profiles and
ferroelectric domains.[22, 24, 42] All these improvements have increased the instrument’s performance, and spatial resolutions of 100 nm have been reported.[40] This
type of instrument can be adapted to image conductors if the samples are coated with
a thin insulating layer.[11]
Most near-field microwave microscopes consist of a sub-wavelength antenna-like
probe, for example the tungsten tip in our experimental setup, which emits and collects the electromagnetic signals. The electrical properties and symmetric geometry
45
Figure 3.2: The picture of cavity of Scanning Near-field Microwave Microscope.
46
of the wire tip are the most important factors in determining the resolution of the
microscope. In order to create a near field to interact with the sample, the tip’s
characteristic size, D, must be much smaller than the wavelength λ. In many cases
where a rough surface or unevenness of the complex permittivity over the surface of
the sample is to be studied, the sharpness and the geometry of the tip are important
factors, which effect the resonant frequency shift and quality factor.
3.1.2
Tip preparation
Sharp wire tips are prepared from tungsten wires by drop-off electrochemical etching,[43,
44] which can be done by alternating-current or direct-current. Each procedure gives
a different geometrical tip shape; ac-etched tips have a conical shape with much larger
cone angles than dc-etched tips. Dc-etched tips, on the other hand, have the shape of
a hyperboloid, and are much sharper than ac-etched tips. The ac-etched tip is usually
prepared by the drop-off method in which etching occurs at the air-electrolyte interface causing the portion of the wire in solution to ‘drop off’ when its weight exceeds
the tensile strength of the etched or necked down part of the wire. When the lower
portion of the etched wire drops off, the etching current through the tip should be
turned off as quickly as possible to prevent blunting. A second etching of the tip can
be used to remove the oxide layer and to further sharpen the tip. A nitric dip is used
to further smooth the tip and remove any remaining oxide layer. This method of tip
preparation is simple, fast, and reliable. Later, the tips are examined with a scanning
electron microscopy (SEM) at high magnification to check their shape and quality as
well as to measure the radius of curvature of the tip. See Fig. 3.3 and Fig. 3.4.
The electrochemical etching reaction involves the anodic dissolution of tungsten
47
Figure 3.3: AC-prepared wire-tip.
Figure 3.4: AC-etched wire-tip after nitric acid dipped.
48
Figure 3.5: Experimental setup of for tungsten wire etching.
in a basic aqueous solution (saturated NaOH).[45, 46] The etching reaction can be
adjusted by changing either the supply voltage or the solution’s concentration.
Fig. 3.5 shows the details of the etching cell, which consists of a beaker containing
approximately 100 ml of saturated NaOH solution and a power supply. In our setup,
the tungsten wire with diameter 0.35 mm is attached to form the tip. A counter electrode is made from a 0.01 mm platinum wire formed into a circular loop of diameter
∼20 mm. This is placed in the electrolytic solution and a tungsten wire electrode is
placed in the center of the loop.
Etching occurs at the air-electrolyte interface when a voltage is applied between
the tungsten wire (anode) and the platinum loop (cathode). It is important to maintain strict vertical, azimuthal symmetry with the loop and wire tip. The overall
electrochemical reaction takes place in about one minute. The reactions are:
49
At the cathode:
6H2 O + 6e− → 3H2 ↑ +6OH − ,
(3.1.1)
W + 8OH − → W O42− + 4H2 O + 6e− .
(3.1.2)
At the anode:
The tungsten oxide is formed once the potential exceeds 1.43 V. Usually, the potential required to drive an electrochemical reaction is higher than the one calculated
from standard electrode potentials, particularly when one of the products is a gas
such as hydrogen. The excess potential is required because most electrode reactions
occur with the potential distributed along the reaction path. The local potential
difference is affected by changes in the concentration of the reactants, products and
other mass transfer processes. We use a potential of 5 V.
3.1.3
Tip-sample distance control
When the etched tip is assembled into the cavity and brought to measurement. Its
distance to the sample must be carefully controlled. To do this, the sample sits on a
3-axis stage which can be moved by three linear positioners in the x, y and z directions
and the tip height is controlled by a piezocrystal. Since near field is confined to the
vicinity of the probe, high accuracy is needed to move the sample a few nanometers
from the probe. Furthermore, in order to image features with sub-micron resolution,
the tip-sample separation has to be as close as possible to the height variation of the
smallest feature in the sample. The precision of any near field measurement is always
related to the precision with which the tip-sample distance can be controlled, so it is
advantageous to have a distance regulation mechanism. Several solutions have been
proposed, including constant-height modulation, electron-tunneling distance control,
50
atomic-force distance control, and shear-force feedback. In constant-height modulation, the tip is positioned at some vertical position corresponding to a point along the
distance vs. frequency curve, see Fig. 3.9, for example. Some microwave microscopes
have reported employing height-modulated control.[47, 48, 49, 50] This kind of control
has no feedback and the scanning is fast, however, some kind of a priori information
about the height variation is required in order to protect both the tip and the sample from damage. Electron-tunneling distance control uses the well-established STM
technique to maintain the tip-sample separation at 1 nm. At this distance, with a DC
dias applied, a quantum mechanical tunnel junction is established, and the current
decays exponentially with tip-sample distance. A number of research groups have
integrated this mechanism into their near field microscopes to maintain a constant
tunnel current during the surface scanning.[40, 51, 52, 53, 54, 55] However, it works
only for metallic samples. In atomic-force distance control, the tip is mounted on
an AFM cantilever, when the tip is brought close to the sample surface, the repulsive forces displace the cantilever, which can be measured by a laser beam reflection
technique. This type of control allows atomic-scale resolution and has been used to
image ferroelectric domains.[56] However, dielectric sensitivity is limited. At present,
the most common method of probe-sample distance regulation relies on the detection
of shear forces between the end of the near-field probe and the sample.[57] The basic
idea is the probe can be mounted with a tuning fork and dithered by an oscillator.
Due to van der Waals forces between the tip and the sample surface, the resonant
frequency of the fork is changed and the dithering amplitude decreases as the probe
approaches the sample. This motion is detected by a feedback loop to control the
distant to a precision of a few nanometers.[27, 58, 59] Shear force feedback allows
51
topography alone, or simultaneous topography, and near field imaging.
3.2
Our SNMM Experimental Setup with temperature control
The SNMM in our laboratory is depicted in Fig. 3.1 and Fig. 3.2. It consists of a
1.75GHz (17cm wavelength) λ/4 copper coaxial resonator. An aperture is drilled in
the center of the bottom plate and the inner conductor inside the cavity is cone shaped
at its lower end. This is essentially an open circuit cavity, in which the maximum of
the electric field is near the end point of the inner conductor. Ideally, the size of the
aperture should be small to shield the far-field propagation components away from the
sample, and keep the cavity quality factor Q as high as possible. This way only the
near-field components interact with the sample and give rise to the imaging signals.
To achieve this configuration and optimize the spatial resolution and sensitivity, a
sapphire disk is inserted into the aperture. The disk has a center hole with the
diameter comparable to the tungsten tip and is coated with a gold layer on the outside
surface with the thickness of the skin depth. Two antennas made from transmission
line are placed inside the cavity at the top. These are weakly coupled with and radiate
the electromagnetic waves into the cavity. The depth to which they are inserted and
the directions they face affect the coupling factor and therefore determine the quality
factor of the resonator. They are thus carefully adjusted to optimize the quality factor
and coupling. The microwave source is an Agilent HP 8753D vector network analyzer
(VNA), which drives the cavity through the antennas. The resonant frequency, in this
measurement, is the lowest supported TEM mode. An etched tungsten wire protrudes
52
through the sapphire disk from the central conductor and provides close coupling to
the sample under study. The EM field at the tip end is well described by near-field
theory. The measured data are based on the transmission and reflection coefficient of
the cavity relative to the source. When a sample is present, the dielectric properties
of the sample distort the electric field near the tip and cause a measurable shift in
the resonant frequency, f, and in the quality factor, Q, of the cavity. These combined
shifts can be used to determine the complex permittivity of the sample with spatial
resolution in the sub-micron range. This spatial resolution is determined by the shape
of the probe tip, rather than by the wavelength λ.[60, 42, 61]
The sample is placed on a stage, which can be moved in three dimensions. The
in-plane displacements along the X and Y axes are driven by two mechanical motor
actuators with a minimum step of 1 µm. The out-of-plane, Z axis, displacement
has to be accurately controlled, so in addition to a mechanical motor actuator, a
piezoelectric actuator is used for fine positioning. The minimum step size for the
piezoelectric actuator is 30 nm. All mechanical motor actuators are controlled by a
Newport motion controller and the piezoelectric actuator is controlled by a Thorlabs
piezo-controller, which is driven by a DAC.
Since one part of the experiment is focused on how to measure the dielectric
constant of the sample undergoing phase transition, it is critical to be able to cool
or heat the sample to a specific temperature. To this end, an HFS91 heating and
freezing stage from Linkam with a temperature range from −196◦ C to 300◦ C is added
to the sample stage. When measuring a phase transition, the sample is mounted on
a copper block to ensure excellent heat transfer and good temperature stability. A
platinum resistor sensor, accurate to 0.01◦ C, is used in the setup to provide precise
53
and stable temperature sensing for a LakeShore 330 PID temperature controller. The
heating is done resistively and the cooling is provided by liquid nitrogen. The cooling
control unit houses a pump and a 2 Liter dewar. The liquid nitrogen flow is precisely
controlled to enable specific cooling rates, and the recycled dry nitrogen gas is used
to purge the sample chamber of water vapor.
One of the most crucial problems to solve in the low-temperature measurement
is to minimize vibration, which is not a problem in the measurement above roomtemperature. The vibration could cause permanent tip damage or lead to artifacts,
since the distance between the tip and the sample is just few nanometers on softcontact and the field varies rapidly as this gap changes. The main source of the
vibration is the thermal expansion which comes from the heater’s on-off control and
the boiling of the liquid nitrogen. The noise from the heater can be eliminated by using
continuous proportional-integral-derivative control instead of on-off control, so that
when the stage approaches equilibrium, the current through the heater is maintained
at a stable level with small deviations. However, the noise from the liquid nitrogen
is not easy to remove. First, the liquid nitrogen runs through the tubing under the
sample stage, and this mechanical vibration exists as long as something is flowing.
Second, the liquid nitrogen vaporizes during cooling, which brings extra vibration.
We tried several means to get rid of these vibrations, including decreasing the flow
rate, using cold nitrogen gas instead of liquid, using a reservoir as damper to stabilize
the gas stream. But none of these worked. Finally, we redesigned the cooling system
by adding an extra copper stage, which mechanically separated the heater from the
nitrogen tubing. See Fig. 3.6.
In this scheme, a soft copper braid provides thermal contact between the sample
54
Figure 3.6: Scanning near field microwave measurement on heating and cooling stage
stage and the cooling block. The sample is placed on top of heating stage during the
measurements and the liquid nitrogen still flows through the cooling block as before.
Since the braid is soft, the vibration generated by the cooling stage is much attenuated
before reaching the heating stage, while the heat still can be exchanged between the
two stages because of the cooper’s high thermal conductance. The improvement from
this change is significant, as can be seen in Fig. 3.7.
In this Figure, both graphs were taken at 0◦ C on the same sample, LaAlO3 . The
upper graph is the one before redesign, and the lower one is after redesign. The
abscissa represents the tip-sample distance with a certain offset, and ordinate is the
first derivative of resonant frequency vs. distance (basically the differentiation of
the top curve in Fig. 3.9 ). A detailed description of the information in this graph
will be given in later, but here, we only want to compare the smoothness of these
two curves. Clearly, the lower one is less noisy and shows a sharper peak than the
upper one, which indicates that the amplitude of the stage vibration is decreased
55
Figure 3.7: The comparison measurements of df /dz vs. tip-sample distance ‘z’ before
and after redesign the temperature stage.The sample measured here is bulk LaAlO3 .
