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Different probing techniques for scanning near-field microwave microscopy

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Different Probing Techniques for Scanning
Near-field Microwave Microscopy
By
Abdolreza Karbassi
A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
(Electrical Engineering)
at the
University of Wisconsin - Madison
2007
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
UMI Number: 3294034
Copyright 2007 by
Karbassi, Abdolreza
All rights reserved.
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© Copyright by Abdolreza Karbassi 2007
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C om m ittee’s Page. This page is not to be hand-written except for the
A dissertation entitled
Different Probing Techniques
for Scanning Near-field
Microwave Microscopy
submitted to the Graduate School of the
University of Wisconsin-Madison
in partial fulfillment of the requirements for the
degree of Doctor of Philosophy
by
A b d o lr e z a K a r b a s s i
Date of Final Oral Examination: September 19, 2007
Committee's Page. This page is not to be hand-written except for the signatures
Month & Year Degree to be awarded:
December 2007
May
A ugust
Approval Signatures of Dissertation Committee
Signature, Dean of Graduate School
TtTfr t ( a A i
j
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Acknowledgements
This thesis is based on a collaboration am ong many people. I wish to thank all o f them,
especially:
• M y academic advisor, Professor D aniel W . van der W eide, from w hom I learned how
to think about research problem s with a positive and open vision. This thesis has been
com pleted through his generous support.
• Hassan Tanbakuchi from A gilent technology, for his generous technical help,
especially for AFM head fixture design, and im pedance analyzer calibration.
• A lexander B.
Kozyrev
who
helped
me
with
theory
and
understanding
of
electrom agnetic modeling.
• Alan D. Betterman for his assistance with experiments, help with m anuscripts, and
gram m ar related advice.
• D aniel R uf for his efforts in preparing micro-fabricated test samples.
• Charles A. Paulson for teaching me the basics o f A FM operation, and his valuable
discussions regarding A FM measurement.
• M udhurin Baneijee for her interactive discussions with me.
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• Yaqiang W ang, from whom I learned the basics o f the AFM probe micro-fabrication
process.
• Sung-jin ho, Kae-Oh Sun and M in-ki choi for the valuable discussions o f m icrowave
circuit modeling.
• Chao Qin for his insightful com m ents on device m icro-fabrication processes.
• Chulki Kim and Hyun Cheol Shin for their help in preparing the SEM micrographs.
• Professor R obert Blick for teaching me quantum electronics, and for inspiration in the
field o f nano-technology, and for serving as a final defense com m ittee m ember
• Professor Zhenqiang M a, for teaching me sem iconductor device physics and serving as
a final defense com m ittee member.
• Professor David Anderson, and Peter Tim bie for their valuable tim e, and for serving on
m y committee.
• M y parents, brother, sister, and her family, for their support and encouragement.
• All my other friends in M adison, especially my girl friend, D asha Cherepanov for her
support, Vladim ir Totolin, O lga Godes, Aby A brisham , Sina, Arash Bahram i, A rezoo
Daneshi, and Ali Shishehgar, for all being such good friends.
This w ork was supported by A ir Force Office o f Scientific Research (AFOSR M URI, grant #:
F 49620-03-1-0420) and an A gilent Technologies research grant.
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Abstract
A
scanning
near-field m icrowave m icroscope
(SNM M ) consists
o f a coaxial
transm ission line (TL) having a sharpened tip (coaxial probe) that is scanned over the surface
of a sample to produce im ages based on the local high-frequency electrom agnetic properties of
the sample. Em erging nanotechnology needs m etrology tools capable o f reliably and
accurately probing surfaces and sub-surfaces o f nano-em bedded structures. Since SNM M was
first dem onstrated in 1972, researchers have continued to increase the instrum ents sensitivity
and resolution. W e have developed a new type o f SNM M probe, called a “Q uadraxial” probe,
which is based on a coaxial probe and has additional, added, co-centric shielding layers. This
new probe, in conjunction with the use o f a differential feeding technique, results in im proved
SNM M m icrowave im aging capability.
The gathering o f simultaneous topography and quantitative m icrowave im ages is made
possible by com bining SNM M with an atomic force m icroscope (AFM ), and em ploying a low
parasitic conductive A FM probe as a m icrowave field emitter. The SNM M /A FM is used to
measure the local capacitance changes occurring between the probe’s tip and the sample. The
m inute change o f this capacitance cannot be resolved using a conventional vector network
analyzer (VNA), thus a half-wave length coaxial resonator is em ployed to increase the
sensitivity o f the capacitance measurement. A new calibration technique is used to obtain a
quantitative dielectric constant m easurem ent o f thin films. The dielectric constant value o f a
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silicon nitride thin film is extracted from measurem ents, and is in good agreem ent with other
reported values, based on the finite elem ent m ethod (FEM).
By modifying a conventional VNA, we have constructed a VNA capable o f m easuring
high input impedance devices, such as conductive AFM probes used in SNM M measurement.
W e call this reconfigurable VNA a “high precision im pedance analyzer” , in which a large
reflected wave from a high input im pedance device is subtracted from a reference wave, and
the difference is then amplified, resulting in an increase in detectability o f small changes in the
reflection coefficient measurement. The m easurem ent results using the SNM M /AFM
im pedance analyzer were found to be in good agreement with simulations.
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Table of Contents
iii
ABSTRACT
CHAPTER 1
Background study for scanning near-filed microwave
microscopy (SNMM)
1.1 Introduction
1
1.2 M axw ell equations 2
1.3 Im age construction in the near-filed
1.4 H istory and applications o f SN M M
1.4.1 SNM M research groups
9
CHAPTER 2
5
8
1.5
1.4.2 SNM M using coaxial tips integrated with AFM
O ther high-frequency near-field techniques 21
1.6
Contribution from author to this area
1.7
References
23
26
Routes to higher field localization through quadraxial
near-field probes
2.1
Introduction
34
2.2 Probe design and fabrication
35
2.3 Electric field localization m easurem ent
CHAPTER 3
17
39
2.4
2.5
Application o f quadraxial probe in SNM M
Conclusion 48
43
2.6
References 49
Appendix A: Differential scattering param eters
52
Experimental measurements with multi-functional
micro-fabricated probes in SNMM
3.1
Introduction
56
3.2
SNM M retrofitted to AFM
59
3.2.1 M icro-fabricated tall coaxial tip integrated with the
cantilever, and coplanar waveguide (CPW )
61
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3.2.2 M echanical design o f the rectangular cantilever o f the
probe 62
3.3 Prelim inary SNM M results for a metallic sample
67
3.4 M odeling o f an ultra-tall coaxial probe tip on cantilever with
its corresponding cable assem bly
CHAPTER 4
68
3.5 A novel m ethod for the resonator o f SNM M
72
3.6 SNM M results for dielectric measurements
75
3.7
Conclusion
77
3.8
References
78
Quantitative SNMM combined with AFM for
dielectric constant measurement
4.1
Introduction
81
4.2 SNM M /AFM m easurem ent setup
4.3 Calibration m ethodology
83
84
4.4 Sensitivity analysis
86
4.5 Capacitance model o f the conductive AFM probe
4.6 Im aging results for thin film Si 0 2 , Si 3N 4
88
93
4.7
N anom eter spatial resolution 96
4.9
Conclusion
4.8
References
100
Appendix B: Capacitance model for conductive AFM probe
99
104
CHAPTER 5
High precision impedance analyzer for broad-band
SNMM application
5.1 Introduction
106
5.2 Sim plified diagram o f one-port reflectom eter
107
5.3 Reconfigurable one-port reflectom eter for high im pedance
m easurem ent (Im pedance analyzer) 109
5.4 Im plem entation o f im pedance analyzer for high impedance
m easurem ent
111
5.5 Calibration o f the im pedance analyzer for m easurem ent of
SNM M based on A FM probe
114
5.6 M easurem ent results for SNM M based on A FM probe using
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impedance analyzer
115
5.6.1 M easurem ent results at 725 M Hz
116
5.6.2 M easurem ent results at 1.55 GHz and 39 dB
am plification
122
CHAPTER 6
5.7 Conclusion
123
5.8 References
124
Summary and future research directions
6.1
Summary and future research directions
125
6.1.1 Quadraxial probe w ith differential feeding technique
125
6.1.2 M ulti-functional m icro fabricated probes for SNM M
126
6.1.3 Calibrated SNM M using very tall probes
127
6.1.4 Reconfigurable netw ork analyzer (Impedance
analyzer)
128
6.2 Future w ork suggestion
129
6.2.1 SNM M on thin-film s deposited on high resistivity
silicon substrate
129
6.2.2 A more sensitive im pedance analyzer for SNM M
application 130
Apendix C: Alternative route for field focusing
C .l Introduction
133
C.2 D ispersion curve engineering
133
C.3 Simulation results in com parison to the
experimental results
137
C.4 Conclusion and future w ork
138
C.5 References 139
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viii
List of Tables
4.1
Changes in mutual capacitances o f A FM probe to the thin-film SiC>2 sample for two
Different probe’s height, but with the same tip apex radius; r=100 nm
97
5.1
M easurem ent and simulation results o f change in tip apex-sam ple mutual capacitance
122
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List of Figures
1.1
Sketch for Fourier optics theory
1.2
Schem atic o f SNM M introduced by Steinhauer et al.
1.3
Sim ultaneous im aging of gold on thin film mica using SNM M integrated with STM
(nm height control)
12
6
10
1.4 SNM M experimental set-up: The sapphire ring was inserted inside the hole to
decrease its size o f the hole to 1 0 0 -2 0 0 pm
13
1.5 Image o f the dielectric constant o f 0 .6 7 (P b M b l/3 N b 2 /3 0 3 )-0 .3 3 P b T i0 3 film grown
On sapphire substrate
14
1.6 (a) M icro-strip line resonator and probe assembly. The field fall-off in the case of
electric-field probe (open circuit) (b) the m agnetic-field probe
16
1.7 (a) Schem atic o f coaxial tip, cantilever and chip body integrated with CPW . (b) Probe
interface to m acroscopic instrum entation: Chip body connects through gold wirebonds to external gold C PW and micro-coax, (c) N ear-field scan over a metal edge
(dark) on quartz
19
1.8 Basic idea o f high frequency electric force microscopy
2.1
22
(a) N ear-field m icrowave/ R F electric field m easurem ent setup. The four conductors
o f the quadraxial probe are numbered, (b) Photograph o f the quadraxial probe
36
2.2 N orm alized iso-lines magnitude o f total electric field (at 600 M Hz) calculated using
Numerical m ethod (FEM ) in h alf cross-section view for (a) the coaxial probe and (b)
the quadraxial probe
37
2.3 Comparison o f the electric field m agnitude fall-off measurem ents. F or coaxial probe,
we m easured S 12 (single-ended transm ission coefficient) from the PN A 23
2.4 U nit cell model o f a RH N LTL used in the ADS simulation
42
2.5 (a) Scanning m icrowave/ R F transm ission m icroscopy m easurem ent setup, (b)
M agnitude o f transm ission Sds2i and S 21 versus position w hile scanning across the
100 p m wide strip line for the quadraxial probe and the coaxial probe, respectively
(c) phase m easurem ent 45
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X
2.6
(a) A simple model for quadraxial probe in m icrow ave/ RF transm ission m icroscopy
m easurem ent setup, (b) Comparison between the simulation and experimental results
for S(js2i using the same setup and param eters as used in figure 2.4 (a) 47
A .l
Incident and reflected waves in a tw o-port netw ork
53
A.2 Incident and reflected waves diagram m ed in a tw o-port network (differential single­
ended)
54
3.1
Incident and reflected w aves in a one-port netw ork
3.2
Reflection coefficient versus load im pedance (Zi)
55
55
3.3 Fig. 3.3: (a) A schem atic o f our SNM M com bined with a commercial AFM . (b) Probe
interface to macroscopic instrum entation
60
3.4
(a) Schem atic o f tall coaxial tip, cantilever, and chip body integrated with coplanar
waveguide. The typical dim ension of the probe is shown (b) SEM m icrograph for an
tall coaxial tip and an integrated CPW structure (c) Close-up o f the apex region
showing a ~3-pm -radius aperture 62
3.5
A sim plified schematic o f SNM M
64
3.6 (a) Cross-section schematic of the coaxial tip that was used for simulations. D ifferent
voltages (V in volts) and lengths are shown in draw ing (not to scale) (b) Tip- sample
capacitance calculated from this model shows a logarithm ic drop vs. distance (h)
63
3.7
(a) M agnitude o f the reflection coefficient near the resonant frequency (fn) when the
coaxial tip was in air, and over the gold sample. Sim ultaneous images using our probe
in system o f figure 3.3 (a): (b) Topography im age; (c) M icrow ave m agnitude image.
67
3.8 Reflection coefficient ( T ) o f tall coaxial tip integrated with CPW on chip body from
45 M Hz to 20 GHz
69
3.9 Equivalent circuit model o f tall coaxial tip probe with its interface to a macroscopic
Instrumentation up to the SM A connector (a) CPW on chip body,
and ultra tall
coaxial tip (b) the CPW , and gold wires (c) M icro-coaxial cable, micro-strip, and the
SMA connector 70
3.10 (a) M agnitude an d (b ) phase o f r eflection coefficient (Si 0 o f the m icro-coax cable
assembly connected to tall coaxial tip probe 69
3.11
Schematic o f the novel resonator used in SNM M
71
3.12 (a) M agnitude o f the reflection coefficient near the resonant frequency (f„i),
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when the coaxial tip was in air, over the silicon sample, and over the 1 |im thick SiC>2
on high
resistivity silicon. Sim ultaneous images were perform ed using the tall
coaxial tip probe using the resonator system o f figure. 3.11: (b) Topography image;
(c) M icrow ave im age
75
4.1 SNM M com bined with a com mercial A FM
84
4.2 (a) Optical im age o f the fabricated capacitive load with arrays o f different patch sizes
(capacitance values) on 44 nm thin film Si 0 2 grown on low resistivity silicon
substrate for calibrating the SNM M based A FM (b) A SEM m icrograph close-up of
7.00 pm by 7.00 pm patch size within an array
85
4.3 Sensitivity curve o f reflected w ave-voltage with respect to different capacitance
values (patch sizes)
87
4.4 Tip-sam ple m utual capacitances model for a conductive A FM tip in a soft contact
with a dielectric thin film grown on conductive silicon substrate
90
4.5 The experim ental and simulation data (FEM ) for the tip apex to thin-film sample
mutual capacitances 91
4.6 Changes in mutual capacitances o f A FM probe relative to the thin-film sample
92
4.7 Sim ultaneous im ages using SNM M based A FM using a very tall probe. The AFM
scan was perform ed in contact m ode at an interface between SiC>2 and SisN 4 which
was previously planarized: (a) topography (b) microwave im age (magnitude of
reflected w ave-voltage). A line was a cut through the images: (c) topography (d)
magnitude o f reflected wave voltage
94
4.8 Tip apex-sam ple mutual capacitance o f A FM probe tip relative to the thin-film
sample for different tip radius 95
4.9
Slope o f apex-sam ple and cone-sam ple capacitance curves o f our very tall AFM
probe tip for S i0 2 thin-film sample and tip radius; r= 100 nm
97
4.10 Sim ultaneous im ages using SNM M based A FM using the very tall probe (lOOnm tip
radius), (a) Topography (b) m icrowave im age (magnitude o f reflected wave-voltage)
98
5.1
A sim plified diagram o f a one-port netw ork analyzer (reflectometer)
5.2
V ector subtraction inaccuracy in the com plex 2-D plane
107
104
5.3 A three dim ensional graph of the suppressed signal versus phase
mism atch o f the tw o vector subtraction
110
and magnitude
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xii
5.4 A sim plified diagram o f the m odified one-port network analyzer ( im pedance
analyzer) for m easuring D U T with high input im pedance
112
5.5 (a) A photograph o f fabricated Baiun on RT/D 6010LM substrate from Roger
corporation (b) equivalent circuit o f the balun in com m on m ode rejection
113
5.6
A schematic o f our im pedance analyzer com bined with a commercial AFM. The
AFM uses an optical detection m ethod to track the topography o f the sample (SUT)
116
5.7
Reflection coefficient
(rDUT) m easurem ent
show ed on the Smith chart for the very
tall conductive A FM probe assembly in N A m ode (High reflection coefficient)
5.8
M agnitude o f the reflection coefficient (|A rDUT| ) for the AFM probe assembly, when
m easured in IA m ode
5.9
117
117
Reflection coefficient ( ArDUX) m easurem ent on Smith chart display for the tall
AFM probe assem bly in IA mode. The frequency sweep range was from
up to 727 M Hz. 118
722 M Hz
5.10 Reflection coefficient ( ArDUT) m easurem ent w hen the AFM probe was scanned (in
contact m ode) on different capacitive loads in IA m ode (Cartesian coordinate)
119
5.11 Change in m agnitude o f reflection coefficient ( |ATdut | ) when the A FM probe was
scanned (in contact m ode) on different capacitive loads measured in IA m ode
5.13 Change in m agnitude o f reflection coefficient
(|ArDUT| ) when the A FM
scanned (in contact mode) on different capacitive loads in IA m ode
6.1
120
probe was
121
Simplified equivalent circuit model for conductive AFM probe tip to thin-film
deposited on silicon substrate (a) W hen the silicon substrate is conductive (b)
Proposed model when silicon substrate is not conductive: R s and Cs are resistance and
capacitance associated with non-conductive silicon substrate (c) An air capacitance
(C0) is added, when AFM is scanned in non-contact/ tapping m ode
129
6.2 Proposed reconfigurable network analyzer configuration for SNM M application
131
C .l
(a) Schem atic o f a unit-cell o f 2-D m ushroom structure. The dimension o f designed
Structure is: T=254 pm , P=145 pm , g=32 pm , C=222 pm , H=85 pm , h=4.5 pm ,
d=35 pm , h+H=89.5 pm (b) A schematic o f 5-section 1-D m ushroom structure, (c)
An equivalent circuit model for a unit cell
134
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C.2 (a) S 21 (transmission coefficient): 1-D 9-section m ushroom type transm ission line Sparam eters measurement, full-wave simulation using HFSS (Driven-mode), and
circuit simulation using ADS. (b) D ispersion curve: full wave simulation using HFSS
(Eign-mode), and a circuit model based by using equation 4.1 [7] (c) Electric and
magnetic field pattern for LH m ode (without patches)
136
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1
1
Background study for scanning near-field microwave
microscopy
This chapter introduces scanning near-field m icrowave m icroscopy (SNM M ) and
provides an overview o f previously published journal articles. A t the end o f this chapter, I will
present our achievem ent in this area.
1.1 Introduction
The new field o f nanotechnology requires nano-tools possessing nanom eter spatial
resolution. These tools m ust capable o f imaging surfaces and sub-surfaces o f m aterials in the
nm regime. Conventional optical microscopy (400 nm to 700 nm is the visible light w ave­
length range) has the resolution o f ~ A/2 (A bbe’s limit), w here A is the light free space’s wave
length [1]. One w ay o f increasing the resolution even further is through the use o f a scanning
microscope. W ith scanning microscopes, an im age is reconstructed by scanning the sample
point by point, which increases the accuracy, but scarifies the speed.
Scanning probe
m icroscopy (SPM) has been more developed in recent years, and it is typically used for
surface topography and m aterials characterization at an atomic scale [2]. These SPM
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2
techniques include, but are not lim ited to scanning tunneling m icroscope (STM ), atomic force
microscope (AFM ), magnetic force m icroscope (M FM ), electrostatic force microscopy
(EFM ), scanning near-field optical m icroscope (NSOM ), and Scanning near-field m icrowave
m icroscopy (SNM M ). In addition to all m entioned scanning microscopes, there has been a
great interest for using w ave length below optical regim e and above m icrowave regim e (30 cm
till 1 mm is the wavelength o f m icrowave) for scanning m icroscopy [3 ,4 , 5],
SNM M is the oldest but still the most active area which is in dem and for localized
material characteristics at m icrowave frequency. The m easured quantities typically include
conductivity (ct), perm ittivity (e), and perm eability (p), if a suitable probe is used. These
quantities are im portant to know as a function o f frequency, and may be temperature. If the
material under the test is inhom ogeneous at the nm length scale, It may need to be
characterized. The resolution o f SNM M in general is on the order o f probe’s tip size, and is
not lim ited by the operating wave length o f m icrowave signal [5]. Such localized near-field
measurem ent can also be em ployed for fault debagging and perform ance observation o f active
integrated circuits(IC’s), especially those involving M M IC [4] (monolithic microwave
integrated circuits). Later in this chapter, I will present the history and a brief review of
SNM M , plus I will present m y own contributions to this field o f research.
1.2 Maxwell equations
The time varying electric and magnetic fields are described using the well known
M axw ell equations (the following equations are re-w ritten from [6 ]):
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3
V-D = p
1.2.1
V -# = 0
1.2.2
to
V X *E = ------dt
1.2.3
d'D
V x # = J/ + -----
1.2.4
dt
The electric flux density (D) is related to the electric field intensity (E) by the following
relationship (for isotropic and linear medium):
*D = ffE —e0£r 'E
1.2.5
W here e is the perm ittivity, £o is the perm ittivity o f free space, and er characterize the effect of
the m olecular and atomic dipoles inside the medium.
