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Quantitative evanescent microwave microscopy

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Quantitative Evanescent Microwave Microscopy
By
Frederick William Duewer
B.S. (California Institute of Technology) 1995
A dissertation submitted in partial satisfaction of the
requirements for the degree of
Doctor of Philosophy
In
Physics
In the
GRADUATE DIVISION
Of the
UNIVERSITY OF CALIFORNIA, Berkeley
Committee in charge:
Co-Chair Doctor Xiao-Dong Xiang
Co-Chair Professor Yuen-Ron Shen
Professor Michael Crommie
Professor Theodore Van Duzer
Fall 2000
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UMI Number: 3001824
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Dedication
I dedicate this thesis to my mother, Theresa E. Suen Duewer. In
particular, I would like to thank her endless patience, persistence, and
encouragement. Thanks. You've always been there for me.
i
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Acknowledgements
I thank my advisor, Xiao-Dong Xiang for his unremitting advice and
encouragement. He's been a good friend and I've learned a lot from him. I'm
grateful to the rest of my committee: Yuen-Ron Shen, Michael Crommie, and
Theodore Van Duzer. In particular, I'm indebted to Professor Van Duzer for his
work. His efforts improved the quality of this thesis. In addition, I owe thanks to
DARPA and the Advanced Energy Projects Division for funding, Young Yoo,
Ichiro Takeuchi, Chen Gao, Yalin Lu, Wang Gang, Hai-Tao Yang, Sheng Liu,
Tyuoshi Onishi, Yi Dong, Jing-Wei Li, Jing Song, Ted Sun, and others too
numerous to name for your friendship and knowledge. Young - I'm done now,
hurry up. :) Cynthia Gong, Jarah Evslin, Jamie Walls, and company for taking
me out of lab. Hauyee Chang in particular for helping me through qualifiers and
pestering me until I finished. One advantage of minor disasters is learning how
many people care for you. Thanks. You kept me working through the rough bits
and the boring ones. You've been a better friend than I ever expected.
ii
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Quantitative evanescent microwave microscopy
I Introduction
1
Overview of Thesis
1
Scanned Probe Microscopy
1
Scanning Tunneling Microscopes
Atomic Force Microscopes
Near-Field Scanning Optical Microscopes
Miscellaneous Microscopes
II
Scanned Microwave Probe Microscopy
4
Applications
8
Quantitative measurement
9
Experimental Details
13
Evanescent Waves
13
Cavity design and dimensions
14
Tip Fabrication
18
Measurement setup
18
Frequency sweep
Analog feedback
AC measurements
iii
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I/Q Mixer
III
Staging
26
Sample preparation
27
Quantitative calculations
29
Overview
29
Insulating materials - complex dielectric constant
29
Thick film
Thin film
Ferroelectric materials - nonlinear dielectric constant
IV
Conductive materials - resistivity
38
Ferroelectric domain measurements
44
Indirect Measurements
44
Direct Measurements
54
Noncontact Measurements
57
V
Measurement of conductive materials
67
VI
Measurement of superconductors
79
VII Measurement of manganites
88
Overview
88
Lai.xCaxMnOa
89
iv
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En.xCaxMnOa
105
VIII Future possibilities
121
Distance control
121
AFM
Frequency
IX
Sum m ary
127
V
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Chapter I: Introduction
Overview of Thesis
The subject of this thesis is quantitative scanned evanescent microwave
probe microscopy. This thesis consists of 9 chapters. The first contains a
general overview of scanned probe microscopy with specific emphasis on
microwave near-field microscopy. The second describes the experimental
setup.. The third describes the quantitative calculation of the electrical
impedance. Chapters 4-7 describe images acquired using the system. Chapters
8 and 9 summarize the work and indicate some possible future directions.
Scanned Probe Microscopies
The optical microscope is the earliest and certainly most successful
method of exploring the microscopic world. However, the optical microscope is
limited in spatial resolution to the wavelength of the probing radiation. The
technique of scanned probe microscopy provides a means of surmounting the
limits to spatial resolution of earlier instruments. Typically, a scanned probe
microscope consists of a microscopic sensor and a mechanism used to move the
sensor relative to the sample. In these instruments, the spatial resolution is
determined by the sensor geometry. Since the microscopy signal is a
combination of topography and physical properties, tip-sample distance
regulation is required. I will describe some previously developed scanned probe
microscopes in the following paragraphs.
1
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Scanning Tunneling Microscopy
The scanning tunneling microscope (STM) is the best-known scanned
probe microscope.1 By measuring the tunneling current when a conductive tip
approaches a sample, the sample topography can be obtained. The tunneling
current is proportional to the overlap between orbitals on the tip and on the
sample. This overlap falls off exponentially over distances characterized by the
size of the orbital, allowing atomic resolution. In addition, by measurement of the
variation of the tunneling current with voltage, local l-V spectroscopy can be
performed. However, the STM is very sensitive to the surface condition and is
generally limited to the study of conductive samples in vacuum environment.
Atomic Force Microscopy
Since the development of the STM, a plethora of scanned probe
microscopes have been invented. The atomic force microscope (AFM) has been
the most widely adopted.2 When the tip contacts a sample, the cantilever
bending provides a direct measure of sample topography. This bending can be
measured optically or by measurement of piezoelectric voltages. The AFM
allows near-atomic resolution of sample topography and can be operated in
ambient conditions. A variety of modifications of the AFM have been developed
to allow measurement of sample properties. For instance, if an alternating
voltage is applied to the tip, the induced electric field will induce alternating
distortions in piezoelectric materials. This effect has been used to allow
measurement of ferroelectric domains.3 In addition, cantilevered or vibrating
2
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probes have been widely adopted as methods of controlling tip-sample
separation.4
Scanning Near-Field Optical Microscopes
Following the development of the preceding microscopes, renewed
attention was paid to the task of improving the spatial resolution of optical
instrumentation. In the far-field, Maxwell’s equations limit the spatial resolution to
the wavelength - the propagating solutions for the field distribution have
wavelength determined by the frequency of the radiation. In the proximity of a
subwavelength boundary, the field distribution must satisfy the boundary
conditions and will contain higher spatial frequencies. These higher spatial
frequencies decay exponentially for distances larger than a wavelength.
Therefore, in order to improve the spatial resolution, the near-field
scanning optical microscope (NSOM) contains a subwavelength feature. The
electromagnetic field distribution near the probe contains high spatial frequencies
and allows spatial resolution better than a wavelength when the probe is moved
relative to the sample. The portion of the field distribution containing spatial
frequencies larger then MX is known as the near-field. Probes using this portion
of the field distribution are called near-field probes. Synge originally proposed
the use of a tapered probe.5 The earliest demonstrations at optical frequencies
were those of Pohl etaP and Lewis et a? in 1983. The most widely adopted
near-field scanning optical microscope (NSOM) consists of a tapered optical
fiber, outside coated with metal, which ends in a subwavelength aperture. The
later implementation of tip-sample distance regulation by means of shear force
3
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detection allows reproducible imaging.8'9 These microscopes have demonstrated
measurements of optical properties at submicron length scales. One particularly
interesting advantage is that they allow spatially resolved spectroscopy.
However, they tend to encounter sensitivity limitations for spatial resolutions
below 50 nanometers since the transmission through the aperture diminishes
rapidly with decreasing size. In addition, they are plagued with difficulties with
interference effects, tip-sample distance regulation, and sample topography.
Miscellaneous microscopes
. Temperature has been measured by means of miniature thermocouples,
while elasticity has been measured by modified atomic force microscopes.10'11
Adaptations of the atomic force microscope to measurements of magnetic
properties are particularly interesting - the magnetic force microscope, in which
the force on a magnetic tip is measured, allows measurement of the spatial
distribution of the magnetization. In addition, by measuring the shift of the
properties of a resonant cantilever, extraordinary sensitivity can be achieved. In
principle, the magnetic field of single nuclear spins may be detectable.12'13
Scanning Microwave Microscopes
Electrical impedance is perhaps the most important and frequently
encountered property of a material. Given the rapid miniaturization of electronic
devices, the measurement of electronic properties at submicron length scales is
a topic of current interest. However, the measurement of the resistivity and
dielectric constant at small length scales has proven difficult.
Most electronic applications
occur at microwave frequencies,
4
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implying that measurements of material properties should be done at similar
frequencies. Unfortunately, the wavelength of electromagnetic radiation at 1
GHz is measured in centimeters, which tends to limit the spatial resolution. This
complicates the measurement of material structures with submicron dimensions.
Several methods for the observation of sample electrical properties have
been developed. At large length scales, lithography can be employed to form
capacitive or resistive structures of known geometry. One advantage of these
methods is that they often allow direct measurements of the properties of the
desired structures. For example, the capacitance of a film capacitor can be
measured directly. However, these methods have poor spatial resolution. In
addition, since they typically require multiple lithographic and deposition steps,
these methods can qualitatively alter the behavior of the sample in question. For
example, complications can come from the interface between electrodes and
materials under investigation.
The scanning near-field optical microscope has demonstrated pure
dielectric contrast on submicron length scales.14 However, the use of this
microscope for measurements of the resistivity and dielectric constant is
prohibited by the large difference in the frequencies of optical radiation and
electrical applications. In addition, many low energy collective effects are
obscured at optical frequencies. For instance, the superconducting bandgap lies
in the microwave region for low Tc materials and in the terahertz region for YBCO
and other high Tc ceramics. The observation of the spatial distribution of such
effects is simplified at low frequencies. A direct probe of sample properties near
5
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the frequencies used in electronic circuitry is needed.
Operating in the near-field region allows subwavelength measurements.
Synge, in 1928, originally proposed using a sharply tapered tube to attain high
resolution at microwave frequencies.5 Fraint and Soohoo independently
demonstrated such measurements at microwave frequencies in 1959 and
1962.15,16 Ash et al17has often been credited the development of this method for
his work published in 1972. These early efforts utilized aperture/tapered
waveguide probes. However, these probes suffer severely from waveguide
decay. They suffer a dramatic tradeoff between resolution and sensitivity. For
subwavelength geometries, the solution to the problem of the propagation of
electromagnetic waves through narrow apertures approaches the static solution.
In addition, if a waveguide is tapered so as to focus the incident radiation and
reduce the problem of reflection, the waveguide diameter near the tip quickly
decreases below the wavelength. For cylindrical waveguides, no propagating
solution exists and the fields decrease exponentially. A linear improvement in
resolution will cause an exponential reduction in sensitivity.16 The use of
cylindrical waveguides has been generally successful for optical radiation, since
the ratio of the desired resolution (which determines the aperture size) to the
wavelength of the radiation is typically greater than 0.01.
However, for microwave radiation, it has proven difficult to attain high
spatial resolution using a tapered probe since (r/X) is less than 10 s for 1 micron
resolution and 1 GHz radiation. A geometry that permits high spatial resolution
with high sensitivity is required.
6
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Since coaxial waveguides have no cut-off frequency for their transverse
electromagnetic (TEM) mode, they suffer no waveguide attenuation. A number
of similar microscopes have been developed. Gunn et al in 1965 demonstrated a
taped coaxial probe with high resolution and signal-to-noise ratio using such
geometry.18 Matey and Blank in 1985 modified a RCA capacitance meter to
measure local sample capacitance.19 However, since their microscope operates
with unshielded microwave radiation, the far-field components of the signal
dominate - prohibiting DC measurements of the electrical impedance. Chu in
1989 suggested using a transmission line probe with a reduced cross-section.20
However, for the proposed geometry, the resolution is mainly determined by the
cross-section. Therefore, submicron resolution involves significant transmission
line decay. In addition, the unshielded far-field components around the tip in the
transmission line probe also limit the resolution significantly and complicate
quantitative analysis. Later authors have further developed this geometry by
integrating a microwave probe onto an AFM.21
In 1995, our group developed a scanning evanescent microwave probe
(SEMP) with a shielding structure designed so that the propagating far-field
components are shielded within the cavity whereas the non-propagating
evanescent waves are generated at the tip. W e described the theoretical
calculations to obtain near-field analytic solutions, which for the first time (owing
to the substantial shielding of far-field components) allowed quantitative
microscopy of microwave electrical impedance of materials with sub-micron
resolution.22'23
7
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The use of a slit to allow high spatial resolution measurements also
deserves mentioning. Golovsky et al have developed a microscope based on a
slit geometry.24 Since the length of the slit is comparable to the wavelength while
the width of the slit can be narrow, micron resolution in one dimension with high
sensitivity can be obtained. Image reconstruction is complex, as the spatial
resolution in the slit direction is poor. However, this method has the advantage
of allowing the polarization of incident radiation. This allows the measurement of
anisotropy in materials.
The scanned evanescent microwave probe (SEMP) is capable of
quantitative measurements of complex dielectric constant and conductivity with
submicron spatial resolution. I have demonstrated quantitative measurements of
the dielectric constants of bulk crystals and thin films. I have also measured
ferroelectric domains and demonstrated tip-sample distance regulation. In
addition, I present resistance variations in manganite thin films. Following this
work, a number of other authors have entered this field.25'26
Applications
The SEMP offers a local probe of sample electrical impedance. It allows
direct, spatially resolved characterization of sample electronic properties in a
nondestructive fashion. It can be used for the imaging of integrated circuits. The
SEMP has demonstrated utility for the direct quantitative measurement of
dielectric constants. It provides a spatially resolved probe rather than an average,
which allows for tests of film uniformity. In addition, it avoids the sort of interface
/ diffusion effect occasioned by growth of capacitor structures. It also allows the
8
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direct measurement of conductivity versus position. As such, it can be useful for
the measurement of dopant distributions in semiconductors.
I have used the SEMP to measure periodically-poled ferroelectrics.27-29 By
comparison to other methods used to study these compounds, the SEMP
provides additional information since it can report the dielectric constant, loss
tangent, and nonlinear dielectric constant. I present measurements of
superconducting YBCO thin films, providing spatially resolved measurements of
the transition temperature or lack thereof.30 In addition, I have measured the
resistivity as a function of composition for thin film continuous phase diagrams of
manganite ststems.31'32
Quantitative measurement
For microscopes based upon electromagnetic radiation, quantitative
measurements have generally not been achieved. For most such microscopes,
the signal is directly dependent on the details of the tip-sample geometry, which
may not be well characterized and may be easily altered. In addition, reflectionbased transmission geometries tend to have backgrounds much larger than the
signal. These probes function by measuring the change in a wave reflected from
a subwavelength aperture as a sample is moved relative to the aperture.
However, the vast change in impedance from the transmission geometry to the
subwavelength aperture results in the vast majority of the wave being reflected,
independent of the sample properties. For unshielded geometries, a large
transmitted far-field’ signal exists. This ‘far-field* signal tends to average over
large portions of the sample and seriously complicates quantitative
9
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measurements since it adds a nonlocal, varying background. In our work, a
shielding structure is used to reduce the far-field signal. This shielding structure
is critical for not only high spatial resolution but also quantitative profiling of
electronic properties.
References
1. G. Binnig, H. Rorher, C. Gerber, and E. Weiber, Phys. Rev. Lett. 49, 57
(1982).
2. G. Binnig, C. F. Quate, and C. Gerber, Phys. Rev. Lett. 56, 930 (1986).
3. R. Wiesendanger, Scanning Probe Microscopy and Spectroscopy (Cambridge
University Press, Cambridge, 1994).
4. F. Saurenbach and B. D. Terris, Appl. Phys. Lett. 56,1703 (1990).
5. E. H. Synge, Philos. Mag. 6, 356 (1928).
6. D. W. Pohl, W. Denk, and M. Lanz, Appl. Phys. Lett. 44, 651 (1984).
7. A. Lewis, M. Isaacson, A. Murray, and A. Harootunian, Biophys. J. 41, 405a
(1983).