56
significantly after redesigning the stage. This modification however, is not perfect
because it weakens the cooling efficiency, and although the cooling block still can be
brought to liquid nitrogen temperature, the lowest temperature we could obtain on
the heating stage was −40◦ C. This limits the range of phase transition that can be
observed. In short, we have to sacrifice the temperature range to obtain a good signal
to noise ratio.
The electronic detection is also done by the HP 8753D vector network analyzer
working in a two port mode. At microwave frequencies, wavelengths become small
compared to the physical dimensions of the devices such that two closely spaced
points can have a significant phase difference. The cavity, antennas, tip and sample
can be considered as a distributed parameter network, where voltages and currents
vary in magnitude and phase over its length. Additional high frequency effects such as
radiation loss, dielectric loss and capacitive coupling make microwave circuits more
complex. Low frequency lumped-circuit element techniques must be replaced by
transmission line theory to analyze the behavior of devices at higher frequencies. For
this reason the S parameters are measured here by a two-channel microwave receiver
to find the magnitude and phase of the transmitted and reflected waves from the
input and output loops of the cavity. The RF source from port 1 is set to sweep over
a specified bandwidth, usually 15MHZ. 1600 data points are evenly placed over the
bandwidth, which provides an accuracy of 0.01MHZ. The data acquisition system is
programmed with Labview 6.0 software package, which controls the analyzers and
actuators via the IEEE-488 (GPIB) bus.
57
3.3
The Measurement
The thickness of the sample is measured before placing it on the stage. Then the
bare copper stage is separated from the tip a distance exactly equal to the sample’s
thickness. At that position, the resonant frequency is recorded as f0 , the background
frequency. The reason we do this is to be consistent with the boundary conditions
set in the numerical simulation. Then the stage is lowered and sample is glued onto
it with thermal cement. After that, the sample is brought to within a few millimeters
of the tip. First, a mechanical actuator is used to move the sample towards the tip at
several microns per step, controlled by a labview Vi program. The resonant frequency,
f, versus tip-sample distance, z, as well as its first derivative, df/dz are monitored
continuously. When the tip is far from the sample, no strong near-field interactions are
observed, so the resonant frequency is nearly constant. But when sample approaches
within few microns, f begins to drop because of the near field coupling to the sample.
Since accurate vertical positioning is critical for our measurement, the mechanical
actuator is stopped when the sample is close to the tip, shown in Fig. 3.8, and a
piezoelectric actuator is used for closer approach. The voltage step applied to the
piezocrystal can be as small as 0.07 V, which corresponds to a 30 nm movement. The
dielectric constant measurement is made by stepping the sample towards the tip in
these small, 30 nm, steps until the sample just contacts the tip. A room-temperature
frequency vs. height plot for a rare earth element thin film on an LaAlO3 substrate
is shown in Fig. 3.9.
Part A in this graph is the curve f (MHz) vs. z ( µm). The abscissa has an offset,
because it is directly converted from the piezo voltage. The background frequency, f0 ,
is noted on graph with sample far away from the tip. However, the soft contact point
58
Figure 3.8: The frequency f, quality factor Q and df/dz vs. z of a thin film sample
by motor approach. Motor stopped at df/dz = 0.08.
59
Figure 3.9: The scanning frequency of a thin film sample in room temperature. The
value of df/dz at the contact point is 7.
60
is determined by the df/dz curve, B. As expected, the strongest interaction occurs
at the contact point, where df/dz reaches its maximum value. After that, the tip
contacts the sample and df/dz drops to 0. We usually allow the actuator to advance
2 or 3 points past the contact point to insure that peak was not caused by noise. Once
the contact point is determined, the frequency shift ∆f is found, as shown on the
graph. For low-dielectric-constant materials, the shift is small, and a typical ∆f /f is
less than 1/1000. Another thing that can be observed from this graph is the effective
range of the near field. The df/dz curve gives very different noise levels in different
distance ranges. When the distance is greater than 4.8 µm, the noise level is less than
±0.2, but when it is closer, the noise level quickly increases to ±1. Assuming the
stage vibration amplitude is constant, this increase in the noise indicates the onset of
near-field interaction. From the graph, we estimate the effective range of the nearfield to be less than 1 µm. This has been verified by the numerical simulation shown
in Fig. 2.3.
The quality factor, Q, is obtained simultaneously. Similarly to f, Q decreases as
the sample approaches the tip. Similarly, Q0 and Qcontact are also be determined in
the same way as f0 and fcontact . But, in general, the dQ/dz curve is more noisy than
df/dz curve. That is because the Q factor is a measure of energy loss and the main
loss in this design is due to the cavity wall current plus tip radiation, which is more
than 20 times larger than the dielectric loss from the sample. Fig .3.10. shows a
typical Q vs. z curve for a thin film sample, from which it is clear that f vs. z is more
appropriate for finding the contact point.
Using the measurement techniques and numerical modeling derived here, we can
accurately measure the complex dielectric constant. Examples of measurements of a
61
Figure 3.10: The scanning Q factor of a thin film sample in room temperature.
number of bulk materials are shown in Table 3.1. Except for the very low dielectric
material, the agreement between our measurements and the literature values is good.
Table 3.1: Table
Sample
Measured Teflon
3.16
Sapphire 9.8
M gO
9.53
STO
278.2
BTO
292.23
T iO2
91.7
of dielectric constant measurement for bulk samples.
literature Measured tanδ Literature tanδ
2.4
3.6E-5
9.5
1.8E-5
1.4E-5
9.8
1.8E-5
1.6E-5
276
1.804E-4
1.6E-4
300
0.513
0.47
86
1.5E-2
1.2E-2
62
Chapter 4
Crystallization temperature and
Ferroelectric phase transition
In this Chapter, two studies that we have made with the SNMM and Femlab simulation for this dissertation research will be described: one is the study relating
crystallization of thin films to their dielectric response and the other is a study of
films composed of alternating layers of ferroelectric and dielectric materials. These
experimental results benefit our understanding of the mechanism for the dielectric
relaxation and for the phase transformations involving reorientation and ordering in
dielectric thin-film materials. Each of these studies are important for potential applications, such as RF filters, micro-electromechanical switches, high-value capacitors
and system-in-package (SiP) devices.
63
4.1
Relationship between crystallization and dielectric response of epitaxial rare-earth scandate films
The motivation for this research is to test materials with potential as alternative gate
dielectrics for future metal-oxide-semiconductor field-effect transistors (MOSFETs).
Candidate materials must simultaneously satisfy the criteria of having a high dielectric constant , chemical stability in contact with silicon under normal processing
conditions,[62] and have a large band gap and large band offsets relative to silicon.
The material system that we investigate is a new family of rare-earth scandates.[63]
Dielectric constants and large optical band gaps have been reported for single crystals and for amorphous films.[64, 65] The crystallization temperature is a critical
issue for films because in applications of amorphous high dielectrics, crystallization
of the material during subsequent device processing must be avoided, whereas other
applications of epitaxial crystalline films are technologically feasible only if the crystallization temperature is sufficiently low. Thus, in order to systematically investigate
the crystallization temperatures of the entire family of known rare-earth scandates, a
full series of Re ScO3 thin films, (where Re =Y, La, Pr, Nd, Sm, Gd, Tb, Dy, Ho, Er,
Tm, Yb, and Lu, i.e., the entire series for which the individual oxides are chemically
stable in contact with Si) were deposited in a temperature-gradient pulsed laser deposition system onto LaAlO3 substrates by our collaborators at Oak Ridge National
Lab.[66, 67] The deposition method was pulsed laser deposition (248nm radiation)
in 80mTorr of O2 . The deposition temperature was 800◦ C, and film thicknesses between 250 and 1000 nm were obtained in each 25000-pulse run. Thicknesses were
64
measured by spectroscopic ellipsometry and a conventional fixed polarizer-samplerotating polarizer system under nitrogen ambient. The thickness values obtained
were very accurate, with correlated errors less than 0.2% of the film thickness.
The dielectric constants were measured at 1.7 GHz by our near-field microwave
microscope. The following resonance frequencies were measured: f0 for the resonator
without a sample, fsample for the resonator in contact with each of the thin film
samples, and fsubstrate for the resonator in contact with a bare LaAlO3 substrate. Every
frequency was measured three times and the average value and standard deviation
were obtained. Care was taken to measure fsubstrate before and after each measurement
of fsample to guard against shifts in the background signal over the course of the
experiment. The tip flatness was determined by scanning electron microscopy after
each series of measurements. Then, the frequency shift ratios were converted to the
film’s value by using the lookup chart from the Femlab simulation. Results are
shown in Fig. 4.1.
In this Figure, the dielectric constant for the series of rare-earth scandates determined by microwave microscopy are represented by open circles with the error
bars representing the standard deviation of three measurements for each sample. For
comparison, bulk dielectric constants measured at 1 MHz are shown as solid circles. Multiple data points indicate the tensor coefficients along the principal axes for
single-crystal samples.[68] The data are ordered by atomic number along the abscissa.
Comparing the thin-film results to the bulk values, it is immediately observed
that the two anisotropic films’ (Gadolinium and Dysprosium) values are closest
to the larger bulk values measured along the orthorhombic c-axis (001 direction).
Considering the electric field polarization of the microscope probe, it is clear that the
65
Figure 4.1: Dielectric constant of rare-earth scan dates deposited at 800◦ C.
66
Figure 4.2: X-ray diffraction in determining the crystal orientation.
measured values are the out-of-plane component of , in another words, the c-axis
of the film is normal to the substrate. This places our results in contradiction with
work done by another research group in Jülich Germany, who reported that the caxis of these films on SrTiO3 and MgO substrates is in-plane.[69] We had the x-ray
structure of one sample, GdScO3 , remeasured, where a pole figure (111 reflection),
Fig. 4.2, indicates that the film crystallizes predominantly with the c-axis normal to
the substrate, with a much weaker component exhibiting an in-plane c-axis. Scans
through the film peaks are consistent with a=5.72 Å and b=5.80 Å (in plane), and
c=8.11 Å perpendicular to the substrate. Thus the fact that our films were deposited
on a different substrate, LaAlO3 led them to take an orientation different from that
of the Jülich group’s.
The x-ray diffraction measurements also serve as a method to determine the films’
crystallization temperature. The onset crystallization temperature (Tonset ) is defined
by the first appearance of a film peak (poor crystallinity), like that 700◦ C curve in
Fig. 4.2, and the temperature for complete crystallization (Tcryst ) is that above which
67
the film’s Bragg peak’s due to the Cuκα1 and Cuκα2 lines are clearly separated, as
in the 800◦ C curve. The crystallization temperatures determined in this way show
that YScO3 and ErScO3 have poor crystallinity at 800◦ C, TmScO3 , YbScO3 and
LuScO3 have a crystallization onset temperature above 800◦ C or no formation of
the perovskite phase (no x-ray peaks). The rest of the samples have crystallization
temperatures that are below 800◦ C.[66]
In Fig. 4.1, the neodymium and samarium films seem to be outliners, whose values are much larger than those reported for their bulk (polycrystalline) morphologies.