Similarly, the magnetic flux density (B) is related to the magnetic intensity (H) by the
following relationship (for isotropic and linear material):
H = ji9{ ~
1 . 2 .6
W here p is the perm eability, po is the perm eability o f free-space, and Pr characterize the
effect o f the m agnetic dipole mom ents of the atoms inside the medium.
For a source-free hom ogenous medium ( p = J = 0 ), taking curl of 1.2.3 gives:
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Using 1.2.4,12.5, and the associated vector identity [6], results in:
d<E
V fV • E j - V Z(E = - j i e — —
dt
1.2.8
U sing 1.2.1,1.2.5, and p=0, results in:
d Z(E
V ZrE = u e dt2
, and V -
1.2.9
I— is the wave velocity in the medium.
Similarly, for the m agnetic field vector:
1. 2.10
V Zt f = u £ ^ - ^ dt2
1.2.9, and 1.2.10 are called the three dim ensional wave equation [6 ], and each o f these
equations can in turn break down into three scalar equations.
I assume sinusoidal time-harm onic fields ( e
), thus from 1.2.9, 1.2.10:
V 2E - - k 2E
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1.2.11
5
V 2H = - k 2H
1. 2.12
k 2 - c o 2/us
1.2.13
The above equations are called three dim ensional helmholtz equations [6 ],
1.3 Image construction in the near-field
O ne way o f explaining near-field im age form ation is by using scalar diffraction theory,
also known as Fourier optics [5, 7].
Figure 1.1 presents an illustration o f the near-field
im aging concept, using Fourier optics. In this Figure, the amplitude o f the field (can be either
magnetic or electric) is consider to be U(x,y,z=0) on the source plane as shown in figure 1.1.
One needs to determ ine w hat the field am plitude (U(x,y,z)) at the image plane(point i) is. In a
tw o-dim ension (2-D) case, the helm holtz vector equations i.e. 1.2.11 (1.2.12) can be expressed
by a scalar function Ez (Hz ) and can be written in term s of other dependent variable [5]. For
instance, in 2-D from 1.2 .1 1 ,1 have:
V 2E z = - k 2E z
d 2E z
^
dx2
1.3.1
1.3.2
+
oy
This m eans a scalar quantity (U) can be a representative o f the field. N ow, by using the
concept o f a Fourier transform across the x-y plane, the function U(x,y,0) corresponds to a two
dim ensional Fourier transform given by [5, 7] as:
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6
oo
oo
A0 (kx , k y ) = J $ U ( x , y , 0 ) e x p ( - j ( k xx + ky y))dxdy
1.3.3
—oo —oo
The function Ao (kx, ky) is called the angular spectrum o f the U (x,y,0) [7].
Now, let’s calculate U (x,y,z) at the image plane (i.e. at point i in figure 1.1). I assume
that A(kx,ky;z) is the angular spectrum of the U(x,y,z), thus by definition:
'D irection of
►wave
propagation
Coordinate system
z=0
Source plane
image plane
Fig. 1.1: Sketch for Fourier optics theory. A scalar field U(x, y, z=0) is assumed to be at source
plane. Fourier optics is used to calculate scalar field U(x, y, z) at image plan (point i) [5,7],
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A(kx , k y ’, z) = J $ U ( x , y , z ) e x p ( - j ( k xx + k y))dxdy
1.3.4
—oo —oo
U(x,y,z) can be written as the following [5,7](Definition o f inverse Fourier transform):
J
OO
oo
U ( x , y , z ) = --------7 J J A(kx , k y -,z)exp(j(kxx + k y y))dkxd k y
(2nf
1.3.5
-co -o o
Conversely, U(x,y,z) m ust satisfy the H elm holtz equation at the source-free points.
Thus from equation 1.3.1:
V 2U + k 2U = 0
1.3.6
One solution for the angular m om entum which satisfies 1.3.6 [5 ,7 ] is as follows:
A(kx , k y ;z) = AQ(kx , k y ) e x p ( j k z z)
1.3.7
where k z = ^ k 2 - { k 2 + k 2 )
1.3.8
W hen kz is real, the solution is a propagating wave (far-field). W hen kz is im aginary, the
solution is a decaying or evanescent waves, this happens when the condition kx
2
+ ky2 > k 2
is satisfied as when in the near-field region. The reason for this is that the contribution due to
kx and ky com ponents cannot be ignored within the near-field region o f the electrom agnetic
source interacting with very electrically small objects, thus providing the basis for the im age
construction in the near-field region. In order to perform high-resolution near-field m icroscopy
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8
an appropriate probe should designed and the distance between the probe and a sample should
be much sm aller than wave length [5, 8 ].
1.4 History and applications of SNMM
Synge [9], proposed the first evanescent electrom agnetic wave m icroscope in 1928, and
Soohoo dem onstrated the first near-field m icrowave m icroscope in 1962 [10]. In 1972, Ash
and Nicolas dem onstrated a near-field m icrowave m icroscope exhibiting a A./60 spatial
resolution, where X is the free-space m icrow ave wave-length. Their w ork is generally credited
as the first experimental near-field m icrow ave microscope in the SN M M literatures. Their
m icrowave microscope consisted o f a resonant cavity with a small hole (aperture - 1 . 5 m.m. in
diameter). This aperture locally interacted with a sample placed underneath it in the near-field
region [11]. O ther researchers used hollow waveguide for their near-filed antenna [12], In
general, tapered aperture probes have the inherent disadvantage o f large attenuation of
electrom agnetic wave radiating from aperture since they are usually operating well-below the
cut-off frequency o f the wave guide [4]. As a means to overcome this, a tapered coaxial
waveguide were em ployed by Bryant et al [13], The inherent advantage o f a coaxial
waveguide is that there is no lim it in cutoff wavelength. Open ended coaxial transm ission line
microscope was proposed by Fee et al. [14]. They achieved a resolution better than A./4000.
There has been much research on m easurem ent o f perm ittivity o f materials using open-ended
coaxial probes on macro-scale. M isra used an open-ended coaxial probe for m easuring the
perm ittivity o f liquids (dispersive medium ) [15]. Berube and his co-w orkers perform ed a
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9
com parative studies o f different models o f open-ended coaxial probes used in broad-band
material measurements [16]. Blackham, et al. w orked on im proving the m aterial m easurem ent
techniques [17]. In recent years, m any research groups have been constructed SNM M and
used them for passive or active sample characterization with pm and sub-pm resolution.
1.4.1 SNMM research groups
Anlage and W ellstood’s research group from the U niversity of M aryland-College Park,
com plied an extensive w ork on near-field microwave m icroscopy and investigated both
metallic and dielectric samples.
Their microwave microscope consisted o f an open-ended
half-wavelength coaxial resonator with end o f inner conductor sharpened. C. P. Vlahacos, et
al. [18] dem onstrated their first near-field measurem ent and determ ined a working resolution
o f 100 pm at 12 GHz. The principle o f operation is based on the shift in resonant frequency of
the coaxial transm ission line resonator that occurs when the coupling capacitance between
coaxial resonator end and a m etallic sample is changed. The m icroscope was used both in
reflection and transm ission mode. Steihauer, et al., [19, 20] developed a frequency following
circuit in order to de-convolve the effect o f the frequency shift and associated changes in Q. In
the case o f metallic samples, losses associated with the metallic samples decreases the quality
factor o f the transm ission line resonator, whereas in the case o f dielectric materials, any
changes in the dielectric constant affects the resonant frequency o f the transm ission line
resonator. The set-up for this version o f microscope, used to study the surface resistivity o f the
metals, as shown in figure 1 .2 . Volahacos, at al., im aged the uneven topography
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10
Micro
Source
I frequency control
Directional
M
lr
Feedback circuit
c o u p le r
.........*&’i
j r
Diode
detector
\
Decoupler
Oscillaloi
3 kH /
P r o b e
Sample
Probe center
conductor
Fig. 1.2: Schematic of SNMM introduced by Steinhauer et al. [19]. The inset shows the model
for interaction between the probe and the metallic sample.
associated with the m etallic surfaces at the constant height mode of microscope. They
calculated the changes from probe’s center conductor end to the sample separation from the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
shift in the resonant frequency o f resonator [21]. Steihauer et al used the sam e set up in figure
1.2 to successfully im age varying perm ittivity o f the bulk and thin film materials such as
SrTi 0 3 , Bao.6Sro.4T i 0 3 with a sensitivity o f A £ r = 2 (for £ r = 5 0 0 ) , by m easuring the shift
in the resonant frequency o f resonator. The dielectric m easurement was found to be a function
o f the applied field and the thickness o f the dielectric sample and the shape o f the probe’s tip.
To determine this relation ship, they created an intensive finite elem ent model (FEM) to
calculate electric field around tip area. The resolution in this case was im proved to l pm by
using a sharpened coaxial inner conductor [22].
In 2005, A. Imtiaz, et al., integrated the SN M M with a STM in order to achieve nm distancefollowing feedback control. They used a constant height scan (< 10 nm ) in order to study the
nm scale variation present in the microscope probe tip geometry. They assum ed that a STM tip
can be m odeled as sphere above conducting plane and observed that a few nm protrusion on
the surface o f a 37 p m STM tip radius, results in a considerable change in the tip to sample
capacitance when tip to sample separation was < 10 nm [23].
In [24], A. Imtiaz, et al.,
studied the effect o f the different shapes and sharpness of com mercially STM tips available
for SNM M . They used a sphere above the metal plane as their STM tips m odel, and observed
that the larger the radius o f sphere, the m ore im age contrast was available in the microscopy
signal. It was from their work that I concluded a supper-sensitive resonator based approach is
required to obtain sufficient contrast in im ages when using very sharp tips (nm radius).
A nother criteria for achieving a high-resolution nm scale image is an appropriate feed-back
control system to regulate the height with the precision in the nm regime. This can be achieved
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12
using either STM (for metallic sample) or A FM for both metals and insulator samples.
Figure 1.3 shows an im age generated by A. Im tiaz o f gold on mica using SNM M com bined
with STM. STM was used as a nm constant height feedback control [24].
0
6
11
17
22
28
0.00 0.03 0 .06 0 .09 0.12
STM topography (n.i no-mctcrs) "V-ymagnitude (Volts)
1
51
10 1
1 50
200
nnigniludc (IvHz)
Fig. 1.3: Simultaneous imaging of gold on thin film mica using SNMM integrated with STM
(nm height control). The bias for the sample is 0.1 V and tunnel current set point is 1 nA. This is a
room temperature experiment performed at 7.48 GHz. (b) The V2/ presented here is proportional to the
Q (-384) of the resonator. Almost no changes in losses are detected, (c)
A /
is related to sample
unevenness. The experiments were performed using etched tip with nm tip sharpness [24].
V. V. Talanov and his co-workers, built a m icro-fabricated balanced probe that confined
the reactive fields to >99% o f the sample volum e using a parallel strip resonator. They
constructed a SNM M using this probe in conjunction with a balanced strip line resonator, and
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13
reported they w ere able to perform localized quantitative m easurem ent o f low perm ittivity thin
film materials on sem iconductor wafers [25, 26].
[vco
Died*
Detector
Amgiiilter
Fig. 1.4: SNMM experimental set-up: The sapphire ring was inserted inside the hole to decrease
its size of the hole to 100-200 pm. The 1 pm thick metal coating on the outside of sapphire disk
reduced the far-field effects [28].
W ei, et al [27] in X iang’s research group from Law rence Berkeley national lab,
California, designed a m agnetically coupled quarter-wave length coaxial resonator having a
sharp center conductor extending beyond the shielding. They achieved the spatial resolution o f
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14
~ 5 pm when im aging a 0.1 pm thick N b (Conductive) thin film on silicon. The tip o f their
m icroscope was in physical contact with the sample (gravitational force). Later, Gao, et al.
[28] introduced an updated version o f their SNM M with an im proved spatial resolution of
445
2339*75
90:
II
s
75
0.1 pun
60J
7.Q
Fig. 1.5: Image of the dielectric constant of 0.67(PbMbl/3Nb2/303)-0.33PbTi03 film grown
on sapphire substrate, (a) resonant frequency image, (b) quality factor image, (c) A line scan profile
from (a) at Y =4.5 pm. The arrow indicates a region with a 100 nm FWHM criteria resolution [28].
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15
100 nm. They decreased the size o f the aperture from 1 m m to 100-200 pm by inserting a
sapphire disk inside the hole, and also sharpened the end o f the tip. Decreasing the size o f the
aperture reduced the far-filed effects considerably, and enabled them to use a simple model to
predict tip-sam ple interaction. They also introduced a soft contact between probe and sample,
im plem ented by pressing the sample against the probe using a cantilever. They perform ed
measurem ent o f ferro-electric dielectric materials around 1 GHz. Figure. 1.4 diagrams the
near-field microwave m icroscope invented by G ao’s and his co-workers. Figure 1.5 shows the
results o f the imaging o f high-value dielectric materials. They formulate their cavity based on
perturbation theory and used a simple tip-sam ple interaction model and calibrated their near­
field microscope with known samples, Based on this, They m easured dielectric perm ittivity o f
different samples with a good accuracy [28]. G ao and his colleagues introduced a model for
their SNM M around resonant frequency of the cavity, and the sensitivity for perm ittivity
Ss
m easurem ent was predicted to be as large as — = l x l 0 -5 [29]. Duewer, et al. introduced the
£
tip-sam ple feedback scheme (for metallic samples) based on regulating the resonance
frequency of a cavity to m aintain a constant separation. They successfully m easured both
topography and resistivity o f the metallic samples (low resistivity metals) [30].
M. Tabib-Azar, et al. w orked on the characterization o f different materials, such as
dielectric, metallic, and even biological samples using half-w ave length m icro-strip line
resonator with capacitive coupling. The end o f m icro-strip signal line was tapered to a point
and then a short thin wire was connected to this point providing the sharp probe tip. The probe
w as used as either a m agnetic field probe or electric probe with non-contact scanning (Fig
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1.6). They reported 100-150 micron resolution for the first version o f their near-field
m icrowave m icroscope [31]. Later, by em ploying a chem ically etched sharpened tip, and
probe-sam ple distance m odulation in 100 Hz, and synchronous detection m ethod using a lockin amplifier, they im proved the resolution o f their microscope to 0.4 p m at 1 GHz [32], They
published a series o f microwave m easurem ent on non-uniform ity detection defects in metals,
semiconductors, dielectrics, com posites, and plants [33],
Insulator (Dureid)
Magrictic-Dipotc Probe / t DEI
Fig. 1.6: (a) Micro-strip line resonator and probe assembly. The field fall-off in the case of
electric-field probe (open circuit) (b) the magnetic-field probe (by shorting the probe end to the back
side of micro-strip) is modeled as a micro-strip line with a short length of current carrying wire [33].
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17
A. Kim, et al., [34] from Seoul National University, Seoul, Korea, built a scanning
microwave microscope sim ilar to the G ao’s W ork, based on X/4 coaxial resonator with a sharp
tungsten tip sticking out o f a small hole in the bottom plate o f the cavity, which interacted with
an underlying sample. They used this m icroscope to m easure the topographical profile o f good
metals. The vertical resolution for their w ork was -5 0 nm around approxim ately a 1.5 GHz
resonant frequency. They show ed for a good metal, the change in resonant frequency can be
used for surface-following control (topography), and the quality factor o f a resonator can be
used to measure the metallic losses (resistivity). In 2005, Park, et al. [35] from the same
research group used their SNM M to image biological samples. They m easured the change in
the resonant frequency and the quality factor o f the resonator, as a function sodium chloride
concentration in water. In their experiment, they used a 240 pm blunt tip in com bination with
a 150 pm thick glass to prevent im mersion o f the tip. They showed that their SNM M could
m easure changes in sodium chloride concentration in water, thus they claim ed that their
microscope can be used for the study of biological samples.
1.4.2 SNMM using coaxial tips integrated with AFM
Since 1984, an introduction o f AFM [36], there has been many new techniques
developed in the area o f scanning probe m icroscope (SPM ) useful for nano-m eter material
characterization. One scanning scheme which one could consider sim ilar to SNM M , is SCM ,
first dem onstrated in 1985 by m atey and blanc [37]. W illiam s, et al., [38] dem onstrated the
SCM on a 25 nm scale and achieved a 10'19F capacitance sensitivity at 915 M Hz operating
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18
frequency. Lanyi, et al. dem onstrated local capacitance measurem ent at low M H z operating
frequency [39]. Som e other research groups have been working on SCM using commercial
conductive AFM probes [40, 41]. The issue here is that a commercial conductive AFM tip
presents a large am ount o f coupling capacitance between the chip body and the sample, i.e.
parasitic capacitance ( in the order o f PF), which lim its the sensitivity o f the local tip to sample
capacitance m easurem ent possible on nm A FM ’s tip [41]. Also, this approach is not practical
for microwave frequencies, since commercial A FM probes are not a suitable candidates for
guiding a m icrowave signal. Integrating coaxial structures with cantilever based A FM probes
enables (room tem perature, and pressure) m ulti-functional probing o f both conductive, and
insulators materials in the sub-micron regim e at m icrowave operating frequencies, and perm its
determination o f both the topography and electrical properties o f a sample [4], van der
W eide[42] constructed his first multi-functional probe by coating com m ercial conductive
silicon AFM probes with a layer o f photo-resist, and then adding a gold layer on the top of
photo resist as a shielding layer. The opening for tip apex was performed by rubbing the tip
apex on a rough substrate very gently. These probes were used for both localized m easurem ent
of electrom agnetic fields and sample topography. The disadvantages o f this technique were 1)
the process of opening the metal layer was not reliable or reproducible, 2) the structure did not
function as a good waveguide at higher m icrowave frequencies, 3) silicon is a lossy substrate
at m icrowave frequencies [4]. In [43], van der W eide, et al., introduced the concept o f the
nano-oscilloscope, w hich refers to the co-m easurem ent o f both topography and AC fields with
nano-m eter level spatial resolution. They presented the initial micro-fabrication results for a
coaxial tip probe. Rosner, et al., [44], showed the com plete fabrication o f cantilever based,
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19
coaxial tip probes which consisted of a coplanar structure to allow efficient microwave signal
transfer onto the cantilever chip body, and from there to the coaxial tip. They also reported
initial result o f m icrowave im aging using these coaxial tips. Figure 1.7 presents the schematic
o f these silicon coaxial tips, and shows a reflection m easurem ent using these probes on the top
o f m etallic sample [44, 45]. W ang, et al. [46] with the van der W eide group reported the
m icro-fabrication o f coaxial tips with height > 50 jxm, an efficient m eans for decreasing the
!Oxide
2.788 V
2232.56 nm
1116.28 nm
Onm
2232,56 nm
Fig 1.7: (a) Schematic of coaxial tip, cantilever and chip body integrated with CPW. (b) Probe
interface to macroscopic instrumentation: Chip body connects through gold wire-bonds to external gold
CPW and micro-coax, (c) Near-field scan over a metal edge (dark) on quartz [44,45].
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20
parasitic coupling between chip body and sample. In order to verify the design, they
em ployed, they a commercial AFM tip in non-contact m ode as the sam ple for the m icro­
fabricated coaxial tall probe, and then im aged the width o f com mercial A FM cantilever with
this ultra-tall coaxial tip probe. They show ed that the lateral resolution o f this probe was at
least, 38 pm (the width o f com mercial cantilever). W ang, et al. in [47] reported the complete
fabrication process o f ultra tall coaxial tips cantilever based probes. The probes were
fabricated on high resistivity silicon in order to have the very low loss necessary for
m icrowave signal propagation. K arbassi, et al. [48] dem onstrated the first A FM Based,
simultaneous m easurem ent o f both a m icrowave signal and a topography signal o f a gold
sample on alumina substrate using a m icro-fabricated coaxial ultra-tall probe and by using
AFM in non-contact mode. They also calculated that chip body to sample capacitance
(parasitic capacitance) is on the order o f ~ 0.5 fF, and that the tip apex to sample capacitance
in the order of ~2 fF for the setup that they used. This w ork will be discussed in m ore detail in
chapter 3. Karbassi, et al. [49], dem onstrated a novel probe incorporating a differential feed
technique for application with SNM M . They dem onstrated the superior field localization and
sensitivity o f these probes, in com parison to conventional coaxial probes, when used in
transm ission experim ent for a metal line. This will be discussed in additional detail in chapter
2.
High frequency m agnetic-field m easurem ent in pm scale is im portant for microwave
m onolithic integrated circuits (M M IC) industry for on circuit probing [45]. Agrawal, et al.
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21
[50] with the van der W eide group, presented the fabrication and dem onstrated a localized
magnetic field m easurem ent using loop tip probes. The loop probe fabricated on the cantilever
had inner diam eter o f 6 pm . A lso in this group, Lagally, et al. [51] fabricated shielded loop
probes as a means to decrease the mutual coupling between the chip body and a sample.