8. R. Toledo-Crow, P. C. Yang, Y. Chen, and M. Vaez-lravani, Appl. Phys. Lett.
60, 2957 (1992).
9. E. Betzig, P. L. Finn, and J. S. Weiner, Appl. Phys. Lett. 60, 2484 (1992).
10. M. Nonnenmacher and H. K. Wickramasinghe, Appl. Phys. Lett. 6 1 ,1 6 8
(1992).
11. Meyer, E., Heinzelmann, H., Grutter, P., Jung, Th., Hidber, H.-R., Rudin, H.,
Guntherodt, H.-J., Atomic Force Microscopy for the study of tribology and
adhesion, Thin Film Solids vol. 181 (16th International Conference on
10
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Metallurgical Coatings, San Diego, CA USA 17-21 April 1989, p. 527-544.
12. J. A. Sidles, Appl. Phys. Lett. 58, 2854 (1991).
13. D. Rugar, C. S. Yannoni, and J. A. Sidles, Nature (London) 360, 563 (1992).
14. A. Lahreeh, R. Bachelot, P. Gleyzes, and A. C. Boccara, The near-field in
the microwave to IR, Optics and Photonics News, Vol. 9, Opt. Soc. America, May
1998, p. 40-5.
15. Z. Frait, Czechoslov. J. Phys. 9, 403 (1959).
16. R. F. Soohoo, J. Appl. Phys. 33 ,12 76 (1962).
17. E. A. Ash and G. Nicholls, Nature (London) 237, 510 (1972).
18. C. A. Bryant and J. B. Gunn, Rev. Sci. Inst. 3 6 ,1 6 1 4 (1965).
19. M. Fee, S. Chu, and T. W . Hanach, Opt. Comm. 69, 219 (1989).
20. J. R. Matey and J. Blanc, J. Appl. Phys. 5 7 ,14 37 (1985).
21. D. W . van der Weide and P. Neuzil, J. Vac. Sci. Tech. B 14,4144 (1996).
22. T. W ei, X.-D. Xiang, W. G. Wallace-Freedman, and P. G. Schultz, Appl.
Phys. Lett. 68, 3506 (1995).
23. C. Gao, T. Wei, F. Duewer, Y.-L. Lu, and X.-D. Xiang, Appl. Phys. Lett. 71,
1872 (1997).
24. M. Golosovsky and D. Davidov, Appl. Phys. Lett. 6 8 ,1 5 7 9 (1996).
25. C. P. Vlahacos, R. C. Black, S. M. Anlage, A. Amar, and F. C. Wellstood,
Appl. Phys. Lett. 69, 3272 (1996).
26. S. M. Anlage, C. P. Vlahacos, S. Dutta, and F. C. Wellstood, IEEE Trans.
Appl. Sup. 7(3), 3686 (1997).
27. Y.-L. Lu, T. Wei, F. Duewer, Y. Q. Lu, N.-B. Ming, P. G. Schultz, and X.-D.
11
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Xiang, Science 76, 2004 (1997).
28. C. Gao, F. Duewer, Y.-L. Lu, and X.-D. Xiang, Appl. Phys. Lett. 73, 1146
(1998).
29. F. Duewer, C. Gao, and X.-D. Xiang, Rev. Sci. Instrum. 71,
30. I. Takeuchi, T. Wei, F. Duewer, Y.-K. Yoo, X.-D. Xiang, V. Talyansky, S. P.
Pai, G. J. Chen, and T. Venkatesan, Appl. Phys. Lett. 71, 2026 (1997).
31. Y.-K. Yoo, F. Duewer, H. Yang, D. Yi, J.-W . Li, and X.-D. Xiang, Nature 406,
704 (2000).
32. Y.-K. Yoo and F. Duewer, T. Fukumura, H.-T. Yang, D. Yi, T. Hasegawa, M.
Kawasaki, H. Koinuma, and X.-D. Xiang, submitted to Physical Review B
12
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Chapter II: Experimental details
I will discuss the design of the SEMP used in this thesis. I begin with a
brief discussion of evanescent waves and underlying principle for the invention. I
then proceed to a description of the experimental setup.
Evanescent Waves
I begin with a discussion of propagating and evanescent electromagnetic
waves. Here, propagating waves refer to waves with real wavevectors.
—=
A
+ k; + k: , where A is the wavelength and it, is the wavenumber in the i th
direction. Evanescent waves refer to waves with imaginary wavevector not
originated from dissipation. Since propagating waves contain only Fourier
components up to A , their spatial resolution is limited to A . Since evanescent
waves contain imaginary components, their wavenumbers can exceed
,
allowing higher spatial frequency components and better spatial resolution.
However, these waves decay rapidly with distance.
Near-field microscopes operate by measuring the disturbance of
evanescent waves by the proximity of a sample. The problem is to generate
evanescent waves while minimizing the generation of propagating waves. In a
near-field optical microscope, the typical solution is to utilize a subwavelength
aperture. This aperture restricts the transmission of propagating waves. Another
solution is to drive a spherical object with an electromagnetic wave. The
curvature of the sphere forces the generation of evanescent waves and thereby
13
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allows resolution comparable to the radius of the sphere. However, in this case,
a shielding structure should be constructed to restrict the propagation of the
driving wave.
The central problem is the shielding of far-field radiation without
generating massive attenuation. To generate evanescent waves, a pointed wire
was attached to the probe. The simplest shielding structure is a metal plate with
an aperture slightly larger than the wire diameter. This structure prevents the
transmission of propagating radiation. However, the hole and wire form a narrow
coaxial cable. The attenuation of this cable increases with decreasing radius. To
minimize attenuation while still allowing shielding of the propagating radiation, we
deposited a 1000
Asilver film on a drilled sapphire disk.
Cavity Design and Dimensions
Cavity signal
To determine the electrical properties of a sample, the SEMP measures the
variation in resonant frequency (fr) and quality factor (Q) of a resonant cavity.
The tip-sample interaction is modeled using the following equivalent circuit
(Figure 2-1 ).1
14
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Coupling loops
/
Sapphire
disk
Fig. 2-1 Resonator and equivalent circuit
The tip-sampie interaction appears as an equivalent complex tip-sample
capacitance {Cn^ samp,e). Given frand Q, the complex tip-sample capacitance can
be extracted.
15
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Af __ Cr_
/o
J O
A
(1)
2C
—
f 1
= ------------+
2C, W
—
(2)
—
C is the cavity
where
capacitance, and /„ and Q0 are the unloaded resonant frequency and quality
factor. The calculation of C,ip.simplt is described in the following chapter.
The resonator can be approximated as a coaxial line terminated by a
varying complex capacitance. (Fig. 2-1) The calculation of the quality factor Qr
and resonant frequency fr of the cavity is straightforward.1 For a coaxial line,
a
a
Here, C and L are the capacitance and inductance per unit length of the
resonator, a c is the attenuation (dB) per unit length owing to losses in the
conductors, b is the outer diameter of the coaxial cavity, a is the inner diameter of
the coaxial cavity, X is the wavelength, Ss is the penetration depth of the
conductors, and e0, er , /*„, fir are the permittivity of free space, the relative
dielectric constant of the interior of the cavity (assumed real, lossless), the
permeability of free space, permeabilty
of free space, and the relative
16
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permeability, respectively. Typical fr and Qr values of the resonator used are 1.2
GHz and 2000.
To maximize the sensitivity, minimize the attenuation. The outer diameter
of the cavity should be chosen to ensure single mode operation of the cavity.
(This assumes that the cavity will be operated only in the lowest TEM mode.)
0.95c
1+ T
* fr
I
^
b
The attenuation also depends on the ratio of b to a. The minimum
attenuation per unit length occurs at roughly — = 4.68. This can be obtained by
a
minimizing the attenuation as a function of b/a.
For operation at 1 GHz, the wavelength is 28 cm. For quarter-wavelength
operation, this corresponds to a 7 cm coaxial cavity. The outer diameter of the
resonator, b, was chosen to be 2.5 cm while the inner diameter, a, was chosen to
be 0.5 cm. The cavity was made of copper. The center conductor was sharply
tapered near the end of the cavity. To minimize vibrational instability, the tip was
fixed by application of a nonconductive resin (soluble in acetone). A drilled
sapphire disk, coated with > 1000 angstroms silver was used as a shielding
structure. (Fig. 2-1) A thin layer of titanium was used to increase the adhesion of
the film. Coupling into and out of the cavity was accomplished by the use of
inductive loops. These loops proved easier to adjust than capacitive coupling. In
addition, the background off resonance was minimal by comparison to a single­
coupling reflection method. The quality factor was roughly 2500. For operation
17
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at 2 GHz, several similar cavities of smaller dimensions were fabricated. A
sapphire filling was used to reduce cavity dimensions.
Tip Fabrication
The tips used were fabricated by etching of 5-10 mil tungsten wire in a
solution of 10% NaOH solution by volume. A 10-20 V potential difference was
used to etch the wires. A platinum wire loop was used as the second electrode
to maximize the uniformity of the etching. Upon completion of etching, the
voltage was immediately switched off and each tip washed in water and acetone.
Depending on the etching voltage, tip opening angles from 20 to 60 degrees
could be obtained. The entire etching process takes a few minutes.
Detection system
By contrast to most types of microscope, SEMP measures a
complex quantity, i.e. the real and imaginary parts of the electrical impedance.
This is realized by measuring the changes in the resonant frequency (fr) and
quality factor (Q) of the resonator simultaneously. I will outline a few methods
used for the acquisition of resonant frequency and quality factor.
Frequency Sweep
A conventional method of measuring these two quantities is to sweep the
frequency of the microwave generator and measure the entire resonant curve.
(Fig. 2-2)
18
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Digital board
RF Source
Analog board
f
Diode detector
Resonator
Fig. 2-2
For each measurement, this can take seconds to minutes depending on
the capabilities of the microwave generator. These measurements are limited by
the switching speed of a typical microwave generator to roughly 20 Hz. With the
use of a fast direct digital synthesizer based microwave source, the throughput
19
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can be improved to roughly 10 kHz, but is still limited by the need to switch over
a range of frequencies. A fast frequency sweep was originally used to measure
the microscope signal. I used a fast digital board (DIO-32F) to sweep the
frequency of the RF source (PTS1000). The PTS1000 uses direct digital
synthesis - which has the advantage of fast switching and high frequency
stability. The transmission of the cavity as a function of frequency was measured
by a diode detector (Pasternack / HP) and read by a fast 16-bit analog board
(A2150). Currently, vastly improved boards exist. To increase the bandwidth,
faster methods for data acquisition were developed.
Analog Feedback
Another method is to implement an analog phase-locked loop for
frequency feedback control. (Fig. 2-3)
20
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PtlMC
t (after
P bue
d c tc c u r
la tc g n to r
Cou p in g
Sapphire
Fq bc o o b
L o c k -in
icaeratar
t m p lin e r
M e ta l *
C o ating
M e ta l
b a c k in g
M oU O B
Co a trailer
Fig. 2-3 Setup for measurement of nonlinear
dielectric constant2
This method tracks the changing resonant frequency in real time and
measure fr and Q quickly. However, one has to use a voltage-controlledoscillator (VCO) as a microwave generator which usually only has a frequency
stability of 10'4. This low frequency stability seriously degrades the sensitivity of
the instrument. Since Ae/e ~ 500 AfA , frequency instability in the VCO will limit
21
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measurement accuracy. Another problem is that interaction between this
frequency feedback loop and the tip-sample distance feedback loop can cause
instability and oscillation, which will seriously limit the data rate. One advantage
of this setup is that AC measurements can easily be implemented, allowing very
sensitive measurements of variations in material properties. This setup was
used for measurements of the nonlinear dielectric constant.
The phase shift of the cavity was used to drive the frequency of the VCO
to the cavity resonant frequency. The resonant frequency could be obtained by
measurement of the VCO frequency while the quality factor could be obtained by
measurement of the transmitted power. The AC variation of the resonant
frequency upon application of the microwave voltage gave the nonlinear
dielectric constant. This will be discussed in detail in chapter 3.
AC measurement
One method for measurement of the resonant frequency is to vary the
frequency of the microwave source and to measure the first harmonic of the
output signal. The first harmonic, for small variations of the source frequency, is
proportional to the derivative of the resonant curve. The resonant frequency may
be measured directly from this signal. Alternatively, an analog feedback loop
may be simply implemented in a fashion similar to the above. However, the use
of a lock-in amplifier limits the achievable bandwidth to roughly 100 Hz. As such,
this method was not used.
I/Q Mixer
W e decided to implement a
novel method. In this method, a direct
22
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digital synthesizer (DOS) based microwave generator is used, which has
frequency stability of better than 10'9. The frequency of the microwave signal is
fixed at the previous resonant frequency and the in-phase (I) and quadrature (Q)
signals are measured simultaneously.
Since the microwave frequency is fixed, the DDS switching speed does
not limit the data rate. By measuring the in-phase and quadrature microwave
signals, fr and Q can be derived. Near resonance, the in-phase and quadrature
signals are given by:
i = A sin 0
q = A cos 0, where Ais the amplitudeof the microwavesignal on
resonance and 0 is the phase shiftof the transmitted wave.Given
i, q, the
current input microwave frequency, and the input coupling constants, the current
fr and Q can be calculated.
For a resonator with initial quality factor Q0, transmitted power A0, and
resonant frequency fo, driven at frequency f,
1
f
A / = — tan 0 —
2
Q
1 s in 0
~ 2 C~ A ~ ’
where c = A A
00
,
23
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Then, f r and Q can be obtained by
Q =
c
fr
q
—
Initially, since the l/Q mixer does not maintain perfect phase or amplitude
balance, these quantities must be calibrated. To calibrate the relative amplitudes
at a given frequency of the i and q outputs of the mixer, the relative values of the
outputs when the reference signal is shifted by 90 degrees are measured. This
can easily be extended by means of a calibration table.
To calibrate the relative phases of the i and q outputs at a given
frequency, the i output of the mixer onresonanceis measured. At resonance,
i/q = 8 , where
8
is the phase errorof the l/Q mixer.
Near resonance,
i = A sin
(0
+ 8 ) = A sin 6 cos 8 + A sin
= A sin
0
+A
8
= A sin
0
+q
8
cos
8
cos
6
0
This allows the correction of the phase error of the mixer.
This method of measurement only requires one measurement cycle.
Therefore, it is very fast and limited only by the DSP calculation speed. To
increase the working frequency range, a digital signal processor (DSP) is used to
control the DDS frequency to shift when the resonant frequency change is
beyond the linear range. This method
allows data rates around 100 kHz-1
24
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MHz (limited by the bandwidth of the resonator) and frequency sensitivity below 1
kHz ( ^
= 1O' 6 to 1 0 7).
fr
Impedance matching for the resonator can be somewhat difficult since the
impedance varies with sample properties. A schematic of the system consisted
of a RF source (PTS3200), a phase shifter, an attenuator, a resonator, an
isolator, and an l/Q mixer. (Fig. 2*4)
25
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Phase Shifter
RF Source
Attenuator
Isolator
Resonator
I/Q M ixer
To computer
Fig. 2-4 Schematic for phase-sensitive detection
of SEM P signal
The source drove one input of the mixer directly through the phase shifter and
attenuator, allowing adjustment of the relative amplitude and phase of the signal.
The source also drove the other input through the resonator and isolator. The
isolator was used to minimize the effects of variation of resonator impedance.
26
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The phase shifter was adjusted so that one output of the mixer was zero on
resonance.