Polycrystalline NdScO3 and SmScO3 is made of many small and randomly oriented
crystallites.[68] Similar to common metals and many ceramics, these crystallites are
often referred to as grains, and the randomness of grains make these bulk materials
isotropic. In order to investigate the difference in the dielectric properties between
the bulk and thin film forms, our collaborators at Oak Ridge made an entire series of
NdScO3 thin film samples at different deposition temperature (from 350◦ C to 800◦ C).
Since the Tonset is 700◦ C for NdScO3 thin films, this temperature range covers both
the amorphous zone and completely crystallized zone. Then, the dielectric constant
and the refractive index were measured, as shown in Fig. 4.3. Clearly, a high value
of = 36 is obtained for well crystallized samples which were deposited above 700◦ C,
while the amorphous samples exhibit a significantly lower = 25, which is the same
value as the bulk. Thus the explanation of the discrepancy in dielectric constant is
due to the difference in sample morphology.
All the films shown in Fig. 4.3. are 400 nm thick. The dielectric constants are
represented by circles on the left-hand scale. Open and solid circles refer to relative
permittivities determined from background substrate measurements made before and
68
Figure 4.3: Dielectric constant of NdScO3 films deposited at different temperature.
after the films are measured. These are placed on the Figure to demonstrate the
reproducibility of the measurements. For comparison, the refractive index values are
plotted on right-hand scale, and they show the same trend as the dielectric constant
measurements.
The data of Fig. 4.1 can then be explained in terms of crystallization. All of the
films in this graph are deposited at 800◦ C. With the exception of ErScO3 (Z=68) and
YScO3 (Z=39), all films are fully crystallized by 800◦ C, and no obvious dependence
of on the atomic number is observed for the films. NdScO3 (Z=60) and SmScO3
(Z=62) have Tcryst ∼ 700 to 750◦ C. Their thin film values are larger than in the
polycrystalline bulk form. GdScO3 (Z=64) and DyScO3 (Z=66) have Tcryst ∼ 800◦ C.
Their thin film values are almost same for single-crystal sample measurement along
69
the c-axis, 33 of the dielectric tensor.
The accuracy of the approach was checked by measurements of two compositions
with significantly different film thicknesses, LaScO3 and PrScO3 . Any errors resulting
from changes in tip geometry during the sequence of measurements or from numerical
artifacts related to different film thicknesses would immediately result in different
values of being found for the samples of each pair. In fact, excellent agreement was
found, as shown in Table 4.1.
Table 4.1: The measurement of rare-earth scandate thin films on LaAlO3 substrates.
By using the fitting curves from the Femlab simulation, the films’ permittivities with
errors are shown on the last column.
Thin film sample Thickness(nm) Film with error
LaScO3
953
32.3 ± 1.5
P rScO3
1000
29.6 ± 1.2
N dScO3
309
47.0 ± 3.0
SmScO3
372
37.3 ± 2.0
T bScO3
312
38.7 ± 4.7
GdScO3
704
31.0 ± 2.0
DyScO3
215
31.3 ± 2.1
HoScO3
226
31.7 ± 1.5
ErScO3
245
19.7 ± 1.5
Y ScO3
500
20.3 ± 1.5
LaScO3
400
32.3 ± 1.2
P rScO3
350
30.0 ± 2.0
This comprehensive study of scandate films on LaAlO3 substrates provides the
most complete set of data for epitaxial scandate films available. Low crystallization temperatures of 700◦ C are found for several scandate materials; the dielectric
constants of the crystalline films ∼ 30 determined by microwave microscopy are
significantly larger than those of their polycrystalline counterparts. In combination
with the large observed band gaps, these results indicate good potential for these
70
materials as high dielectrics in field-effect transistor applications.
4.2
Ferroelectric phase transition study in KTN
Perovskite thin films
4.2.1
Ferroelectricity in Perovskite Oxides
Ferroelectricity is a collective phenomenon, where ferroelectric crystals exhibit spontaneous electric polarization and hysteresis effects in the response of their dielectric
displacement in an externally applied electric field.[2] Usually, they have very high
dielectric constants with nonlinear properties, and the direction of polarization can
be switched between equivalent states by the application of an external electric field.
This behavior is mostly observed in temperature ranges delimited by transition points
(Curie temperatures), above which the crystals are no longer ferroelectric and show
normal dielectric behavior.
KNbO3 , KTaO3 and their solid solutions and superlattice mixtures are the materials which will be discussed in this section. These ferroelectric materials are members of a large group of compounds, called perovskites. The parent member of this
group is CaTiO3 (calcium titanate), which is a relatively rare mineral occurring in
orthorhombic (pseudocubic) crystals. It was discovered in the Ural mountains of
Russia by Gustav Rose in 1839 and named for Russian mineralogist, L. A. Perovski
(1792-1856). The general formula for the perovskite oxide is ABO3 , where A is a
monovalent, divalent or trivalent metal and B is a pentavalent, tetravalent or trivalent element.[2] The general crystal structure is a primitive cube, with the A-cation in
71
Figure 4.4: Symmetry of perovskite crystal structure ABO3 .
the middle of the cube, the B-cation in the corner, and the anion, commonly oxygen,
in the center of the face. The structure is stabilized by the 6-fold coordination of the
B-cation (octahedron) and 12-fold coordination of the A cation. See Fig. 4.4.
The crystal shown in Fig. 4.4 is the prototypical perovskite ferroelectric, BaTiO3 ,
Barium titanate. It is, to date, the most extensively investigated ferroelectric material, because of its simple structure and its chemical and mechanical stability. It is
easily prepared in the form of ceramic polycrystalline samples. Above its Curie temperature, 120◦ C, the symmetry of the non-polar phase is cubic and the crystal shows
paraelectric behavior. Below that temperature, the symmetry of the polar phase is
tetragonal, and the Ti and Ba sublattices shift relative to the negatively charged
oxygen atoms, as shown in picture B and C for up and down polarizations. This
shift breaks the cubic symmetry and brings the crystal into the ferroelectric phase.
Upon cooling, the crystal undergoes two more phase transitions to orthorhombic and
rhombohedral (not shown in this picture). One can apply an external electric field
to switch between the symmetry-related variants, and the crystals remain polarized
when the field is removed.
After the discovery of ferroelectricity in BaTiO3 in 1943 by Wainer and Salomon
72
in United States[70], Ogawa in Japan[71], and Wul and Goldman in Russia[72], considerable effort was expended to search for other ferroelectric materials in the same
family. This led to the discovery of SrTiO3 , PbTiO3 , KNbO3 and KTaO3 . Now after
60 years, these ferroelectric oxide materials have been found to exhibit many interesting properties, such as nonzero remnant polarizations, high dielectric constants, and
electric-field-dependent dielectric constants. These varied properties can be applied
to make unique devices. For example, their switchable electric polarization can be
used for memory storage and integrated microelectronics devices,[20] the electric field
dependent dielectric constant is ideal for developing microwave phase shifters, frequency agile filters, and tunable high-Q resonators.[73, 74] Furthermore, the modern
electronics industry demands even shorter switching times and smaller devices, which
are harder to achieve with conventional semiconductor designs. But for ferroelectric
oxides, their polarization and bound charges are all produced by displacements in each
individual atom, which, in principle, can make devices that operate on atomic scales.
However, it was not until recently that breakthroughs in the synthesis of complex
oxides have removed hurdles to material progress and brought the field to an entirely
new level. Now it has became possible to make complex artificial heterostructures
of oxides with atomic-level precision. Film thickness have been achieved down to 1
lattice constant with advanced vapor techniques, such as Molecular Beam Epitaxy
(MBE), Pulsed laser deposition (PLD) and Chemical vapor deposition (CVD). These
heterostructures can be used to create high-quality ferroelectric film devices. Several research efforts aimed at elucidating the electrical response of films have shown
behavior significantly different than in bulk.[75, 76, 77]
73
4.2.2
Ferroelectric Phase transition
A phase transition is the transformation of a thermodynamic system from one ordered state to another. The distinguishing characteristic of a phase transition is an
abrupt, sudden change in one or more physical properties with a small change in
temperature. This thesis will particularly focus on the ferroelectric transition, which
is normally marked by a maximum in the dielectric constant and by the appearance
of a spontaneous or remanent polarization.
According to the nature of the ferroelectric phase change at the Curie point,
ferroelectric crystals can be classified into two categories. One group undergoes an
order-disorder transition, such as KH2 PO4 and tri-glycine sulfate. In these materials, each unit cell has a dipole moment, and at high temperatures all these moments
point in random directions. But upon lowering the temperature through the phase
transition, all moments within a domain will point in the same direction and form
an ordered phase. The other group undergoes a displacive type transition. Barium
titanate is a typical ferroelectric of this type, as are most of the perovskite oxides. In
these materials, the central ion is displaced from equilibrium slightly, and the force
from the local electric fields due to the ions in the crystal increase faster than the
elastic-restoring forces. This leads to an asymmetrical shift in the equilibrium ion
positions and hence to a permanent dipole moment. The lattice-dynamical treatment of displacive phase transitions leads naturally to a soft-mode model, in which a
particular phonon is involved in phase transition, but the phonons disappear at the
transition temperature.
From the aspect of thermodynamics, any ferroelectric crystal can have its free
energy expressed in terms of strains and polarization A(X,P). If no stress exists, the
74
free energy A is determined by even-order terms in P.
A(P ) = χP 2 + ξP 4 + ζP 6 ,
(4.2.1)
where the coefficients are functions of temperature.
It is possible for the free energy to have two equal minima at some temperature:
one for P = 0 and the other for P 6= 0. The stable state will jump from the one
with P = 0 to the other with P = P0 discontinuously. At some temperature, the two
phases will coexistent in equilibrium, and thus this is the transition temperature for a
first order transition. Such first order phase transitions always involve latent heat, and
during such a transition, the system either absorbs or releases a fixed (and typically
large) amount of heat. Because the energy cannot be instantaneously transferred
between the system and its environment, first-order transitions are often associated
with ‘mixed-phase regimes’ in which some parts of the system have completed the
transition and others have not. Mixed-phase systems are difficult to study, because
their dynamics are so complex. The ferroelectric transitions in bulk BaTiO3 and
KNbO3 are examples of first order transition, and the ferroelectric transitions in
KNbO3 are shown in Fig. 4.6. It is also possible for the polarization P to change
from 0 to a finite value continuously at the transition (for some combinations of
coefficients). This is a second order transition. It has no associated latent heat. The
ferroelectric transition in KH2 PO4 is an example of second order transition,[2, 78]
and its temperature-dependent dielectric constant is shown in Fig. 4.5.[1]
Ferroelectric phase transitions often take place between phases with different lattice symmetry. Generally speaking, if the transition is from one more symmetrical
phase to a less symmetrical one, it is called a symmetry-breaking process, which occurs
75
Figure 4.5: Temperature dependence of dielectric constant of KH2 PO3 at 9.2GHz.
The curve of a is for field along the a-axis, the curve of c is for field along the
c-axis.[1]
76
in both first order and in second order transitions. The presence of symmetry-breaking
is important to understand the behavior of phase transitions, as the paraelectric to
ferroelectric transition is an example of a symmetry-breaking transition, where the
crystal lattice is changed from a cubic to a less symmetrical tetragonal or rhombohedral phase. Typically, the more symmetrical phase is on the high-temperature side
of a phase transition, and the less symmetrical phase on the low-temperature side.
Considering system’s Hamiltonian, one can account for all the possible symmetries
of the system. At low temperatures, the system tends to be confined to a low-energy
state which lacks some of these symmetries, while at higher temperatures, thermal
fluctuations allow the system to access states in a broader range of energies, and thus
more of the symmetries of the Hamiltonian. It was pointed out by Landau that,
given any state of a system, one may unequivocally say whether or not it possesses
a specified symmetry. Therefore, it cannot be possible to analytically deform a state
in one phase to a phase possessing a different symmetry.