Previously, Tabib-Azar, et al. [52], reported fabrication o f SNM M coaxial probes having a
height o f 10 pm . They also used their probes in conjunction with AFM and sim ultaneously
recorded microwave im age and topography o f a silicon nitride grid sample. They claim ed to
achieve a 50 nm spatial resolution in microwave im age at ~1 GHz operating frequency.
1.5 Other High frequency near-field probing techniques
There are other techniques that have been used in high-frequency near-filed m icroscopy
research [45]. Here, I will briefly introduce high frequency electrostatic force microscopy
(HFEFM ). W ith this technique, A high frequency electrical signal is dow n-converted to a
low er frequency mechanical force signal, which is m easured by using a scanning force
m icroscopy technique. Bloom , et al., [53] used this technique to measure the wave forms
on
the top o f ICs. In figure 1.8, if a voltage difference U exists between the conductive SFM tip
and a sample device, a force will act on the cantilever as the following equation [54]:
F =- ^ U
2z
sp2
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1.5.1
22
SFM-Probe
U s p - U
M M IC
o
Fig. 1.8: Basic idea of high frequency electric force microscopy. The difference in potential
between the tip and MMIC signals generates a force that bends the cantilever and can be detected via
standard SFM detection techniques [54].
where A is the effective tip area, and z is the effective tip to sample separation. The
dependence o f the force on the square o f the voltage results in the cantilever acting like a
mixer, if the difference between the high frequency signal which is applied to the probe and
the high frequency signal present in the sample device lies below the fundam ental mechanical
resonant frequency o f the cantilever, the cantilever can track this low frequency signal [54],
A nother alternative for sub-wave length resolution imaging, em ploys negative refractive
index (NRI) m edia [55]. N RI m edia expressed theoretically by Veselago in 1968 [56]. The
first experim ental verification o f negative refractive index reported by Shelby, et al., in 2000
[57]. They constructed the negative refractive index m edia or left handed m edia (LHM ) in
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23
com parison to conventional m edia or right handed m edia (RHM ) by com bining split ring
resonators and wire strips on fiber glass circuit board.
These com posite material were
resonant based, thus a negative refractive index was dem onstrated only in a very narrow
microwave frequency range. Recently, O ther research groups have built LH M based on
transm ission line techniques (LH-TL) which is a non-resonant technique. Eleftheriades, et al.
[58] and Itoh, et al. [59] published series o f papers on novel m icrowave devices using LHT L ’s. Kozyrev, et al. [60] introduced a capacitance based non-linearity in LH -TL by using
varactors, and studied novel non-linear wave phenom ena in non-linear LH TL (NL-LHTL),
such as novel m icrowave circuits based on NL-LH TL such as phase shifters, and harmonic
generators. From the same research group, Chao, et al. [61] designed and micro-fabricated
LH -TL based N R I m edia operating in the 50 GHz regime.
1.6. Contributions to this area from author
In chapter 2, we introduce the novel probe (quadraxial) em ploying a differential feed
technique. It has the advantage o f greater field localization in com parison to the conventional
coaxial probe. To carry out the work, I designed, and simulated, and fabricated the described
probe. The validity o f simulation was confirm ed by experim ental results.
In chapter 3, we explain the design and application o f multi-functional micro-fabricated
SNM M /AFM probe. The new cantilevered based tall coaxial tip capable o f being used in
conventional AFM has been designed with a CPW on the top o f chip body to m inim ize the
coupling from the probe’s chip body to the sample. These micro-fabricated cantilever based
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24
coaxial probes are high input im pedance devices. This means they act like an electrically small
antenna having an input im pedance o f a few k-ohm up to M-ohm. A conventional vector
netw ork analyzer doesn’t have sufficient sensitivity to m easure small changes in the input
im pedance of such probes. Thus, a new systematic technique was im plem ented to increase the
sensitivity of the probe measurement. To achieve this goal, we developed a coupling m ethod
to create a resonator for the probes measurement. To dem onstrate utility, we have applied this
new resonant technique to the tall coaxial probes, and constructed a fixture suitable for
perform ing localized m icrowave measurement.
W e have measured the local capacitance
changes on metal surface and thin film dielectric sample. In order to calculate dielectric
constant from a localized capacitance measurement, we also need to have the topography of
the sample. Thus, it is im portant to perform sim ultaneous topography and a capacitance
m easurement o f the thin film dielectric sample on pm /nm scale.
In chapter 4, we introduce a calibration technique by using a new ly developed
calibration kit for quantitative capacitance m easurem ent using SNM M based AFM . This
technique is successfully used to measure dielectric constant o f thin films on low resistivity
silicon. Numerical simulation using finite elem ent m ethod (FEM) used to extract dielectric
constant o f thin films from m easured capacitance values.
Chapter 5 presents the theory and im plem entation o f a m odified netw ork analyzer
(impedance analyzer) architecture for broad-band high-precision m icrowave reflection
coefficient m easurem ent o f high im pedance loads. As an application o f the im pedance
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25
analyzer, we use it to m easure m inute changes in capacitance between the very tall conductive
AFM probe’s tip and thin film dielectric sample in SN M M based AFM probe experiment.
In chapter 6, w e summarize our achievem ent in the field o f SNM M and present the
future research direction.
Appendix C presents an alternative approach to focus electrom agnetic waves in
microwave regim e using left handed m edia w hich exhibits negative refractive index property
under certain conditions. W e im plem ented this m edia as one dimensional transm ission line for
50 GHz operating frequency.
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26
1.7. References
[1] E. Abbe, “Beitrage zur Theorie des M ikroskops und der m ikroskopischen W ahm ehm ung,”
Archivfuer mikroscopische anatomie und entwicklungsmechanik 9 ,4 13 (1873).
[2] D. Darid, Scanning Force Microscopy with applications to electric, Magnetic, and Atomic
Forces, Oxford university press, 1991.
[3] G. C. Cho, H. Chen, S. Kraatz, N. Karpowicz, and R. Kersing, ” Apertureless terahertz
near-field m icroscopy,” Semiconductor science and technology, 20, S286 (2005).
[4] B. Rosner, “Near-field m icroscopy from the m icrowave regim e to the visible,” Ph. D.
thesis, University of Delware, New ark, Delw are, 2002.
[5] A. Imtiaz, “Quantitative m aterials contrast at high spatial resolution with a novel
near- field scanning m icrowave m icroscope,” Ph. D. thesis, U niversity o f M aryland,
College Park, M D, 2005.
[6] S. Ramo, J: R. W hinnery, and T. van Duzer, “Fields and w aves in com munication
electronics,” Third Edition, John W iley & sons, 1993.
[7] J. W. Goodman, Introduction to Fourier optics, M cG raw-H ill, 1988
[8] C. Gao, F. Duewer, and X.-D. Xiang, ’’Quantitative m icrowave evanescent
m icroscopy,” Applied physics letters, 75, 3005 (1999).
[9] E. H. Synge, "A suggested m ethod for extending the microscopic resolution into the
ultramicroscopic region," Philosophy magazine 6, 356(1928).
[10] R. F. Soohoo ”A m icrowave m agnetic m icroscope,” Applied physics letters, 33,
1276 (1999).
[11] E. A. Ash, and G. N icholls, ’’Supper-resolution aperture scanning m icroscope,”
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Nature (London) 237, 510(1972).
[12] M. Golosovsky, E. M aniv, D. Davidov, and A. Frenkel,” Near-field o f a scanning
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34
2
Routes to higher Held
quadraxial near-field probes
localization
through
In chapter 2, w e propose and dem onstrate a m iniaturized quadraxial probe that em ploys a
differential feed technique, for use in near-field RF/m icrow ave transm ission microscopy. O ur
quadraxial probe electric field measurements show higher electric field localization than a
conventional coaxial (monopole) probe. The higher electric field localization that we observed
in the experiments is corroborated by simulations o f coaxial and quadraxial probes.
The
details, regarding our m otivation, and experimental design, an associated simulations and
measurem ents, will also be discussed in depth in this chapter.
2.1 Introduction
Scanning near-field m icrowave (SNM ) m icroscopy can quantify the local highfrequency electrom agnetic properties o f m aterials and devices. The near-field probe is the
m ost critical elem ent in a SNM microscope, since it determ ines the ultimate resolution o f the
entire system. Coaxial near-field probes have been investigated by several groups [1-10], and
because o f this, these structures have been previously electrically well characterized.
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35
However, non-zero currents and electric fields on the coaxial shield ultimately lim it the field
resolution of coaxial probes.
Here, we describe a probe design for im proving the spatial resolution of SNM m icroscopy
measurements. In conventional m iniaturized coaxial probes, electric field localization depends
on the length o f the protruding center conductor and also on the diam eter o f shield that
surrounds the center conductor. In order to decrease the effect of the coaxial shield on the
sample in SNM m icroscopy and to further localize the fields, we have developed a quadraxial
probe (i.e. four conductors arranged around one axis), plus we also employ a differential feed
technique (subtracting tw o signals induced on the center conductor and the third conductor of
the quadraxial probe, w hile the second and fourth conductors are grounded). D ifferential
measurements are a well-established technique to suppress noise in electronic circuit
measurements [11, 12].
2.2 Probe design and fabrication
W e fabricated a quadraxial probe from starting with a coaxial probes, and then adding two
separate concentric m etallic shield layers and tw o dielectric spacer layers (Figure 1.1). A
separate coaxial probe w as used either as an em itter, or a detector, and was fabricated from a
50 £2, semi-rigid coax (outer diam eter = 508 pm , inner diam eter = 112 pm ), w here the
protruding center conductor length was 300-500 p m long. This length was chosen to be
com parable to, or shorter than, the shield diam eter [2]. The dielectric material used was Teflon
(er = 2 . 1).
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36
In order to design the quadraxial probe for m axim um electric field localization around the
center conductor in the near-field, we first need to find the field pattern. In the case o f a simple
geom etry such as an ideal dipole having a know n current distribution, one can calculate the
field pattern around it by simply solving for the (magnetic) vector potential (A) using the wave
equation 2 .2 . 1 .
V 2 A
H
=
/ueA =
C
O
2
+
—
-jAj
2 .2 .1
V x A
2 .2 .2
M
Q uadraxial
Protruding
center
conductor
(b)
Fig. 2.1: (a) Near-field microwave/ RF electric field measurement setup. The four conductors
of the quadraxial probe are numbered, (b) Photograph of the quadraxial probe.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
37
£
=
— !—
jcoe
( VxH- J)
2.2.3
, and then calculating magnetic field (H) or electric field (E) from 2.2.2 and 2.2.3, respectively
[14]. B ut in the case o f the com plex geom etry o f quadraxial probe, we em ployed finite
elem ent m ethod (FEM) software (Ansoft H FSS[15]) to optim ize the near-field pattern o f the
0.27
300 fan
1
2
(a)
0.032
.031
300 f# t
1 2
3
4
(b)
Fig. 2.2: Normalized iso-lines magnitude of total electric field (at 600 MHz) calculated using
numerical method (FEM) in half cross-section view for (a) the coaxial probe and (b) the quadraxial
probe.
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38
quadraxial probe.
In our model, the probe was surrounded by air with radiation boundary
conditions, and a PM L (perfect m atched layer) was placed in a plane normal to, but far away
from, the probe’s tip. Simulations were run using the adaptive mesh technique with
As < 0.02 at discrete frequencies between 0.3 and 1 GHz. By simulation, we found that the
electric field on the protruding center conductor tip could be maximized, and the electric field
on the first shielding conductor could be m inim ized if conductors 1 and 3 w ere driven
differentially (two signals having the same magnitude, but a with 180 degree phase
difference), and if the thickness o f the three dielectric layers were all made equal (which is the
ideal geometry). In the quadraxial probe fabricated, the thickness o f the dielectric layer
between conductors 1 and 2 was 135 pm , between conductors 2 and 3 was 216 pm , and
between conductors 3 and 4 the thickness was 241 pm . These thicknesses approxim ate the
ideal geometry.
Figure 2.2 shows the iso-lines magnitudes o f total electric field simulation results for the
coaxial probe (Fig. 2(a)) and for the quadraxial probe with the above dim ension (Fig. 2(b)) in
the near-field when each probe is simulated as an emitter. From Figure 2(b), we calculated that
the normal electric near-field com ponent (Ez) above conductor 2 (at the point m arked by a
circle) o f the quadraxial probe was suppressed by a factor o f ~ 2.6 when com pared to E z above
the coaxial probe at the same point (as seen in Figure 2(a)).
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39
2.3 Electric field localization measurement
The electric field in the near-field zone o f the quadraxial probe was m easured by a coaxial
(near-field) probe using an Agilent N 5230A (PNA series) 4-port vector netw ork analyzer
(VNA) configured as diagram ed in Figure 1(a). Here, the quadraxial probe used a protruding
center conductor length o f -3 0 0 pm. The coaxial probe is sensitive to the norm al com ponent
of the electric field due to its axial sym m etry [2, 7]. Tw o ports o f the VNA were used as the
differential port 2 for excitation of conductors 1 and 3 of the quadraxial probe. The third port
of the VNA w as connected to the near-field coaxial probe as the single ended port 1 in order to
measure the voltages induced on it from the electric field in the near-field zone o f the
quadraxial probe. This voltage is approxim ately proportional to Ez (the proportionality factor
depends on the frequency [7 ,1 6 ]) o f the quadraxial probe.
U sing a sim ilar procedure as in [16], we determ ine the relationship between the induced
voltage on the near-field coaxial probe and m easured electric field. A ssum ing a quasi-static
approxim ation in near-field o f the probe (static analysis), we determ ined the charge Q induced
on the exposed face o f center conductor (A) o f the near-filed coaxial probe. From G auss’s law
[17]:
\A D d s = Q
2.3.1
A ssum e a cylindrical symmetry o f probe’s center conductor, thus
£qE z A i + £0E r A,2 = Q
2.3.2
Ai is area o f the bottom of the center conductor, A 2 is outer cylinder area, E z is normal
com ponent electric field on Ai and Er is the normal com ponent on A 2 in cylindrical coordinate
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40
(not shown in figure 2.1 (a), but it coincides with the Cartesian coordinates shown in the
figure). Each field com ponent on the corresponding face was approxim ated to be alm ost
homogenous. Note that near to the center conductor o f quadraxial probe, the Er Com ponent of
the field is much sm aller than E z, thus 2.3.2 can be approxim ated as:
£0E z A i =Q
2.3.3
On the other hand:
Q = I I.d t; or I = jCOQ
2.3.4
Form 2.3.3 and 2 .3 .4 ,1 have:
I ~ j(oe0E z Ai
2.3.5
W here, I is the induced current on the near-field coaxial probe’s center conductor. If we also
assume that V = Z QI (matched probe), (V: voltage induced on the center conductor o f near­
field coaxial probe), we have:
|v| ~ (oe0Z 0E z A\
2.3.6
As we can see, the voltage induced on coaxial probe’s center conductor is approxim ately
proportional to the Ez o f the near field of quadraxial probe, when we probe on the proxim ity of
the quadraxial probe’s center conductor.
The m easurem ent used a single-ended, differential m ode transm ission coefficient from the
VNA, Ssdi 2 [11-13]. If we assume a thevenin equivalent voltage E 2,Thfor differential port 2,
and Zo2d as its intrinsic im pedance, the transm ission coefficient SSdi2 can be calculated in a
m anner sim ilar to that o f single-ended S 12 [18], as follows:
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41
Ssdi
& 7\
W
E2,Th
2.3.7
where, V is the voltage induced in port 1 from the differential port 2. From 2.3.6, and
2.3.7, we concluded that Ssdi 2 is approxim ately proportional to E z near the quadraxial probe’s
center conductor.
D uring measurements, the quadraxial probe rem ained motionless while the coaxial probe
w as scanned laterally (y-axis) in ~ 6 pm steps. W hile scanning, m easurem ent o f Ssdi 2 was made
from 100 M Hz up to 1 GHz. The vertical distance between the two tips was ~ 6 pm when they
were co-linear and facing each other.
The results of the electric near-field scanning measurements described above are shown in
Figure 3. For com parison, this figure also shows results o f the same measurements carried out
for triaxial and coaxial probes (as em itters), while using the same near-field coaxial probe as a
detector. For these m easurem ents, the transm itting coaxial probe’s size was identical to the
receiving coaxial probe’s size, except for the protruding center conductor lengths which were
-3 0 0 pm for transm itting coaxial probe, and -1 m m for the detecting near-field coaxial probe.
The triaxial probe was fabricated by adding a metallic shield layer and a dielectric spacer layer
concentrically around the original coaxial probe. The thickness o f the dielectric layers between
conductor 1 and 2 was 135 pm , and between conductors 2 and 3 was 216 pm , which are the
sam e thicknesses as those discussed earlier for the quadraxial probe. This choice approxim ated
an equal separation between any neighboring pair o f conductors in the probe structures (both
triaxial and quadraxial).
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42
1
1
1
Triaxi
Coaxial
Quadraxial
E i s im u la te d fo r q u a d ra x ia l
s im u la te d fo r c o a x ia l
1
------------------------- ----------------------t -
0
300
----------- 1------------ r 0
600
900
1200
Distance along line scan-y axis (pm)
Fig 2.3: Comparison of the electric field magnitude fall-off measurements. For coaxial probe,
we measured S J2 (single-ended transmission coefficient) from the PNA. For the triaxial probe, we
measured Ssdi2 (single-ended differential mode transmission coefficient) from the PNA. For the
quadraxial probe, we measured Ssdi2 from the PNA. We calculated Ez (normal electric field component)
of coaxial, and quadraxial probes from FEM simulations. All measurements and simulations were
performed at 600 MHz.
The triaxial probe’s electric near-field m easurem ent also used Ssdt2, obtained from the VNA.
The coaxial probe electric near-field m easurem ent used a conventional single-ended
transm ission coefficient, S 12 [IB], which is proportional to the induced voltage on the near­
field probe (Vi), as follows:
2.3.8
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O ur transm ission coefficient results show that the electric field in the near-field zone o f the
quadraxial probe falls off much faster with lateral distance r from the center conductor when
com pared to the conventional coaxial probe. Figure 3 indicates that the lateral spatial extent of
the m easured electric field in the near-zone of quadraxial probe was sm aller by ~ 2 com pared
to the m easured electric field in the near-zone of coaxial probe, which is in a good agreement
w ith the simulation result. The sim ulation data presented in figure 3 is a line cut through the E z
data o f figure 2, across the lines labeled X. It can be seen that the decay o f S 21 and Ssdi2 with
distance agree with the Ez simulation results near the center conductor o f coaxial probe and
quadraxial probe, respectively. However, the m easured probe signal is larger than the
sim ulated result obtained further aw ay from the center conductors. This is due to probe
response which depends on Ex and Ey when two probes are not co-axially aligned. Thus, we
need to use a near-field probe with a m etallic plate to shield from the effect of the transverse
electric field com ponent [19]. The effect o f mismatch between the quadraxial probe and 50
ohm VNA feeding netw ork was found negligible because the operating frequency was
low (several M Hz), and thus the quadraxial probe length was electrically small, but in higher
operating frequencies, we have to m ake appropriate correction for phase and magnitude o f the
signals in order to have balanced signals at the end o f the probe.
2.4 Application of Quadraxial probe in SNMM:
W e used the quadraxial probe to im age a 100 pm wide metal (gold) strip on a glass substrate
by m easuring the spatial field perturbation caused by the presence of the metal and monitoring
the transm ittance between the probe and a coaxial antenna (Figure 4(a)). Here, the metal strip
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44
on glass substrate was m oved laterally (y-axis) relative to the quadraxial probe (used as the
receiver) that was placed in close proxim ity to the sample (~ 6 pm). A large, blunt coaxial
antenna (outer diam eter = 6 mm, inner diam eter = 1.64 m m ) was used as an em itter in order to
generate a reasonably hom ogenous field around the sample. The effect of third and fourth
conductors of quadraxial probe on the metal strip sample may be m odeled as mutual
capacitances which are negligible in com parison to m utual capacitance between the center
conductor tip and the metal strip. For comparison, a coaxial probe was also used as a detector
(instead o f the quadraxial probe) in a separate experim ent, where all other param eters were
held constant. The coaxial probe had the same protruding center conductor length, the same
inner conductor diam eter (1 1 2 pm ) and the same first shielding layer diam eter as that o f the
quadraxial probe. Figure 4(b) shows the magnitude o f Sds2i, and S 21 for the quadraxial and
coaxial probes, respectively. U sing a differential scattering-param eter relation to single-ended
ones, we found [13]:
(Appendix A. gives m ore details about this calculation):
S d s2 1 = (S 2 1 -S 3 i)/c , C -
in which
S 21
V2
2.4.1
and S 3) refer to single-ended scattering param eters. Port 2 and 3 were connected
to conductor 1 and 3 o f the quadraxial probe, while port 1 was connected to the coaxial
antenna, as in figure 2.4(a). The spatial resolution can be deduced from the im age profile of
figure 2.4(b); the full width at half m axim um for the quadraxial probe case was 151 pm , while
for the coaxial probe it was 174 pm , a -12+1
increase in spatial resolution. Figure 2.4(c)
shows that the phase m easurem ent for the quadraxial probe (phase of SdS2i) has a larger
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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(D
L_
OJ
CD
TJ
-66
C o a x ia l
o -64 H
|
OQ
400
x
-6 8 1
600
R3
s
(b )
70
69
Quadraxial probe
68 H
67
66
0
Coaxial probe
200
400
600
CL □ s tan ce along line scan-y axis (pm )
(c)
Fig. 2.4: (a) Scanning microwave/ RF transmission microscopy measurement setup, (b)
Magnitude of transmission Sd^i and S2i versus position while scanning across the 100 pm wide strip
line for the quadraxial probe and the coaxial probe, respectively (c) phase measurement. All
measurements were made at 100 MHz. The metal stripe on glass sample is sketched at the top of this
Figure, where the width of the strip has the same scale as the y-axis of the plot.