Staging
I collaborated in the construction of two scanned evanescent microwave
microscopes. The first, designed primarily for long-range travel, used a simple
mechanical stage and servo motors / controllers from Ealing Electro-Optics. The
motor controller was a repackaged product from Precision MicroControl (DCXPC-100). Motion in z was accomplished over long ranges by a mechanical
screw. For short-range tip-sample distance control, I used a piezoelectric pusher
from Burleigh (PZS-035), which allowed 35 micron scan range. This setup was
simple. However, there were several difficulties. First, the servo motors / stage
resonances limited the scan rate to approximately 1 line /1 0 sec. Second, the
piezoelectric pusher, while convenient to use because of its long range, had a
low (-200 Hz) resonant frequency, rendering it unsuitable for high-frequency
acquisition of the sample topography.
To alleviate these problems, a smaller resonator was designed and
mounted on a piezoelectric tube. Owing to the large resonator mass, the tube
mechanical resonant frequency was still limited to -1 kHz, but the use of the tube
did allow an increase in the scan rate to several lines per second. One
disadvantage of the current system is that it does not allow large displacements
(at most a few microns) in z. A number of designs for longer range motion in z
are currently being tested.
Sample Preparation
27
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Ultrasonic cleaning
I used an ultrasonic cleaner to remove accumulated dust and residues. A
reasonable recipe for cleaning samples is:
15 minutes trichloroethane
15 minutes acetone
15 minutes methanol
These solvents are widely available and are chosen for their ability to take
various residues into solution. Methanol is chosen last owing to its generally
higher purity.
References
1
. D. M. Pozar, Microwave Engineering (Addison-Wesley, New York, 1990).
2. C. Gao, F. Duewer, Y. Lu, and X.-D. Xiang, Appl. Phys. Lett. 73, 1146 (1998).
28
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Chapter III:
Quantitative calculations
Overview
A number of quasistatic models can be applied to the calculation of the
probe response to dielectric, nonlinear dielectric and conductive materials. 1' 4
Here, we describe the application of these models to the calculation of the
complex dielectric constant, nonlinear dielectric constant, and conductivity.
We describe in detail the procedures necessary to accomplish the above:
a) Modeling of the cavity response for dielectric materials
b) Modeling of the cavity response for nonlinear dielectric materials
c) Modeling of the cavity response for conductive materials in the low and high
conductivity limits.
Insulating materials - complex dielectric constant
To allow the quantitative calculation of the cavity response to a sample
with certain dielectric constant, a detailed knowledge of the electric and magnetic
fields in the probe region is necessary. The most general approach is to apply
an exact finite element calculation of the electric and magnetic fields for a timevarying three dimensional region. This is difficult and time consuming. Since the
tip is sharply curved, a sharply varying mesh size should be implemented. At the
tip, the mesh size should be much less than the tip-radius. However, further from
the tip, the mesh size can increase without loss of accuracy since the field
distribution will not vary sharply with position. As the spatial extent of the region
of the sample-tip interaction is much less than the wavelength of the microwave
29
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radiation used to probe the sample (X- 28 cm at 1 GHz, -1 4 cm at 2 GHz), the
quasistatic approximation can be used, i.e., the wave nature of the electric and
magnetic fields can be ignored. This allows the relatively easy solution of the
electric fields inside the dielectric sample. First a finite element calculation of the
electric and magnetic fields under quasi-static approximation for the given tipsample geometry can be applied. A number of other approaches can be
employed for the determination of the cavity response with analytic solution,
which is much more convenient to use. We outline the calculation of the relation
between complex dielectric constant and SEMP signals for bulk and thin film
dielectric materials by means of an image charge approach.
Thick Films
For films thicker than the tip radius, the complex dielectric constant
measured can be determined by an image charge approach if signals from
SEMP are obtained. By modeling the redistribution of charge when the sample is
brought into the proximity of the sample, the complex impedance of the sample
for a given tip-sample geometry can be determined with measured cavity
response (which is described below). A preferred model is one that can have an
analytic expression for the solution and is easily calibrated and yields
quantitatively accurate results. Since the tip geometry will vary appreciably
between different tips, we require a model with an adjustable parameter
describing the tip. Since the region close to the tip predominately determines the
sample response, we can model the tip as a metal sphere of radius Ro. Figure
3-1 illustrates the infinite series of image charges used to determine the tip30
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
sample impedance. For dielectric samples the dielectric constant is largely real
£
(— < 0.1), where er and
5
are the real and imaginary parts of the dielectric
constant of the sample, respectively. Therefore, the real portion of the tip-sample
capacitance, Cr , can be calculated directly and the imaginary portion of the tipsample capacitance,
cc ,
cv,
can be calculated by simple perturbation theory.
£
e c„
» — c
The tip-sample capacitance may be calculated by the superposition of two
series of image charges. The first arises from the presence of the dielectric
sample. The second arises from the presence of the spherical tip. Iterating this
process of action and reaction results in an infinite sum of image charges. We
find that the tip-sample capacitance is given by: 1
C,=4TO0* 0£ - ^ - ,
(3)
where tn and an have the following iterative relationships:
«» =1+ <*'--— r
and
l + « + a «-i
(4)
'■ = 1 + b,' t
(5)
with ai =
1
e —£
d
+ a , ti = 1 , b = ------ , and a = — , where c is the dielectric constant
e+e0
R
of the sample,
£0
is the permittivity of free space, d is the tip-sample separation,
and R is the tip radius.
31
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This simplifies to:
Cr = 4jc£qRq
ln(l - b )
(6)
+1
as the tip-sample gap approaches zero.
Since the dielectric constant of dielectric materials is primarily real, the
loss tangent (tan5 ) of dielectric materials can be determined by perturbation
theory. The imaginary portion of the tip-sample capacitance will be given by:
C, = Cr tan S .
(7)
Given the instrument response, Clip.s<mple, the complex tip-sample
capacitance, and therefore the complex dielectric constant of the sample can be
determined. Figure 3-1 illustrates agreement between the calculated and
measured frequency shifts for variation of the tip-sample separation.
32
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
954.0
▲ : Measured
: Best fitting
953.8
953.6
-N
|
953.4
Fit parameters:8=9.5
953.2
~ 0 953.0
952.8
Sample: MgO
952.6
952.4
0.0
2.0
4.0
6.0
8.0
10.0
tip-sample distance(pm)
12.0
14.0
Fig. 3-2 Measured and theoretical fitting
of the resonant frequency versus tipsample separation1
The dielectric constant and loss tangent can be determined from Ctip_samplt
by a number of methods. One simple approach is to construct a look-up table
which yields the complex dielectric constant corresponding to a given Ctip_samplt.
Another way is to directly calculate from the signals using the Eqn. (6). Table
33
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3-1 shows a comparison between measured and reported values for dielectric
constant and loss tangent.
Material
Measured e Reported
r
Measured tanfi
Reported tanS
YSZ
30.0
29
1.7x1 O'3
1.75x1 O'3
LaGaO3
23.2
25
1.5x1 O'3
1.80x10‘3
CaNdAIC)
4
T iO ,
18.2
-19.5
1.5x1 O'3
0.4-2.5X10’3
86.8
85
3.9x1 O'3
4x1 O’3
BaTiOg
296
300
0.47
0.47
Y A '°3
16.8
16
-
8.2x1 O'5
SrLaAIO.
4
LaALOg
18.9
20
-
-
25.7
24
-
2.1x10‘5
MgO
9.5
9.8
-
1.6x10 s
LiNbO^X-cut)
32.0
30
-
-
Table 3-1 Single Crystal Measurement1
Electrical properties for single crystals are comparatively easy to obtain.
The measured values agree to within -2% for dielectric constant and ~15% for
34
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loss tangent.
The perturbed electric field inside the sample is: 2
rer + (z + an)e.
(8 )
where q = 47ce0RV0, V0 is the voltage, and er and e, are the unit vectors along
the directions of the cylindrical coordinates r and z , respectively. This formula
will be used later in the calculation of the nonlinear dielectric constant.
Thin films
The image charge approach can be adapted to allow the quantitative
measurement of the dielectric constant and loss tangent of films thinner than the
tip-radius. Strictly speaking, the image charge approach will not be applicable to
thin films due to the divergence of the image charges. However, if we can model
the contribution of the substrate to the reaction on the tip properly, the image
charge approach is still a good approximation. We expect that all films can be
considered as bulk samples if the tip is sharp enough since the penetration depth
of the field is only about R. It is obvious that the contribution from the substrate
will decrease with increases in film thickness and dielectric constant. W e model
this contribution by replacing the effect of the reaction from the complicated
image charges with an effective charge with the following formula:2
\
—b2a + (fcl0
£
—£
^20
)exP
£
(9)
0.18
—£
where b1J0 = - 2 — l , h = —— - , e, and e. are the dielectric constants of the
£2
e,+£0
35
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film and substrate, respectively, a = — , and d is the thickness of the film.
R
This formula reproduces the thin and thick film limits for the signal. The
constant 0.18 was obtained by calibrating against interdigital electride
measurements at the same frequency on SrTiC>3 thin film. Following a similar
process to the previous derivation, we have : 2
'20
n + 1 + 2mna
n=l m=0
f
1
A*
+ 2 (m + 1 )na
(10)
f
tan5,
n=l /n=0
n+
^
Y\
'2 0
n + 1 + 2 mna
(n + 1 + 2 (m + 1 )na)
2 £ l £ 2 t<M l
(e2 + e, Xe2 + eo) (« +
1
+ 2(m + 0 * * )
(11)
where blt =
e2 -£ i
, tan 5, and tan^t are the tangent losses of the film and
e2 +e,
substrate. Thin film measurements using the SEMP and interdigital electrodes
at the same frequency
(1
GHz) have been done.5 The estimates for microwave
dielectric constant are within
10
%, demonstrating that this model allows
quantitative measurements of thin film dielectric constant.
Ferroelectric materials
The nonlinear dielectric constant can be imaged by measurement of the
variation in fr upon application of a bias voltage. For example, the lowest order
nonlinear dielectric constant ejjk can be imaged
The detailed knowledge of the field distribution in Eq.
8
allows quantitative
36
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calculation of the nonlinear dielectric constant. The component of the electric
displacement D perpendicular to the sample surface is given by:
^3 = ^3 + £33(Pi + Em)"*■ 2 ^3330^/ ^
) + ^^3333(^/
^m ) "*
0 2)
where D 3 is the electric displacement perpendicular to the sample surface, P3 is
the spontaneous polarization, etj ,eiJk, eijU,... are the second-order (linear) and
higher order (nonlinear) dielectric constants, respectively.
Since the field distribution is known for a fixed tip-sample separation, we
can estimate the nonlinear dielectric constant from the change in resonance
frequency with applied voltage. For tip-sample separations much less than R,
the signal mainly comes from a small region under the tip where the electric
fields (both microwave electric field Emand low frequency bias electric field El) are
largely perpendicular to the sample surface. Therefore, only the electric field
perpendicular to the surface needs to be considered.
From Eq. 12, the effective dielectric constant with respect to Em can be
expressed as a function of £,:
«„(£<) = I f 5- = * 3, + em (£, + E .) +
(E .+ E J + ...,
(13)
(It
and the corresponding dielectric constant change caused by E, is:
Ae = e m E, +
2 £ 3333 &1
+ ---
0^)
The change in f r for a given applied electric field Et is related to the change in
the energy stored in the cavity. Since the electric field for a given dielectric
constant is known and the change in
the dielectric constant is small, this can
37
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be calculated by integrating over the sample:
I AeEldV
-
fr
J l f m E, * ' £ „ „ £ / + . . f c d V
______________
\{e E l+ v H l)d V
(1 5 )
\{£ E l+ » H l)d V
V,
V,
v »
where Vs is the volume of the sample containing electric field, H xis the
microwave magnetic field, and V, is the total volume containing electric and
magnetic fields, possibly with dielectric filling of dielectric constant e . En is
given by Eq.
8
. The application of a bias field requires a second electrode
located at the bottom of the substrate. If the bottom electrode to tip distance is
much larger than tip-sample distance and tip radius, Eq.
8
should also hold forE,.
The numerator can be calculated by integrating the resulting expression. The
denominator can be calibrated by measuring the dependence of f r versus the
tip-sample separation for a bulk sample of known dielectric constant. If the tipsample separation is zero, the formula can be approximated as:
C,(V) = C ,(V = 0 ) + 4j K„/ i - ! - —
32 £ 3 3 R
2£q
(16)
where V is the low frequency voltage applied to the tip. Higher order terms
exist, but are typically small. This calculation can be generalized in a
straightforward fashion to consider the effects of other nonlinear coefficients and
thin films.
Conductive materials
For materials with low conductivity, the complex tip-sample capacitance can
38
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be calculated exactly. The dielectric constant of a conductive material at a given
frequency f may be written as:
e = er + ~ j ~ ' where er is the real part of the permittivity and a is the
conductivity. The quasistatic approximation should be applicable when the
wavelength inside the material is much larger than the tip-sample separation.
For R0 - 1 urn and A = 14cm,
( A V
““
\
Rn
For a «
0
~ 2 x l0 10 or <r
max
/
f£
1
= iinsL =: 2 x l0 7 — 5— .
2
(17)
Q -c m
, the quasistatic approximation remains valid. Ctip, sample can
be calculated by the method of images. Each image charge will be out of phase
with the driving voltage. By calculating the charge (and phase shift) accumulated
on the tip when it is driven by a voltage V, frequency f, one can calculate a
complex capacitance.
Using the method of images, we find that the tip-sample capacitance is
given by:
(18)
tip-sam ple
where tn and an have the following iterative relationships:
\+a+a
(19)
and
(20)
1+ a, + fln_1
39
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
with ai =
1
+ a , ti = 1 , b = e c° , and a = — , where e = e r -ie ,e is the complex
£+£„
R
dielectric constant of the sample, £o is the permittivity of free space, d is the tipsample separation, and R is the tip radius.
By writing b = br + ib{ = \b\e'*, we can separate real and complex
capacitances.
C , = 4 TOA X f f C ° s M 8 - and
* * * '* '.
ai + a n
S
where g, =
1
(2 1 ,
(2 2 )
and g„ is given by:
S . = T - 4 = 7 ------ ■
1
<2 3 >
+ fl
This calculation can be generalized in a straightforward fashion to thin
films.
40
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3.8e-5
5e-6 -
C(pf)
Im C(pf)
2.8e-5-,
0.1
1000
1000
0.1
Ei
Fig. 3-2 Calculated tip-sample capacitance for R0 = 1 pm, d
= 1 pm versus imaginary e, real e = 10. Max loss: sheet
resistance- 30 kQ-Sq
Figure 3-2 illustrates fr and A(l/Q) as a function of conductivity. The curve peaks
approximately where the imaginary and real components of e become equal.
For those plots, the complex dielectric constant is£ =
1 0
2(J
+ 1—
41
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
For <t >
, the magnetic field should also be considered. The real
portion of Clip_samplc can be derived using an image charge approach. This is
identical to setting b = I.
Cltp-sample = 4 ne0R0J
^ -^ — ,
(2 4 )
where tn and an have the following iterative relationships:
+
t.
with ai =
— 7
and
+ «„-i
(25)
(26)
= — 7 7 -----1+ a + anA
1
+ a , ti =
1
£ —£
d
, b = ------ 1 , and a = — , where £ is the dielectric constant
£ + £ 0
R
of the sample, eo is the permittivity of free space, d is the tip-sample separation,
and R is the tip radius.
In this limit, the formula can also be reducedto a sum of hyperbolic sines. 1
C r = 4 keqR0 sinh
(2 7 )
^ s in h n a
where a = c o s h (l + a ) .
The magnetic and electric fields at the surface of the conductor are given
by:
E s(r)
=
(28)
^
^ 0
[ r 2 +(fl^/?0) 2]
= - i y - ' L l n [r2 +-^ nRo)Z-] - ~ n\
[r2H a nRo)2]h
(29)
42
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The radiation loss owing to the film may be calculated by integration of the
divergence of the Poynting vector over the film.