These broadly stated principals do not translate well to detailed, analytical description of ferroelectric behavior. Thus, although ferroelectrics have been in use
for decades and many theoretical and experimental treatments have been applied to
them, the nature of the phase transition in some ferroelectric crystals is still not well
understood.
4.2.3
Ferroelectric phase transition in a coupled KTN superlattice
The ferroelectric thin film is one of the most highly studied candidates for random access memory, because of its power consumption, the lowest of all kinds of
77
semiconductor memories, and because it has nonvolatile and random access characteristics. Among those studies, periodic heterostructures consisting of paraelectric and ferroelectric perovskite titanate or niobate layers have been given special
attention.[79, 80, 81, 82]
Ferroelectricity in KNbO3 was first discovered by Matthias in 1949,[83] following
which, a large amount of research was done on its dielectric and optical properties.
KNbO3 is the only ferroelectric crystal that exhibits the same phase symmetries and
the same sequence of transitions as BaTiO3 . It undergoes a paraelectric to ferroelectric phase transition at a Curie Temperature, Tc , of 435◦ C (708K). Upon continuous
cooling, it undergoes two more structural transitions at 225◦ C (498K) and −10◦ C
(263K), while still maintaining the ferroelectric phase. The crystal symmetry above
435◦ C is cubic, and below it is tetragonal. The symmetry becomes orthorhombic at
225◦ C and finally rhombohedral below −10◦ C.[84, 85] The dielectric response of the
different phases of KNbO3 is shown in Fig. 4.6.[2]
KTaO3 bulk crystals have lattice parameters and electronic band structure very
similar to that of KNbO3 , but their ferroelectric properties are quite different. Hulm
et al. measured the temperature dependent dielectric constant of KTaO3 down to 1.3
K and found that it obeys Curie-Weiss law only to 52 K.[86] Later, Barrett pointed
out that quantum effects play an important role in the low temperature region.[87]
Today, KTaO3 is considered as an ‘incipient ferroelectric’, which means it remains
in the cubic paraelectric state down to 0 K without undergoing any of the phase
transitions, shown in Fig. 4.6
The material studied in this section will be the KTN superlattice. Generally, a
superlattice is a structure with periodically alternating layers of several substances.
78
Figure 4.6: Temperature dependence of dielectric constant of KNbO3 in different
symmetries and of ‘incipient ferroelctric’ KTaO3 .[2]
79
Such structures possess periodicity both on the scale of each layer’s crystal lattice and
on the scale of the alternating layers. In X-ray diffraction patterns, this leads to the
appearance of characteristic satellite peaks. In this kind of thin film, the superlattice
crystal structure is generally ‘clamped’ to that of the substrate, which causes strain
and size effects to alter the ferroelectric properties of the film.[88] In the literature,
the physical ferroelectric properties of thin films with superlattice structures have
been reported, including spontaneous polarization, Curie temperature, and dielectric
susceptibility. These are often much different from those of similar bulk materials.
[89][90][4]
The superlattices we have studied are stacked layers of alternating KTaO3 and
KNbO3 thin films grown one upon another. The commonly notation KTN nxm
means each period has n layers of KNbO3 atoms and m layers of KTaO3 atoms. The
thickness of each layer need not be the same. Since small lattice structural differences
of the constituent layers are known to strongly influence their dielectric response,
small structural changes such as size effects, strain effects and long-range ferroelectric
interactions between layers can drastically alter the ferroelectric properties.[4]
All samples describe here are grown by pulsed laser deposition on KTaO3 (001)
substrates at the University of Florida by Jennifer Sigman. Excellent film flatness
and crystallinity are observed in these films and the interface is found to be compositionally sharp on an atomic scale. TEM scanning shows that the in-plane KNbO3
lattice spacing is identical to that of the KTaO3 substrate, so the superlattices are
uniformly strained in-plane but with few dislocations. Fig. 4.7 shows a typical KTN
4x3 superlattice with a perfect lattice continuity.
From this figure, the KTN displays a very little change in lattice spacing from one
80
Figure 4.7: Atomic resolution Z-contrast scanning transmission electron micrograph
of KTN superlattice.[3]
81
Figure 4.8: The contact frequency vs. temperature of KTN 1x1 superlattice.
layer to the next. This minimizes the defect density in the system, which makes it
ideal for studying size effects and the influence of strains.[3] It is important to note
that, except for in a thick KNbO3 film, which have an orthorhombic structure, the
cubic KTaO3 lattice is well matched to the thin-film forms of cubic and tetragonal
KNbO3 . The pseudo-cubic lattice parameter of KNbO3 (a=4.014 Å) differs from that
of KTaO3 by only 0.6%.
The first sample we discuss is the KTN 1X1 superlattice with 200 periods, and a
film thickness of 160nm. Its temperature dependent dielectric constant is shown in
Fig. 4.8 and Fig. 4.9.
The upper graph is the contact frequency, f, versus temperature (from 260o C to
-20o C). The lower graph is the dielectric constant, converted as described in Chapter 2. For comparison, the relative permittivity of the KTaO3 substrate is 208 at
82
Figure 4.9: The temperature dependent dielectric constant of KTN 1x1 superlattice.
The peak at 195o C coincides with XRD structural transition, but the peak at 137o C
does not.
83
room temperature.[91, 92, 93] Here we find three distinct regions separated by two
transitions. Above 210o C, the sample is in the paraelectric phase with the dielectric
constant around 260. A transition at 195o C with a peak dielectric constant about 495
occurs to put the sample into the anti-ferroelectric phase.[94] Upon further cooling,
a second phase transition centered at 137o C with a peak dielectric constant around
475 signals the onset of the ferroelectric phase. Below 130o C, the dielectric constant
is nearly temperature independent.
X-ray diffraction (XRD) measurements of the out-of-plane lattice parameter as
a function of temperature of an identical sample have been performed.[90, 4] These
indicate that the thermal expansion has an turning point at 200o C, pointing to a
tetragonal-to-tetragonal phase transition, which is consistent with Fig. 4.9. Thus
our higher temperature peak is evidence for the structure transition, with additional
information concerning the quantitative value of the permittivity.
In order to test of the value of the dielectric constant measured experimentally,
we compared our results to the theoretical simulation of Stephanovich, et al. [5] They
postulate that since the periodicity of the KTN 1X1 superlattice is much smaller than
a ‘critical length’ (5nm),[4] the KTN 1x1 superlattice must be in the strong coupling
regime, where the neighboring ferroelectric KN bO3 layers interact by a depolarization
field to penetrate the paraelectric KTaO3 layers. Here the whole superlattice acts like
a single ferroelectric. By solving the electrostatic equations with a Ginzburg-Landau
functional, the theory predicts the best fit ⊥ is 500, which matches with our result
quite well.
For the second transition at 137o C, we anticipate it is the transition from antiferroelectric to ferroelectric phase, because the phase development of ferroelectricity
84
is typically observed at temperatures below the anti-ferroelectric regime. This is
consistent with Sigman’s observation,[94] and Stephanovich’s model.[5] Specifically,
Sigman’s capacitance measurements at low frequency showed that above 140o C, the
dielectric constant increases upon the application of a dc electric field, which is consistent with anti-ferroelectric behavior.[94] Based on the geometry of the low frequency
measurements, the Sigman experiment measures mainly the in-plane dielectric response, however our measurement by scanning near-field microscopy is mostly sensitive to the out-of plane component. Thus, measurements by the two methods can
give different results if the sample is dielectrically anisotropic. By comparing our
data to the low frequency data, we find that although they are very similar, but
with some differences. For example, the onset of the phase transition is at 140o C in
both data sets, but the maximum of the in-plane positive dc bias tunability at low
frequency is at 175o C, which indicates the center of in-plane phase transition. At this
same temperature, the transition of the out-of-plane component, at high frequency,
has already finished. This means that the in-plane transition is broader than the
out-of-plane, which gives insight into the natural of the heterostructure. Since these
epitaxial structures were deposited layer by layer, it is reasonable to expect a thin
film of the bulk isotropic material to become anisotropic due to biaxial stress and to
a relatively weaker out-of-plane vs. in-plane coupling. Nevertheless, our data in the
antiferroelectric regime suggest fairly strong coupling between individual polar layers.
The KTN 1X1 is the simplest KTN superlattice, and x-ray diffraction data from
the literature show that the phase transition temperature, Tc , for other symmetrical KTN superlattices depends on the layer thickness. As the superlattice period
decreases, Tc is reduced. For symmetric superlattices with periodicities of 5nm or
85
Figure 4.10: X-ray study of KTN superlattice phase transition by H. M. Christen.(a)
is the plot of lattice parameters of both in-plane and out-of-plane (normal)directions.
(b) is the plot of paraelectric to antiferroelectric transition temperatures at different
layer thicknesses.[4]
less, their Tc s are identical, which demonstrates significant long-range ferroelectric
coupling across the KTaO3 layers.[4] This phenomenon has been recently examined
by molecular-dynamics simulations. The simulations predict that the Curie temperature should decrease for small superlattice modulation lengths, and when the
modulation length is sufficient short, the superlattice responds as a single artificial
structure.[95, 96, 97] This is consistent with the data of Fig. 4.10.
Fig. 4.10 shows both the in-plane and normal lattice parameters for KTN superlattices between room temperature and 700o C.[4] Clearly, the in-plane lattice parameter
expands identically with the substrate, and no anomaly is observed, confirming that
the superlattice is a perfectly ‘clamped’ film. For all measured superlattices, the outof-plane thermal expansion coefficients change sign at the antiferroelectric transition
86
Figure 4.11: Domains and depolarization field in coupled ferroelectric layers in a
ferroelectric/paraelectric superlattice close to Tc . Plot (a) shows the weak coupling
for thicker layers and plot (b) is the strong coupling for thinner layers (below critical
thickness).[5]
temperatures. This result is summarized in panel B.
Thermodynamic considerations lead us to expect that the thinner layers will undergo a phase transition at lower temperature because the free energy associated with
electrostatic field penetration is reduced. This idea is manifested in the well known
effect of a ferroelectric slab or thin film separating into domains to reduce the free
energy. As the domain thickness decreases, the ferroelectric fields are more easily
coupled with each other, making it favorable to form smaller domains and hence
decrease the free energy.[98, 99] This effect is illustrated in Fig, which shows the
simulated patterns of domains formed in the ferroelectric layers.
The simulations indicate that surface charges are produced by the discontinuity
and nonuniformity of the polarization close to the layer surface. Then these charges
produce a depolarization electric field, which allows domains to interact with neighboring ferroelectric layers across the paraelectric layers. But, this effect is critically
dependent on the thickness of the paraelectric layers, and the depolarization field’s
87
penetration length δ is roughly proportional to the domain size d. Kittel studied
the energy of the equilibrium domain structure in ferromagnetics and derived the
√
famous Kittel formula d ∼ a, where ‘a’ is the layer thickness and ‘d’ is the domain
size.[100] If the distance between ferroelectric layers is larger then the penetration
length δ, the inter-planar interaction is small and confined essentially to the ferroelectric/paraelectric interfaces, which makes it is almost independent of the paraelectric layer’s thickness. This is the weak coupling regime realized for long superlattice
periods, where each ferroelectric layer has its own domain structure and acts as an independent thick film embedded in the paraelectric media. The numerical calculation
shows that the transition temperature Tc is inversely proportional to layer thickness
until reaching the bulk value. In the opposite limit of the domain size exceeding the
superlattice period, the depolarization field penetrates the paraelectric layers and so
couples the domains in the neighboring ferroelectric layers. Now the size of the ferroelectric domains is defined by the global depolarization field of the sample, which results in a dramatic increase of the domain width. This is the strong coupling regime,
in which the superlattice behaves effectively as a uniform ‘composite’ ferroelectric
with no depolarizing surface charges at the ferroelectric/paraelectric interfaces. In
this regime, the transition temperature is insensitive to the layer thickness and stays
at a certain value below a critical layer thickness.