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46
change (by a factor ~3) than the same m easurem ent (phase o f S 21) using the coaxial probe,
which equates to a m ore sensitive phase measurement.
W e m odeled our quadraxial probe as two-decoupled transm ission lines that only couple to
each other ju st at the very end o f the tw o lines, the tip. The coupling between the quadraxial
probe and coaxial antenna in figure 2.4 (a) was modeled as: 1) coupling capacitor (Cci )
between the end o f one transm ission line which consists o f the center conductor and the first
shielding conductor o f the quadraxial probe and coaxial antenna, 2 ) coupling capacitor CC2
between the end o f the other transm ission line which consists o f the third conductor and the
forth conductor o f quadraxial probe, and coaxial antenna, and finally 3) coupling capacitor C c3
between the ends o f these tw o transm ission lines. The model of quadraxial probe and its
coupling capacitors as described above is shown in figure 2.5 (a). Figure 2.5 (b) shows the
com parison between the m easured and sim ulated Sds2i. This model was sim ulated in ADS
software from Agilent technologies [21]. The calculated values for the three coupling
capacitors are shown in figure 2.5 (a). Figure 2.5(b) shows a good agreem ent between theory
and experim ent up to ~ 600 MHz. But, above 600 M Hz, the effect o f the m ism atched between
differential port-2 to quadraxial probe will degrade the performance. A nother reason in this
perform ance degradation is that the phase between two ends of differential quadraxial probe is
altered from 180 degree.
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47
Differential Port-2
Quadraxial
Probe model
TL
TL
Open-ended coaxial
TL
Single-ended Port-1
(a)
-40
m -5 0
TJ
— ADS Model
• Expriment
-90,
0.2
0.8
0.4
(b)
Fig. 2.5: (a) A simple model for quadraxial probe in microwave/ RF transmission microscopy
measurement setup, (b) Comparison between the simulation and experimental results for SdS2i using the
same setup and parameters as used in figure 2.4 (a).
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48
2.5 Conclusion
In summary, we have dem onstrated a quadraxial electric field probe for use in scanning
near-field RF/m icrowave microscopy. Unlike other designs, our probe is driven differentially,
which allows it to suppress the electric field near the first shielding layer around the center
conductor of the probe, thus decreasing the effect o f stray capacitance from the first shielding
conductor to the sample. W e have dem onstrated higher spatial resolution and larger phase
changes in com parison to conventional near-field coaxial probes. Figure 2.3 predicts that by
adding aditional metal layers to the quadraxial structure, and by driving the adjacent layer 180
degrees with respect to each other, we can localize the electric field even further.
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49
1.6 References
[1] B. T. Rosner, T. Bork, V. A grawal, and D. W. van der W eide, ’’M icro-fabricated silicon
coaxial field sensors for near-field scanning optical and m icrowave m icroscopy,” Sensors
and actuators A (physical), 102, 185(2002).
[2] R. Kantor, and I. V. Shvets, "Method o f increasing spatial resolution o f the scanning near­
field m icrowave microscopy", Journal o f applied physics, 9 3 ,4 9 7 9 (2003).
[3] M. Fee, S. Chu, and T. W. H ansch,” Scanning electrom agnetic transm ission Line
microscope with sub-wavelength resolution,” Optics communications, 69, 219 (1989).
[4] Y. W ang, and M. Tabib-Azar, “D esign and fabrication of scanning near-field
m icrowave probes com patible with atomic force m icroscopy to im age em bedded
nanostructures,” IEEE Transaction on microwave theory and techniques, 52, 971
(2004).
[5] Y. W ang, C. A. Paulson, G. Ning, and D. W. van der W eide, “Fabrication and
M easurem ents using near-field ultra-tall silicon coaxial tips,” IEEE International
M TT-S microwave symposium digest, pp. 2147 (2005).
[6 ] C. P. Vlahacos, R. C. Black, S. M. A nlage, A. Amar, and F. C. W ellstood, ’’N ear­
field scanning microwave m icroscope with 100 pm resolution,” Applied physics
letters, 69, 3272 (1996).
[7] Y. Gao, A. Lauer, Q. Ren, and I. W olff, “Calibration o f electric coaxial near-field
probes and applications,” IEEE Transactions on microwave theory and techniques, 46,
1694(1998).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
50
[ 8 ] F. Keilmann, D. W. van der W eide, T. Eickelkam p, R. M erz, and D. Stockle, “Extrem e
sub-wavelength resolution with a scanning radio-frequency transm ission m icroscope”
Optics communication, 129, 15 (1996).
[9] M . Ferrara, and O. Odoradi, “A scanning capacitance m icroscope based on a coaxial
resonator: High sensitivity and cheap and com pact design without the use of lock-in
detection systems,” Review o f scientific instruments, 74, 2735 (2003).
[10] C. Gao, F. Duewer, and X.-D. X iang, ’’Quantitative microwave evanescent
microscopy,” Applied physics letters, vol. 75, 3005 (1999).
[11] D. E. Bockelman, W . R. Eisenstadt, “Combined differential and com m on-m ode scattering
parameters: theory and sim ulation,” IEEE Transactions on microwave theory and
techniques 43, 1530 (1995).
[12] Agilent technologies application note : literature no: 5988-2924 EN, pp. 1373-6.
[13] Dallas semiconductor/Maxim Application Note: H FA N -5.1.0 Rev 0.
[14] W. L. Stutzman, Gary A. Thiele, Antenna theory and design, second edition, john wiley
& sons, 1998.
[15] ANSOFT corporation, Pittsburgh, PA.
[16] S. K. Dutta, C. P. V lahacos, D. E. Steinhaur, A. S. Thanawalla, B. J. Feenstra, F. C.
W ellstood, and S. M. A nlage,” Im aging m icrowave electric fields using a near-field
Scanning m icrowave m icroscope,” Applied physics letters, 7 4 ,1 5 6 (1999).
[17] S. Ramo, J. R. W hinnery, and T. van Duzer, “Fields and w aves in com m unication
electronics,” Third edition, John Wiley & sons, 1993.
[18] G. Gonzalez,” M icrow ave transistor am plifier,” Second Edition, Prentice Hall, 1996.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
51
[19] J. S. Dahele, and A. L. Cullen,’’Electric probe measurem ent on m icrostrip,” IEEE
Transactions on microwave theory and techniques, M T T -28, 752 (1980).
[20] D. Baudry, A. Louis, and B. M azari,” Characterization o f the open -en d ed coaxial probe
used for near-field m easurem ent in EM C applications, ” Progress In electromagnetic
research, PIER 60, 311—333( 2006).
[21] Agilent technologies, Santa Rosa, CA.
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52
Appendix A: Differential scattering parameters
Here, I sum m arized the basic concepts o f scattering param eters. In general, voltage and current
on any location, x, on a transm ission line can be written as [18]:
V ( JC) = V + (JC) + y - ( j c )
A .l
Lq
B y definition:
A.3
a(x) is called a norm alized incident voltage wave
b(x)s— M -
AA
b(x) is called a norm alized reflected voltage wave
Thus,
a (i) = - J
= [V U )tZ J (i)]
A .5
2^57
b(x) = — j = [ V ( x ) - Z „ l ( x ) l
A .6
2V z J
Figure A .l shows a general tw o port network. A tw o port network can be represent by
scattering parameters through a incident wave ai(li) and reflected wave bi(li) at port 1 , and
incident wave a 2(l2) and reflected w ave b 2(l2) at port 2 as following:
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53
Output
port
Input
port
a ifr)
-01
3
fii)
JMX)
Two-port
network
K % ( \2 )
b2(l2)
a2(x)
b2(x)
-02
/V >
Port 1
Port 2
X i= li
X 2 l2
Fig A .l: Incident and reflected waves in a two-port network [18].
P 2 ( h )_
~sn
$12
_S 2l
^22
S n , S 12 , S 2 1 , S 22
A.7
_
are called
_a 2 ( h )_
S
param eters m easured at specific location on port 1 and port 2 as
shown in figure A .l. F or exam ple, S 21 can be calculated as following by term inating the out
put port properly:
J 21=
^
A .8
" i t f l ) „ ,( / ,) = 0
This is called a single-ended transm ission coefficient.
In 1995, Bockelman, et al. [11] generalized the scattering param eters concepts to the
m icrowave differential circuits by introducing differential and com m on-m ode scattering
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54
param eters (mixed-mode). For example, consider a three port network shown in figure A.2.
A ssum e that Port 2 is a differential port and port 1 is a single-ended port.
Two-port
Input
port
altfx)
' a $ 1)
Differentialsingle
ended
or
AW
-
Port 1 single­
ended X^l.,
jw y
a2(x)
b2(x)
o2
/\ >
Port 2
X2-l2
Port 2 differential
^d2
Three-port
Single
ended
-od2
b3(l3)
a3(x)
b3(x)
”o3
Port 3
^ 3 “ *3
Fig. A.2: Incident and reflected waves diagrammed in a two-port network (differential single­
ended). Port 1 is single-ended and port 2 is differential. Differential port 2 is shown in terms of the
equivalent two single-ended ports.
L et’s define the differential-m ode voltage, and current at node x [11]:
^ 2W
^ 2W -^ W
Id2(x) = \ v 2( x ) - h ( x ) )
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A.9
A.10
55
The differential norm alized voltage waves at port 2 based on A .5 and A .6 definition becomes
[ 11]:
1
arf 2 (*) =
Wd 2 ( x) + z0d2I d2( x))
A .l l
Wd2(x)-Z0d2Id2(x)]
A.12
2-\jzod2
b dl i x) =
^yjzod2
A .13
Zod2 —2Z02 ~
Sds21 =
bd2(ld 2 >
ad2(ld2) = °
A .14
U sing A.9, A.10, A.12, and A .13 in A .14, gives:
Sds2l =
1 b2 (l 2)-b3(l3)
a /2
a \ (l\)
0 2
( h ) = ®’a3( h)
A.15
Thus, SdS2i in terms of 3 port single-ended scattering param eters can be described as:
s d s 2 l = ^ ( s 2 l ~ s 3l)
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A .16
56
Experimental measurements with multi-functional
micro fabricated probes for SNMM
In this chapter, I propose and dem onstrate a solution for increasing the sensitivity o f the
reflection coefficient m easurem ent when using micro-fabricated coaxial probes in a SNM M
based application. I em ployed this technique in order to im plem ent a SNM M measurement
system co-integrated with a AFM system.
The details o f m y motivation, design approach,
prelim inary m easurem ent results, and future plans will be discussed in more depth in this
chapter.
3.1 Introduction
O pen-ended coaxial probes have been previously used for perm ittivity m easurem ent of
materials on macro scale using reflection coefficient m easurem ents [1, 2], Reflection
coefficient measurem ent for material permittivity characterization has the advantage o f a
m inim um sample preparation, and is a non-destructive process. However, as the dimension of
a coaxial probe shrinks from m acro scale to m icro scale, the sensitivity of reflection
coefficient m easurem ent also decreases. This can be mathem atically proven as follows:
From appendix o f Chapter 2 , 1 have:
The reflection coefficient is defined as:
V+
a(x)
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3.1.1
57
W here V", V+ are the voltage reflected waves, and the voltage incident wave at node x,
respectively. From definition:
a(x) = — j = \ V ( x ) + Z 0I ( x )]
\ o
3.1.2
b(x) = - j = [ V ( x ) - Z 0I(x)]
y Z Q
3.1.3
21')
+
^1)
^ V (,)
o
Port,
One-port
network
X =l'
Fig. 3.1: Incident and reflected waves in a one-port network. Input im pedance is Z\, and
the reflection coefficient at port x=l is T ( l ).
A ssum e Zj is the input im pedance o f the one-port netw ork at x=l which is shown in figure 3.1:
Z, =
iu >
Substituting 3.1.2 and 3.1.3 in 3.1.1:
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3.1.4
58
nn
b{l) _ V { l ) - Z om
a(l )
V(l) + ZQI(l)
Substituting 3.1.4 in 3.1.5:
T(/) =
Zj Zp
Zi + Z0
3.1.6
0.8
0.6
'iVi'iir
i t».»•
I 04
|
0.2
£
o
0
1 -0.2
£
® -0.4
oc
-
0.6
0.8
Load im oed an ce/oh m l
Fig. 3.2: Reflection coefficient versus load impedance (Zi). In the region (a), the reflection
coefficient r(Z) is very sensitive to changes in Zj which is very close to Zq=50 ohm. In regions (b), and
(c), the reflection coefficient F (/) has a low sensitivity to Zi changes which corresponds to either a
very high impedance: region (b), or a very low impedance: region (c).
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59
W hich is the reflection coefficient at the port x=l. N ow, assume for simplicity that Z jis a pure
resistive load. The reflection coefficient ( T ( /) ) versus the load im pedance (ZO is shown in
figure 3.2. In region (a), T(l) is very sensitive to Zi changes which is close to Zo=50 ohm
(since the slope o f the curve in figure 3.2 is very steep around 50 ohm). In the regions (b), and
(c) T(l ) has a low sensitivity to Zi change, (since the slope o f the curve in figure 3.2 is
flattened out when distant from the 50 ohm im pedance). M iniaturized coaxial probes perform
as electrically small antennas and exhibit very high input im pedances at microwave
frequencies [3], thus a reflection coefficient m easurem ent using these probes will have a very
poor sensitivity using 50 £2 equipments. Now the question is, what is the solution to increase
the sensitivity, which is very important for material characterization in m icro/nano scale
regime. The solution is a resonator. In the follow ing sections, different ways to im plem ent a
sensitive resonator suitable for our fixture are proposed and implemented.
3.2 SNMM integrated with AFM
In order to perform simultaneous material m easurem ents and surface topography in
micro/nano scale, we have constructed a SNM M com bined with a com mercial AFM. B jom
Rosner from our lab, built the first version o f this microscope [4], An A FM measures the
topography of either insulators, or metallic sam ples using a sharp nm size tip at the free end of
a cantilever. The forces between the tip and sample surface cause the cantilever to bend. The
cantilever’s deflection is m easured via an optical detection m ethod in order to track the
topography o f the sample. In m y opinion, there are m any crucial factors to build a SNM M
com bined with an AFM . These include but are not lim ited to 1) design and fabricate a suitable
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60
probe 2) design and fabricate a sensitive m easurem ent system for SNM M 3) design and
fabricate a suitable fixture to m easure the quantity of interest, for exam ple in order to m easure
the dielectric constant o f bulk/ thin films, an electric field probe is needed. Coaxial near-field
probes having a tapered, protruding center conductor, are good candidates for SNM M since
they do not exhibits a cut off frequency for long-wave length and can provide localized
m icrowave m easurem ent [5]. Because o f these attributes, m icro-fabricated cantilever’s coaxial
tip probe are ideal for perform ing SNM M integrated with AFM . These micro fabricated m ulti­
functional probes are connected through a micro-coaxial cable to the external circuitry such as
Laser diode
Mirror;
Photo I
detector
P iezo
controller
SNMM
M icro-coax
Probe
Microwave
Sweeper A
De-coupling
.Ref.
Sam ple under test (SUT)
AFM
Directional
/ Coupler
DAQ
Lock-in
Amp
Microwave
S w eep er B
(M
)
Fig. 3.3: (a) A schematic of our SNMM combined with a commercial AFM. The AFM uses an
optical detection method to track the topography of the sample. By adding a microwave signal to the
probe tip, simultaneous microwave and AFM topographical imaging of SUT is made possible, (b)
Probe interface to macroscopic instrumentation: the chip body of the probe connects to a coplannar
waveguide (CPW) with gold wires, and CPW connects to a low-loss micro-coaxial cable, and a SMA
connector (not shown in the picture).
a reflectom eter [6 ]. A reflectom eter is used to separate the reflected voltage wave from the
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61
incident voltage wave. The inform ation contained in a reflected voltage w ave can be utilized
to perform localized m icrowave measurements of m aterials. Figure 3.3 (a) shows our SNM M
m easurem ent system retrofitted for use on a Topom etrix™ A FM system. The electric field at
a near-field of the probe apex is perturbed by the region o f the sample under test (SUT) which
is scanned. W e m easured this perturbation by m onitoring the reflected voltage wave. The
reflected voltage wave contains inform ation about the SUT, such as resistivity, permittivity, or
doping level in the case o f semiconductors.
The m icro-coax com ponent o f the SNM M
consists of an approxim ately 17 cm long, coaxial cable (AW G-42 from Gore), o f which one
end is connected through a CPW and gold wire bonds to the coaxial tip probe, while the other
end is coupled to a m icrowave source through a mism atch (de-coupling capacitance). The
reflected wave, delivered from the directional coupler, is then down-converted using a m ixer
and fed to the input o f a lock-in am plifier (LIA) w hose reference signal is derived from downcon version of the reference signal [7]. The output of the LIA, which is proportional to the
reflected wave, is then delivered to an input o f the A FM system to create an image.
3.2.1 Micro-fabricated tall coaxial tip integrated with the cantilever
, and coplanar waveguide (CPW)
The short tip height (-1 0 fim) o f currently available SNM M coaxial A FM probes creates
an unwanted electrom agnetic interaction between the chip body and the sample, resulting in
large parasitic capacitances [4 ,7 , and 8 ]. One w ay to decrease this parasitic capacitance and to
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62
M
Cr/Au
[U lSi
— ^ Si02
2 m m 45-50 p m f jjl
Fig. 3.4: (a) Schematic of tall coaxial tip, cantilever, and chip body integrated with coplanar
waveguide. The typical dimension of the probe is shown (b) SEM micrograph for an tall coaxial tip and
an integrated CPW structure (c) Close-up of the apex region showing a ~3-pm-radius aperture.
achieve a more localized microwave m easurem ent (perhaps quantitative m easurem ent) is
tofabricate taller tips.
To dem onstrate this approach, we have designed and fabricated tall
coaxial tips (having heights up to 50 fim) and integrated them with the cantilever and a
coplanar waveguide (CPW ) on the chip body, as shown in figure 3.4 [9].
3.2.2 Mechanical design of the rectangular cantilever of the probe
The spring constant o f a cantilever’s probes is the critical com ponent for determ ining the
mechanical response o f the cantilever. The resonant frequency, f 0, of the cantilever, when it is
vibrating, is given by /„ = ( 1 / 2 7t)(klm Q) , w here k is the spring constant and m0 is the
effective mass o f the cantilever. A sensitive force detection requires a low spring constant and
consequently a low resonant frequency. The m ore useful equation in calculating the first
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63
resonant frequency o f a cantilever, when supported on one side and vibrating along its vertical
direction, is given by [ 10 ]:
3.2.1
W here p is the mass density o f the cantilever material (silicon), I is the length o f
cantilever, Ac is the cross-section area o f the cantilever and I is the m om ent o f inertia and E is
Y oung’s (elastics) m odulus o f the cantilever material (silicon) [10]. The cantilever o f our
SN M M probe was designed with the sim ilar dim ension as the com mercial AFM probes used
in our system in order to be com patible with our system. U sing the dim ension in figure 3.4 (a),
and the silicon mechanical property ( £ = 9.8 x 1 0
10
^
N /m , p - 2.329x10 kg/m
^
), a resonant
frequency for cantilever (~ 5 pm cantilever thickness) was calculated as / q = 72.4 kH z.
3.2.3 A simple model for our SNMM
C onsider the SNM M part o f our system in figure 3.3(a). The SN M M ’s operating frequency
is considered
< 2.6 GHz . The m icro-fabricated probe is connected by a micro-coaxial
transm ission line to a m ism atched m icrowave source (sweeper A with a voltage Vs (CO) ) as it
is shown in figure 3.5. In this figure, co is the angular frequency, Zth is the source im pedance
plus de-coupling capacitance.
C c.pr0be
is the coupling capacitance between the probe and the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
sample. Now, we can calculate the reflected voltage wave V' at the m icro-coax input
mismatch. First, let’s define the load im pedance as:
zs
Cd
Vs
Micro-coax,
Zo
Fig. 3.5: A sim plified schematic o f SNM M : coupling capacitance betw een the probe and
the sample
sample
( C c.probe)
( C p) ,
consists of parasitic coupling capacitance between the chip body and the
and the coupling capacitance betw een the tip’s shielding and the sample
( C Sh-s) ,
and the coupling capacitance between the tip and the sample (Cts).