Since the physical properties are all calculated from the fr and Q and their
derivatives, the temperature stability of the resonator is crucial to ensure the
measurement reproducibility. In fact, the sensitivity is dependent on the
temperature stability of the resonator. Efforts to decrease the temperature
variation of the resonator using low thermal-coefficient-ceramic materials to
construct the resonator should be useful to increase the sensitivity of the
instrument.
References:
1. C. Gao, T. Wei, F. Duewer, Y. Lu, and X.-D. Xiang, Appl. Phys. Lett. 71,
1872 (1997).
2. C. Gao and X.-D. Xiang, Rev. Sci. Instrum. 69, 3846 (1998).
3. C. Gao, F. Duewer, and X.-D. Xiang, Appl. Phys. Lett. 75, 3005 (1999).
4. C. Gao, F. Duewer, Y. Lu, and X.-D. Xiang, Appl. Phys. Lett. 73,1146
(1998).
5. H. Chang, I. Takeuchi, and X.-D. Xiang, Appl. Phys. Lett. 74,1165 (1999).
43
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Chapter IV:
Ferroelectric Domain Measurements
We have used the SEMP to image ferroelectric domains - both indirectly
by measurement of the variations in dielectric constant and loss tangent induced
by the domains1 and directly by measurement of the sign of the nonlinear
dielectric constant. 2 ,3
Indirect measurements
Crystals with periodic and quasi-periodic ferroelectric domain structures,
such as LiNb0 3 , LiTa03, and KTi0 P 0 4 superlattice crystals4 , 5 (either in bulk form
or as thin-film waveguides), have attracted considerable interest and found
important applications in quasi-phase-matched nonlinear optics6 and in
acoustics. 7 Currently, a destructive method involving optical imaging of
differentially etched surfaces is commonly used to characterize the domain
structures. Several other (mainly charge- or polarization-sensitive) techniques
have also been developed8, but none allows nondestructive, high-resolution
imaging of ferroelectric domains over a large area. Moreover, there is no
effective technique to analyze the variations in dielectric constant corresponding
to variations in dopant concentration, which could give rise to domain formation,
in these materials and related devices.
We developed a scanning-tip microwave near-field microscope with 5-pm
resolution9 and have now improved its performance to nondestructive^ image
surface dielectric constant and microwave loss with submicrometer spatial
resolution and a scanning range of over 2.5 cm. Through images of dielectric
constant, we have observed submicrometer-resolution profiles of periodic
44
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
dielectric constant and evidence of a lattice-edge dislocation in a yttrium-doped
LiNb0 3 superlattice crystal. The profiles of microwave energy loss yield
nondestructive images of ferroelectric domain boundaries. By correlating the
images of both dielectric constant and microwave energy loss, a growthinstability-induced defect in the periodic domain structure has been identified.
These studies should contribute significantly to our understanding of the growth
mechanism of the crystals. In addition, the imaging technique used here should
prove useful in analyzing other ferroelectric and dielectric materials and thin-film
devices.
Yttrium doping has proven to be an effective method of introducing a
periodic ferroelectric domain structure (superlattice) in the as-grown LiNb0 3
single crystal. 10 For this study, an average of 0.5 weight % yttrium-doped
LiNb0 3 superlattice single crystal was grown along the a axis (x axis in Fig. 4-1)
with the Czochralski technique at the Nanjing University. 10 A 3-mm-thick wafer
cut normal to the y axis and polished on both surfaces was used. The surface of
the crystal was examined with a profilometer to confirm the optical quality
smoothness. In this configuration, the polarizations of the ferroelectric domains
alternate along the c axis (z axis in Fig. 4-1).
45
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Scanning
• P rim a r ily u s e fu l in
n o n lin e a r o p tics
• P e rio d ic re v e rs a l o f
p o la riz a tio n d ire c tio n
o v e r m ic ro n s a llo w s
e ffic ie n t h a rm o n ic
g e n e ra tio n
• D if f ic u lt to m e asu re
o p tic a lly
M
m
r
\
L iN b O j Superlattices
Fig. 4-1 Periodically-poled ferroelectrics1
Two domain laminas with opposite polarization give rise to the same
surface charge configuration (because there is no y-axis component of
spontaneous polarization), simplifying the interpretation of our experimental
results (charge-sensitive techniques fail to image the structures in this
configuration). The microscope tip scans over the xz plane of the sample
approximately along the x axis (with a typical scan speed of 1 to 5 pm/s in this
46
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
study). The electrochemically etched tungsten tip (with typical radius of -0.1 pm)
was kept in contact with the surface of the sample with a soft spring (with a
spring constant of 4 N/m and a typical contact force of <20 pN). No surface
damage was observed with this scanning format. Neither the scanning format nor
the detected signal is sensitive to topography (that is, material features with
identical dielectric constant and tangent loss but different height are not
distinguishable in the image).
A periodic variation in the concentration of yttrium was generated by
growing the LiNb0
3
crystal along the x-axis in an asymmetric temperature
field.4 ’ 11 The periodic variation in dopant concentration induces (through the
internal space-charge field) the formation of ferroelectric domains with alternating
polarization when the crystal undergoes a para-to-ferroelectric phase transition.
Although alternating polarization does not modulate its second-rank tensor
dielectric constant (LiNbOa belongs to the 3m point group, for which the nonzero
second-rank tensor dj does not change its sign under the transformation of 180°
rotation around the x axis), the periodic variation of dopant level should result in a
periodic change in the dielectric constant. Therefore, in the Czochralski-grown
LiNb0 3 superlattice crystals, two periodic structures (that is, dielectric constant
and ferroelectric domain) should coexist, although the former has never been
observed previously. We have observed both periodic structures through
microwave imaging (Fig. 4-2).
47
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
^2889.56
- “2889.52
foiMHz)
2889.50
860
°
0.0 5.0 10.015.020.025.030.0
8.0
x(nm)
0.0
z(pm)
Fig. 4-2 1 Alternating domains in LiNb0 3 single crystal (a)
Resonant frequency (MHz) as a function of position (b)
Quality factor (c) Slice in X direction of image for Y-cut
LiNb03 single crystal
The image of resonant frequency fr reflects variations in the dielectric
constant associated with changes in dopant levels, whereas the image of Q
corresponds to losses in microwave energy, which are large at the ferroelectric
domain boundaries (primarily as the result of movement of the domain walls
48
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
under the influence of the microwave field). The change in resonant frequency is
Afr/fo = gAer, where e = £r+ iei is the complex dielectric constant and g was
measured to be - 7 x 10' 5 in this configuration. The total dielectric variation in
Fig. 4-2 is estimated to be -0.25 with a noise level of 0.03, which is the current
sensitivity of the microscope. The change in cavity quality factor is A(1/Q) = gAe;
and was calculated to be 7.1 x 10'2.
The total loss tangent variation in Fig. 4-2 is estimated to be about 1x1 O'2.
Detailed theoretical analysis and experimental calibration of the sensitivity and
accuracy were performed. 13 The fr profile mainly indicates variations in dielectric
constant when the change in loss is not large, and the Q profile indicates the
microwave energy loss of the sample surface. Fig. 4-2b indicates the fr and Q
profiles of (A) along the x axis at z = 4.0 pm. The diffuse and sharp domain
boundaries are marked B1 and B2, respectively.
The domain boundaries with relatively low loss (thin yellow stripe in Fig.
4-2a marked as B2) are located at the dielectric constant minima (fr maxima in
Fig. 4-2b), whereas the domain boundaries with high loss (wide dark-red stripe in
Fig. 4-2a marked as B1) are located at a critical value of the gradient in dopant
concentration (where the polarization changes its direction) near (but not at) the
dielectric constant maxima (max). The full width at half maxima of B1 (-2.5 pm)
is greater than that of the B2 (-1 pm). This result is consistent with the previous
finding of sharp and diffuse boundaries by x-ray energy dispersive spectral
analysis in these crystals14 and the proposed scenario that diffuse domain
boundaries do not occur at dopant maxima15 if we assume that the doping of
49
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
yttrium in the LiNb0 3 crystals monotonically increases the dielectric constant.
Because diffuse domain boundaries are not always located at the dopant
maxima (corresponding to Enuw). alternating polarization domains separated by
domain boundaries (as observed in images of Q) may also have different widths.
The width ratio of the two domains in one period of 7.0 pm (Fig. 4-2) is ~0.87.
Because dopant (dielectric constant) and domain boundary profiles can be
imaged simultaneously, it is possible to gain information about the growth
mechanism of the crystal by correlating the detailed features of the profiles. For
example, we observed a defect in the periodic domain structure (that is, an
island-like domain caused by solid-liquid interface instability during growth) ("I" in
Fig. 4-3a).
50
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
B)
A)
2889.44
-2889.42
2889.7
Eo2889.40
2889*;
ta2889J*
2889JO
-2889J
I
K
880.0-
uu.u
0
20
40 60
z(Hm)
80 100
x(Mm)
Fig. 4-31 (a) Resonant frequency and Q versus position I
denotes an islandlike domain (b) Slice at z = 73 microns.
This type of defect is common in Czochralski-grown U N b0 3 crystals and is
destructive to the periodic domain structure. 12 As is evident from the fr profile,
the dopant modulation is more diffuse near the island, and the overall dopant
level is higher in the island region. A corresponding transition from sharp (low
51
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
loss) to diffuse (high loss) domain boundaries is clearly observable in the Q
profile.
These observations indicate that dopant aggregation, caused by
temperature instabilities in the island region during the crystal growth, is the
origin of the island-like domains. A gradual transition from sharp to diffuse
domain boundaries is again evident in Q and fr profiles at z = 73 pm (Fig. 4-3b),
and the domain structure gradually loses its original periodicity approaching the
island domain. For stable growth, the periodic modulation of dopant level should
have a single frequency (which is the frequency of crystal rotation)4,11, whereas
disturbances in the system, such as melt or air convection and power
fluctuations, may introduce additional modulations with different frequencies.
When these additional frequencies are close to the crystal rotation frequency
(with a period of ~7 pm in Fig. 4-3), interference eventually drives the solid-liquid
interface out of stability, causing dopant aggregation and island domain formation
during crystal growth (Fig. 4-3b).
A "butterfly" image of the dielectric constant is an indication of a lattice
"edge dislocation" defect (Fig. 4-4a).
52
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
z( Urn)
z(pm)
Resolution = 0.30 Hm - Kj/1® 5
Fig. 4-41 (a) Resonant frequency
image of a dislocation defect (b)
Slice through the defect
The large difference in dielectric constant adjacent to the dislocation is
caused by compressive and tensile stresses in lattice structure induced by the
dislocation. The stress-induced contour is qualitatively consistent with the
theoretical prediction for a lattice "edge dislocation" defect. 1 6 The profile of a line
scan along the z axis at x = 38.7 pm (Fig. 4-4b), which cuts through regions with
53
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
compressive and tensile stresses, shows large opposite changes in on either
side of the edge defect. This profile also indicates that our spatial resolution in
this configuration (mainly limited by the radius of the tip) is better than
1
pm,
because between the two points (0 . 6 pm apart), changed from the minimum to
near maximum. Lattice dislocations in LiNbOa crystals are not observable by
optical microscopy with polarized light because of the crystal's large
birefringence.
Direct Measurements
Methods capable of imaging ferroelectric domain structures are useful for
potential ferroelectric memory applications and the development of ferroelectric
optoelectronic devices. One example is crystals containing periodic ferroelectric
domains. Such crystals have found applications in nonlinear optics as frequency
doublers and parametric oscillators thorough quasiphase-matching. Their
electrical and optical properties are often dominated by their domain structure.
We previously described indirect measurements of ferroelectric domains by
means of measurement of the change in loss tangent and dielectric constant at
domain boundaries. However, direct measurements of nonlinear dielectric
constant are preferable. Using the analog feedback setup described in Chapter
2
, we can measure the nonlinear dielectric constant.
Figure 2-3 illustrates the setup used to measure the nonlinear dielectric
constant.
54
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
To measure
£ 333 ,
an oscillating voltage VQ, of frequency fa, is applied to
the silver backing of the sample and the output of the mixer is monitored with a
lock-in amplifier (SR 830). This bias voltage will modulate the dielectric constant
of a nonlinear dielectric material at fa- By measuring frand the first harmonic
variation in the phase output simultaneously, sample topography and
£333
can be
measured simultaneously.
Figure 4-5 shows images of dielectric constant and
£333
for a periodically
poled single-crystal LiNbOs wafer.
55
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A8
0.19
X(Jim)
SensitivityAs/ E ~ 10" 3
^333(1^0
3.6E-19
X(Hm)
Fig. 4-6 (a) image of dielectric constant
variation and (b) Nonlinear dielectric
constant for LiNb03 z-cut single crystal
The crystal is a 1 cm x
1
cm single crystal substrate, poled by periodic
variation of dopant concentration. The poling direction is perpendicular to the
plane of the substrate. The dielectric constant image is essentially featureless,
with the exception of small variations in dielectric constant correlated with the
variation in dopant concentration. The nonlinear image is constructed by
56
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
measuring the first harmonic of the variation in output of the phase detector using
a lock-in amplifier. Since Ejjk reverses when the polarization switches, the output
of the lock-in switches sign when the domain direction switches. The value (-2.4
x 10' 19 FA/) is within 20% of bulk measurements. The nonlinear image clearly
shows the alternating domains.
Noncontact measurements
The regulation of the tip-sample separation is a consistent problem for
most scanned probe microscopes. Since the signal is determined both by
material properties and the tip-sample separation, a means of regulating that
separation is necessary to allow reproducible measurements. 3 W e previously
developed a scanning evanescent microwave probe (SEMP) for ferroelectric
domain imaging, capable of quantitative measurements of linear and nonlinear
dielectric properties with submicron resolution. The microscope operated in a
soft contact mode in which we kept samples in soft contact with the tip to
maintain a constant tip-sample distance. This contact can damage the tip or
sample and limit the resolution. It is desirable to implement a tip-sample distance
regulation so that topographic and physical properties image can be obtained
independently. Previously, shear force and atomic force have been used to
provide independent topographic imaging in scanning near-field optical
microscopes. 1 7 18 W e now report a tip-sample distance control mechanism based
on regulation of the tip-sample separation using the microwave signal. By
monitoring fr and Q as the tip scans over the sample surface, we measure the
electrical properties of the sample.
57
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In principle, following the calculations in Chapter 3, the relationship
between tip sample distance, electrical impedance and measured signals (fr and
Q as function of sample difference, bias fields and other variables) is known
precisely, at least when the tip is very close to the sample.
If measured fr and Q
(and their derivatives respect to electric or magnetic fields, distance and other
variables) signal points are more than unknown parameters, the unknowns can
be uniquely solved. If both tip-sample distance and electrical impedance can be
determined simultaneously, then the tip-sample distance can be easily controlled,
so that the tip is always kept above the sample surface with a desired gap (from
zero to microns). Both topographic and electrical impedance profiles can be
obtained. The calculation can be easily performed by digital signal processor or
any computer in real time or after the data acquisition, in order to regulate the
tip-sample separation, we used a previously developed analytic model to
correlate the microwave response as a function of tip-sample separation for a
material of a given dielectric constant.
Given a distance dependence of the frequency response to a substrate of
known dielectric constant, we can fit the distance dependence and extract the tip
radius (Fig. 3-2). For materials with constant e, the tip-sample distance (d) can
be controlled by adjusting the distance to maintain a constant frequency shift.
This method allows for control of the tip-sample separation over micron to
angstrom length scales.
Since the field distribution is known for a fixed tip-sample separation, we
can estimate the nonlinear dielectric constant from the change in resonance
58
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
frequency with applied voltage. The analysis shows that, for tip-sample
separations much iess than R, the signal comes from a small region under the tip
where the electric fields (both microwave electric field £ mand DC electric field £,)
are largely perpendicular to the sample surface. Therefore, only the electric field
perpendicular to the surface need be considered. The component of the electric
displacement D perpendicular to the sample surface is given by:
^3 ~ A
+ ^33
+
) + 2 ^333 ( ^ / +
6 ^3333 ( ^ / +
^ m)
^
0 )
where D 3 is the electric displacement perpendicular to the sample surface,
is
the spontaneous polarization, eiy,eijk,eijU.... are the second-order (linear) and
higher order (nonlinear) dielectric constants, respectively.