As in the study of KTN superlattice, Christen et al. found this critical lattice
period to be 50Å, when the domain size equals to the layer thickness. In this case,
the KNbO3 ferroelectric layers are strongly coupled with each other which leads to
a transition temperature with a single value, as in the KTN 1x1, 4x4, 6x6 shown in
88
Figure 4.12: Measurement of KTN 8x8 superlattice. Plot of the temperature dependence of the dielectric constant.
Fig. 4.10. Furthermore, a clear difference between the superlattices and solid solution KNb0.5 Ta0.5 O3 is observed, where the solid solution has a much lower transition
temperature. This is further evidence that the structure of the ferroelectric phase
is dictated by the local environment and by long range electrostatic interactions,
as distinct from structural difference that exits between the solid solution and the
superlattice.
In order to systematically study the KTN superlattice system, we measured KTN
8x8, 16x16, 1x7 and 2x16 superlatties, all on KTaO3 substrate with 160nm film thicknesses. The first two are symmetrical and the last two are asymmetrical superlattices,
and on none of these systems has vs. T been previously measured. The resulting
data are shown in Fig. 4.12 - Fig. 4.15.
89
Figure 4.13: Measurement of KTN 16x16 superlattice. Plot of the temperature dependence of the dielectric constant.
90
Figure 4.14: Measurement of KTN 1x7 superlattice. Plot of the temperature dependence of the dielectric constant.
91
Figure 4.15: Measurement of KTN 2x16 superlattice. Plot of the temperature dependence of the dielectric constant.
92
Figure 4.16: Tc of KTN with different periods. The red and pink dots of superlattice
samples represent the temperature of structural transition, the green dots represent
the temperature of antiferroelectric to ferroelectric transition. For comparison, the
data of KT a0.5 N b0.5 O3 solid solution is also included.
The KTN 8x8 superlattice shows two phase transitions at 145o C and at 250o C.
The KTN 16x16 has only one phase transition at 151o C. The KTN 1x7 shows two
phase transitions at 120o C and at 221o C, while the KTN 2x16 shows only one phase
transition at 132o C. This is summarized in Fig. 4.16, where all of the transition
temperatures are plotted.
In Fig. 4.16, the red dots are Tc s of the structural transitions, inferred by comparing to the data on Fig. 4.10. The green dots are Tc s of ferroelectric-antiferroelectric
transitions. The structural transition is a change of the crystal’s lattice constant
(tetragonal-to-tetragonal in this case), which coincides with the transition from the
paraelectric phase to the antiferroelectric phase. In symmetrical superlattices, n=m,
93
and we recall the Stephanovich approach to obtaining the transition temperature,
Tc , numerically. For KTN8x8, the calculated Tc ≈ 240o C while our measurement
is at 250o C, which matches well to the theory. For KTN 16x16, the calculated
Tc ≈ 400o C, a temperature is far above the upper limit of this setup’s measurement range (0 − 300o C) but consistent with Fig. 4.10. Thus this transition was
not observed in our dielectric measurements. For the asymmetric superlattice, little
modeling has been done, so it is difficult to compare experimental data with simulations. However, it is still possible to adapt the previous approach of qualitative
analysis and give a tentative explanation. For the KTN 1x7, there is a weak structural
transition at 221o C. Unlike the symmetric superlattice, the peak dielectric constant
value of this transition is only 320, which we believe is an indication of weak coupling among ferroelectric layers. Although its superlattice period is still less than the
critical period 50Å, its relatively thin ferroelectric layer does not provide a strong
enough depolarization field to penetrate the much thicker paraelectric layers. This
weak coupling makes the KTN1x7 act more like an attenuated KTN7x7. It is worth
noting here that the calculated Tc for KTN7x7 is ∼ 230o C, consistent with Fig. 4.14.
A similar thing happens in the KTN2x16, where the 2 unit cells of KNbO3 layers can
not couple with each other through the 16 unit cells of the KTaO3 layers, because
of the confined depolarization field. Just as the KTN 16x16, our measurement does
not show any peak of dielectric constant which might be associated with structural
transition, neither does the KTN 2x16 in the range of 0 − 300o C.
Hao et al. studied the spontaneous polarization behavior of a series of asymmetric
KTN 1xm superlattices (one unit cell of KNbO3 and m unit cells of KTaO3 ), using
first-principles calculations. Their model shows that as the number of m increases, the
94
spontaneous polarization drops rapidly, as does ∆E (the free energy difference between
the ferroelectric phase and paraelectric phase). When m ≤ 2, the ferroelectric phase
is energetically favorable; When m > 3, ∆E becomes zero, the paraelectric phase
is energetically stable, and the paraelectric to ferroelectric phase transition will not
occur.[101] These simulations are consistent with our measurements in the sense that
as the thickness of paraelectric layers increases, the ferroelectric layers are separated
further. The transition does become weaker for KTN1x7 and eventually disappears
for KTN2x16.
As for the ferroelectric to anti-ferroelectric transition, few theoretical calculations
have been reported. For instant, Hao et al. reported the formation of an antiferroelectric phase in KTN 1X1 directly from the atomic relaxation simulations, which
demonstrates a lower free energy than that of the paraelectric phase, but a higher
one than that of the ferroelectric phase. The relaxed geometry of the antiferroelectric
phase is centrosymmetric, with antiparallel movement of the Nb and Ta atoms along
the modulation direction. The potassium and oxygen atoms do not move during the
transition. Hao et al. also calculate the phonon frequencies at the Brillouin zone
boundary by solving the Hessian Matrix, and find that one frequency is imaginary,
which indicates an antiferroelectric instability.[101] However, they do not report the
temperature dependence of the ferroelectric - antiferroelectric transition as a function
of layer thickness. It is intriguing that the antiferroelectric to ferroelectric transition
occurs at the same temperature in our measurements, regardless of composition.
Sigman et al. also discuss possible mechanisms for this antiferroelectric behavior after measuring the nonlinear dielectric response for KTN 1xm and KTN mx1
95
structures. Their results show a significant difference between these 2 sets of superlattices. Since neither KTaO3 , KNbO3 , nor their alloys exhibit antiferroelecticity,
they believe this is the result of the artificial Ta-Nb ordering created in the superlattice structure. The nature of ferroelectric materials is that a reorientation of one
always results in a reorientation of the other. The structural change creates changes
in lattice parameters and thereby creates strain in the layers. These strains alter the
polarization of the Nb atoms’ dipole moments in the KNbO3 layer to favor antiferroelectric coupling. Interfacial effects may also play a role in the dc-field response of
these superlattices.[102]
Christen et al., suggest that we can understand antiferroelectric ordering between
individual ferroelectric layers in a superlattice from energy consideration. In ferroelectric crystals, the energy is associated with long-range electric dipolar interaction.
When this ferroelectric ordering is mediated by elastic (short-range) interactions, the
energy is reduced and smaller domains are formed. In equilibrium, the size of a domain is related to the layer thickness. At a critical point, the domain size might
become larger than the film thickness, and the equilibrium energy is no longer determined by the properties of the individual layers, but by the totality of the material
within a domain. Thus the material under certain conditions of elastic and dipolar
interaction strengths might undergo the formation of antiferroelectric ordering.[103]
However, it is not easy to measure the domain size and the free energy experimentally in our setup, so it is still unclear how the antiferroelectric coupling could
be interface related. Here, we can only say that from our measurements of KTN
superlattices, all antiferroelectric to ferroelectric transitions occur within the range of
120o C to 150o C, which is consistent with Sigman et al.’s measurement. But further
96
understanding of this antiferroelectric coupling will require additional x-ray diffraction measurements as well theoretical simulations of the dipole moment interactions.
4.2.4
Temperature dependent dielectric response of KTN solid
solution
A solid solution is a solid-state mixture of one or more solutes in a solvent. Such a
mixture is considered a solution rather than a compound when the crystal structure
of the solvent remains unchanged by addition of the solutes, and the mixture remains
in a single homogeneous phase. In this way, it is analogous to a metallic alloy.
One attractive material system for both understanding and manipulating ferroelectric properties is K(Nbx Ta1−x )O3 , the KTN solid solution. The solvent is single
crystal potassium tantalate KTaO3 , which is cubic at all temperatures with a room
temperature lattice parameter a300k = 3.9885Å. In the perovskite structure of KTaO3 ,
the oxygen atoms are positioned on the cube faces, a Ta atom is at the center of the
cube, and the K atoms are located on the cube corners.[104, 105, 106]. Its isovalent sister compound, KNbO3 , the solute, has similar lattice parameters and ionic
radii, but KNbO3 is ferroelectric below 708K. This difference is due to their different
electronic structures, namely KNbO3 has a smaller band gap and greater transition
metal-oxygen covalency.
The behavior of mixture can be very different from the solvent and particularly
so when the substituted ion, Nb in this case, is randomly distributed and in offcenter positions. As a result, this ion forms a dipole moments in what can be called
random dipole ferroelectrics. According to the theoretical model of Girshberg and
Yacoby, [107] a strong non-uniform electron-phonon interaction leads to off-centre
97
displacements of the Nb ions in KTN. On the other hand, ab initio calculations
predict narrow a potential well for Nb atoms in KTN without off-centre minima.[108]
The conclusion about the centrosymmetric position of Nb in KTN (above the phase
transition) has been drawn from an experimental NMR study. [109]
The KTN solid solution exhibits low losses and at the critical Nb concentration
x = 0.008, KTN undergoes a ferroelectric phase transition with Tc = 0 K.[105] As
the composition is further altered, the solid solution exhibits a continuous transition from a paraelectric to a ferroelectric material. When the concentration x is
greater than 0.047, the solid solution’s Curie temperature can be chemically tuned
by adjusting the Ta/Nb ratio from 0 K to 708 K, given by an empirical formula
Tc (K) = 676x + 32(x > 0.047).[110] The exhibited phase transition is a first-order,
similar to KNbO3 . So the K(Nbx Ta1−x )O3 system can be optimized over a wide temperature and compositional range for specific applications involving holographic data
storage, parametric oscillators and pyroelectric detectors.
For our study, the solid solution thin film sample is KTa0.63 Nb0.37 O3 on MgAl2 O4
substrate, the Nb concentration x is 0.37. The film is grown at 7500 C in the oxygen
pressures of 50mTorr. Then the sample is annealed at 10000 C for 2 hours in air. The
film thickness is 340 nm. Its bulk crystal shows a phase transition at 283 K with the
peak dielectric constant around several thousand. In general, this is very useful in
most microwave electronics applications because the material is in the paraelectric
phase at room temperature and above to avoid losses associated with domain-wall
motion. But in thin film form, things look a little bit different. We anticipate that
the thin film would have a similar dielectric properties as the bulk, so we chose to
measure its temperature dependent dielectric constant by cooling sample from 300 K
98
Figure 4.17: Temperature dependence of dielectric constant of KT a0.63 N b0.37 O3 thin
film on M gAl2 O4 substrate.
down to 250 K, a range which includes the structural transition temperature. The
step size is initially 2 K and is reduced to 0.5 K when close to the transition point.