Z / = - ------
>
3.2.2
Jcc—probe®
By substituting 3.2.2 into 3.1.6:
1
Jcc—probe®
i cc-probe®
-ZO
3.2.3
+ Z0
Assum ing thata)ZoCc- p robe « 1 , equation 3.2.3 is simplified to:
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65
3.2.4
H O ~ exP( 2 JZ qCc—probe
N ow, using the standard transm ission line equations [11], with a sim ilar procedure as in [12],
we calculated:
L
1
<N
•s'
+2
1
J
1
z*
(N
lsL
+
L
1
i
V'
z*
3.2.5
•cos( 2 (/ 2L + Z qCc—pfpfeCoy)
J
CO
w here L is approximately the length of m icro-coax, and (3 - —
the phase velocity in the transm ission line).
is the phase constant (Vp is
Equation 3.2.5 shows that a reflected voltage
w ave is a function o f the probe to sample coupling capacitances. Thus, we can use the
reflected wave to image changes in coupling capacitance betw een the probe and a sample.
3.2.4 Coaxial tip probe simulation
W e m odeled our probe tip as a cone above a floating metallic sample in figure 3.6(a). Because
the probe tip height above the sample was much sm aller than the wavelength o f the operating
frequency, the near-field o f the probe is well approxim ated using a static analysis [ 11 , 13].
Thus, I solved the poisson equation to determ ine the m utual capacitances values between the
coaxial probe’s apex and the m etallic sample [13] with appropriate boundary conditions
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showed in figure 3.6(a). O ur simulation was perform ed using ANSOFT™ M AX W ELL 2D
software. The simulation results is shown in figure. 3.6(b) dem onstrate a logarithm ic decrease
(a)
Projected
0h
Tip
Floating sampl
0 2 nm
1
10
100
h=tip height above the sam ple (nm)
Fig. 3.6: (a) Cross-section schematic of the coaxial tip that was used for simulations.
D ifferent voltages (V in volts) and lengths are shown in draw ing (not to scale) (b) Tip- sample
capacitance calculated from this model shows a logarithm ic drop vs. distance (h).
o f tip-sample capacitance versus tip to sample separation distance. This shows a qualitative
agreement with an approxim ate analytical model w here the probe’s tip is m odeled as a sphere
above an infinite conducting plane [13, 14]. The sim ulations also show that both the shield to
sample capacitance and the shield to tip capacitance is fairly insensitive to the nanom eter scale
variation in tip height. Thus, the changes in the capacitance o f the tip apex for nm changes in
height, is m ostly due to the changes o f tip-sam ple capacitance.
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67
3.3 SNMM results for a metallic sample
W e used the tall coaxial probe to scan the 90 degree bend region o f a C PW waveguide
sample (consisting o f a gold conductor on an alum ina substrate). W e perform ed a non-contact
mode A FM scan at a cantilever mechanical resonance frequency of 98 kHz, and a scan rate o f
— Gold
(a)
— Air
2.061
2.064
2.067
Freq(GHz)
0.57V
0
100pun
0.46V
50 100 pun
Fig. 3.7 (a) Magnitude of the reflection coefficient near the resonant frequency (f„) when the
coaxial tip was in air, and over the gold sample. Simultaneous images using our probe in system of
figure 3.3 (a): (b) Topography image; (c) Microwave magnitude image.
lpm /s. In figure 3.3 (a), W e set the pow er level o f sweeper A at 18 dBm and its operating
frequency was chosen to be close to the resonant frequency o f our unperturbed resonator in air
(fig. 3. 7(a)). Sw eeper B was offset by 100 M H z from sweeper A and at a pow er level o f 0
dBm in order to provide a reference for LIA. The data presented in figure. 3.7(a) shows the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
68
m agnitude of the reflection coefficient
(S n )
and clearly dem onstrates that the resonant
frequency of the resonator decreases when the probe tip m oves from air to the gold (metal)
surface, since the capacitance between the tip and sample increases [15]. Figure 3.7 (b), (c)
shows the simultaneous topography and m icrowave (reflected wave-voltage) im ages o f the
sample, respectively. In figure 3.7 (c), the areas o f the im age exhibiting a brighter color
correspond to the gold and darker areas to the alum ina substrate. The spatial variation in the
m icrowave image confirm s that a localized capacitance m easurem ent has been made. The
problem with the resonant system described above was a low sensitivity o f the system. The
quality factor o f resonator (Q) o f the above system in the air was achieved as high as 680
which it may not be sensitive enough to m easure the small changes in er (dielectric constant)
o f thin film materials in nm scale. A nother difficulty was the frequency dependence o f the de­
coupling capacitor in order to create resonances. Thus, I propose a novel technique to over­
com e these problems. I also present a model o f the tall coaxial probes that models the probe
and its transition to the SM A connector. This model is needed to perform quantitative local
capacitance measurement.
3.4 Modeling of the tall coaxial probe and its transition to a SMA
Connector
RF/M icrowave characterization o f the tall coaxial tip cantilever based probes was
perform ed using ground-signal-ground (GSG) probes (Cascade M icrotech), and an Agilent
8510C vector network analyzer (VNA). A reflection coefficient m easurem ent
(S n )
for the
ultra-tall coaxial tip probe was m ade at frequencies ranging from 45 M Hz to 20 GHz as shown
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69
in figure.3. 8 . Figure 3.8 shows that the probe response is similar to a lossy open-ended
transm ission line i.e. at low er frequencies the probe looks like an open circuit), and then by
increasing the operating frequency, the S n response m oves around the circum ference o f the
smith chart. In order to perform calibrated m easurem ent at the end o f the
probe, a model of
the probe is required that incorporates its transition to micro-coax, and then to the SM A
1.0j
2.0j
0.5j
0.2j
5.0j
-0.2j
-5.0j
-0.5j
-2.0j
-1.0j
Fig. 3.8: Reflection coefficient
(S n )
of the tall coaxial tip integrated with CPW on chip body
from 45 MHz to 20 GHz. Measurement showed on Smith chart which acted like an open ended very
short transmission line.
Connector at the frequencies of interest. W e m odeled the probe, gold wire bonds, gold CPW ,
low loss micro-coax, and the SM A connector up to a frequency o f 1.2 GHz using ADS
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70
CPW on chip
body
Ld1
L=37.16 pH
R=0
Rd1
R =38.52 Ohm
Rd2
R=38.52 Ohm
C=333.339 fF
Transition from
:...
micro-coax to the CPjW
c
Coaxial tip
Ld4
Ij
L=37.16£>H
R=0
i!
R=10.436 kOhm
CPW on glass
Ld2
L=0.457 nH
R=0
L=1475 nH
Cp2
^C=212fF
C3
R=0
C =0.46500069 pF
Cp1
C =119fF
Gold wire bond
Lw
L=1.3 nH
R=0
Ld3
R
L=0.457 nH
Rp2
R=0
R=tl534 kOhm
(b)
Transition from SMA
to microstrip
Micro-strip
Transition from micro-strip to
Micro-coax
micro-coax
MLIN
j L
L2
Terml
L=5.81052
nH
SubsW'MSubr!
R=0
W=1.252 mm
L=6.6 mm
!
L=1.475 nH
'Q Z >
COAX
C
eg
' C4497.1232fF
I
C=378.16fF
_
TL5
Di=63.5 urn
Do=0.147625 mm
L=204.0704 mm
Er=1.3
TanD=0.0047
Rho=1
(C)
Fig. 3.9: Equivalent circuit model of tall coaxial tip probe with its interface to a macroscopic
instrumentation up to the SMA connector (a) CPW on chip body, and ultra tall coaxial tip (b) the CPW, and
gold wires (c) Micro-coaxial cable, micro-strip, and the SMA connector (see figure 3.3 b ).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
71
simulation
m easurm ent
-
10,
0.2
0.4
0.6
0.8
Freq(GHz)
1.2
(a )
200
100
-100
Simulation
m easurm ent
-200
1.5
0 5 Freq(GHz)
(b)
Fig. 3.10: (a) Magnitude and (b) phase of reflection coefficient
(S n )
of the micro-coax cable
assembly connected to tall coaxial tip probe. This figure shows both ADS simulation using the model,
and the measurement results performed using Agilent E 8364A PNA.
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72
software [16] and em ployed the optimization technique to find the optim um values o f circuit
elem ent’s model.
Figure 3.9 shows the equivalent circuit model for the probe with its
transition up to SM A connector. Figure 3.9 (a) shows equivalent circuit model o f the ultra tall
coaxial tip probe, figure 3.9 (b) shows the model o f transition from micro-coax to gold CPW
on glass, and the wire bond to
the probe’s chip body. Figure 3.9(c) shows the model of
transition from SM A connector (microwave source) to the low loss micro-coax. The low loss
m icro coax is A W G -42 was obtained from Gore Co. The micro-strip line was built on RO4003 C substrate from R oger corporation. W e made a circuit model based on the individual
m easurem ent o f every com ponent. Then, we m easured the reflection coefficient (S n ) of the
micro-coax cable assem bly connected to the tall coaxial tip probe. Figure 3.10 shows both the
m agnitude and the phase o f the measurement, as well as the circuit model simulation results,
which show a good agreem ent between the model and the measurement. This model is very
im portant for the de-em bedding process. The de-em bedding process is the process used to
m ove the reference plane o f S-parameters m easurem ent from the SMA connector to the end of
the probe’s tip which is necessary for the w ide-band measurement.
The m easurem ent for
figure 3.9(a) was perform ed using Agilent 8510C vector network analyzer (VNA).
M easurem ents in figure 3.9 (b), 3.9 (c), 3.10 were perform ed using a A gilent E 8364A PNA.
3.5 A novel method for the resonator of SNMM
Consider figure 3.5, instead o f creating a m ism atch using a de-coupling capacitor, a 50 ohm
shunt im pedance can be used in parallel with the m icro-coax cable assembly in order to create
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73
a half- wave length resonator. Figure 3.11 shows a diagram o f the basic idea o f this resonator
structure. This coupling m ethod is frequency independent, in com parison to capacitive
coupling [17]. N ow, w e derived an equation to relate the changes in the resonant frequency of
the resonator caused by changes in the capacitance at the end of the coaxial transm ission line
(for simplification, the effect o f the transition between the micro-coax and SM A is not
z
L
5
<---------------------------------------------------------------------------------
X
r
z1
Micro-coax,
c-probe
-o
Fig. 3.11: Schematic of the novel resonator used in SNMM: Shunt Z\ = 50£2 was used as a
coupling method for creating half-wave length resonator.
considered here, which still provides a good approxim ation at low operating frequency o f
SNM M around 1 GHz). The input im pedance o f a low loss transm ission line is [11]:
^
l + r z exp(-2y^L)
^in ~ ^ 0 , ^
, ^ . o T\
1 - T; e x p (- 2 y/?L)
By substituting 3.2.4 in 3 .5 .1 ,1 found:
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_
3.5.1
74
. - O ') ,
2 i/)d . i x °c '
,
zc P
z,„ = Z0
l- e x p ( - 2 j ^ ( L +
3.5.2
^ >robe )
Equation 3.5.2 shows the probe to sample coupling capacitance increases the effective length
o f transm ission line by:
^0
Cc-probe®
3 5 3
P
The resonant frequencies o f a half-wave length transm ission line, without coupling at the end
o f it is given by [15]:
fn = -^= 2yjer L
n = l, 2 ,....
3.5.4
W here c is the speed o f the light in vacuum, and 8r is dielectric constant o f material filled the
transm ission line. The change in resonant frequency, ( A / = /
- /
), after adding C c.Probe,
was calculated by using equation 3.5.3 in 3.5.4 following algebraic manipulation:
Af _
/« ,
2 Z qC c-
probe fn x
n
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3 5 5
75
3.6 SNMM results for dielectric measurements
W e used the tall coaxial probe to scan a 1 pm thick SiC>2 deposited on high resistively
silicon substrate (5 kQ. cm ) having a thin gold metal layer in the back side o f the silicon
(Depositing metal on the back side o f the silicon was used to shield the far-filed effects). W e
2, -62
ico -67
* -72
I
S
“7 7
-82
I -87
1.0036
1.004
1.0045
freq(GHz)
(a)
0.6 fF
1.26 urn
(b) Ah
<«> AC,c-probe
0(i.m
50fxm
Fig. 3.12: (a) Magnitude of the reflection coefficient near the resonant frequency (f„i), when the
coaxial tip was in air, over the silicon sample, and over the 1 pm thick S i0 2 on high resistivity silicon.
Simultaneous images were performed using the tall coaxial tip probe using the resonator system of
figure. 3.11: (b) Topography image; (c) Microwave image (capacitance was calculated using equation
3.5.5).
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76
perform ed a contact m ode A FM scan at scan rate o f lp.m/s. In figure 3.3 (a), we added the
50 Q. shunt resistance to ground after micro-coaxial cable to create a resonator. A matching
circuit (series stub, and variable capacitance) after the 50 £2-to-ground was em ployed to
achieve a better coupling to a 50 Q source, and thus a more sensitive system. Pow er level of
sweeper A was set at 3 dBm and its operating frequency was chosen to be close to the resonant
frequency of our unperturbed resonator in air (figure 3.12 (a)). Sweeper B was offset by 100
M H z from sweeper A and at a pow er level o f 15 dBm in order to provide a reference for the
LIA. The data presented in figure. 3.12(a) shows the m agnitude o f the reflection coefficient
(S n ) and clearly dem onstrates that the resonant frequency o f the resonator decreases when the
probe tip moves from air to silicon oxide, and then to the silicon surface, since the capacitance
between the tip and sample increases (equation 3.5.5). An Agilent E 8364A PN A was used to
perform the measurements. Figure 3.12 (b), (c) shows the simultaneous topography and
m icrowave im ages o f the sample, respectively. In figure 3.12 (c), the areas o f the image
exhibiting a brighter color correspond to the silicon-dioxide and the darker areas to the silicon
substrate. The change in capacitance in figure 3.12 (c) was calculated using equation 3.5.5.
W e also performed the simulation o f whole probe and its assembly (figure 3.9), and the
resonator system (using ADS software) to validate sim plified equation 3.5.5 for our probe
assembly resonator. There was a good agreem ent between the simulation and equation 3.5.5 at
~ 1GHz operating frequency (< 6 % error).
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77
3.7 Conclusion
W e have dem onstrated localized m icrowave measurem ents using SNM M cantilevers
integrated with ultra-tall coaxial tips on both metallic and dielectric samples.
O ur results
dem onstrate im proved electrom agnetic field confinem ent with enhanced im m unity to the
parasitic capacitive coupling that is typically associated with SNM M im aging using cantilever
based probes.
The next step is to calculate a dielectric constant o f the thin film materials
(especially, the newly developed low dielectric constant m aterials [18]). In order to perform
this m easurem ent quantitatively, the system should be calibrated with a m aterial having known
dielectric constant such as therm ally grown dioxide on silicon substrate. A model o f tip to thin
film sample interaction also needs be developed. A m easured capacitance of an unknown
material is a function of both the dielectric constant and the topography o f the sample
(geometry). In the next chapter, a systematic calibration technique for dielectric constant
m easurem ent is presented.
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78
3.8 References
[1] D. M isra, “On the m easurem ent of the com plex perm ittivity o f m aterials by an open-ended
coaxial probe,” IEEE Microwave and guided wave letters 5, 161 (1995).
[2] D. Blackham, R. D. Pollard, “An im proved technique for perm ittivity measurements
using a coaxial probe” IEEE Transactions on instrumentation and measurement,
4 6 ,1 0 9 3 (1997).
[3] R. Kantor, and I.V. Shvets, "M easurement o f electric field intensities using scanning near­
field m icrowave microscopy", IEEE Transaction on microwave theory and techniques,
51, 2228 (2003).
[4] B. T. Rosner, T. Bork, V. Agrawal, and D. W. van der W eide, ’’M icro-fabricated silicon
coaxial field sensors for near-field scanning optical and microwave m icroscopy,” Sensors
and actuators A (physical), 102, 185(2002).
[5] M. Fee, S. Chu, and T. W. H ansch,” Scanning electrom agnetic transm ission line
m icroscope with sub-wave-length resolution,” Optics communications, 69, 219 (1989).
[6 ] D. M. Pozar, Microwave Engineering, 2nd edition, pp. 414-416 (1998).
[7] Y. W ang, C. A. Paulson, G. Ning, and D aniel van der W eide, “Fabrication
and m easurem ents using near-field ultra-tall silicon coaxial tips,” IEEE International
MTT-S microwave symposium digest, pp. 2147 (2005).
[8 ] Y. W ang, M . Tabib-Azar, “D esign and fabrication o f scanning near-field
m icrowave probes com patible with atom ic force microscopy to im age em bedded
nanostructures,” IEEE Transaction on microwave theory and techniques, 52,9 7 1
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
79
(2004).
[9] Y. W ang, A. Betterm ann, et. al, “Scanning N ear-field m icrowave M icroscope Probes with
Integrated Ultra-tall Coaxial Tips,” accepted fo r publication in journal o f vacuum
sciences.
[10] G. Y. Chen, R. J. W arm ack, T. Thundat, D. P. Allison, and A. Huang, ’’Resonance
response o f scanning force m icroscopy cantilevers,” Review o f scientific instruments, 65,
2532 (1994).
[11] S. Ramo, J. R. W hinnery, and T. van Duzer, Fields and waves in communication
electronics, Third edition, John W iley & sons, 1993.
[12] C. P. Vlahacos, R. C. Black, S. M. Anlage, A. Amar, and F. C. W ellstood, ’’N ear­
field scanning m icrow ave microscope with 100 pm resolution,” Applied physics
letters, 69, 3272(1996).
[13] A. Imtiaz, M. Poliak, S. M. Anlage, J. D. Barry, and J. M elngailis,” Near-field
microwave m icroscopy on nanom eter length scales,” Journal o f applied physics, 97,
44302 (2005).
[14] D. Sarid, Scanning force microscopy with application to electric, magnetic, and atomic
forces, Oxford U niversity Press, pp. 133 (1991).
[15] C. P. Vlahacos, D. E. Steinhauer, S. K. Dutta, B. J. Feenstra, Steven M. Anlage,
and F. C. W ellstood,” Quantitative topographic im aging using a near-field
scanning m icrowave m icroscope”, Applied physics letters, 7 2 ,1 7 7 8 (1998).
[16] Agilent technologies, Santa Rosa, CA.
[17] H. Tanbakuchi, “ H ighly sensitive scanning capacitance microscope” , NSTI Nanotech
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Conference (2007).
[18] V. V. Talanov, R. L. M oreland. A. Scherz, A. R. Schwartz, and Y. Liu,
“Scanning near-field m icrowave probe for in-line m etrology for low -K dielectrics,”
Materials research society symposium, 812 (2004).
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81
4
Quantitative SNMM combined with AFM for
dielectric constant measurement
In this chapter, I propose and dem onstrate a novel solution for explicit calibration of
SNM M based AFM , which was introduced in chapter 3. I em ployed a novel calibration
methodology with a new fabricated calibration standard in order to m easure localized
dielectric constant o f thin film s on low resistivity silicon substrate using a SNM M combined
with A FM system. The details o f design approach, and the first quantitative microwave
measurem ent based on A FM setup is discussed in more depth in this chapter.
4.1 Introduction
The use of low -k dielectric material in m odem integrated circuits (ICs) operating at
higher frequencies make on-wafer m icrowave frequency electrical evaluation im portant in
process developm ent and control o f IC fabrication [1, 2]. The scanning near-field microwave
microscope (SNM M ) was first used to quantify the local high-frequency electrom agnetic
properties of m aterials in 1972 [3]. Since then, many researchers have been working on
increasing the resolution and sensitivity o f SNM M . Fee et al. dem onstrated a coaxial cable
near-field m icrowave microscope with a A/4000 spatial resolution (several pm for spatial
resolution) [4]. M ore recently, SNM M w ith sub-micron spatial resolution on high contrast
m aterials as a sample has been reported based on sharp tips which act as evanescent sources
[ 5 , 6 ],
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Em erging nanotechnology is dependent on m etrology tools capable o f reliably and
accurately probing surfaces and sub-surfaces o f em bedded structures. The atomic force
m icroscope (AFM) [7] has been found to be extrem ely useful for characterizing the mechanics
o f surface material at the nanom eter level. Y et only a few groups have been w orking on
SNM M com bined with A FM [ 8 , 9] which enables the A FM to be com bined with localized
microwave measurement. In general, a conductive A FM tip has been used as an evanescent
source. However, a reliable quantitative microwave m easurem ent using a conductive AFM
probe has not been reported. One o f the challenges o f m aking quantitative microwave
measurem ent based on conductive A FM probes is the large parasitic capacitance associated
with conventional A FM probes (i.e. the unintended interaction from the cantilever/chip body
to the sample) [10, 11] which makes a robust and repeatable m easurem ent very difficult. W e
previously showed that a reduction in parasitic capacitances (from the cantilever and the chip
body to the sample) is possible by fabricating ultra-tall conductive probes [ 10 , 11 ] or by
em ploying a more com plicated geom etry like a quadraxial probe with differential feed
technique [12]. Here, w e present a solution for a quantitative microwave m easurem ent by
com bining a new netw ork analyzer calibration m ethod with a low parasitic conductive AFM
probe (i.e. a very tall probe) for an accurate thin film dielectric constant measurement.