The effective dielectric constant with respect to Emcan be expressed as a
function of £,:
and the corresponding dielectric constant change caused by E, is:
A e — £ 333£ , +
(3)
2 ^ 3 3 3 3 ^ 1 ■*■•••
Since the electric field for a given dielectric constant is known and the change in
the dielectric constant is small, the change in
f r
for a given applied electric field
£, can be calculated by integrating over the sample: 11
(4)
fr
V
59
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
where V is the volume containing electric field.
The design of our microscope is based on a previously constructed
SEMP. From the calibration curves, a reference frequency fref is chosen to
correspond to some tip-sample separation. To regulate the tip-sample distance,
we employ a phase-locked loop. (Fig. 4-6)
Phase
shifter
I vcof'
Phase
dclcctoi
Intcgratoi ■■■ >
Coupling
loops
A/D
Converters
Diode
dctcciof1
Sapphire
Sample
'
Amplifiei
Coaxial A/4
resonator
Mctal^
Coating
Piezoelectric
Crystal ”
Sample
x-y-z stage
Motion
controller
Computer
Image
Display
Fig. 4-7 Schematic of system set-up for tip-sample distance
regulation3
60
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A microwave signal of frequency fref is input into the cavity and the cavity
output is mixed with a signal coming from a reference path. The length of the
reference path is adjusted so that the output of the mixer is zero when the
resonance frequency of the cavity matches fref. The output of the phase detector
is fed to an integrator, which regulates the tip-sample distance by changing the
extension of a piezoelectric actuator (Burleigh PZS-050) to maintain the
integrator output near zero. For samples with uniform dielectric constant, this
corresponds to a constant tip-sample separation. To measure £3 3 3 , an oscillating
voltage Vq, of frequency fn, is applied to the silver backing of the sample and the
output of the phase detector is monitored with a lock-in amplifier (SR 830). This
bias voltage will modulate the dielectric constant of a nonlinear dielectric material
at f . Since fn exceeds the cut-off frequency of the feedback loop, the high
frequency shift in e from Vn does not affect the tip-sample separation directly.
Using the calibration curves (fr versus d), a resonance frequency that
corresponds to a specific tip-sample separation is chosen for the cavity (Fig. 32). The resonance frequency chosen is fed into the cavity and the output of the
phase detector is used to regulate the applied voltage to the piezoelectric
actuator. Sample topography is measured by monitoring the variation in voltage
applied to the actuator. By measuring the applied voltage to the piezoelectric
actuator and the first harmonic variation in the phase output simultaneously,
sample topography and
£333
can be measured simultaneously.
In principle, changes in the polarization direction do not induce changes in
the linear dielectric constant (for most crystal symmetries). If the variations in
61
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
dielectric constant are sufficiently small, the frequency shift may be used to
regulate the tip-sample separation while £ 3 3 3 is simultaneously imaged. Since Cjjk
is a third-rank tensor, it reverses sign when the polarization switches, providing
an image of the domain structure. For the case of the periodically poled LiNbOs
single crystal imaged in this work, small variations in the linear dielectric constant
occur due to presumably piezoelectric-induced strain at the boundary or other
process-related defects. 5 However, these changes are in the range of 10 ‘ 2 in
dielectric constant and will induce negligible variations in the tip-sample
separation on the order of nanometers.
Fig. 4-8 shows images of topography and
£333
for a periodically poled
single-crystal LiNb0 3 wafer.
62
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Height
Nonlinearity
X(|im)
Fig. 4-83 Simultaneous topography and
ferroelectric domain imaging of poled
LiNb03
The crystal is a 1 cm x
1
cm single crystal substrate, poled by application
of a spatially periodic electric field. 19 The poling direction is perpendicular to the
plane of the substrate. The topographic image is constructed by measuring the
voltage applied to the piezoelectric actuator. It is essentially featureless, with the
exception of a constant tilt and small variations in height correlated with the
63
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
alternating domains. The small changes are only observable if the constant tilt is
subtracted from the figure. The nonlinear image is constructed by measuring the
first harmonic of the variation in output of the phase detector using a lock-in
amplifier. Since £jjk reverses when the polarization switches, the output of the
lock-in switches sign when the domain direction switches. The nonlinear image
clearly shows the alternating domains.
Ferroelectric thin films, with their switchable nonvolatile polarization, are
also of great interest for the next generation of dynamic random access
memories. One potential application of this imaging method would be in a
ferroelectric storage media. A number of instruments based on the atomic force
microscope have been developed to image ferroelectric domains either by
detection of surface charge or by measurement of the piezoelectric effect.21'22
The piezoelectric effect, which is dependent on polarization direction, can be
measured by application of an alternating voltage and subsequent measurement
of the periodic variation in sample topography. These instruments are restricted
to tip-sample separations less than 10 nanometers because they rely on
interatomic forces for distance regulation, reducing the possible data rate. Since
our microscope measures variations in the distribution of an electric field, the tipsample separation can be regulated over a wide range (from nanometers to
microns).
64
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References
1. Y. Lu, T. Wei, F. Duewer, Y. Lu, N.-B. Ming, P. G. Schultz, and X.-D. Xiang,
Science 276, 2004 (1997).
2. C. Gao, F. Duewer, Y. Lu, and X.-D. Xiang, Appl. Phys. Lett. 73,1146 (1998).
3. F. Duewer, C. Gao, and X.-D. Xiang, Rev. Sci. Instrum. 71, 2414 (2000).
4. D. Feng and N. B. Ming, Appl. Phys. Lett. 37, 607 (1980).
5. N. B. Ming et al., J. Mater. Sci. 1 7 , 1663 (1982); G. A. Magel, et al., Appl.
Phys. Lett. 56, 108 (1990); Y. L. Lu and N. B. Ming, ibid. 69,1660 (1996).
6. J. A. Armstrong and N. Bloembergen, Phys. Rev. 127,1918 (1962).
7. See, for example, S. D. Cheng, Y. Y. Zhu, Y. L. Lu, N. B. Ming, Appl. Phys.
Lett. 66, 291 (1995), and references therein.
8. T. Ozaki, et al., J. Appl. Phys. 80, 1697 (1996); R. LeBihan and M. Maussion,
J. Phys. 33, C2-215 (1972); P. J. Lin et al., Philos. Mag. A 48, 251 (1983); F.
Saurenbach and B. D. Terris, Appl. Phys. Lett. 56,1703 (1990).
9. T. Wei, X. D. Xiang, P. G. Schultz, Appl. Phys. Lett. 68, 3506 (1996).
10. Y. L. Lu, Y. Q. Lu, N. B. Ming, ibid., p. 2781.
11. Y. L. Lu, L. Mao, N. B. Ming, ibid. 59, 516 (1991).
12. H. A. Bethe and J. Schwinger, Perturbation Theory of Cavities (National
Defense Research Committee, Washington, DC, 1943), pp. D1-117.
13. C. Gao, T. Wei, F. Duewer, X.-D. Xiang, Appl. Phys. Lett. 7 1 , 1872 (1997).
14. J. Chen, Q. Zhou, J. F. Hong, W. S. Wang, D. Feng, J. Appl. Phys. 66 , 336
(1989).
65
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
15. Y. L. Lu, Y. Q. Lu, N. B. Ming, Appl. Phys. Lett. 68, 2642 (1996), and
references therein.
16. J. P. Hirth and J. Lothe, Theory of Dislocations (Krieger, Malabar, FL, ed. 2,
1 9 9 2 ).
17. E. Betzig, P. L. Finn, and J. S. Weiner, Appl. Phys. Lett 60, 2484 (1992).
18. R. Toledo-Crow, P. C. Yang, Y. Chen, and M. Vaez-lravani, Appl. Phys. Lett.
60, 2957 (1992).
19. Y. Cho and K. Yamanouchi, J. Appl. Phys. 61, 875 (1987).
20. C. Gao, F. Duewer, Y.-L. Lu, and X.-D. Xiang, Appl. Phys. Lett. 73,1146
(1 9 9 8 ).
2 1 . F. Saurenbach and B. D. Terris, Appl. Phys. Lett. 5 6 , 1703 (1990).
2 2 . P. Guthnerand K. Dransfeld, Appl. Phys. Lett. 6 1 , 1137 (1992).
66
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
V
Measurement of conductive materials
Direct Measurements
Measuring the resistivity of conductive materials is complicated by the
necessity of implementing some sort of distance control. W e began by
measuring film losses at constant tip-sample separations. (Fig. 5-1)
67
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
f0(Mhz)
6500
6500
1222.5
x(M-m)
948.5
1209.5
0 + -
3402
898
y (lim )
Fig. 5-1 Cu-Mn Non-contact Scan - Cu on bottom,
Mn on top
The copper and manganese films offer essentially no contrast in resonant
frequency, but show clear differences in quality factor. One advantage of SEMP
68
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
is the large penetration depth (microns). Fig. 5-2 is an image of resonant
frequency and quality factor for a niobium coil.
fr(M hz
m
- 949.160
-949 1 40 ;
•1188000
- 949.120
xOuH
- 949.100
1
-1188000
-949 080
-1184 000
-949060
• 949.040
1182.000
- 949.020
-949000
55
60
85
70
7!
50
55
80
85
70
Fig. 5-2 Image of niobium spiral(~2 pm
lines) (a) Image of resonant frequency fr
and (b) Q for niobium spiral
69
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The conductor under the coil is clearly visible. Most other microscopies are
limited to the first few atomic layers of a sample. However, to achieve reliable,
reproducible results, some sort of tip-sample distance control is necessary.
Tip-sample distance control for conductive materials
We have developed a means of tip-sample distance regulation for a scanning
evanescent microwave microscope over conductive samples.1 Changes in
resonant frequency and quality factor are measured, where changes in resonant
frequency are related to the tip-sample capacitance and changes in quality factor
are related to microwave absorption. With the analytical expression of the tipsample capacitance as a function of tip-sample distance, we can quantitatively
regulate the tip-sample separation. W e demonstrated simultaneous noncontact
imaging of topography and surface resistance with high spatial resolution. With
the rapid development of the electronics industry, methods for imaging the
electrical impedance with high spatial resolution have become increasingly
important. For dielectric samples, this interaction is dependent on the dielectric
constant and tangent loss of the nearby sample. For metallic samples, the
interaction depends on the surface resistance of the sample. Since the
microscope response is highly sensitive to the tip-sample geometry for all
scanned probe microscopes, tip-sample distance regulation is crucial in
achieving high spatial resolution, accuracy, and reliability. In previous works, to
minimize the effect of topographic variations, we have maintained a constant tipsample geometry by keeping the tip in a soft contact with sample.2 This soft
contact can introduce tip distortion, decreasing the spatial resolution, and can
70
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
damage the sample and/or tip. For conductive samples, the shift in frdiverges as
the tip-sample separation decreases to zero. To avoid this difficulty, we have
coated metallic samples with insulating layers. However, variations in the
properties of the insulating layer will be convolved with the sample properties.
These layers complicate the tip-sample geometry, making quantitative analysis
difficult. Previously, shear force, atomic force, and tunneling current have been
used to provide independent topographic imaging in various scanned probe
microscopes.3'5 To enable noncontact imaging, we have developed a tip-sample
distance control by means of regulating fr of the cavity to maintain a constant
separation. For this purpose, we used a previously developed analytic model to
correlate the microscope response as a function of tip-sample separation for
metallic materials. The fractional shift in resonance frequency as a function of the
tip-sample separation is derived in a manner similar to that of Ref. 5. Figure 53 shows the fit of the experimental data for Cu with the theory, determining Rq
and the absolute gap width.
71
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
954
X
re
Fit parameters:
953
R0 = 8.1 pm
A = 0.00258
Vi
©
a
Si
s
?
•Q
S
re
3
952
951
.
Measured
Best fit
950
I
S?
949
948
10
0.1
Tip sample distance (mm)
Fig. 5-36 Frequency shift vs tip-sample separation for
Cu.
Since these expressions are independent of conductivity for good metals, we
can use the frequency shift as a distance measure and control. This solution
should be generally applicable to a wide class of scanned probe microscopes
72
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
that include a local electric field between a tip and a conducting sample. It should
prove widely applicable for calibration and control of microscopes such as
scanning electrostatic force and capacitance microscopes.
For conductive materials, the tip-sample separation and microwave
resistivity can be measured simultaneously in a similar fashion. Since the tipsample capacitance is independent of conductivity for good metals, we can use
Ctipsanpit as a distance measure and control. This solution should be generally
applicable to a wide class of scanned probe microscopes that include a local
electric field between a tip and a conducting sample. It should prove widely
applicable for calibration and control of microscopes such as scanning
electrostatic force and capacitance microscopes.
From the calibration curves, a frequency f n[ is chosen to correspond to
some tip-sample separation. (Fig. 5*3) We then regulate the tip-sample
separation to maintain the cavity resonance frequency at f nf. This can be
accomplished digitally through the use of the digital signal processor described in
attachment A. W e have also used an analog mechanism. The phase-locked
loop described in Figure 4-7 has also been used to regulate the tip-sample
separation. Figure 5-4ab illustrates the measurement of topography with
constant microwave conductivity.
73
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
x( (Am)
Fig. 5-41 Silver squares on 2.1 pm
silver film (a) Topographic image (b)
Loss image
The figure consists of several silver squares of thickness 1000, 2000, and 4000
deposited on a uniform silver film. Variations in the sample topography are
74
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A
clearly evident while variations in the loss are negligible. Figure 5-5ab illustrates
the measurement of conductivity variations.
(a) 5 9 0 : Topographic image |
:0.5
Cr
3.48e3.46e-5
3.44e-5
3.42e-3
— 3.40e-5l
(b )
Loss image
Cr 3
Zr
' 3.38e-5|
3.36e-:
A
3.34e-S
3.32e2
x( M-m)
3
4
5
p(mQ-cm)
590
Fig. 5-51 Conductivity Profiling 2500 A thick squares on
750 A Pt film (a) Topograpic image (b) Loss image (c)
Measured resistivity vs loss
75
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2500 A squares of Mn, Cr, and Zr were deposited on a thin platinum film. The
microwave loss is clearly determined by the resistivity of the films. The sample
topography shows height variation between the different films, but those height
variation are consistent with those measured by profilometer.
Doped semiconductors can be measured using this approach.
76
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Heighten)
Fig. 5-6 Simultaneous topography and conductivity profiles of
doped silicon. The dopant level is -1 0 15 cm 3. The line
spacing is 10 pm. (a) Image of topography versus position,
(b) Image of loss versus position
Fig. 5-6a is a topographic image of doped silicon lines. Fig. 5-6b is a loss image
of the doped lines. The doping level is ~1015/cm3. Periodic variations in the
doping level are clearly visible in the loss image.
77
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This method allows submicron imaging of the conductivity over large
length scales. This method has the advantage of allowing distance regulation
over a wide length scale (ranging from microns to nanometers) giving rise to a
capability analogous to the optical microscope’s ability to vary magnification.
References
1. F. Duewer, C. Gao, I. Takeuchi, and X.-D. Xiang, Appl. Phys. Lett. 74, 2696
(1999).
2. T. Wei, X.-D. Xiang, W. G. Wallace-Freedman, and P. G. Schultz, Appl. Phys.
Lett. 68, 3506 (1995).
3. E. Betzig, P. L. Finn, and J. S. Weiner, Appl. Phys. Lett. 60, 2484 (1992).
4. F. Houze, R. Meyer, 0 . Schneegans, and L. Boyer, Appl. Phys. Lett. 69,1975
(1996).