The substrate’s relative dielectric constant is 8.325 and its loss tangent is in the order
of 10−3 ,[111] which turns out to be a large contrast with the film. As expected, the
penetration E field decreases rapidly and is mainly concentrated inside the film. This
implies that the tip is sensitive to the response of the film and particularly sensitive
to the out-of-plane component of the dielectric constant of the film. The measured
dielectric constant vs. temperature is plotted in Fig. 4.17.
From the Figure, the transition temperature of the thin film is 274 K, about 9 K
below the bulk case. The dielectric constant has an abrupt change at the transition,
from 850 to 650, which is much smaller than in the bulk. The reason for this is still
99
Figure 4.18: The phase diagram of bulk KT a0.64 N b0.36 O3 vs. pressure. At atmosphere
pressure 1 bar, the crystal undergoes 3 phase transitions.[6]
unclear. What we can point out here is this jump reveals the discontinuous polarization at the transition, which makes it a typical first-order transition. For comparison,
a phase diagram of a similar bulk KTN solid solution measured by Samara is shown
in Fig. 4.18, the concentration x of this sample is 0.36.[6]
Fig. 4.18 reveals 3 phase transitions from cubic paraelectric to tetragonal ferroelectric to orthorhombic ferroelectric and to rhombohedral ferroelectric at atmospheric
pressure. This raises a question that our measurement is actually the paraelectric
to ferroelectric transition or just a transition between different ferroelectric phases?
To test this, we asked our collaborator, Syed Qadri in Naval Research Laboratory,
100
Figure 4.19: X-ray diffraction pattern of the solid solution thin film at the temperature
both above and below the transition.
to measure the crystal structure by x-ray diffraction at the temperatures above and
below the transition. Surprisingly, the x-ray data shows that the solid solution is polycrystalline cubic both above and below the transition. A plot of the x-ray diffraction
data is shown in Fig. 4.19.[112]
Clearly, in this case, the structural transition does not occur simultaneously with
the dielectric transition, which is unusual but not unheard of behavior.
Literature research shows that we are not the only group observe the ferroelectric
transition above the Curie-Weiss temperature Tc . In 1958, Triebwasser reported
different temperatures for the dielectric transition and structural transition of triglycine sulfate.[113] Lin et al. reported the co-existence of difference phases in a
K0.5 Na0.5 NbO3 (KNN) crystal.[114] Thonhauser et al. of Rutgers University reported
101
a theoretical study of the structural instability in a cubic BaBiO3 crystal.[115] J.
Toulous reported off-center shifts of Nb ions in a cubic KTN solid solution.[116] G.
Samara reported pressure-induced ferroelectric-to-relaxor crossover in a KTN crystal,
which remains cubic down to 0 K.[117] Still, thus far, there is no direct report of the
co-existence of difference phases and structural instability in KTN solid solutions. In
this context, we will only discuss possible explanations based on off-center ion shifts
and on a ferroelectric-to-relaxor crossover.
Explanation 1: Off Center Shift
The niobium atom in KTN solid solution is randomly substituted for tantalum and
occupies an off-center position in the unit cell. It has been shown that, even above
Tc , Nb resides off-center in one of eight < 111 > positions.[7] A recent study by
Samara further reveals that Nb is actually not static but hops between off-center
positions.[117] An illustration of this shift is shown in Fig. 4.20.
Rankel et.al studied the quadrupole Nb93 NMR spectra of a KTN single crystal
doped with 15% Nb,[116] and found that Nb ions at all temperatures reside in offcenter positions along < 111 > directions and that the local breaking of the cubic
symmetry is of rhombohedral nature. Other direct evidence comes from extended
x-ray absorption fine structure (EXAFS) data, which have convincingly shown that
the Nb ions are off-center in a sample containing 9% Nb.[118] Much more evidence
is indirect, showing that the system’s symmetry at the microscope scale is broken
even in the paraelectric phase at the temperatures well above the phase transition.
For example, the linear dielectric constant’s response departs from a Curie-Weiss law,
a finite remnant polarization is measured, coercive fields, and polarization overtones
102
Figure 4.20: The off-center of the Nb cation in KTN solid solution. Eight diagnose
< 111 > directions are shown.[7]
103
have been reported.[119] These anomalies clearly indicate a lowering of the local symmetry on a nanometer scale while x-ray diffraction still gives the average symmetry
as cubic.
As a result of the off-center position of the Nb atom, a permanent dipole moment
is created. This moment can couple dynamically or statically to the surrounding
host unit cells. As the temperature is decreased, this impurity-host coupling grows
stronger and polar regions can form/condense when two or more Nb cells (along with
their neighboring host cells), interacting in pairs or groups, begin to respond as a
single polar unit. An essential feature of these regions is the local distortion that
accompanies their local spontaneous polarization and leads to the modification of the
normal characteristics of the paraelectric phase. The corresponding alignment of the
local distortion fields gives rise to a macroscopic strain. In the KTN solid solution,
these regions can be very easily reoriented [120] and the ferroelectric transition is not
displacive: the main change at the transition is that the off-center displacements of
the Nb ions change from a disordered to orientationally ordered arrangement. [121]
Explanation 2: Ferroelectric-to-Relaxor Crossover
The atomic-level origin of ferroelectricity in perovskites has been unclear until recently when first-principles electronic structure and total-energy calculations have
made it more explicit. These calculations have shown that all Coulomb lattices are
unstable with respect to off-center ferroelectric ionic displacements, and the shortrange repulsive forces tend to stabilize the lattice against such displacements.[122]
In the perovskites, hybridization of the oxygen 2p states and the d states of the
transition metal ion reduces the short-range repulsion between metal ions leading
104
to off-center displacements. In the absence of strain, or for small strain, the lowest
energy off-center displacements will be along the < 111 > direction.
As there is an intimate connection between ferroelectricity and lattice strain,
Samara used pressure to study the ferroelectric properties and the result is shown
in Fig. 4.18 and in Fig. 4.21.[6] It is clear from the Figure that the temperature
range of stability for the ferroelectric phases decreases with increasing pressure. Extrapolation of the data to higher pressures (dashed lines) suggests that the tetragonal
and orthorhombic phases should vanish around v 25 kbar, and at higher pressure the
crystal should transform directly from the cubic to the rhombohedral phase.
The more interesting pressure effect observed in KTN with low Nb concentration
(2.3%), is a crossover from a normal ferroelectric to a glass-like relaxor state. Where
ferroelectrics are characterized by macro-size polar domains, sharp equilibrium phase
transitions at their Curie temperatures, and frequency independent dielectric properties (up to microwave frequencies), relaxors are characterized by nano-size domains
associated with the disorder, the absence of a macroscopic phase transition and strong
frequency dispersion in their dielectric properties. See Fig. 4.21.
Up to 4 kbar, the responses in Figure are characteristic of ferroelectric behavior.
This is because at low pressure, the presence of dipolar entities induces polarization (or dipoles) in adjoining unit cells on cooling, forming a dynamic polarization
‘cloud’ whose extent is determined by the polarizability, or correlation length for
dipolar fluctuations, rc . These polarization clouds are effectively polar nanodomains.
With decreasing temperature, the rapidly increasing rc increases the size of these
nanodomains and couples them into rapidly growing polar clusters thereby increasing
their Coulombic interactions. Ultimately, these clusters percolate throughout most
105
Figure 4.21: Temperature dependence of the dielectric constant of KTN solid solution
under different pressure. The concentration is x=0.023. The transition vanishes at
9.2kbar.[6]
106
of, if not the entire, sample and precipitate a static, cooperative long-range ordered
ferroelectric state at Tc .
With applied pressure, the character of the response changes to that of a relaxor with its strong frequency dispersion and absence of a macroscopic symmetry
change, at sufficiently high pressures, 9.2 kbar in this case, the transition pattern even
vanishes, because the pressure causes a large decrease in rc . When increasing the pressure, the clusters still increase in size in the paraelectric phase, but do not become
large enough to overlap and permeate the whole sample to precipitate a ferroelectric
transition upon decreasing the temperature. Rather, the clusters exhibit a dynamic
‘slowing down’ of their fluctuations leading to the observed relaxor behavior.[123]
Meanwhile, the crystal remains cubic down to the lowest temperature.
This high pressure dielectric response is very similar to our measurement in Fig.
4.17, except that our measurement was made at atmospheric pressure. However, one
may reasonably assume that the thin-film ‘feels’ the equivalent pressure from the
substrate. If the thermal expansion coefficient of substrate is greater than that of the
thin-film’s, the ‘clamped’ thin-film will have to contract more than it does in bulk
case, and this will bring about an abnormal change of the lattice parameter in the
film. In the condition where this thermal expansion coefficient difference between
the film and substrate is considerably large, the thin-film may show relaxor type
behavior at temperatures above the structural transition point. Thus, the key issue
in explaining this dielectric transition is to determine the temperature dependent
strain of the sample, which is a future step for our research.
107
Chapter 5
Conclusion
The goals of this thesis are several. The first aim is to contribute to the understanding
of near fields generated by the scanning microwave microscope, especially in the submicron range. It is the fundamental part of this work. We simulate the tip-sample
response by a 3D finite element calculation, which reveals interesting phenomena in
the near-field region. A static approximation is used because the tip size and the
tip-sample distance are much smaller than the wavelength. We model the tip as a
cylinder capped by a cone with a flat end, all held at a constant potential. We also
calculate the field polarization from this geometry. The change of resonant frequency
is converted to the real part of the permittivity by using perturbation theory, and the
imaginary part of the permittivity is obtained from the change of the quality factor
via a high frequency power dissipation model.
The second aim is to study the microscope itself. For instance, what is possible
to measure with such an instrument and what does the tip really detect? These are
crucial questions. We introduce the basic concepts of designing, and present a detailed
description of the complete scanning near field microwave microscope system.
The third aim is to apply this integrated technique to practical applications, such
108
as studies of semiconductor crystallization and ferroelectric phase transitions. This
is the main interest of this work. In the first project, we study the crystallization
temperatures and dielectric constant of epitaxial rare-earth scandate films. The entire
series of rare-earth scandates are deposited by a pulsed laser deposition onto LaAlO3
substrates at 800◦ C. The films’ thicknesses vary from 200 nm to 1 µm. We use x-ray
diffraction to determine the films’ onset crystallization temperatures and crystalline
orientations. For the films whose crystallization temperatures are greater than 800◦ C,
we observe how dielectric constants consistent with those of polycrystalline materials.
For the films which are well crystallized by 800◦ C, we find that their thin-film values
are significantly larger than those in the polycrystalline bulk form. Even for the films
just starting to crystalize at the deposition temperature, we find their values to be
almost the same as the single-crystal bulk values along the c-axis, 33 . In combination
with the large observed band gaps, these results indicate the potential for application
of these materials as high-κ dielectrics for field-effect transistors. Our comprehensive
study of scandate films on LaAlO3 substrates provides the most complete set of data
for epitaxial scandate films available.
The other project is the study of ferroelectric phase transitions in thin film KTN
superlattices and solid solutions. For superlattice thin films, our measurements show
that the antiferroelectric to ferroelectric transition occurs in every sample within a
fairly narrow temperature range. But the paraelectric to antiferroelectric transition
temperatures is revealed to be determined by inter layer coupling. Measurements
of symmetric superlattice are well matched to previously reported data from x-ray
diffraction and capacitance measurements. 2 asymmetric superlattice are measured
for the first time and reported in this thesis as well. Our results for the values of z
109
near paraelectric to antiferroelectric transition are consistent with theoretical models. However, the mechanism of the antiferroelectric to ferroelectric transition in
this heterostructure is still not fully understood. For the solid solution thin film,
our measurement shows an abnormal dielectric transition compared to the bulk case.