4.2 SNMM/AFM measurement setup
Figure 4.1 show our SNM M m easurem ent system retrofitted for use on a Topom etrix™
A FM system as previously described in chapter 3, and figure 3.3. The m icrowave
m easurem ent fixture used on our AFM head was designed and manufactured with technical
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
83
support o f Agilent Technologies. The differences between figure 4.1 and figure 3.3 are a 50
Q shunt resistor to ground (it was described in previous chapter and in [13]) was em ployed as
a coupling m echanism, and the probe was a very tall conductive (tungsten) A FM probe which
is com m ercially available from M ulti-probe Co Santa Barbara, CA. This probe is more
suitable for contact m ode operation o f A FM than m icro-fabricated silicon based probe,
because it is less lossy and m ore robust to the mechanical damage. In figure 4.1, small changes
of the probe’s im pedance are to be m easured by the reflectom eter in the high reflection
coefficient domain where m easurem ent sensitivity is very low. The increase in m easurem ent
sensitivity o f the reflectom eter reported here was obtained using a half wavelength coaxial
cable resonator and a 50 £2 shunt resistor to ground. The electric near-field o f the probe tip is
perturbed by the local m icrowave properties o f the sample under test (SUT). W e measure this
perturbation by m onitoring the reflected w ave voltage (or reflection coefficient using a
conventional vector netw ork analyzer (VNA)) at a frequency close to the resonance o f the
resonator. O ur goal was to extract the dielectric constant o f thin films. The resonator
com ponent o f the SNM M consists o f approxim ately 3 inch section o f a low-loss micro-coaxial
cable assembly (4LS01ZST0030 from Gore), o f which one end is connected to a very tall
tungsten tip epoxied to stainless steel cantilevers, while the other end is coupled to a
m icrowave source via a 50 Q shunt resistor.
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84
L a s e r d io d e
Mirror,
P h o to |
d e te c to r
SNMM
P ie zo
controller
M icro-coax
Tall p ro b e
D irectional
c o u p le r
(fn
50 Q S h u n t
S a m p le u n d e r te s t (SU T)
AFM
re sisto r
DAQ
Microwave
Sweeper A
NRef.
Lock-in
Amp
)
M icrow ave
Sw eeper B
(fr> + # )
Fig. 4.1: SNMM combined with a commercial AFM. This setup is similar to the one in figure
3.3. The only differences are the coupling technique is 50 Q load to ground instead of de-coupling
capacitor, and the probe is very tall conductive AFM probe (commercially available) instead of micro­
fabricated ones from previous chapter.
4.3 Calibration methodology
A standard VNA calibration aims to calibrate the instrum ent for m easurem ent of
im pedance over an extrem ely large range (from short to open). In contrast, a SNM M is
designed to m easure a given load im pedance and small variations around this im pedance
during localized measurem ents. The calibration protocol we developed used m easurem ent o f a
graded series o f know n standard capacitances. To create the standard, arrays o f repeating
square metal patches w ere fabricated over a very thin, therm ally grown silicon dioxide layer
(Er=4.0) [2] deposited on the surface o f a low resistivity (<=5 m Q.cm ) silicon. This substrate
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85
(b)
Fig. 4.2: (a) Optical image of the fabricated capacitive load with arrays of different patch sizes
(capacitance values) on 44 nm thin film Si02 grown on low resistivity silicon substrate for calibrating
the SNMM based AFM (b) A SEM micrograph close-up of 7.00 pm by 7.00 pm patch size within an
array (-40.68 fF mutual capacitance to the ground, was calculated from Agilent ADS simulation
software). The smallest patch size was 3.00 pm by 3.00 pm (-11.82 fF).
is highly conductive at the m icrowave operating frequencies used; w e approxim ated the
substrate as a metal [1, 2, 14]. The fabricated capacitive loads, having different capacitance
values (patch sizes), are shown in Figure 2(a). T he close-up of one patch (within an array) is
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
shown in Figure 3(b). The thickness of dielectric layer was about 44 nm as m easured by
optical reflectom eter (F 20 from Film m etrics Co.). The metal patches were 170 nm thick (20
nm Ti and 150 nm gold). ADS sim ulation software from Agilent technologies was used to
calculate the capacitances o f these patches to the ground (low resistivity silicon). W e also
perform ed an identical calculation by using ANSOFT™ M A X W ELL 2D software, which is
based on finite elem ent m ethod (FEM).
4.4 Sensitivity analysis
W e m odeled the resonator o f the SNM M (Figure 1) as a series resonant circuit, of which
the input impedance is close to its resonance frequency, as given by [15]:
^ in
^ resonator 0-
4.3.1
_
U
U sing a sim ilar procedure as outlined in ref. 16, a change in the m agnitude o f reflection
coefficient (AT)can be written as:
4.3.2
If coZ 0 A C c_pro|)e « 1 ,
4.3.3
w here Q is the quality factor o f our resonator (typically several hundred), Zo is the
characteristic im pedance o f the system (50 £2), f0 is the resonant frequency o f the system
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(-1.788 GHz), and Cc.pr0be is the sum o f coupling capacitances between the probe and sample,
as shown in Figure 4.1. The individual capacitances are the tip-sam ple m utual capacitance
(Cts), cantilever-sam ple mutual capacitance (Cp), and chip body/holder to sample mutual
capacitance (Cp). Equation 4.3.2 states that the change in magnitude o f reflection coefficient
(and thus magnitude o f reflected wave-voltage: |AVr | = |AT • V) | , w here V; is the incident
w ave-voltage [15]), is linearly proportional to the changes in capacitive coupling occurring
between the probe and sample if CDZoACc.prob e « l is satisfied [ 11 ]. In a sim ilar manner, for the
magnitude of reflected wave-voltage, we have:
0)
TJ
13 0■)
0-5 1
♦ Expriment
— Fitting cun/e
y = 0 .03 6 x-0.184
R2 = 0.9765
10
15
20
Patch capacitance(fF)
Fig. 4.3: Sensitivity curve of reflected wave-voltage with respect to different capacitance values
(patch sizes). As we expect, there is a linear relationship between a change in reflected wave-voltage
magnitude from LIA and patch mutual capacitances to the ground (slope: k -0.036 V/fF).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
W here, AV r =
A l ia
in Figure
Figure
4 .3 .1 .
• AVr , and
4 .3
k
includes the gain associated with the LIA
( A l ia ) ,
as shown
dem onstrates the sensitivity o f our system. In this case, the AFM
was operated in contact mode. Shown are the changes in m agnitude o f reflected w ave voltage
with respect to different capacitive loads, at an operating frequency near the resonant
frequency o f the resonator. This graph reveals a linear relationship (slope:
k ~ 0 .0 3 6
V/fF)
between that obtained from LIA ( AV r ) and the coupling capacitance present between the
probe and the sample (ACc-Probe)- This result confirm s the validity of our calibration approach,
and also dem onstrates its usefulness to set up the A FM based SNM M measurement.
4.5 Capacitance model of the conductive AFM probe
Because the conductive AFM probe length is much sm aller than m icrowave
wavelength o f operating frequency, a quasi-static approximation was found sufficient for
perform ing a field distribution analysis between the tip and the sample (sim ilar to chapter.
3 ).
A lum ped-element circuit consisting of several capacitances can be em ployed as an
equivalent model [1, 11, 17, 18, 19]. Previous efforts in analytical m odeling o f mutual
capacitances between an AFM conductive probe and m etallic surface
[2 0 -2 3 ]
have m odeled
the capacitance o f a conductive AFM tip apex above a m etallic surface as a metallic sphere
above a conductive ground plane. In a later report, H udlet et al.
[2 2 ]
derived an approxim ate
analytical form ula for predicting the capacitance o f a conductive AFM tip above a m etallic
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
89
surface which incorporated m ore of the geom etry o f an actual AFM probe. In particular,
they considered the effect o f canonical contributions to an AFM tip, and they used their
model to predict capacitive forces existing between a conventional conductive AFM tip and
a metallic surface in electrostatic force microscopy. Law et al., [23] extended the Hudlet
model, em ploying an additional capacitive term , and due to more data collection, they were
able to more effectively validate their model. Figure 4.4 depicts the geom etry and a model of
tip-to-sample interaction that occur when an A FM tip is in soft contact with a thin film
dielectric deposited on the surface of a highly conductive silicon wafer. W e m odeled our
AFM tip-to-sample interaction as that of a mutual capacitance occurring between a half
sphere (apex) and a truncated cone, with respect to the planar dielectric sample [22]. The
param eters used in our m odel were: a tip height (h) o f ~ 250 pm , one half o f the angle o f the
tip cone (0 ~ 7°), and the tip apex radius (r) o f few microns. The thin-film (dielectric)
thicknesses (z) available on our calibration standard varies from ~3 nm to -6 0 nm. The
opening angle (0), and tip height (h), were m easured from SEM m icrographs, while film
thicknesses were m easured using optical reflectom etry. W e ju st need to consider the effect r
o f the tip apex-sam ple mutual capacitance, because the effect o f other coupling capacitances
between our very tall conductive AFM probe and the sample are negligible during nm height
variations of thin film dielectric sample [11, 24].
In order to model the tip apex-to-dielectric thin film sample mutual capacitance, we
em ployed a FEM based sim ulation [11]. After calibration, our SNM M produced a sensitivity
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90
Cone
X / ' I r v f(1-sin©o)
Apex
Dielectric (er)
.... ....
Vsi t j
;
apex
Conductor
Fig. 4.4: Tip-sample mutual capacitances model for a conductive AFM tip in a soft contact
with a dielectric thin film grown on conductive silicon substrate (C ts in Figure 1). The tip-sample
mutual capacitance consists of two parallel capacitances: apex to sample and truncated cone to
sample (C apex> and Ccone). The parameters of model were: one half of the angle of the tip cone (0O) ~
7°, tip apex radius (r) ~ few microns, film thickness (z) varies from 3 nm to 60 nm, and tip height h ~
250 pm.
curve sim ilar to that shown in Figure 4.3. W e experim entally determ ined the tip apex
geometrical effect by conducting A FM scans in contact mode on several different
thicknesses o f SiC>2 thin films. Then em ploying a FEM simulation, we numerically
calculated that RtiP~8.5 pm , corresponding to the best fit o f the m easurem ent results shown
in Figure 4.5.
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91
Tip rad iu s, r= 6 0 0 0 nm
15
Tip rad iu s, r=8500nm
▲ E xperim ent
Tip rad iu s, r=11000nm
0
0
20
40
60
Filmthickness(nm)
Fig. 4.5: The experimental and simulation data (FEM) for the tip apex to thin-film sample
mutual capacitances. Experimental data was calculated from calibrated reflected wave-voltage
(figure 3) on various thicknesses of S i0 2 (Sr=4.0) on a conductive silicon wafer. Based on the best
curve fit of the simulation to experimental data, a tip apex radius r~8.5 pm was calculated for our
very tall AFM probe.
W e will now discuss the effects of other m utual capacitances (parasitic capacitances) that
exist between our very tall conductive probe and our thin-film dielectric sample. O ur model
incorporated four mutual capacitances, in parallel, between the conductive A FM probe and
the sample. They are the tip apex to sample capacitance, truncated cone to sample
capacitance, cantilever to sample capacitance, and chip-body/ holder to sample capacitance.
The last two mutual capacitances are shown in Figure 4.1 as parasitic capacitances (Cp).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 6 shows that the changes in mutual capacitance o f the tip apex-sam ple with nm height
variations on the sample surface is dom inant in com parison to the other mutual
m
£
Film thickness(nm)
50
100
Cone, Cantilever,
Chipbody/holder
Fig. 4.6: Changes in mutual capacitances of AFM probe relative to the thin-film sample. The
thin film was thermally grown dioxide on low resistivity silicon substrate with dielectric constant
Sr=4.0. The tip apex-sample mutual capacitance was calculated using FEM simulation. Cone-sample,
cantilever-sample, and chip-body/holder-sample mutual capacitances were calculated using
approximate analytical formula in appendix [22, 24], The tip apex-sample mutual capacitance is
found to be the dominant when the tip experiencing nm height variations.
capacitance contributions. This confirm s that the cone to sample, cantilever-sam ple, chipbody/holder-sam ple mutual capacitances can all be considered as constant parasitic
capacitances (Cp), as previously assum ed [24]. Because of the height o f the probe, we have
constant and very low parasitic capacitances in com parison with conventional conductive
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
AFM probes. The tip apex to sample mutual capacitance was calculated using FEM
simulation, while cone to sample, cantilever to sam ple, and the chip-body/holder to sample
m utual capacitances w ere calculated using the analytical form ula in the appendix [22, 24], as
shown in Figure 4.6. The probe’s cantilever and chip-body’s capacitances, with respect to
the sample, were m odeled as a rectangular plane tilted relative to a conductive surface and
having dim ensions 95 pm wide x 1230 pm long for the cantilever and 1075 pm w ide x 1564
pm long for the chip body. The total parasitic capacitance (which included the cone to
sample, cantilever to sample, and the chip-body/holder to sample mutual capacitances) were
m easured to be Cp~ 57 fF when probing the 44 nm thick SiC>2 thin film. The holder
capacitance was m odeled with the same width as the chip body, but using a length o f 1460
pm . The tilt o f the AFM head (AFM probe) with respect to the sample was a ~ 9°, as shown
in Figure 4.1.
4.6 Imaging results for thin film Si02, Si3N4
W e performed an A FM scan in contact m ode at a scan rate of 5pm /s on an interface
between the SiC>2 and Si 3N 4 thin-film. The interface between nitride and dioxide was initially
planarized using a chem ical-m echanical polishing (CM P) technique [25], perform ed at the
Cornell nanoscale science and technology facility (CNF). In figure 4.1, the pow er level of
sw eeper “A ” was set at 9 dBm, and its operating frequency was chosen to be close to the
resonant frequency o f the unperturbed resonator in air. Sweeper “B” was offset by 100 M Hz
from sweeper “A ” and set at a pow er level o f 1 dBm, in order to provide a reference for the
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0 nm
15 Jim
5-828V o
15 Jim
32
24
16
(c)
8
x
0
5
10
15
Distance along line
scan(micron)
5.844 i
_ 5.838
o
— -Microwave signal
I ” 5.832
5.826
o
5
10
15
D istance along line
scan(m icron)
FIG. 4.7: Simultaneous images using SNMM based AFM using a very tall probe. The AFM scan
was performed in contact mode at an interface between Si0 2 and Si3N4 which was previously planarized:
(a) topography (b) microwave image (magnitude of reflected wave-voltage). A line was a cut through the
images: (c) topography (d) magnitude of reflected wave-voltage. The height variation had a slope because
the thin-film sample was tilted. There was little local height variation at the interface between Si0 2 and
Si3N 4 (~lnm). A change in capacitance was calculated to be ~0.42±0.009 fF at the interface between two
thin films (-38 nm) by utilizing the sensitivity curve shown in Figure 4.3. A dielectric constant of £r~ 6.2
(ASr= 0.1) was calculated for Si3N4 layer, using FEM numerical simulation.
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LIA. Figure 4.7 shows simultaneous topography and m icrowave images. Figure 4.7(a) shows
the topography at the interface between SiC>2 and Si 3N 4. Figure 4.7(b) shows the magnitude of
the reflected w ave voltage at 1.788 GHz. In Figure 4.7(b), the lighter region corresponds to
Si 3N 4, and darker areas to SiC>2. Figure 4.7(c) shows a line cut through the topography image
(Figure 4.7(a)), which shows very little height variation when transitioning from Si 3N 4 to SiC>2
(less than 1 nm). Figure 4.7(d) shows the magnitude o f reflected wave voltage variation along
the same line. This result clearly indicates a decrease in the m agnitude o f reflected wavevoltage when probe’s tip moves from Si 3N 4 to SiC>2. This change in the m agnitude o f the
reflected w ave-voltage wave corresponds to a capacitance change o f -0 .4 2 fF±0.009 fF
(measured with a bandw idth o f 1 kHz) at the interface between the two thin films. A dielectric
constant (based on FEM simulation) o f £r~ 6.2 (A£r= 0.1) is calculated for Si 3N 4, identical to
that reported in [2], The film thicknesses o f Si 3N 4 and SiC>2 were m easured -3 8 nm using the
reflectometer. W e also m easured the dielectric constant o f a thin-film alum ina (AI2 O 3 -2 0 nm
thick) to have a £r= 9.8 (A£r=0.1). This value is within the range o f previously reported
dielectric constant values obtained for alumina at m icrowave frequencies [15].
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96
4.7 Nanometer spatial resolution
As probe’s tip apex size scales down from few pm to nm, mutual capacitance o f tip apexsample is decreasing from several to a few aF as shown in figure 4.8. For instance, a 100 nm
_ 400
i t
IB
(A
Tip radius,
r=1000nm
Tip radius,
r=500nm
Tip radius,
r=100nm
300
o
c
i B 200
8 .0
« 2.100
Q. TO
ill o
100
200
Filmthickness(nm)
Fig. 4.8: Tip apex-sample mutual capacitance of AFM probe tip relative to the thin-film sample
for different tip radius. The thin film was S i0 2- The tip apex-sample mutual capacitance was
calculated using FEM simulation. The tip apex-sample mutual capacitance is found to decrease from
several to few aF, when the tip apex size scales down from one pm to nm.
tip radius o f AFM tip needs a very low parasitic probe. Table 4.1 shows a com parison between
our very tall probe and the conventional A FM probe used in [24] when probe tip is scanned
over a - 4 0 nm step height o f the SiC>2 thin film. Based on table 4.1, changes in parasitic
capacitances o f the conventional probe
( A C Ch ip -b o d y
and A C c a n t i i e v e r ) are com parable
to
tip-
sample capacitances (ACts ; A C Cone and A C apex). but our very tall probe with the same tip
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97
A C Ch ip -b o d y
Az=40nm(Change in film thickness;Si02)
Conventional AFM probe [24](h=20 pm,
r=100nm)
Very tall AFM probe (h=250 pm,
r=1 OOnm)
(aF)
A C c a n tiie v e r
A C COn e
A C a pex
(aF)
(aF)
(aF)
10
2
9.2
30
1
0.4
11
30
Table 4.1: Changes in mutual capacitances of AFM probe to the thin-film Si0 2 sample for two
different probe’s height, but with the same tip apex radius; r=100 nm. Changes in parasitic capacitances
of the conventional (shorter tip height) probe
tip-sample capacitances
( A C COn e
( A C Ch ip -b o d y
and A C c a n tiie v e r) are comparable to changes in
and A C a p e x ) . but changes in parasitic capacitances of our very tall probe
is much smaller in comparison with changes in tip-sample mutual capacitances. Thus, they can be
considered constant parasitic capacitances for our very tall probe.
50
10 0
150
200
0
<D
U
C
*?
£ -5
« oT
*5 £
-0.5
1
—A pex
-C o n e
-1.5
2
«
Q. 3
u
O
-2.5
(/)
-3
Film thickness(nm)
Fig. 4.9: Slope of apex-sample and cone-sample capacitance curves of our very tall AFM probe
tip for S i0 2 thin-film sample and tip radius; r=100nm. If z « r (z is film thickness), changes in apexsample mutual capacitance is dominant, but if z » r cone-sample mutual capacitance is found to be
dominant.
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98
radius as the conventional probe (1 0 0 nm tip radius) has much smaller parasitic capacitances
in com parison to tip-sam ple capacitance, thus parasitic capacitances for the very tall probe
(height: 250 pm , tip radius=100 nm) is considered constant, which means it is a good
candidate for quantitative SNM M based A FM m easurem ent with nm spatial resolution. Figure
4.9 shows slope o f cone-sam ple, and apex-sam ple curves for our very tall probe with tip radius;
r=100 nm. W hen z / r « l , tip apex contribution is dominant, and cone contribution can be
ignored. W hen z / r » l , cone contribution is dom inant and tip apex can be ignored. O ther wise,
both tip apex and canonical contribution should be considered [2 2 ].
W e conducted an im aging experim ent sim ilar to the one in section 4.6 using our very
0
5 fun
o
5 (am
Fig. 4.10: Simultaneous images using SNMM based AFM using the very tall probe (lOOnm tip
radius), (a) Topography (b) microwave image (magnitude of reflected wave-voltage). The AFM scan
was performed in contact mode on a high resistivity silicon substrate having ~450nm deep, 2 pm pitch
line (from Nano-devices). Since the film (wafer) thickness (z) is much bigger than tip radius(r); i.e.
z » r , the changes in cone-sample mutual capacitance is dominant (from figure 4.9). A change in
capacitance was calculated to be ~920±9 aF from calibrated reflected wave-voltage.