5. B. Knoll, F. Keilmann, A. Kramer, and R. Guckenberger, Appl. Phys. Lett. 70,
2667 (1997).
6. C. Gao, F. Duewer, and X.-D. Xiang, Appl. Phys. Lett. 75, 3005 (1999).
78
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Chapter VI:
Measurement of superconductors
We explored the low temperature capability of a scanning-tip microwave
near-field microscope to study superconductors.1 The shift in the quality factor of
the microscope resonator can be used to obtain the temperature dependence of
the surface resistance of a local region under the tip. Patterned YBa 2 CU 3 O 7 -2 *
films were scanned at various temperatures and surface resistance mapping was
performed with high spatial resolution at 1.2 GHz. Superconducting transitions at
different positions on a film can be detected. Edge-region defects in wet-etched
patterns were observed and were shown to be nonsuperconducting at microwave
frequencies at 80 K.
High spatial resolution and high sensitivity imaging tools with different
electrical properties have become increasingly important. Over the years, a
variety of scanning probe techniques have been developed to probe local
variations of properties and the structures of materials.2 Microwave near-field
microscopes, in particular, are invaluable for nondestructive electrical impedance
measurements. We have previously reported on the development of a scanned
evanescent microwave probe (SEMP)3 and its application to submicron
microwave impedance microscopy of dielectric materials.4 The microscope
utilizes evanescent waves from a metal tip similar to ones used in scanning
tunneling microscopes (STM) to obtain microwave information with submicron
spatial resolution (A/106 ). W e have extended our capability to low temperature
measurements.
79
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The low temperature microscope is useful for studying temperature
dependent microwave properties of a variety of materials. Thin film microwave
devices made of high Tc superconductors such as YBa2 Cu3 0 7 .2 x (YBCO) offer
numerous advantages over those made with conventional materials because of
their low surface resistance and high current carrying capability. Early high Tc
microwave devices have displayed low insertion loss and higher performances
compared to their counterparts.5 Unfortunately, YBCO films have also exhibited
nonlinear effects at moderate power levels. Such effects include power
dependence of surface resistance Rs , intermodulation distortions, and harmonic
generation.5 While there may be nonlinear effects stemming from the intrinsic
nature of the superconductor,6 it is generally believed that most of the effects
arise from extrinsic factors such as the existence of Josephson weak links and
other defects in the films. Patterning processes for device fabrication might also
contribute to this. Identifying the origin of the nonlinearity is a formidable
challenge and an important direction of research in pursuing the microwave
applications of high Tcsuperconductors.
Properties of YBCO films are highly sensitive to slight changes in
stoichiometry. A small variation in oxygen content can lead to a drastic change in
Tc. Microwave device applications require very large area uniformity in thin films.
Thus it is of crucial importance to study large area homogeneity of microwave
properties such as Rs of thin films. While most of the standard measurement
techniques can give Rs values of the films with reasonable accuracy, the values
obtained are averaged over the entire area of the films.5 Since local off-
80
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stoichiometric regions can occur at very small scales, it is highly desirable to be
able to "map out” microwave properties of the film with high spatial resolution.
Previous experimental attempts have had limited spatial resolution.7 It is
necessary to utilize a microwave measurement technique that can resolve
features at very small scales (submicron) as well as scan a very large area
(inches). The microwave measurement is performed in a transmission mode, and
f r and Q are obtained by fitting a resonance cun/e and are recorded as the tip is
scanned over a sample surface. Thus, images of /rand Q are obtained
simultaneously. Although the shift in
frdirectly measures the dielectric constant of
dielectric materials, it is not very sensitive to the physical property differences in
metals. However, the Q of the resonator changes very sensitively as a function of
the surface resistance of the film under scan, and the tip of the microscope can
be positioned at different locations of a film to map the local surface resistance.
The samples scanned here were high quality c-axis oriented YBa2 Cu3 0 7 -2 *films
of 3000
Aon low loss substrates of LaAI
0 3
or NdGa0 3 with Tc of 90 K measured
by dc resistive transitions. Films were patterned into arrays of 100 pm by 100 pm
square patches photolithographically using dilute nitric acid. Samples were then
covered with
2000 Aof Si
0 2
to avoid direct electrical contact of the tip with
conducting samples. In order to obtain a reference surface resistance for
comparison, a Cu film -3 000
Athick was also patterned into similar square
arrays and covered with Si02. The samples are mounted over a cantilever spring
made of thin tungsten tape. The estimated force of the spring is 20 pN, and this
provides a “soft” constant contact of the tip with the surface. This configuration
81
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keeps the tip at a constant distance, determined by the thickness of the S i0 2
layer from the superconductor, and it assures that the measurement is mostly
insensitive to tip-surface distance changes which would be caused by the
topographic features of the surface. The tip radius is 1000
Aprior to contact, but
it increases to an order of pm after contact. The spatial resolution of the
microscope is currently limited by the tip radius. With careful tip preparation and
adjustment of the tip-sample distance, submicron features have been clearly
resolved using a similar ambient room temperature measurement setup. But in
the present experiment, we did not make attempts to image submicron features.
For the low temperature setup, the sample holder is top loaded to a resonator
that faces upward. The setup is mounted inside a cryogen bath LN2 or LHe
which is located inside a LN2 Dewar, separated by a vacuum jacket. Cooling is
via exchange He gas. (Fig. 6-1)
82
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• 2 5 m m x 25 m m
scan range
• 4 .2 K to 300 K
• Easy sam ple
exchange
• R T X - Y scan
• C ontact scanning
m ode
Fig. 6-11 Implementation to low temperatures
The resonator is mechanically connected from the bottom of the cryostat by a
stainless steel rod to a room temperature XY-scanning stage with which up to
approximately a 1-in. diameter area can be scanned.
83
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e
3
t
flj
> 0.6
o.:
/«— s
O'
75
Superconducting
80
85
T (K )
90
95
100
JMiiSL
Fig. 6-21 Q images of a patterned YBa 2 Cu3 0 7 -2 * film at room
temperature.
Seen is a large area scan showing multiple patterned
squares. The inset is a detail of one of the squares Q scale
on the left. Local variations in Q on the order of several pm
can be seen.
Figure 6-2a is the Q plot of a sample on LaAIC>3 at room temperature. A clear
difference between the substrate and the YBCO film is evident. The inset of Fig.
6-2 is a detail of the scan showing one of the patterned squares. There are
84
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visible local variations of the size on the order of several |im in Q -slightly lower
reflecting inhomogeneous Rs ■The largest changes in Q correspond to a
variation in Rs by a factor of 1 .5. We can place the tip over a fixed position and
monitor the temperature dependence of f0 and Q. Figure 6-2b shows the data
from one such run while the temperature was varied from 78 to 98 K with the tip
over a fixed region of a square. There is little change observed in fQbecause it
depends primarily on the reflection of the electromagnetic field, which does not
vary significantly for different conducting materials. On the other hand, a drastic
change near Tc in Q is observed which represents the change in microwave
absorption in the film. The broadened transition is presumably due to processing
induced damage in this region. This demonstrates that we can use the SEMM to
detect superconducting transitions of different local positions in a film at
microwave frequencies. Below 80 K, the magnitude of the relative Rs becomes
comparable to that of the measurement resolution. This indicates that we are
approaching the sensitivity limit of the current setup. In order to improve the
sensitivity, we are planning to implement a superconducting resonator that will
have a much higher Q value than that of the present Cu resonator. We have also
measured a patterned Cu film of the same thickness for comparison. Average Rs
values of YBCO films at 1.2 GHz are found to be two to three times greater than
the average Rs values of Cu at 300 K and slightly lower than those of Cu at 80 K.
It is convenient to plot
1 /Q
ybco
*
1 / Q j n io a d e d
= A(1/Q) since this quantity is
proportional to Rs. . Figures 6-2a and b show A(1/Q) images of one patterned
square on LaAI0 3 taken at room temperature and at 80 K, respectively, with the
85
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same contrast scale. In sharp contrast to the image taken at room temperature,
the 80 K image clearly shows regions near the physical edge of the square ~5
pm where Rs is significantly higher (by a factor of >10) than the inner regions
(darker regions indicate higher Rs). These pronounced higher Rs or lower Q
regions are only seen in patterned YBCO films near this temperature and are
often not continuous around the edge. We believe this edge effect is neither due
to an artifact at patterned edges nor due to current flowing at the edge. Rather,
this is an indication that these regions are not superconducting at 80 K measured
at 1.2 GHz. This is likely caused by the wet-etching process resulting in oxygen
off-stoichiometric regions around the film edges.
The bright white "spots” seen in Fig. 6-2A arise from regions where the
S i0 2 layer has peeled off and the tip has come into direct contact with the YBCO
film. This results in substantial displacement of the resonance peak, and thus the
image of the region is out of scale. To eliminate these imaging defects, we are
searching for a better low loss material to be used as the cover layer.
In conclusion, we have demonstrated surface resistance mapping of YBCO films
using a low temperature SEMP. Superconducting transitions and surface
resistance of local regions can be probed with high spatial resolution.
Since microwave devices carry rf currents at the edges, a further study of
the state of patterned edges should be helpful in evaluating the performance of
the devices as well as investigating the cause of the nonlinearity.
86
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References
1. I. Takeuchi, T. Wei, F. Duewer, Y.-K. Yoo, and X.-D. Xiang, Appl. Phys. Lett.
71,2026 (1997).
2. See, for example, Scanning Tunneling Microscopy //I, edited by R.
Wiesendanger and H.-J. Guntherodt -Springer, Berlin, 1993; Scanned Probe
Microscopy, edited by H. Kumar Wickramasinghe, AIP Conf. Proc. 241 (1991).
3. T. Wei, X.-D. Xiang, W. G. Wallace-Freedman, and P. G. Schultz, Appl. Phys.
Lett.
6 8
, 3506 (1996).
4. Y. Lu, T. Wei, F. Duewer, Y. Lu, N.-B. Ming, P. G. Schultz, and X.-D. Xiang,
Science 276, 2004 (1997).
5. Z. Y. Shen, High Temperature Superconducting Microwave Circuits, Artech
House, Dedham, MA, 1994.
6
. T. Dahm and D. J. Scalapino, Appl. Phys. Lett. 69, 4248 (1996).
7. J. Gallop, L. Hao, and F. Abbas unpublished; M. Golosovsky and D. Davidov,
Appl. Phys. Lett.
8
6 8
, 1579 (1996).
. C. P. Vlahacos, R. C. Black, S. M. Anlage, A. Amar, and F. C. Wellstood,
Appl. Phys. Lett. 69, 3272 (1996).
87
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Chapter VII
Measurement of manaanites
Manganese oxides belong to a class of highly correlated electronic
systems with a strong interplay between electronic and magnetic order. They are
of technological interest owing to the large magnetoresistance and/or spin
polarization observed in some compositions, and scientific interest owing to their
complex behavior as a function of composition and temperature. For example,
upon variation of the carrier density, Lai-xCaxMn0 3 passes through a variety of
magnetically ordered phases at low temperatures. These states include:
F: F-type ferromagnetism, in which neighboring manganese ion spins are
aligned
A: A-type antiferromagnetism, in which neighboring manganese ion spins
are ferromagnetic in-plane and anti-ferromagnetic out of plane
C: C-type antiferromagnetism, in which neighboring manganese ion spins
are anti-ferromagnetic in-plane and ferromagnetic out of plane
G: G-type antiferromagnetism, in which neighboring manganese ion spins
are anti-ferromagnetic in three dimensions
Furthermore, some recent studies suggest highly anisotropic patterns of
charge carriers in these manganese oxides systems. Charges tend to segregate
into a pattern formed by a pair of Mn^Oe stripes separated periodically by stripes
of
Mn4+06
octahedra. 1,2
Also,
metallic
(ferromagnetic)
and
insulating
(antiferromagnetic or charge-ordered) domains co-exist in a large scale, namely
phase separati'j*3. Such phenomena are of general interest to the physics of
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condensed matter. The self-organization of charge carriers into highly anisotropic
patterns is not restricted to manganites. For example, electrons in high T c
cuprates (strongly correlated two-dimensional electronic systems) self-organize
into one dimensional charge stripes embedded in a co-existing magnetic
structure (spin stripes): metallic in one direction and insulating in the other.4 ' 8
Lai.xCaxMn03
To explore the correlation between different physical phases, we studied a
continuous phase diagram (CPD) of Lai.xCaxM n 0 3 in epitaxial thin film form
where x is changed continuously from
0
to
1 .9
Detailed studies of the
compositional dependence for these materials typically require difficult and timeconsuming single crystal growth. Probing the manganites through a continuous
doping ’’window” allows us to tune and study the complex correlation between
physical properties as function of band-filling in a precisely controlled manner. At
room temperature, we observed clear phase boundaries of electronic nature,
which coincide with those of long-range magnetic order at low temperatures.
Furthermore, a narrow conducting singular phase region within a non-magnetic
semi-conducting phase region was identified for the first time. W e believe that
these boundaries arise from remnant orbital order. Remnant short-range orbital
order can change the resistivity of the material by changing the overlap of the
manganese d orbital and the oxygen p orbital. Earlier studies of the Lav
xCaxM n0 3 observed a monotonic dependence of the resistivity on doping, x. 10
However, these studies were conducted on ceramic samples rather than on
oriented thin filmr and would be unlikely to detect resistivity variations due to
89
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anisotropy in the thin film samples.
W e fabricated the CPD of Lai.xCaxMn0 3 on (100) LaAI0
3
. As illustrated in
Fig 7-1 , the CPD was synthesized from gradient depositions of three precursors
using a high precision in situ linear shutter system (Fig. 7-1) and subsequent ex
situ post-annealing.
90
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Automated In Situ Shutter System
8 Target Carousel
......................
Homogeneous mixing
Crystalline C P D
Gradient depositions o f amorphous precursors
o f three precursors
Fig. 7-19 High precision in situ shutters in a pulsed laser
deposition system.
Eight-target carousel allows
uninterrupted depositions without breaking the vacuum.
Vertical shutters are used to define the width of each phase
strip on a substrate while precise gradient profiles of three
precursors are deposited with horizontal shutters moving
across the phase strip at constant speed.
First, a gradient of La203 is deposited at the bottom.
Moving the
horizontal shutter at a constant speed from one edge to the other edge of the
substrate defined by two vertical shutters during deposition (Fig 7-1) generated a
linear thickness gradient (1226
Ato 0 A).
Afterwards, gradients of Mn3 0 4 (774
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A
to 0 A) and CaMnOa (OA to 1537 A) were deposited as a middle and top layer
similarly. The precursor films in this study were deposited at room temperature
with a high vacuum (~10 ‘ 7 torr) pulsed laser deposition (PLD) system.
The
forward expanding plume in high vacuum coupled with scanning of laser beam
across the
2
” x 2 ” targets during deposition results in a thickness uniformity of
better than 1.5% over a 15 mm x 15 mm area. This ensures accuracy in local
stoichiometry easily controlled by the shutter. Following deposition, the sample
was annealed at
2 0 0
hours followed by
2
°C for several days before it was annealed at 400°C for 30
hours sintering at 1000°C.
Low temperature annealing is
necessary to allow homogeneous mixing of precursors into an amorphous
intermediate before crystallization at higher temperatures. We found that the
lateral diffusion is insignificant at tens of micron scale.
By careful choice of
annealing conditions, epitaxial growth can be obtained as indicated by x-ray
diffraction studies shown in Fig. 7-2.
92
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i—
r
(b)
(101) planes o f La0 7Ca0 3M n O 3
FW HM 0.2°
(101) planes o f L a A 1 0 3
20
40
60
80
0
50
20 (degrees)
100
150
200
250
300 350
4> (degrees)
Fig. 7-29The 0/20 XRD pattern and the <|>-scan of the (101)
plane of a La o.7Cao.^n 0 3 (LCMO) thin film made from
three precursors of La 2 O3 , Mn 3 0 4, and CaMnO 3 on (001)
LaAI03.