Furthermore, the x-ray diffraction data shows that this dielectric transition is distinct from any structural transition. Two possible models are applied to explain this
observation. First, this transition could be an order-disorder transition instead of a
displacive transition as the off-center shift in correlated units and form polar regions
at low temperatures. Alternatively, the large difference in the thermal expansion coefficients between the film and the substrate induces a temperature-dependent strain
upon cooling. This strain drives the film from a ferroelectric transition behavior to a
relaxor behavior.
In summary, the near-filed microwave microscope is a novel tool that takes advantage of the non-propagating near-field focused at the apex of a conducting tip.
This technique is sensitive to dielectric region as small as few hundred nanometers.
It is non-destructive, requires no sample preparation, which can help researchers in
studying dielectric response and ferroelectric phase transitions in new materials systems, especially thin films. Despite the experimental challenges, we measure a variety
of dielectric thin films by scanning near-field microwave microscopy, and we perform
a theoretical model analysis for quantitative characterization of the thin films’ microwave properties with sub-micron resolution. Some results, like dielectric properties
of rare-earth scandate thin films and paraelectric to antiferroelectric transitions in
KTN superlattice, are highly consistent with previous repoted measurements. Some
110
results, like ferroelectric to antiferroelectric transition in KTN superlattice and abnormal dielectric transition in KTN solid solution, are first time observed. We have
brought some possible theoretical explanations for these results. However, to fully
understand these newly found phenomena is the work for future research.
111
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Appendix A
Manual for SNMM operation and
Femlab simulation
This document describes the procedures for measuring and calculating the sample’s
dielectric constant by SNMM and Femlab.
A.1
Procedures of experiment and numerical calculation
1. Prepare a tip with the method given at chapter 3 section 3.1.2 .
2. Bring the tip to Scanning Electron Microscope (SEM). Take a picture, and then
measure the cone angle of the tip, as shown in Fig.A.1
The normal SEM parameters are:
system vacuum< 2e − 4Pa
Chamber vacuum< 4.8e − 1Pa
EHT starts from 5KV
Filament target 3.31A
WD range 10 to 12mm
122
Figure A.1: A SEM picture of a tip with 22◦ cone angle.
123
3. Attach the tip through the sapphire disk at the bottle of the cavity.
4. Find the resonant peak at the front screen of VNA, normally, the resonant
frequency is around 1.7GHZ to 1.8 GHZ. Set the span to 50 MHz.
5. Do not put sample on the stage, measure the sample’s thickness, d, with a
caliper. Most samples in this research are 0.5mm thick.
6. Move the stage towards the tip by actuator. Since the stage is made of copper,
an extreme lossy metal, the resonant frequency will be shift a lot in soft contact.
Find this soft contact point, then move the stage away from the tip by ’d’, record
the background resonant frequency and quality factor at this position as f0 and
Q0 .
7. Lower the stage, mount the sample on it with Omega thermal grease. Set the
temperature with lakeshore temperature controller.
8. Wait 30 minutes, let the grease dry before doing any measurement.
9. Approach the sample to tip by actuator, monitor the frequency shift with labview program, f oQV − z motor.vi. When df/dz exceeds 0.5, stop the actuator
and the vi. As shown in Fig.A.2
10. Approach the sample to tip by piezocrystal, monitor the frequency shift with
labview program, contactbypiezo.vi. Stop the piezocrystal and the vi about 3
points after the peak, by this, one can make sure the peak is the real soft contact
point instead of noise. Record the contact frequency and quality factor at the
peak. As shown in Fig.A.3
11. Repeat the measurements for a few more times to check the consistency. Typically, the results of first few measurements are different because the tip is getting
deformed. This is normal and inevitably. Once the tip reaches 3 or 4um flatness, the results will become very repeatable. Record the contact frequency and
124
Figure A.2: A motor approach of a thin film sample, recorded by f oQV − z motor.vi.
quality factor at the peak as f and Q, and the shifts of frequency and quality
factor ∆fsamp =f0 − f , ∆Qsamp =Q0 − Q.
12. Lower the stage, set the new temperature. And then repeat the previous three
steps for temperature dependent measurements.
13. Replace the sample (assume studying thin film sample) with its bulk substrate.
Repeat the measurement at room temperature (assume the bulk substrate has
no phase transition within the temperature range). Get the shifts of ∆fsubstrate
and ∆Qsubstrate .
14. Take the ratio of ∆fsamp /∆fsubstrate and ∆Qsamp /∆Qsubstrate .
15. Take the tip off, take a picture with SEM again, record its flatness valve.
16. Load Femlab with Matlab from Linux environment by manually typing in commands:
125
Figure A.3: A Piezocrystal approach of a thin film sample, recorded by contactbypiezo.vi. The piezocrystal voltage step is set to be -0.1V, corresponding to about
30nm.
126
Figure A.4: Define tip and sample geometry in femlab GUI.
cd /usr/local/comsol32/license/glnx86
./lmgrd -c ../license.dat -l /var/tmp/comsol32.log
/usr/local/comsol32/bin/comsol matlab
17. Open a Quasi-static mode for real part permittivity calculation (frequency),
or a High-frequency mode for imaginary part permittivity calculation (quality
factor).
18. Draw the geometry in femlab GUI with exactly the tip cone angle and flatness,
as shown in Fig.A.4
19. Define the boundary conditions. As shown in Fig.A.5
20. Define the subdomain properties (mostly .) As shown in Fig.A.6
127
Figure A.5: Define boundary conditions of tip and sample in femlab GUI.
128
Figure A.6: Define subdomain properties of sample in femlab GUI. The figure shows
the thin film dielectric constant is defined as 26, same as substrate. In this case, the
sample is equivalent to a bulk material.
129
Figure A.7: Generate mesh in femlab GUI. Notice in this figure, the mesh area
around the contact point has been refined for several times. Make sure the actual
mesh element size there is less than 30mn. That is critical for an accurate solution.
21. Generate mesh and then solve the geometry. As shown in Fig.A.7
22. Export the solution to Matlab as ‘fem1’. As shown in Fig.A.8
23. Save a copy of the geometry, reset all subdomains’ values to 1. Add another
application with dependent variable ‘W’. leave all boundary conditions intact.
As shown in Fig.A.9 and Fig.A.10.
24. Don’t have to solve the new geometry, just export it matlab as ‘fem2’.
25. Run wpert.m in matlab. This will do the domain energy integration.
26. Do (film’s value -1)*energy integration in film domain +(substrate’s value
-1)*energy integration in substrate domains, this gives out the perturbation
130
Figure A.8: Export the solution (electric field) to matlab.
131
Figure A.9: Add another application through femlab multiphysics menu.
132
Figure A.10: Define the second mode as ‘Electrostatics’, define its dependent variable
as ‘W’, in order to consistent with matlab program.
133
energy.
27. Saving the model as matlab M-file, and modify film’s value, one can make a
reference curve at a given film thickness. The source code is list below.
28. Match the experimental ratio with reference curve to determine thin film’s permittivity.
A.2
Source code of simulatation of a 200nm thin
film reference curve on LaO (=26) substrate
This is a matlab program, but it needs both the matlab and femlab environments, because many functions used in the program are provided by femlab. In main program,
the program defines an array, SubEps, of thin film’s dielectric constant, in this case,
it varies from 1 to 300. When the film’s dielectric constant is set to 26, which is the
same as the substrate’s, the boundary between the film and substrate vanishes, and
this leads to a bulk system. By calling 2 sub-modules: Thk200 and Thk200freesp,
the main program simulates the finite element analysis as femlab does and gives out 2
sets of electric field solutions: fem1 (sample present) and fem2 (free space, no sample
present). Finally main program calls Wpert to integral the product of two fields over
the entire space and then calculates the perturbed energy. The results are exported
into a text file, Thk200Thinfilm.txt. The advantage of this program is it can be iterated as many times as the size of the array, SubEps, while femlab GUI can only do
one round of calculation. It even can be modified to handle different film thickness,
which makes it possible to generate a group of reference curves by an over nights
calculation, as shown in Fig.2.5.
% Main Program
134
clear;
SubEps=[1 3 5 7 9 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48
50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260
270 280 290 300];
for Subindex=1:length(SubEps);
fid = fopen(‘Thk200F00ThinFilm.txt’, ‘a+’);
fprintf(fid, ‘SubEps: % 1.2e\t’, SubEps(Subindex));
% f00
Thk200;
fem1=fem;
clear fem;
Thk200freesp;
fem2=fem;
clear fem;
wpert;
D1 = postint(f em2,0 epsilon0 es ∗ pi ∗ r ∗ (V r ∗ W r + V z ∗ W z)0 ,0 dl0 , [3]);
D2 = postint(f em2,0 epsilon0 es ∗ pi ∗ r ∗ (V r ∗ W r + V z ∗ W z)0 ,0 dl0 , [1, 2]);
I=(SubEps(Subindex)-1)*D1+(26-1)*D2;
fprintf(fid, ‘F00PertEngy: %1.4e\n’, I);
clear fem1 fem2 D1 I;
% Close data file
fclose(fid);
end;
% Thk200.m, called by main program
% FEMLAB Model M-file
135
% Generated by FEMLAB 3.0a (FEMLAB 3.0.0.228, Date : 2004/04/0518 : 04 :
31)
% Some geometry objects are stored in a separate file.
% The name of this file is given by the variable ’flbinaryfile’.