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99
tall probe with tip radius; r=100 nm. A simultaneous SNM M /AFM scan was perform ed using
AFM in contact m ode at a scan rate o f 0.2|im /s on a high resistivity silicon substrate which has
~450nm deep, 2 pm pitch line (from Nano-devices Co.). A change in capacitance was
calculated to be ~920±9 aF from calibrated reflected wave-voltage. B y decreasing the
bandw ith (B) o f LIA from 1 KHz to 12 Hz, 1 aF resolution in capacitance m easurem ent w ould
be attainable, since (Johnson) noise voltage, Vn
V b [26].
4.9 Conclusion
W e have dem onstrated a new and robust calibration technique for quantitative
localized dielectric measurements at m icrowave frequencies, useful for quantifying SNM M
based A FM results gathered using low -parasitic, tall tip, conductive probes. O ur results
dem onstrate localized m icrowave (capacitance) m easurem ents are possible on relatively low-k
thin films. W e have successfully extracted dielectric constants o f tw o thin-film s Si3N 4, and
AI2O 3 as a dem onstration o f the instrum ent and approach. Scaling down the tip apex size to
nm is also discussed. Perform ing dielectric spectroscopy at m icrowave frequencies when the
silicon substrate is not as conductive requires m ore com plicated analysis, and this will be a
topic for future work.
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100
4.8 References
[1] V. V. Talanov, A. Scherz, R. L. M oreland, and A. R. Schwartz, “ N on-contact dielectric
constant metrology o f \ow-k interconnect film s using a near-field scanned m icrowave
probe,” Applied physics letters, 8 8 , 192906 (2006).
[2] Z. W ang, M. A. Kelly, Z. Shen, G. W ang, X. X iang, and Jeffery T. W etzel, “Evanescent
microwave probe m easurem ent o f low-£ dielectric film s” Journal o f applied physics, 92,
808 ( 2 0 0 2 ).
[3] E. A. Ash and G. N icholls, “Super-resolution Aperture Scanning M icroscope” Nature, 237,
510 (1972).
[4] M. Fee, S. Chu, and T. W . Hansch," Scanning electrom agnetic transm ission line
microscope", with sub-wavelength resolution", Optics communication 69, 219 (1989).
[5] C. Gao, T. W ei, F. D uew er, Y. -L. Lu, and X.-D. Xiang," High spatial resolution
quantitative m icrowave im pedance m icroscopy by a scanning tip microwave near-field
microscope", Applied physics letters, 71, 1872 (1997).
[6 ] C. Gao, and X.-D. X iang,"Q uantitative m icrowave near-field microscopy o f dielectric
properties ” , Review o f scientific instruments, 69, 3846 (1998).
[7] G. Binning, C. F. Quate, and Ch. Gerber, “Atomic Force M icroscope", physical
review letters, 56, 930(1986).
[ 8 ] B. T. Rosner, and D. W. van der W eide, "High-frequency near-field microscopy",
Review o f scientific instruments, 73, 2505 (2002).
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101
[9] M. Tabib-A zar and Y. W ang," Design and fabrication o f scanning near-field
microwave pobes com patible with atomic force m icroscopy to image em bedded
nanostructures", IEEE Transaction on microwave theory and techniques, 52, 971
(2004).
[10] Y. W ang, A. D. betterm ann, and D. W. van der weide, "Process for scanning near-field
microwave m icroscope probes with integrated ultratail coaxial tips", Journal o f vacuum
science and technology B, 25, 813 (2007).
[11] A. Karbassi, C. A. Paulson, Y. W ang, A. D. betterm ann, and D. W. van der W eide,"
Localized M icrow ave M easurem ent using A FM -Com patible Scanning near-field
M icrowave M icroscope Cantilever with Ultra-tall Coaxial P robe", IEEE AP-S
International symposium, 11-15 June, 2007.
[12] A. Karbassi, C. A. Paulson, A. B. Kozyrev, M. Banerjee, Y. W ang, D. W . van der
W eide, “Q uadraxial probe for high resolution near-field scanning RF/m icrowave
microscopy” , Applied physics letters 89, 153113 (2006).
[13] H. Tanbakuchi, “ H ighly sensitive scanning capacitance microscope”, NSTI nanotech
conference (2007).
[14] V. V. Talanov, A. Scherz, R. L. M oreland, and A. R. Schwartz,"A near-field scanned
microwave probe for spatially localized electrical metrology", Applied physics letters,
8 8,134106 (2006).
[15] D. M. Pozar, Microwave engineering, John W iley & sons Inc., New York, NY, 1998.
[16] M. Tabib-Azar, D. Akinwande, G. Ponchak , and S. R. LeClair, "Novel physical sensors
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102
using evanescent microwave probes", Review o f scientific instruments, 70, 3381 ( 1999).
[17] A. Imtiaz, M. Poliak, S. M. Anlage, J. D. Barry, J. M elngailis,"N ear-field microwave
m icroscopy on nanom eter length scales", Journal o f applied physics, 9 7 ,0 4 4 3 0 2 (2005).
[18] A. Imtiaz, T. Baldwin, H. T. Nembach, T. M. W allis, and P. Kabos," N ear-field
m icrowave m icroscope measurements to characterize bulk material properties",
Applied physics letters, 90, 243105 (2007).
[19] A. Tselev, S. M. Anlage, Z. M A, and J. M elngailis,"Broadband dielectric microwave
m icroscopy on micron length scales", Review o f scientific instruments, 78, 044701
(2007).
[20] D. Sarid, Scanning Force Microscopy with Applications to Electric, Magnetic, and
Atomic forces, Oxford University Press, 1991.
[21] H. W . Hao, A. M. Baro, and J. J. Saenz, “Electrostatic and contact forces in force
m icroscopy” , Journal o f vacuum science Technology B, 9, 1323 (1991).
[22] S. Hudlet, M. Saint Jean, and C. Guthmann, and J. Berger,’’Evaluation o f the capacitive
force between an atomic force m icroscopy tip and a metallic surface”, European physics
journal B, 2, 5 (1998).
[23] B. M. Law, and F. Rieutord H , " Electrostatic forces in atomic force m icroscopy", Physics
Review B, 66, 035402(2002)
[24] L. Fum agalli, G. Ferrari, M. Sampietro, I. Casuso, E. M artinez, J. Samitier, and G.
Gomila, N anotechnology 17,4581(2006).
[25] S. W olf and R. N. Tauber, Silicon processing fo r the VLSI Era, vol. 1,
Lattice press, Sunset Beach, CA, 2000.
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103
[26] D. M. Pozar, Microwave and RF design o f wireless systems, John W iley and sones, 2001.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
104
Appendix B: Capacitance model of conductive AFM probe
An analytical capacitance model for canonical contribution o f a tip, a tilted cantilever, and
chip-body of a conductive A FM probe to a conductive surface, is presented here. In general,
these mutual capacitances have been m odeled as constant parasitic capacitances for our very
tall probe. The canonical contribution o f probe tip to a conductive surface has been m odeled as
a truncated cone, as shown in figure 4. U sing a sim ilar approach as Ref. 22 and the result o f
Ref. 21, we derived:
Ccone
ln[tan ( 0 o)]
[(zA - z B) - z c • ln(— )]
zn
B .l
where:
zA= z + h
B.2
z B = z + r ( l - s i n ( 0 o))
B.3
B.4
The m utual capacitances from chip-body/holder and cantilever to the conductive surface can be
m odeled as a tilted, rectangular plane capacitor. The general expression for the capacitance o f
a rectangular plane shape with length L, width w, and tilt angle a is given by [24]:
B.5
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W here d is ~ h (the cone height), and z is the thickness of the thin film. In both capacitance
m odels, the effect o f the very thin film (nm thick) dielectric deposited on the conductive
silicon surface was ignored since thin film thickness was much smaller than probe’s tip height
( z « h ) [24],
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106
5
High precision impedance analyzer for broad-band
SNMM application
In this chapter, the theory and im plem entation necessary for carrying out a w ide-band
measurem ent technique that is useful for determ ining the reflection coefficient o f a high
im pedance load is described. This technique is a solution for increasing the sensitivity o f a
broad-band, high-input im pedance reflection coefficient measurement. The system built based
on this approach was constructed by m odifying a conventional netw ork analyzer and is
referred to as the “im pedance analyzer” . As a dem onstration o f proof o f principle, and for the
first time, an im pedance analyzer is used for the purpose of making a SNM M probe based
measurement.
5.1 Introduction
In chapter 3, section 3 .1 ,1 described the basis o f the problem o f low sensitivity that is
encountered when m aking a reflection coefficient ( T ) m easurement o f a high im pedance load.
I presented an approach em ploying resonators as a solution for increasing the sensitivity o f
measurements made at discrete frequencies. By modifying the architecture o f a traditional
reflectom eter, it is possible to also increase the sensitivity o f w ide-band m easurem ents [ 1 ].
This new architecture can be used for characterization o f new m icro/nano-fabricated devices
that typically exhibit a very low or very high im pedance, such as is the case with SNM M
probes.
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107
5.2 Simplified diagram of one-port refiectometer:
In order to accurately m easure the reflection coefficient ( T ) o f a m ism atched load (i.e.
the ratio of reflected wave to incident wave), a circuit (refiectometer) is needed which
separates and isolates the reflected w ave (b) from the incident wave (a). Traditionally, a
refiectom eter, which is at the heart o f a network analyzer, employs directional couplers [2]. A
low cost refiectom eter m ay be built using cheap, resistive T-bridges, instead o f expensive
directional couplers, but results in a low er perform ance [3]. A simplified block diagram of a
refiectom eter configured for use as a one port netw ork analyzer is diagramed in figure 5.1[1],
Directional C oupler-2
RF S o u rce
Directional Coupler-1
DUT
K \y
Mixer-2
ReceiverRef
A/D
A/D
Mixer-1
Fig. 5.1: A simplified diagram of a one-port network analyzer (refiectometer). The reflection
coefficient is calculated as: T = —. All the resistors (R) were 50 Q,.
a
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108
A reflection coefficient can be calculated as T = — , following a calibration procedure
a
[2].A ssum ing the device under test (DUT) im pedance shown in figure 5.1 is Z\, (from 3.1.6),
the reflection coefficient ( T ), becom es:
r = Z / ~ Z°
5.2.1
Zl +Zo
I have shown (mathematically) in figure 3.2 o f chapter 3, that the sensitivity o f the reflection
coefficient used for measuring the changes in load im pedance drops when the load im pedance
deviates from Zo=50 ohms, tow ard larger values. For exam ple, when Z p 3 0 0 k-ohm, then
r = 0.999666722, and when Zi=320 k-ohm, then T = 0.999687548, thus the changes in the
reflection coefficient when the load im pedance changes from 300 k-ohm to 320 k-ohm, is only
0.000020827. This change can not be resolved using a 14 bit A/D (analog to digital) converter
(the typical resolution o f a conventional A/D converter found in m odem network analyzers
[1]). Assum ing that the analog input into a 14 bit A/D converter ranges from 0 to a 1 volt, the
A/D resolution is — = 0.0000610351 volt. This lack of resolution is because the reflection
2
coefficient undergoes a com pression (equation 5.2.1 or figure 3.2) for higher values of load
im pedance. Thus, the problem is how to measure the small changes occurring within a large
reflected wave,
b, when operating close to the m axim um input limit o f an A/D converter.
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109
5.3 Reconfigurable one-port reflectometer for high impedance measurement
(Impedance analyzer)
In order to m easure high im pedance load changes using a reflection coefficient
measurement, or in another words, measuring a small change occurring within the larger
reflected wave (b), w e subtract the reflected wave (b), from a known m odified clean signal:
for exam ple a m odified incident wave (aO, in order to null the reflected wave r= b -a i, then
am plify the resultant vector (ri):
rl = g ' { b - a l)
5.3.1
b+AbZ<f>+A®
rZ9
Fig. 5.2: Vector subtraction inaccuracy in the complex 2-D plane. The phase inaccuracy A<t>, and
magnitude inaccuracy A b , results in finite suppression vector, rZ. 0.
where g ( g Z a ) is the am plification’s gain. At this point, a small change in the reflected wave
(b), results in large changes in ri, which can be easily m easured at the input of a typical 14 bit
A/D converter. From the theory, a reflected wave having a m odified incident wave (aO, can be
perfectly nulled [4] (perfect signal cancellation). This means that the suppressed signal
magnitude, r, can be made ~ 0, or R = - ° ° dB ( R = 2 0 1 o g jo (r ) )• In practice, a finite am ount of
suppression can be achieved, which depends on how closely the two signals (ai, b) can be
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110
m atched (non-ideal signal cancellation) [4]. Figure 6.2 shows the vector subtraction o f two
signals in a com plex 2-D plane. The reflected voltage wave,
b = & Z O , the m odified incident
w ave a i =b + A&ZO + AO , and the suppressed signal, r = r Z 0 ; where Ad>, and Ab are
inaccuracies in the phase m atching, and the magnitude m atching o f the m odified incident
w ave with the reflected wave, respectively. The question now becom es w hat degree of
accuracy in the phase and the m agnitude o f a m odified incident wave is required to achieve
Suppressed signal (R, dB)
0
A& <0.0264 dB, AO < ±0.05°
£ *-50
dB
-80J cl"....
Phase mismatch
degree)
Magnitude mismatch
(Ab, dB)
Fig 5.3: A three dimensional graph of the suppressed signal versus phase and magnitude mismatch
of the two vector subtraction.
a finite am ount o f suppression, such as for R=-50 dB. For simplicity, assume that b = lZ 0 .
From figure 6.2, r can be calculated as [4]:
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111
r 2 = (l + Ab)2 + l - 2 - ( l + Afc)cos(AO)
5.3.2
Equation 5.3.2 shows that r, the suppressed signal, is a function o f two variables; the phase
inaccuracy (A<I>), and the magnitude inaccuracy (A b). A plot o f the suppressed signal R (in
dB), versus A<t> (in degrees), and Ab (in dB) is shown in figure 5.3. As an exam ple, in order
achieve a -50 dB suppression, a AO < ±0.05°, and a Ab< 0.0264 dB, is necessary. Practically,
this means we need to incorporate an equalizer (i.e. a w ay to change the m agnitude and the
phase o f the m odified incident w ave) exhibiting the degree o f accuracy outlined in the above
m entioned example.
5.4 Implementation of impedance analyzer for high impedance
measurement
In order to im plem ent this technique, a m odification to the conventional reflectom eter
(one port network analyzer) shown in figure 5.1 was performed. Figure 5.4 diagrams this new
reflectom eter architecture (or im pedance analyzer). In figure 5.4, when the RF switch is set to
path no. 1, we have an im pedance analyzer (IA mode), but when the RF switch is changed to
path no. 2, we have norm al network analyzer operation (NA mode). Let us consider the IA
mode: The m odified incident voltage wave (a*) was created by using a variable R F phase
shifter to change the phase o f the signal, and a broad-band continuously variable attenuator to
change the magnitude o f the signal. This signal ( a i ) was subtracted from the reflected w ave (b)
using a broad-band balun (a 1 -to -l im pedance ratio transform er which is used to reject the
com m on mode signal between b and a i) to generate the suppressed signal (r), which was then
am plified (ri) before going to the input o f the A/D converter. Now, if the im pedance o f D U T
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112
changes slightly, r i will undergo a large change, thus the A/D converter can accurately
measure this change. In order to measure calibrated, high reflection coefficients
Directional Coupler-2
RF S ource
,—
f/\/
(rd u t),
Directional Coupler-1
, -I
DUT
50 Q
50 Q
Pow er divider,
50 0 ^
50 Q
Baiun
Variable
P h a se shifter
Variable
Attenuator
Ref
A/D
Mixer-2
rv )
Amo
ReceiverA/D
^ RF Switch
50 Q
RF Sw itches
Mixer-1
lo
Fig. 5.4: A simplified diagram of the modified one-port network analyzer (impedance analyzer)
for measuring DUT with high input impedance. For impedance analyzer operation, the RF signal was
switched on path no. 1(IA mode), but for a normal network analyzer operation the RF signal switched
to path no. 2 (NA mode).
such as those obtained from the com bined A FM /SN M M probe described in chapter 4, a novel
calibration technique has been developed to use with the im pedance analyzer shown in figure
5.4 which will be discussed in section 5.5. The different com ponents used in figure 5.4 were: a
pow er divider, with an operating frequency up to 12 GHz, and 17 dB m inim um isolation (PS4101, M icrowave com m unications laboratories, Inc, Saint Petersburg, FL), a continuous voltage
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113
variable (analog) attenuator: operating frequency: 5M Hz-2.0 GHz (TG 9025, from Spectrum
microwave, Palm Bay, FL). The coaxial phase shifter had an operating frequency from 1GHz5.0 GHz (model 3752, from N arda M icro-line), the am plifier was a ZX 60-2534M with an
operating frequency o f 0.5 GHz-2.5 GHz), the balun, obtained from M acom ™ (ETC 1-1-13)
had an operating frequency up to 2.0 GHz and was built on RT/D 6010LM substrate (£r=10.2)
from Roger corporation. The R F switches in figure. 5.4 were m anually operated. Finally, the
netw ork analyzer used was a refurbished N5230A obtained from Agilent technologies
(operation frequencies from 300 kHz up to l3 .5 GHz). Figure 5.5 shows the fabricated circuit
board, with its equivalent circuit model.
° u ,Pu ,cv - t l / v w O -
-o in ,
-o in .
_ET
(b)
Fig. 5.5: (a) A photograph of fabricated Balun on RT/D 6010LM substrate from Roger
corporation (b) equivalent circuit of the balun in common mode rejection.
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114
5.5 Calibration of the impedance analyzer for measurement of
SNMM based on AFM probe
Here, the general steps for a calibration procedure o f the im pedance analyzer are
presented. First, we calibrate the system o f figure 5.4 in netw ork analyzer m ode (NA m ode) at
the D UT reference plane using a conventional open, short, and load, one port calibration
technique [2], After calibration, w e m easure reflection coefficient of D U T and we call it:
r ref= ~
a
Second, we switch the system o f figure 5.4 from N A mode to im pedance analyzer mode
(IA mode). By equalizing the magnitude and phase o f a reference wave (ai) to the reflected
wave from DUT (b), we get b=ai at any frequency o f interest, which means the am plified
suppressed wave ri is ~0 (equation 5.3.1). W e divided the two side o f equation 5.3.1 by
incident wave (a):
* -» •£ -* )
a
a
5.5.1
a
from definition o f one port reflection coefficient, and
ArDUT = — , we have:
a
^DUT ~ & ' (FDUT ~ ^ref )
5.5.2
W here g is the gain o f am plifier in figure 5.4. thus,
A T Du t
is ~0 in IA mode. This means,
almost no reflected w ave enters the input port o f netw ork analyzer, and thus the center of
smith chart is equal to
T o u t)
rref, and not
changes slightly, since
r ref
50 Q any more. N ow, if the impedance o f D U T (and thus
is not changing (which is equal to
T Du t
before the change),
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115
thus
ATdut
changes m uch bigger than
T Du t
itself, and thus it can be m easured very precisely
using the im pedance analyzer in IA mode[5].
The tall AFM probes with its assembly that w ere introduced in chapter 4 were used as a
D UT for carrying out an im pedance analyzer perform ance test.
T Du t
o f the A FM probe was
derived in equation 3.2.3. From equation 3.2.3, and a ssu m in g ry Z o Q -p ro fe « 1 , we derived:
^ d u t ~ 1_ 2 jZ 0 C c_probety
5.5.3
From equation 5.5.2, and 5.5.3, we derived (assum ing am plifier phase shift: a ~ 0 ) :
AT = jg 2Z0a)ACc_probe
5.5.4
or:
|AT| = g 2 Z 0 tyACc_probe
5.5.5
This means there exists a linear relationship between |AT|, and ACc-probe, having a slope of
k = g 2 Z 0a) i.e.:
|Ar| = k • ACc_probe
5.5.6
The sensitivity o f reflection coefficient curve for the AFM probe in im pedance analyzer mode
was measured with respect to different capacitive loads (shown in figure 4.2). A fter the above
calibration is perform ed, any changes in Cc.probe can be measured from the sensitivity curve
sim ilar to section 4.3 in chapter 4.
5.6 Measurement results for SNMM based on AFM probe using
impedance analyzer
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116
I used the im pedance analyzer com bined with AFM in figure 5.6 to perform a SNM M
experim ent with the same tall conductive probe used in section 4.3 in chapter 4 (calculated tip
radius; RtjP =8.5 p m ) at tw o different operating frequencies (these frequencies were chosen
arbitrarily in operating frequency range o f the setup, currently from 500M Hz up to 2GHz).
Laser diode
M irror/
.
PhotO
detector
DUT: Tali probe ^
/r
Piezo
controller
Micro-coax
50
Q SMA
Impedance
Analyzer
c-probe
Fig. 5.6: A schematic of our impedance analyzer combined with a commercial AFM. The
AFM uses an optical detection method to track the topography of the sample (SUT). The probe is the
same as the tall conductive AFM probe and micro-coaxial cable is the same as one used in previous
chapter.