We have confirmed similar epitaxial growth in many different compounds
with different hole doping levels.
We also have confirmed the entire CPD
compositions have perovskite crystal structure.
To map the electronic properties
of the CPDs, we chose two very
93
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different energy (by 106) scale probes: visible light and microwave. Fig. 7-3a is
the CCD visible color photograph (optical reflection image) of the film under white
light.
(a )
300
*
e
a
s
250
200
150
%
3.5
3.0
X
2.5
S
2.or
<
-A U f^ x lO 3
(b )
1J
Fig. 7-39(a) The CPD of Lai.*Ca*Mn0 3 and the electronic
phase diagram from single crystal study [19-21]. The
various states are: paramagnetic insulator (PI),
ferromagnetic insulator (FI), ferromagnetic metal (FM),
charge ordered insulator (COI), anti-ferromagnetic insulator
(AFI). (b) Line scan profiles of microwave loss and
frequency shift for Lai.jCajMnOa film as a function of Ca
concentration x measured with a scanning evanescent
microwave microscope at room temperature.
j
increases with increasing microwave loss and
ft
increases with increasing tip-sample capacitance.
The colors roughly indicate the electronic bandwidths of the compounds.
94
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In order to map the electrical impedance at microwave frequencies, we used a
scanning evanescent microwave probe operating at 2.2 GHz . 11' 1 3
Briefly, the
instrument measures the complex impedance of the probe by monitoring the
changes in resonant frequency (fr) and quality factor (Q) of the cavity.
The
proximity of a sample to the tip changes the complex impedance of the probe,
changing fr and Q.
The shift in fr is governed, for bulk samples (qualitatively
correct for thin film case involved here), by: 11
Af r _
_
y
b tn
fr
+ < *,
where an = 1+ — -
, and b =
8
where g is the tip-sample gap, Ro is the tip radius, e0 is the permittivity of free
space and er and e, are the real and imaginary parts, respectively, of the
complex dielectric constant of the sample at fr. For highly insulating samples
Er), the a(1/Q) increases while fr decreases with increasing e*.
conductive samples
4710
(ei
(5
<
For highly
> £r), both a ( 1 /Q) and fr decrease with increasing & =
/(0 . Fig. 7-3b maps the microwave loss A(1/Q) and change in fr as a function
of composition.
To map the magnetic phases at low temperature, we employed a
scanning superconducting quantum interference device microscope (SSQM).14
A miniature SQUID ring with a diameter of 10 pm was scanned over the sample
surface. In this geometry, the SSQM senses a local magnetic field perpendicular
95
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to the surface, 8?. Since the magnetic moments of the ferromagnetic phase are
in plane, the SSQM is limited to the measurement of the field leaked from the
magnetic domains in the material. An oscillating signal indicates the presence of
magnetic domains. The periodicity (or width d) is correlated with the domain size
while the amplitude (ABZ) is related to the presence of ferromagnetic ordering.
The SSQM measurements were performed at 3-7 K without an external magnetic
field. Fig. 7-4 is a line scan of \Bzl taken with the SSQM along x.
96
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i n
40 p-n
(a)
r-,
F
30
3
CD
<
I
m
1 1 1 1 1 i i
| 111
i i 1 1 i 11 i i » 1 1 1 1 1 1
_
"
20 —
10
0
(b)
p n i
3 i
k
IB
.39.30 *.649
_09_30_:
09.30..17.3G
□9 30 :s 43
* * *
___ -
Fig. 7-4® (a) Line scan profiles of perpendicular magnetic field
above Lai.*Ca*Mn03 film as a function of Ca concentration x
measured with a scanning SQUID microscope at 3 K without
external magnetic field, (b) Magnetic domain structures of Lai*Ca*Mn03 film at 7 K. A color bar indicates the measured
perpendicular magnetic field, Bz. Scanned areas are 300 * 300
pm2 in all images. The Ca content x increases linearly with the
horizontal distance in each image. Domain structures of
ferromagnetic films with in-plane magnetization. In the present
thin film sample, magnetic moments lie in plane. Thus, SSQM
looks at the magnetic field flowing in or out of domain boundaries.
Red and purple regions in the SSQM images correspond to the
domain boundaries. Nominal value of xfrom left to right in each
image and AS? are: (A) 0.078-0.099 and 52.8 pT, (B) 0.330-0.351
and 59.3 pT, (C) 0.449-0.470 and 48.3 pT, (D) 0.868-0.888 and
17.2 pT, respectively.
The optical and microwave properties at room temperature (well above the
onset of magnetic order) are clearly correlated with the magnetic properties at
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
low temperature. The complexity of the various magnetic and electronic phases
can be understood naturally as arising from different orbital ordered states in this
system.15 Maezono modeled the orbital order of Lai.xCaxM n03 as a function of x
using a generalized Hubbard model.16 They find four types of spin order:
G-type: x - 0, x > 0.9 (3-dimensional antiferromagnetic ordering)
A-type: 0 < x < 0.1, 0.25 < x < 0.45
(ferromagnetic in x-y plane,
antiferromagnetic in the z direction)
F-type: 0.1 < x < 0.25 (ferromagnetic)
C-type: 0.45 < x < 0.9 (ferromagnetic in the z direction, antiferromagnetic
in the x-y plane)
The experimental data are summarized in Fig. 7*3. Starting at x = 0, the
material
is
insulating.
This
behavior
is
consistent
antiferromagnetic order suggested by Maezono et al.
with
the
G-type
As doping x is slightly
increased from 0, there is an abrupt increase in conductivity which indicates the
predicted transition from G-type order to A-type order.
Such increase in
conductivity compared to G-type region (x - 0) is expected since A-type ordering
is ferromagnetic in-plane.
In 0 < x < 0.2, d and ABz are relatively large and
increasing, which is consistent with the FI phase found in previous single crystal
study.2
A broad dip in A(1/Q) centered at x - 0.15 indicates increase in
conductivity and presumably beginning of transition from A-type to F-type orbital
ordering. Since F-type ordering can conduct electrons both in-plane and out of
plane, it attains higher conductivity than A-type ordering. For 0.2 < x < 0.5, dand
ABz are relatively large but decreasing, which is also consistent with results from
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the single crystal study suggesting ferromagnetic phase approaching charge
ordered insulating phase. A second broad peak in A(1/Q) centered at x - 0.35
corresponds to a decrease in conductivity indicating a return to A-type ordering.
At x = 0.5, A 0 ; is substantially suppressed and an abrupt change is visible in the
optical image as well.
These abrupt changes corresponds closely to the 1/2
charge ordering in single crystal studies.2 For x > 0.5, bJBz almost recovers its
value while d decreases monotonically. We observe a broad plateau in A(1/Q)
centered at x~ 0.75, which we may associate to a transition to a new orbital
ordering, presumably C-type.
At x = 0.67, we observed a sharp boundary (white colored) (Fig. 7-3a)
which, we believe, is the onset of the 2/3 Jahn-Teller type stripe phase.3 For x >
0.82 (~ 4/5), A(1/Q) decreases while fr decreases, indicating an increase in
conductivity. Simultaneously, ABZ reaches a minimum. Around x = 0.85, ABZ is
abruptly reduced to -1 0 |iT (note that the observed ABZ for x = 0.88 is one order
of magnitude larger than the noise level). At x = 0.93, the optical image changes
abruptly.
The changes in A(1/Q) and fr indicate an abrupt transition to an
insulator. This change strongly suggests a transition to G-type antiferromagnetic
ordering.
For x = 0.97, a narrow highly conducting semi-conducting (determined by
the temperature dependence of the conductivity) region appears while adjacent
regions are highly insulating. Such a drastic change in conductivity over a small
range in doping has not been observed before in doped Mott insulators. This
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phenomenon may be common to many highly correlated systems. To study this
type of phenomenon, CPDs are apparently a powerful tool since they allow the
precise exploration of minute ranges of composition. The transport property of
this region has been measured by 4-point probe. Fig. 7-5 shows the resistivity
versus temperature.
100
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10 1
10 1
0
50
100
150
200
250
300
Temperature (°K )
Fig. 7-59 Resistivity of conductive strip vs temperature
As temperature decreases, the conductivity initially decreases slowly and
then rapidly below 100K. The room temperature resistivity is < 1 Ci-cm and no
magnetoresistance was observed.
101
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Orbital ordering results from the strong spin-orbital coupling in these
systems. Simply, the coupling of neighboring orbitals depends on a combination
of direct coulomb and exchange interactions. The orbital coupling determines
the overlap of neighboring orbitals and thereby determines the exchange
interactions between neighboring spins. The
dorbitals of the manganese ions
are split into eg and t2 g orbitals. The two eg orbitals, in the absence of a JahnTeller distortion, are degenerate. The system can be represented by a simplified
Hamiltonian of the form:16
where T is a pseudo-spin operator denoting the occupancy of the eg orbitals, T =
1/2
for di2 yl and T =
- 1/2
for
r, , J ij is the exchange interaction between
neighboring manganese ions, and K{j is the interaction between neighboring
orbitals. The system has been extensively modeled and a variety of orbital
ordered phases are shown to arise. Typically, the orbital ordered state is stable
at temperatures greater than those for magnetic order. The modification of the
exchange interaction induced in these orbital ordered phases then determines
the spin ordering of the material. X-ray and optical methods have observed
these phases.17'21
For Lai.xCaxMn0
3
, extensive modeling has been performed using a mean-
field approximation by Maezono.16
In addition to the orbital term, they also
include a Jahn-Teller distortion, which is needed to stabilize some of the
observed states.
They find, depending on the bandwidth and doping of the
102
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system, a wide variety of orbital ordered states.
Depending on the doping and
bandwidth of these systems, several different orbital ordered states can arise
such as,
d fl_yl
2
: Generally associated with F-type ferromagnetism
d t, v, : Generally associated with A-type antiferromagnetism
d^_ r. : Generally associated with C-type antiferromagnetism
d [2 yl
alternating
with
d^ , r2:
Associated
either
with
F-type
ferromagnetism or G-type antiferromagnetism.
The coincidence between electronic phase boundaries at room temperature
and the long-range order phase boundaries at low temperatures is clearly
revealed by studies of three different probes on CPD and previous single crystal
studies of phase transitions in Lai.xCaxMn0 3 2 2 ' 2 3 (Fig. 7-3a). This correlation
between the room temperature resistivity and the low temperature magnetic
behavior is consistent with the observation of remnant orbital states. In addition,
the CPD of Lai.xCaxMn0 3 revealed an unexpected narrow semiconducting region
that has not been previously observed.
required.
Further study of this narrow region is
If the (La, Ca) ions are appreciably ordered, hopping may be
enhanced if the local environment of some manganese ions is ordered in such a
way to match their energy levels, i.e. energy splitting between adjacent Mn3* and
Mn4+ ions may approach zero, allowing increased hopping.
happen in the presence of charge ordering.
This may also
Here, the Mn3* and Mn4* ions
provide the local distortions. For compositions outside the conductive strip, the
103
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ordering would be broken and the material would rapidly become insulating.
Another possibility is that two directions of charge ordering compete with each
other at the conductive strip and one dominates on each side.
Two-dimensional images shown in Fig. 7-4b of selected compositions
indicate the presence of phase separation in the compounds. Around x = 0.1
(Fig. 7-4b(A)), the domain size is as large as 50 pm and domains have almost
identical widths, implying that a homogeneous ferromagnetic state is established
in this region.
At x ~ 0.33 (Fig. 7-4b (B)), smaller domains dominate.
The
domain width is widely distributed in Fig. 7-4b (B), compared to Fig. 7-4b (A). In
this case, the charge-ordered phase is believed to be embedded around the
green region of the image, where magnetic field is weak, resulting in the
appearance of domains with smaller Bz contrast. At x around 0.46 (Fig. 7-4b
(C)), the domains become smaller, and more interestingly, non-magnetic
domains of about 100 pm coexist, as can be seen in the right hand side of the
image.
This non-magnetic domain may represent AFI (CO) or, at least, AFI-
dominant region, which includes FM droplets smaller than the spatial resolution
of SSQM, -5 pm. It is noted that the size of the non-magnetic region seen from
Fig. 7-4b (C) is much larger than the length scale of phase separation previously
reported in this compound.23 Such a large non-magnetic region has not been
seen for x greater than 0.5, however ferromagnetic ordering seems to be in a
shorter range as x increases, considering the decrease of the domain size with
increasing x.
Nevertheless, there are several regions with weak Bz contrast,
suggesting coexistence of a charge ordered phase.
For 0.51 < x< 0.88, the
104
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domain size gradually decreases.
Fig. 7-4b (D) includes a boundary at the
center of the image, across which the amplitude of Bz is substantially
suppressed. Even in the region for x > 0.88, however, finer domains are still
visible. This may originate from slightly canted magnetic moments in the AFI
phase.
In a similar system of high Tc superconductors, one also finds a rich phase
diagram versus doping and temperature. It would be interesting to see whether
such
phenomena occur in the
high temperature superconductors. The
continuous nature of CPD samples allows comprehensive measurements of the
compositional
dependence
of
electronic
properties.
Unlike
previous
measurements of discrete single crystals, this method can indicate and study the
existence of narrow 'singular1phases.
EiVxCaxMnOs
Understanding of complex systems is one of the major challenges of
science for 21st century. In condensed matter physics, highly correlated
electronic systems, such as doped Mott insulators of transition metal oxides, are
such complex systems. These systems exhibit rich, fascinating yet puzzling
phenomena, including high temperature superconductivity, colossal
magnetoresistivity. Recent studies suggest that in these systems collective
electronic phenomena prevail and arise from long range Coulomb interaction and
magnetic effects.24*26 The qualitative behavior of these systems is strongly
dependent on charge filling (doping) and lattice constant. The ability to perform
experiments that will provide global and systematic information on complex
105
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systems is crucial for further theoretical understanding. W e describe here an
experimental approach - mapping continuous phase diagrams (CPDs) in the
epitaxial thin film form, to study phase diagrams (of physical properties) of
condensed matter systems. We discovered surprising evidences that suggest
phase boundaries of electronic origin at room temperature in manganese oxides,
a class of doped Mott insulators exhibiting colossal magnetoresistive effect.
Numerous studies have been conducted in the past to understand the
relationship between charge dopant level n (or ionic radii) and physical properties
of complex systems, concentrated on a variety of long range order
thermodynamic phase transitions at low temperatures. Two of the well-known
studies were conducted in high temperature superconducting cuprates27 and
colossal magnetoresistive manganites28'30. These phase diagram studies are
very time consuming and deficient because they usually involve growing single
crystal samples of discrete (rather than continuous) compositions and
subsequently measuring physical properties at various temperatures. CPD
method allows the continuous mapping of physical properties versus
composition, which is much more time-efficient, thorough and systematic than
the conventional approach.
The stability of the perovskite manganites against chemical substitutions
provides an opportunity to study the effect of varying the parameters of a highly
correlated electronic/magnetic system. In this study we control the charge filling
range n of Mott insulator by continuous charge doping over the entire range and
hopping integral t through the substrate-induced anisotropic strain effect in
106
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epitaxial films and the average ionic radius of A site.31 Our initial study
uncovered several extremely narrow (in composition) phases with dramatically
different electronic properties. These narrow, unexpected phases would be
difficult to be identified using conventional methods. Such global phase diagram
provides a new and systematic way to study complex materials.
CPDs of perovskite manganites (REi.xAxMn0 3 ) in epitaxial thin-film form were
made, where RE = Eu, Gd, Tb, Er, Tm, Yb, and A = Ca, Sr, Ba and x is varied
continuously from 0 to 1 (Fig. 7-6).