flclear fem
% Femlab version
clear vrsn
vrsn.name = ‘FEMLAB 3.0’;
vrsn.ext = ‘a’;
vrsn.major = 0;
vrsn.build = 228;
vrsn.rcs = ‘N ame :’;
vrsn.date = ‘Date : 2004/04/0518 : 04 : 31’;
fem.version = vrsn;
flbinaryfile=‘Thk200.flm’;
clear draw
g6=flbinary(‘g6’,‘draw’,flbinaryfile);
g10=flbinary(‘g10’,‘draw’,flbinaryfile);
g2=flbinary(‘g2’,‘draw’,flbinaryfile);
g11=flbinary(‘g11’,‘draw’,flbinaryfile);
draw.s.objs = g6,g10,g2,g11;
draw.s.name = ‘CO1’,‘R2’,‘R3’,‘R1’;
draw.s.tags = ‘g6’,‘g10’,‘g2’,‘g11’;
fem.draw = draw;
fem.geom = geomcsg(fem);
% Initialize mesh
fem.mesh=meshinit(fem, ... ‘hmaxfact’,0.8);
% Refine mesh
136
fem.mesh=meshrefine(fem, ... ‘boxcoord’,[-3.4956446825782966E-5 0.0010332093096393667
-5.137446723959587E-4 -1.5494830722118927E-5], ... ‘rmethod’,‘regular’);
% Refine mesh
fem.mesh=meshrefine(fem, ... ‘boxcoord’,[-5.884513786493965E-5 9.717812469672495E4 -5.273953529897626E-4 -2.5732841167471767E-5], ... ‘rmethod’,‘regular’);
% Refine mesh
fem.mesh=meshrefine(fem, ... ‘boxcoord’,[-3.154377667733202E-5 9.154721895178086E4 -5.256890179155371E-4 -4.108985683550111E-5], ... ‘rmethod’,‘regular’);
% Refine mesh
fem.mesh=meshrefine(fem, ... ‘boxcoord’,[-4.860712741958681E-5 4.155160127697435E4 -5.256890179155371E-4 -9.569257921071642E-5], ... ‘rmethod’,‘regular’);
% Refine mesh
fem.mesh=meshrefine(fem, ... ‘boxcoord’,[4.018653321759396E-4 9.052341790724559E4 -3.597085161282466E-5 2.5239977593128124E-4], ... ‘rmethod’,‘regular’);
% Refine mesh
fem.mesh=meshrefine(fem, ... ‘boxcoord’,[-6.055147293916515E-5 8.45512451474564E4 1.4151714905030798E-5 7.188681005601536E-4], ... ‘rmethod’,‘regular’);
% Refine mesh
fem.mesh=meshrefine(fem, ... ‘boxcoord’,[-1.2197953561128238E-4 8.318617708807602E4 2.2683390276158192E-5 7.188681005601536E-4], ... ‘rmethod’,‘regular’);
% Refine mesh
fem.mesh=meshrefine(fem, ... ‘boxcoord’,[-9.809084457212567E-5 5.008327664810173E4 6.192909698334427E-5 7.171617654859281E-4], ... ‘rmethod’,‘regular’);
% Refine mesh
fem.mesh=meshrefine(fem, ... ‘boxcoord’,[5.008327664810173E-4 8.386871111776622E4 -2.3387656727929672E-5 2.09770552019328E-5], ... ‘rmethod’,‘regular’);
% Refine mesh
137
fem.mesh=meshrefine(fem, ... ‘boxcoord’,[-1.569594046329048E-4 8.429529488632257E4 2.1805875627497573E-4 9.449574978950297E-4], ... ‘rmethod’,‘regular’);
% Refine mesh
fem.mesh=meshrefine(fem, ... ‘boxcoord’,[-5.05000148594616E-6 5.310808107877037E5 4.451764822805012E-6 1.4289450430767017E-5], ... ‘rmethod’,‘regular’);
% Refine mesh
fem.mesh=meshrefine(fem, ... ‘boxcoord’,[-5.628688874649802E-6 2.0414548953042225E4 -1.233016944960074E-5 -3.3605149246942144E-6], ... ‘rmethod’,‘regular’);
% Refine mesh
fem.mesh=meshrefine(fem, ... ‘boxcoord’,[-2.735251931131569E-6 6.641789101895425E5 -1.7827699642285395E-5 -8.858045117378856E-6], ... ‘rmethod’,‘regular’);
% Refine mesh
fem.mesh=meshrefine(fem, ... ‘boxcoord’,[6.381379776978783E-5 1.4540871957700207E4 -1.811704333663722E-5 -7.700670339971571E-6], ... ‘rmethod’,‘regular’);
% (Default values are not included)
% Application mode 1
clear appl
appl.mode.class = ‘Electrostatics’;
appl.mode.type = ‘axi’;
appl.assignsuffix = ‘ es’;
clear bnd
bnd.V0 = 0, 0, 0, 1;
bnd.type = 0 ax0 ,0 V 00 ,0 cont0 ,0 V 0 ;
bnd.ind = [1,2,1,3,1,3,4,4,3,2,3,2,3,3,3,3,2,2,2,4];
appl.bnd = bnd;
clear equ
equ.epsilonr = 1, 26, SubEps(Subindex);
equ.ind = [2,2,3,1,1,1];
138
appl.equ = equ;
fem.appl1 = appl;
fem.sdim = 0 r0 ,0 z 0 ;
% Multiphysics
fem=multiphysics(fem);
% Extend mesh
fem.xmesh=meshextend(fem);
% Solve problem
fem.sol=femlin(fem, ... ‘solcomp’,’V’, ... ‘outcomp’,’V’, ... ‘nonlin’,‘off’);
% Save current fem structure for restart purposes
fem0=fem;
% Plot solution
% Thk200freesp.m, called by main program
% FEMLAB Model M-file
% Generated by FEMLAB 3.0a (FEMLAB 3.0.0.228, Date : 2004/04/0518 : 04 :
31)
% Some geometry objects are stored in a separate file.
% The name of this file is given by the variable ’flbinaryfile’.
flclear fem
% Femlab version
clear vrsn
vrsn.name = ‘FEMLAB 3.0’;
vrsn.ext = ‘a’;
vrsn.major = 0;
vrsn.build = 228;
vrsn.rcs = ‘N ame :’;
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vrsn.date = ‘Date : 2004/04/0518 : 04 : 31’;
fem.version = vrsn;
flbinaryfile=‘Thk200freesp.flm’;
% Geometry
clear draw
g6=flbinary(‘g6’,‘draw’,flbinaryfile);
g10=flbinary(‘g10’,‘draw’,flbinaryfile);
g2=flbinary(‘g2’,‘draw’,flbinaryfile);
g11=flbinary(‘g11’,‘draw’,flbinaryfile);
draw.s.objs = g6,g10,g2,g11;
draw.s.name = ‘CO1’,‘R2’,‘R3’,‘R1’;
draw.s.tags = ‘g6’,‘g10’,‘g2’,‘g11’;
fem.draw = draw;
fem.geom = geomcsg(fem);
% Initialize mesh
fem.mesh=meshinit(fem, ... ‘hmaxfact’,0.8);
% Refine mesh
fem.mesh=meshrefine(fem, ... ‘boxcoord’,[-3.4956446825782966E-5 0.0010332093096393667
-5.137446723959587E-4 -1.5494830722118927E-5], ... ‘rmethod’,‘regular’);
% Refine mesh
fem.mesh=meshrefine(fem, ... ‘boxcoord’,[-5.884513786493965E-5 9.717812469672495E4 -5.273953529897626E-4 -2.5732841167471767E-5], ... ‘rmethod’,‘regular’);
% Refine mesh
fem.mesh=meshrefine(fem, ... ‘boxcoord’,[-3.154377667733202E-5 9.154721895178086E4 -5.256890179155371E-4 -4.108985683550111E-5], ... ‘rmethod’,‘regular’);
% Refine mesh
fem.mesh=meshrefine(fem, ... ‘boxcoord’,[-4.860712741958681E-5 4.155160127697435E4 -5.256890179155371E-4 -9.569257921071642E-5], ... ‘rmethod’,‘regular’);
140
% Refine mesh
fem.mesh=meshrefine(fem, ... ‘boxcoord’,[4.018653321759396E-4 9.052341790724559E4 -3.597085161282466E-5 2.5239977593128124E-4], ... ‘rmethod’,‘regular’);
% Refine mesh
fem.mesh=meshrefine(fem, ... ‘boxcoord’,[-6.055147293916515E-5 8.45512451474564E4 1.4151714905030798E-5 7.188681005601536E-4], ... ‘rmethod’,‘regular’);
% Refine mesh
fem.mesh=meshrefine(fem, ... ‘boxcoord’,[-1.2197953561128238E-4 8.318617708807602E4 2.2683390276158192E-5 7.188681005601536E-4], ... ‘rmethod’,‘regular’);
% Refine mesh
fem.mesh=meshrefine(fem, ... ‘boxcoord’,[-9.809084457212567E-5 5.008327664810173E4 6.192909698334427E-5 7.171617654859281E-4], ... ‘rmethod’,‘regular’);
% Refine mesh
fem.mesh=meshrefine(fem, ... ‘boxcoord’,[5.008327664810173E-4 8.386871111776622E4 -2.3387656727929672E-5 2.09770552019328E-5], ... ‘rmethod’,‘regular’);
% Refine mesh
fem.mesh=meshrefine(fem, ... ‘boxcoord’,[-1.569594046329048E-4 8.429529488632257E4 2.1805875627497573E-4 9.449574978950297E-4], ... ‘rmethod’,‘regular’);
% Refine mesh
fem.mesh=meshrefine(fem, ... ‘boxcoord’,[-5.05000148594616E-6 5.310808107877037E5 4.451764822805012E-6 1.4289450430767017E-5], ... ‘rmethod’,‘regular’);
% Refine mesh
fem.mesh=meshrefine(fem, ... ‘boxcoord’,[-5.628688874649802E-6 2.0414548953042225E4 -1.233016944960074E-5 -3.3605149246942144E-6], ... ‘rmethod’,‘regular’);
% Refine mesh
fem.mesh=meshrefine(fem, ... ‘boxcoord’,[-2.735251931131569E-6 6.641789101895425E5 -1.7827699642285395E-5 -8.858045117378856E-6], ... ‘rmethod’,‘regular’);
% Refine mesh
141
fem.mesh=meshrefine(fem, ... ‘boxcoord’,[6.381379776978783E-5 1.4540871957700207E4 -1.811704333663722E-5 -7.700670339971571E-6], ... ‘rmethod’,‘regular’);
% (Default values are not included)
% Application mode 1
clear appl
appl.mode.class = ‘Electrostatics’;
appl.mode.type = ‘axi’;
appl.assignsuffix = ‘ es’;
clear bnd
bnd.V0 = 0,0,0,1;
bnd.type = ‘ax’,‘V0’,‘cont’,‘V’;
bnd.ind = [1,2,1,3,1,3,4,4,3,2,3,2,3,3,3,3,2,2,2,4];
appl.bnd = bnd;
clear equ
equ.epsilonr = 1,1,1;
equ.ind = [2,2,3,1,1,1];
appl.equ = equ;
fem.appl1 = appl;
fem.sdim = ‘r’,‘z’;
% Multiphysics
fem=multiphysics(fem);
% Extend mesh
fem.xmesh=meshextend(fem);
% Solve problem
% Save current fem structure for restart purposes
fem0=fem;
% Plot solution
% (Default values are not included)
142
% Application mode 1
clear appl
appl.mode.class = ‘Electrostatics’;
appl.mode.type = ‘axi’;
appl.assignsuffix = ‘ es’;
clear bnd
bnd.V0 = 0,0,0,1;
bnd.type = ‘ax’,‘V0’,‘cont’,‘V’;
bnd.ind = [1,2,1,3,1,3,4,4,3,2,3,2,3,3,3,3,2,2,2,4];
appl.bnd = bnd;
clear equ
equ.epsilonr = 1,1,1;
equ.ind = [2,2,3,1,1,1];
appl.equ = equ;
fem.appl1 = appl;
% Application mode 2
clear appl
appl.mode.class = ‘Electrostatics’;
appl.mode.type = ‘axi’;
appl.dim = ‘W’;
appl.name = ’es2’;
appl.assignsuffix = ‘ es2’;
clear prop
prop.weakconstr=struct(‘value’,‘off’,‘dim’,‘lm2’);
appl.prop = prop;
clear bnd
bnd.type = ‘V0’,‘cont’;
bnd.ind = [1,1,1,2,1,2,1,1,2,1,2,1,2,2,2,2,1,1,1,1];
143
appl.bnd = bnd;
fem.appl2 = appl;
fem.sdim = ‘r’,‘z’;
% Multiphysics
fem=multiphysics(fem);
% Wpert.m, called by main program
% Compute W in fem2 (assign V from fem1 to W)
fem2.equ.init=‘0’,‘V’;
% Update extended mesh with new intial value
fem2.xmesh = meshextend(fem2);
% Create solution with zeros for V and fem1’s V for W
fem2.sol = asseminit(fem2,‘u’,fem1);
% ”Fill in” the values for V by solving only for V
fem2.sol = femlin(fem2,‘init’,fem2.sol,‘solcomp’,‘V’);
% Now we can integrate Wr and Wz in fem2
%D2 = postint(fem2,’epsilon0 es*pi*r*(Vr*Wr+Vz*Wz)’,’dl’,[2]);
%D3 = postint(fem2,’epsilon0 es*pi*r*(Vr*Wr+Vz*Wz)’,’dl’,[3]);
% This should be the same as the original integral
%I2 = postint(fem1,’epsilon es*pi*r*(Vr*Vr+Vz*Vz)’,’dl’,[3,4,5])
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