107
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a
El iQ 9501
O K 110)
Gd i0 938)
Tb iO 923i
Er rO 831)
Tm :0 37)
Trr ;0 37
tb (0 953)
NdGaO
a - 5 42A
b =5 49A
Eu
>3d
On 1100)
Tb
i
SrTiO,
Er
Tm
a - 3 91A
Vb
Ca (0.99)
/
Ba (1.321
Sr(1.12»
"
Fig. 7-631 The room temperature CCD color photograph
(photo-reflection image) of 36 CPD’s on two different
substrates under white light (4.2~7.8x1014 Hz). The
parentheses indicate the crystal ionic radii of the elements
[29]. The commensurate doping points of singular phases
are denoted. The effective unit cell of (110) NdGa03 has
a’= b’= 3.86 A (a’= b’ = £ ± 3 ) ; .
2
Different phase diagrams were made on six 15 mm by 15 mm substrates of two
different single crystals, (100) SrTi03, and (110) NdGa03. As illustrated in Fig 71a, a gradient of RE oxide is deposited at the bottom where Eu20 3, Gd20 3,
108
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Tb4 0
7
, Er20 3, Tm 2 0
3
, and Yb20 3 were used for the targets. The precursor films
in this study were deposited at room temperature with a high vacuum (~10'7 torr)
pulsed laser deposition (PLD) system. The forward expanding plume in high
vacuum coupled with scanning of laser beam across the 2" x 2" targets during
deposition results in a deposition thickness uniformity of better than 1.5% over a
15 mm x 15 mm area, therefore ensures the accuracy in stoichiometry (easily
controlled by shutter). Following the deposition, the sample was annealed at
200°C for several days before it was annealed at 400°C for 30 hours followed by
2 hours sintering at 1000°C. Low temperature annealing is necessary to allow
homogeneous mixing of precursors into an amorphous intermediate before
crystallization at higher temperatures.
Each phase diagram was synthesized from gradient depositions of three
precursors using a high precision in situ linear shutter system (Fig. 7*1) and
subsequent
ex situ post-annealing.
Compared to the previously used
composition spread method by co-sputtering of multiple targets32'34, this
technique can easily generate precisely controlled stoichiometric profiles within a
small area.
This advantage is crucial when various single crystal substrates
need to be used (as in this study) for high quality epitaxial film growth. Despite
the fact that the crystalline compounds are formed ex situ rather than in situ, we
found that appropriate post-annealing processes can give rise to high quality
epitaxial thin films.
Fig. 7-2a is a 6/26 XRD pattern for Lao.7Ca0.3M n03 on LAO substrate
indicating the film is (100) oriented (in order to study the crystalline quality of the
109
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samples, we made individual samples of various compositions selected from the
phase diagrams under identical fabrication and processing conditions). Even in
logarithmic scale, we can hardly see any other impurity phase. Fig. 7-2b is <|>scans of the (101) planes indicating that the film is in plane aligned with the
substrate.
We have confirmed similar epitaxial growth in many different
compounds with different hole doping levels.
To map the electronic properties of the CPDs, we choose two very
different energy (by 106) scale probes: visible light and microwave. Visible
photographs, optical-reflection image under white lights (4.2~7.5x1014 Hz), of
CPDs were obtained as a function of temperature (Fig. 7-6).
110
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iA
>.
</)
c
2 \
H
JB
ic
--------------1
---------------.... —...... i1-------------------------- V
j
--------- 77 K
430 K i
430 K
X
Fig. y-^Th e room temperature CCD photograph of
Tm1.xCaxMn03 on SrTi03 substrate under white light.
Three doping regions are selected for blue light
(6.2-7.8X1014 H z ) photo-reflection at three different
temperatures.
In order to map the electrical impedance at low frequencies, we used a
scanning evanescent microwave probe operating at 2.2 GHz.36,37 The measured
shift in fr and Q can then be used to calculate the real and imaginary parts of
111
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complex dielectric constant at fr, er and ev(= 4no/o> for conducting samples) (Fig.
7-7 and ref. 38).
3.1
o 2.9
2.7
< 23
23
1
10
100
-i
-i
Conductivity(Q -cm )
Fig. 7-731 Microwave loss (A(1/Q)) versus DC conductivity
(Q‘1-cnv1). Solid circles are obtained from the measured
microwave and corresponding DC responses. The line is the
theoretical fit.
The details of the instrument and quantitative analysis methods were
published previously.37
112
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Due to the large size of data and limited space, we present here detailed
room temperature microwave impedance of a most intriguing CPD of ErixCaxMn0 3 +d (Fig. 7-8) and room temperature optical images for the complete
sets of CPDs for comparison (Fig. 7-6).
■IA
3.0
i -1.15
-u
10
0.93
X 2.0
-135
1.0
-1.75
0.0
0.0
03
0.4
0.6
0.8
1.0
Composition(x)
Fig. 7-931 The microwave response of the Eri.xCa xM n 03 CPD on NdGa0 3 .
Microwave loss (A(1/Q)) and frequency shift (Afres/fres) are measured versus
composition (x). The dashed lines are used to indicate phase boundaries. The
inset is a zoomed portion near the commensurate singularity.
First, we observed many clear boundaries in both optical and microwave
impedance images (dashed line 1, 2 ,4 , 5, 6, 7 from right in Fig. 7-8). Microwave
loss peaks at these boundaries. X-ray data shows no evidence of different
structural phases beside perovskite throughout CPDs. The existence of these
boundaries suggests complex electronic orderings (with fundamentally different
113
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physical properties) occur as a function of x.
Second, a very narrow insulating (or more accurately semiconducting)
strip within a highly conducting region (appearing as grayish blue strip within
black band in Fig. 7-6) was observed at commensurate charge filling point x =
7/8. The strip is so narrow in phase-width that the difficulty is apparent in finding
it using the conventional practice of studying samples of discrete composition
points. We believe that this commensurate singularity point is related to the
static charge (or spin) orderings observed in recent experimental and theoretical
studies in both cuprates24-26,39-46 and manganites47-51 We expect to see small
structural anomaly associated with the singularity due to the strong coupling of
charge ordering with the lattice at the commensurate point.
In Fig. 7-6 different doping ranges are selected for temperature dependent
comparisons of blue light (6.2~7.8x1014 Hz) reflection images. In strip A, as the
temperature is lowered, the suppression of intensity at the middle portion is clear.
This tends to sharpen the phase boundary toward the right side (less doping) of
the dark conducting phase. The conducting phase around x = 1/2 weakens at
low temperature. In other manganite systems such as Ndi.xSrxMn0 3 , a similar
phase near the Vfe doping point was found to have two or more intrinsic
competing phases
(phase separation) with one phase dominant as the
temperature is lowered. Since the signal intensity is proportional to the amount
of scattered light, the disappearance of this dip at low temperature indicates that
a homogeneous phase dominates at 77 K.
In strip B, as the temperature is
raised, there is clear evidence of broader phase boundaries (blurring of the
114
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phase), indicating the effect of thermal fluctuation. Increased temperature allows
the co-existence of different phases over a broader range in composition. It also
shows the high-energy scale of the electronic phases observed. In strip C, the
boundary (shown as a peak in intensity) disappears in high temperature.
Finally, the gradual change in ionic radii also induces unexpected abrupt
changes in phase patterns. This effect can be observed in substitutions of both
rare earth and different divalent alkali-earth elements. The effect of substrate
induced stress can also be observed. W e see relatively smooth phase patterns
and absence of sharp transition boundaries in the Sr doped system, and this
trend furthers in the Ba doped system.
The clear effect of the small t or
bandwidth and associated localization of electron waves is observed by rich and
diverse phase patterns in the Ca doped system.
The doped Mott insulators are strongly correlated electronic system, and
the doped carriers tend to self-organize in highly anisotropic pattern (stripe phase
for example). It is believed that the dynamical nature of these self-organizations
in doped Mott insulators plays a crucial role in the origins of some of heavily
researched phenomena, namely colossal magnetoresistance in manganites and
high Tc superconductivity in cuprates. A possible scenario for the current
observation of phase boundaries can be the electronic self-organizations of
different types. Note that around x = 1/2 and 7/8, we would expect
commensurate locking of electrons to the lattice, which would be an insulator.
Instead, we have an increase in conductivity (both low and high frequency) which
signifies the delocalization of electrons.
115
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If we assume the commensurate singularity at 7/8 (as such from
ErvxCaxMnOs) is due to commensurate locking of charge orderings to the lattice,
the existence of highly conducting adjacent regions, presumably with
incommensurate charge orderings, then suggests a preliminary evidence of
electronic phase transitions in this doping range, possibly similar to electronic
liquid crystal phases recently proposed by Kivelson, Fradkin and Emery.24 The
smeared out critical divergence in microwave loss at the boundaries observed in
this study also leads to this possibility. The thermodynamic nature of the
transition is clearly evidenced by the temperature dependent broadening or
disappearance of the phase boundaries at higher temperature.
The systematic experimental data provided here should help identify new
phenomena and elucidate the underlying physics of this complex system. The
impact of the current study is clear in that we can continuously and spatially
realize doping dependence of complex materials instead of extrapolating over
discrete doping points. For example, if the kind of doping dependent electronic
boundaries are also present in high Tc cuprates such as La2 -xCux0 4 , the detailed
study of the system at low temperature can be carried out and reveal suspected
quantum critical behavior at the critical doping point such as x = 1/8.
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shift in quality factor is double-valued, approaching zero as the sample becomes
insulating or highly conducting. The shift in quality factor has been calibrated
against the measured DC conductivity and is consistent with the theoretical
analysis. Figure 7*8 provides a conversion function from microwave signal to
DC conductivity.
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
VIII Future possibilities
For samples with varying dielectric constant, fr changes with e.
To extract the dielectric constant and topography simultaneously, an additional
independent signal is required. This can be accomplished in several ways:
i)
Measuring more than one set of data for fr and Q at different tip-sample
distances. This method is especially effective when the tip-sample
distance is very small. The models described in Chapter 3 can then be
used to determine the tip-sample distance and electrical impedance
through DSP or computer calculation. In this approach, as the tip is
approaching the sample surface the DSP will fit the tip-sample distance,
dielectric constant and loss tangent simultaneously. These values should
converge as the tip-sample distance decreases. Therefore, this general
approach will provide a true non-contact measurement mode, as the tip
can kept at any distance away from the sample surface as long as the
sensitivity (increase as tip-sample distance decreases) is enough for the
measurement requirement. We call this mode a non-contact tapping
mode. During the scan, at each pixel the tip is first pulled back to avoid
crashing before lateral movement. Then, the tip will approach the surface
as the DSP calculates the dielectric properties and tip-sample distance.
As the measurement value converges to have less error than specified or
the calculated tip-sample distance becomes less than a specified value,
the tip stops approaching and the DSP records the final values for that
121
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pixel. In this approach, a consistent tip-sample moving element is critical,
i.e., the element should have a reproducible distance versus, e.g., control
voltage. Otherwise, it increases the fitting difficulty and measurement
uncertainty. This requirement on the z-axis-moving element may be hard
to satisfy. An alternative method is to independently encode the z-axis
displacement of the element. Capacitance sensor and optical
interferometer sensor or any other distance sensor may be implemented
to achieve this goal.
ii)
In particular, when the tip is in soft contact (only elastic deformation is
involved) to the surface of the sample, there will be a sharp change in the
derivatives of signals (fr and Q) as functions of approach distance. This
method is so sensitive that it can be used to determine the absolute zero
of tip-sample distances without damaging the tip. Knowing the absolute
zero is very useful and convenient for further fitting of the curve to
determine the tip-sample distance and electrical impedance. A soft
contact 'lapping mode” (as described in above) can be implemented to
perform the scan or single point measurements. The approaching of tip in
here can be controlled at any rate by computer or DSP. It can be
controlled interactively, i.e. changing rate according to the last
measurement point and calculation.
iii)
Method described in i) can be alternatively achieved by a fixed frequency
modulation in tip-sample distance and detected by a lock-in amplifier to
reduce the noise. In here, the lock-in detected signal will be proportional
122
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to the derivative of frand Q as function of tip-sample distance. A sharp
decrease in this signal can be used as a determination of absolute zero
(tip in soft contact with sample surface without damaging the tip). Using
relationships in Chapter 3, any distance of tip-sample can be maintained
within a range that these relationship is accurate enough.
Details
At a single tip-sample separation, the microwave signal is determined by
the dielectric constant of the sample. However, the microwave signal is a
function of both the tip-sample separation and the dielectric constant. Thus, the
dielectric constant and tip-sample separation can be determined simultaneously
by the measurement of multiple tip-sample separations over a single point.
Several methods can be employed to achieve simultaneous measurement of tipsample separation and dielectric constant. First, the derivative of the tip-sample
separation can be measured by varying the tip-sample separation. Given a
model of the tip-sample capacitance, the tip-sample separation and dielectric
constant can then be extracted. We have successfully modeled the dependence
on tip-sample separation and dielectric constant using a modified fermi function.
C = 4 r e R
C'
l nQ - b)/b + 1_____
exp{G(£ )(ln a' - (e)]}+1
(1)
where b = (e - e 0)/(e + e0). Furthermore, G(e) and jc0(c) can be fitted well with
rational functions as:
123
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X 9 .5 7 x 1 0
+ 2 .8 4 x 1 0 " £ + 3 . 8 5 x 1 0
£
G (e ) = - - - - - - - - - - - - - - - - - - - - - =- - - - - - - - - - - - - - - - ---------1+ 4 .9 9 x 1 0
,
v
e + 1 .0 9 x l0
£
(2)
5 . 7 7 x 1 0 - ’ + 1 . 3 1 x 1 0 - , £ + 3 .5 5 x 1 0 " 4£ 2
X QI £ ) = ------------------------------------ z---------------------
0
1+ 3 .6 8 x 1 0
£ + 5 .1 6 x 1 0
r - r : ---
£
Figure 8*1 illustrates the agreement between the calculated tip-sample
capacitance and Eqn. 2.
124
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0.00001
0.0001
0.001
0.01
Fig. 8-1 Comparison of exact and approximate formula for the
capacitance of a spherical tip. Solid lines are from Eq. 2, points
are from the exact formula for the capacitance of a spherical tip.
C. Gao, F. Duewer, X.-D. Xiang, Appl. Phys. Lett. 75, 3005
(1999).
Eqn. 2 is suitable for rapid calculation of the tip-sample separation and
dielectric constant. The tip-sample separation and dielectric constant can also
be extracted by construction of a look-up table. Figure 8-2 shows the variation
125
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of the derivative signal versus tip-sample variation.
po>
■o
4.6
1
(Q
o
■o
0
10
0
Fig. 8-2 Derivative signal (pf/pm) vs tip-sample separation
(mm) (a) Calculated value for dC/dg vs tip-sample separation
(b) Measured value for dC/dg vs tipi-sample separation. The
dashed line denotes the zero for tip-sample separation.
Figure 8-2a is modeled assuming a 10 urn spherical tip. Figure 8-2b is a
measured curve. Maintaining the tip-sample separation at the maximum point of
the derivative signal can also regulate the tip-sample separation. This has been
126
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demonstrated by scanning the tip-sampie separation and selecting the zero slope
of the derivative signal. It can also be accomplished by selecting the maximum
of the second harmonic of the microwave signal.
IX
Summary
To summarize, I have describe a scanned evanescent microwave
microscope capable of submicron resolution of microwave dielectric constant and
conductivity.
I have also present images taken using this microscope of a
variety of insulating and conducting materials. I have demonstrated the ability to
image quantitatively both the complex microwave dielectric constant and
conductivity. The ability to spatially resolve the low frequency electronic
properties of materials in a nondestructive fashion is useful for studies of
mesoscopic materials and has not previously been available.
127
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