5.6.1 Measurement results at 725 MHz
In case o f the set up in figure 5.6, 725 M Hz frequency is chosen to be at non-sensitive
point on reflection coefficient curve o f the (tall conductive) AFM probe (far from resonance
frequency, i.e. A./2 ), but some what close to the frequency which corresponds to ~ X/4 length of
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117
1.0j
0.5j
2.0j
0.2j
5.0j
-0.2j
-5.0j
-2.0j
-0.5j
-1.0j
Fig. 5.7: Reflection coefficient
(rDUT) measurement showed on the Smith chart for the very tall
conductive AFM probe assembly in NA mode (High reflection coefficient). The frequency sweep was
from 722 MHz up to 727 MHz. The probe was in the air.
724
726
OQ
TJ.
-10 O
-20
T3
-30
-40
-50
Freq(MHz)
Fig. 5.8: Magnitude of the reflection coefficient (|A rDUT| ) for the AFM probe assembly, when
measured in IA mode. The null could be set at any arbitrary operating frequency of the impedance
analyzer. The amplification gain was G=11 dB.
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118
the low-loss m icro-coax, and thus a probe im pedance at this frequency looks like short, and
the 50 Q load in figure 5.6 can be approxim ately ignored for the calibration purpose. After
calibration in the N A m ode to the end o f SM A connector (shown in figure 5.6), the
m easurem ent reference plane was shifted from the end o f SMA connector to the probe
assembly, by simply adding an electrical delay on the netw ork analyzer to account for a length
1.0j
2.0j
0.5j
0.2]
5.0j
015
-0.2j
-5.0]
-0.5j
-2 .0]
-1 .0]
Fig. 5.9: Reflection coefficient ( A rDUT) measurement on Smith chart display for the tall AFM
probe assembly in IA mode. The frequency sweep range was from 722 MHz up to 727 MHz. The high
reflection coefficient (impedance) of the AFM probe’s assembly shown in NA mode (figure 6.7) was
mapped to the center of the Smith chart using IA mode.
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119
o f micro-coax. Figure 5.7 shows the reflection coefficient o f the AFM probe (P out) with the
assembly m easured using the impedance analyzer in NA mode. The frequency sweep was
from 722 M Hz up to 727 M Hz. Figure 5.8 shows the reflection coefficient magnitude after the
instrum ent was nulled in IA mode. The am plification gain was lld B . This figure shows that
the reflected w ave was nulled at the arbitrary frequency o f 725 MHz. Figure 5.9 shows that
T o u t was m apped to the center of the chart, m eaning that the center o f Smith chart no longer
represents 50 Q any more, and in fact, it represents the im pedance o f the AFM probe and its
assembly.
The sensitivity curve for our im pedance analyzer was found with respect to different
0.08
0.06
*o
a> 0.04
N
"ra
E 0.02
„. . . /
///•
Hi
0
¥A
C -0.02
o»
A -0.04
-0.06
-0.08
i
---- Null
---- 40.68 fF
---- 66.82 fF
82.31 fF
— ■ 160.41 fF
............
M
l i
............... JE.L....................
/;
-0.05
0
0.05
Real(Normalized)
Fig. 5.10: Reflection coefficient ( ArDUT) measurement when the AFM probe was scanned (in
contact mode) on different capacitive loads in IA mode (Cartesian coordinate). Imaginary part of
ATdut was changed with respect to different capacitive loads.
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120
capacitive loads introduced in chapter 4. Figure 5.10 shows the reflection coefficient of the
AFM probe in IA mode, with respect to different capacitive loads. This figure shows that the
im aginary part o f the reflection coefficient was mostly changed with respect to different
capacitances, as was expected (equation 5.5.4). Figure 5.11 shows the change in the reflection
coefficient m agnitude with respect to different capacitive loads. A linear relation ship was
found between the load capacitances and the change in the reflection coefficient magnitude
(equation 5.5.6). The slope o f the linear fit to the experim ental data was k=0.393*10'3l/fF . A
change were m easured in the reflection coefficient when the AFM probe was scanned (in
contact mode) over a step in height o f -4 0 nm com prised o f a therm ally grown SiC>2 fabricated
>
«•—{*■
o o
o
~u *
■
3
c
'E a>
O) 'o
»
£ <L
O
c O
01 c
o> o
->
c 4o
(8 _a>
.£ «#o <D
80
70
y = 0.393x + 8.5024
R2 = 0.9876
eo
♦ Exprimentat
725 MHz
— Fitting curve
60
40
30
20
20
70
120
170
Patch capacitance(fF)
Fig. 5.11: Change in magnitude of reflection coefficient (|A rDUX|) when the AFM probe was
scanned (in contact mode) on different capacitive loads measured in IA mode. Change in reflection
coefficient was linear with respect to different capacitive loads with slope k=0.393*10'3l/fF.
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121
thin film deposited on the surface of low resistively silicon, similar to what has been reported
in chapter 4. It was found that |ArDUT| = 1.6 ± 0 .1 * 10"3, which is equal to ACapex = 4 ± 0.25 fF
based on the sensitivity curve given in figure 5.11. Simulation result for the change in apex to
sample mutual capacitance for 40 nm height change o f thin film SiC>2 w asA C apex = 3.81 fF
which was calculated from ANSOFT™ M A X W ELL 2D software (FEM simulation) with the
same probe geom etry as the one used in chapter 4 (calculated tip radius; r=8.5 pm). The
m easurem ent is in a good agreement with simulation result.
5.6.2 Measurement results at 1.55GHz and ~ 39 dB amplification
A sim ilar experim ent to the one described in section 5.6.1 was conducted at operating
»*—
150
•
y=11.585x-61.809
R = 0.9605
o o
® r 130
T3
3 +S C
e ®
O) o
to
E
®
c
o
c o
®
U) o£
c
«
J=
+*
&
®
O ‘Si
♦ Exprimentat
1.55 GHz
110
90
— Fitting curve
70
50
10
15
20
Patch capacitance(fF)
Fig. 5.13: Change in magnitude of reflection coefficient (|ArDUT|) when the AFM probe was
scanned (in contact mode) on different capacitive loads in IA mode. The amplification gain was 39 dB.
Change in reflection coefficient was linear with respect to small values in capacitive loads with slope
k=11.585*10'3 1/fF.
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122
frequency ~ 1.55 GHz. The am plification gain was increased to 39 dB, and the sensitivity
curve was determ ined w ith respect to different capacitive loads, as shown in figure 5.13. The
slope o f the sensitivity curve was increased proportional to amplification gain, as was
expected from equation 5.5.5. A change in reflection coefficient m agnitude was m easured as
the AFM probe was scanned (in contact mode) over the 40 nm high step o f therm ally grown
SiC>2, on low resistivity silicon, as |AFDUT[ = 4 4 ± 6 x l 0 -3 , which was equal to an
ACapeX = 3 .8 ± 0 .5 f F , based on the sensitivity curve in figure 5.13. Simulation result o f apex
to thin film sample using FEM was calculated as ACapex = 3.81 fF. Table 5.1 sum m arized the
tip apex-sam ple m utual capacitance m easurem ent at 725 M Hz and 1.55 GHz in com parison to
sim ulation results. M easurem ent result shows a good agreem ent with simulation especially at
1.55 GHz, because the system is m ore sensitive (Figure 5.13; higher amplification gain), but it
is m ore noisy.
Operating frequency of impedance analyzer
725 MHz
1.55 GHz
ACapex -measured
4.0 ± 0.25 fF
3 .8 ± 0 .5 fF
ACapex -simulation
3.81fF
3.81fF
Table 5.1: Measurement and simulation results for changes of tip apex-sample mutual
capacitance (ACapex), when the impedance analyzer used in SNMM/AFM set-up. AFM was scanned (in
contact mode) over a thin film SiC>2having 40 nm step in height.
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123
5.7 Conclusion
A new reconfigurable netw ork analyzer (im pedance analyzer) was constructed by
modifying a conventional netw ork analyzer. A detailed theory o f the im pedance analyzer
operation was discussed , and perform ance o f the im pedance analyzer for accurately
m easuring high im pedance loads (SNM M based on A FM tall probe) were tested at two
arbitrary operating frequencies (725 M Hz and 1.55 GHz). The current version o f the
im pedance analyzer works from 500 M Hz up to 2 GHz (the frequency restriction comes from
the external com ponents used in the im pedance analyzer set-up). Sim ilar results were obtained
com pared to the resonator based SNM M described in chapter 5, dem onstrating the capability
of our instrum ent to m easure small changes in high im pedance loads, such as AFM probes
em ployed in SNM M experiments. The noise level in our im pedance analyzer was much higher
than for the resonator based SNM M described in chapter 5, with an even low er IF bandwidth.
An accuracy o f 0.25 fF in capacitance at a 725 M Hz operating frequency was obtained with
our current im pedance analyzer set-up (11 dB gain and IF band w idth=150 Hz), which was
much poorer perform ance than the 0.009 fF (IF band w idth= l kHz) achievable using the
resonator based (narrow band technique) and lock-in amplifier. The noise level (accuracy) of
im pedance analyzer can be decreased by em ploying a low er noise R F amplifier, and a low er
noise variable attenuator. The advantage o f our im pedance analyzer is its capability to perform
broad band reflection coefficient measurements, in com parison to the resonator based
approach.
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5.8 References
[1] H. Tanabakuchi, L. Betts, D. Blackham ,” Converting vector network analyzer to a high
precision im pedance analyzer ” U. S. patent, Agilent technologies, 2006.
[2] D. M. Pozar, “M icrowave Engineering,” 2nd edition, pp. 414-416 (1998).
[3] M. K. Choi, M. Chao, S. C. Hagness, D. W. van der W eide, "Compact m ixer-based
1-12 GHz reflectom eter", IEEE Microwave and wireless components letters, 15,
781(2005).
[4] N. Pothecary, “Feedforw ard Linear Pow er A m plifiers” , Artech House, 1999.
[5] A. Karbassi, H. Tanbakuchi, L. Betts, D. Blackham , R. Stancliff, “High precision
im pedance analyzer” Summer internship presentation, Agilent technologies, 2006.
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125
6
Summary and future research direction
In this chapter, we present a sum m ary o f our w ork and achievem ents in the field of
scanning near-filed m icrowave m icroscopy (SNM M ) and the future direction o f our research is
also discussed.
6.1 Summary and future directions for research
6.1.1 Quadraxial probe with differential feed technique
In order for our new ly developed quadraxial probe to be com patible with the AFM
platform, careful m icro-fabrication of the quadraxial probe tip and integration with the
cantilever and chip-body is required to achieve high spatial resolution and an operating
frequency o f several gigahertz. The quadraxial micro-fabricated probe requires additional
shield layers on the tip, and a balanced coplanar transm ission line structure (fabricated on the
chip-body and cantilever), in com parison to a micro-fabricated coaxial probe tip. Thus, a
sim ilar m icro-fabrication process as the one used in [ 1 ] can be used to construct a quadraxial
probe. This design offers more sensitivity and resolution, in comparison to the coaxial design,
thus, it is a good candidate for use with localized SNM M /AFM .
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126
6.1.2 Multi-functional micro fabricated probes for SNMM
T he new ly developed m icro-fabricated tall coaxial tip SNM M /AFM probe has many
advantages over conventional scanning near-field microwave m icroscopes. Because it uses an
A FM platform , simultaneous m icrowave and surface topography o f the sam ple, with pm /nm
spatial resolution, is made possible. A lthough we can achieve a high m icrow ave frequency (up
to 20 GHz), low-parasitic, SNM M /AFM probe, it still suffers from various limitations. The tip
has a very thin deposited layer o f gold/titanium (0 .2 pm ) [ 1 ], that acts as a signal line, which it
wears off quickly as the tip is scanned on the sample using the contact m ode o f operation. As a
result, one can expect a significant reduction in signal sensitivity after perform ing a few AFM
scans with this type o f probe. Thus, this probe is better used for non-contact AFM operation.
In order to increase the sensitivity o f the high input im pedance (large reflection coefficient)
m easurem ent from the probes, we em ployed a half-wave length resonator. W e introduced a
novel frequency independent coupling technique (50 Q -to-ground) to effectively couple to the
resonator. W e dem onstrated the quantitative capability o f our probes to im age the localized
m icrowave interaction (capacitance) occurring between the probe’s tip and various samples
(Both for metallic sample and non-metallic sample) at microwave frequencies. A model o f our
probe based on scattering param eters was developed and found to be in very good agreement
with the experim ent data. W e perform ed simultaneous microwave/ topography im aging up to
2.6 GHz, although the probe can operate up to 20 GHz. The limitation in operating frequency
is because o f the limitation o f our hom e-m ade connector for the A FM probe. One w ay to
im prove the transition from AFM probe to reflectom eter is an integration o f both reflectom eter
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127
and the probe on one chip, thus allowing the integrated probe/ reflectom eter to operate at
much higher m icrowave frequencies and with a robust frequency response.
6.1.3 Calibrated SNMM using very tall probes
W e performed a quantitative SNM M experim ent using a very tall conductive AFM
probe to measure the dielectric constant o f thin-film s deposited on a conductive silicon
substrate. W e successfully m easured dielectric constant o f couple of thin films (Si 3N 4 and
AI2 O3 ). Sharper tips in nm range can provide nm spatial resolution, but at the cost o f lower
sensitivity in dielectric constant measurement. To overcom e this issue, we need to increase the
sensitivity of our resonator (Quality factor). As the probe’s tip size shrinks down from a few
pm to nm, the change in parasitic capacitance (a com bination contributed from the truncated
cone, cantilever, and chip-body) becom es com parable with the apex to thin-film sample
m utual capacitance, a quality which is not desirable in quantitative SNM M /AFM . One
solution to this problem is to cover the chip body o f a probe with metal shielding on the side
which is facing the sample. Thus, construction o f a very sensitive SNM M /AFM system,
having a few nm ’s spatial resolution, can be made possible by em ploying a high quality factor
resonator in combination with a probe-tip having a radius o f a few nm.
6.1.4 Reconfigurable network analyzer (Impedance analyzer)
A reconfigurable netw ork analyzer was built by m odifying a conventional network
analyzer, which we called a high-precision im pedance analyzer. This im pedance analyzer is
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128
suitable for m easuring very high input im pedance/low input im pedance loads, unlike
conventional netw ork analyzers. The perform ance o f the im pedance analyzer was successfully
dem onstrated using a conductive AFM probe as the device under test (high input im pedance
device). The current set-up operates from 500 M H z up to 2 GHz. The frequency limitation
com es from the external circuitry used in the im pedance analyzer. The noise’s (Johnson noise)
pow er level inherent with the im pedance analyzer is higher than with a conventional network
analyzer. This additional noise is generated in the external circuitry. By em ploying more
broad-band and sm aller noise-figure com ponents, we can achieve a higher frequency range of
operation, and low er noise levels, but at a higher cost. A chieving broad-band, high-precision
m icrowave m easurem ent of lossy m aterials (especially dispersive media) is another im portant
area for future research.
6.2 Future work suggestions:
6.2.1 SNMM on thin-films deposited on high resistivity silicon
substrate
In chapter 4 and 5 thin-film s were deposited on low resistivity silicon substrate, m odeled
as a conductive substrate. W hen a silicon substrate is not conductive enough, a more
com plicated model needs to be employed. Figure 6.1 (a) shows the equivalent circuit model
used for conductive A FM tip to thin-film sample electrom agnetic m odeling (thin-film
deposited on low-resistivity/conductive silicon), as was described in chapter 4. Figure 6.1 (b)
shows the proposed equivalent circuit model for the same system, but with a non-conductive
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Fig. 6 .1: Simplified equivalent circuit model for conductive AFM probe tip to thin-film deposited on
silicon substrate (a) When the silicon substrate is conductive (b) Proposed model when silicon substrate
is not conductive: Rs and Cs are resistance and capacitance associated with non-conductive silicon
substrate, respectively (c) An air capacitance (C0) is added, when AFM is scanned in non-contact
mode.
silicon substrate; thus the resistivity (Rs) and capacitance (Cs) of the silicon substrate should be
included in the model. Cp and Cts can be m easured in our setup using the same calibration
m ethodology as described in chapter 4. In order to measure R s and Cs , we need to calibrate
both for m agnitude and phase o f the reflection coefficient (or reflected wave) based on the
known bulk samples having known conductivity and permittivity, such as a high resistivity
silicon w afer and fused silica [2]. For less dam age to an A FM tip (or a sample), an A FM tip
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130
m ay be scanned in non-contact or tapping mode. The equivalent circuit, in this case, is shown
in figure 6.2 (c). The only difference with figure 6.2 (b) is that an air capacitance (CD) needs to
be added to the model in figure 6.2 (b). This capacitance can be easily m easured by scanning
the A FM tip in non-contact m ode on the surface o f a conductive silicon substrate. The quasi­
static field distribution was the main assumption o f all our lum ped elem ent modeling. In very
high microwave frequencies, this assumption w ouldnot be valid any m ore, thus a distributed
circuit model needs to be considered [3].
6.2.2 A more sensitive impedance analyzer for SNMM application
As we have seen in chapter 5, by increasing the am plification gain, the overall sensitivity
increases. This needs a high-gain broad-band am plifier which is typically very expensive in
microwave frequencies. A nother way to im plem ent an im pedance analyzer is by nulling the
reflected wave at the IF frequency, instead o f the R F frequency as shown in figure 6.2 [4, 5].
This is practical, since the phase and am plitude o f the reflected wave in IF frequency is
preserved, because the LO and R F sources are phase locked. The IF nulling configuration has
several advantages over an R F configuration, such as em ploying a very high gain, low noise IF
am plifier and requiring a cheaper, narrow -band IF variable phase shifter and variable
attenuator in com parison to the R F im pedance analyzer implementation. The disadvantage of
this approach is that a drift exist from the two mixers in figure 6 .2 .
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131
Directional coupler-2
Directional coupler-1
DUT
RF Source Y \ s
Mixer-2
Mixer-1
Differential
Receiver-
Amp
Ref
A/D
50 Q
50 Q
Variable phase
shifter
Variable —
attenuator
Fig. 6.2: Proposed impedance analyzer configuration for SNMM application. The only difference with
figure 5.4 is that the nulling of reflected wave happens to be in IF frequency, not in RF [4]. This
configuration has several advantages: employing variable attenuator, phase shifter, and differential
amplifier is in IF frequency which is cheaper in compare to RF nulling of reflected wave. Since a very
high-gain low noise IF amplifier can be employed, sensitivity could be boosted up in compare to RF
nulling configuration.
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132
6.3 References
[1] Y. W ang, A. D. betterm ann,
and D. W. van der W eide, “Process for scanning near­
field microwave m icroscope probes with integrated ultra tall coaxial tips”, Journal o f
vacuum science technology B, 25, 813 (2007)..
[2] A. Imtiaz, T. Baldwin, H. T. Nembach, T. M. W allis, and P. Kabos, “N ear-field
microwave m icroscope m easurem ents to characterize bulk material properties” ,
Applied physics letters, 90, 243105 (2007).
[3] A. Tselev, S. M. Anlage, Z. M A, and J. M elngailis,” Broadband dielectric m icrowave
m icroscopy on micron length scales” , Review o f scientific instruments,7 8 ,0 4 4 7 0 1 (2007).
[4] H. Tanabakuchi, L. Betts, D. Blackham ,” Converting vector network analyzer to a high
precision im pedance analyzer ” U. S. patent, Agilent technologies, 2006.
[5] A. Karbassi, H. Tanbakuchi, L. Betts, D. Blackham, R. Stancliff, “High precision
im pedance analyzer” Summer internship presentation, Agilent technologies, 2006.
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133
Apendix C: Alternative route for field focusing
An alternative route to focus electrom agnetic fields is by using a (left handed) m edia
which has negative refractive-index (NRI) property. I briefly report on the design o f a one­
dim ensional
mushroom-type
transm ission
line
having
an
operating
frequency
of
approxim ately 50 GHz. This structure is proven to behave as a NRI m edia in low er frequency
band under specific conditions [1]. By dispersion curve engineering, we w ere able to scale
down the original structure to operate in higher frequencies. The result o f our m easurem ent of
a m icro-fabricated 1-D m ushroom structure is reported.
C.1 Introduction
N RI m edia (exhibiting both negative perm ittivity and perm eability) has not be found to
exist in nature [2], but can be made as an artificial m edia (meta-material) [3]. One of the
properties o f a N RI m edia is that it can be used to focus the electrom agnetic field if fabricated
as a flat lens [4]. A m ushroom type periodic structure had been built by em ploying circuit
board technology. The structure showed a N R I when operating at a frequency o f -3 .7 GHz [1].
This structure was first introduced by Sievenpiper [5] and subsequently used as a high
im pedance surfaces for antenna applications. W e designed a m ushroom type structure which
exhibits itself like a left handed-transm ission line (LH-TL). In a LH -TL, the phase and group
velocities are anti-parallel (i.e. they have opposite sign), whereas in conventional transm ission
line (RH-TL) they are parallel (i.e. they have the same sign) [6 ].
C.2 Dispersion curve engineering
The unit cell o f a 2-D periodic m ushroom structure is shown in figure C .l. This
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134
structure consists o f metal caps (black) connected to a conductor ground plane on the bottom
of the unit cell through a via post conductor. The metal patches (gray) are used to enhance the
(a)
Fig. C.l: (a) Schematic of a unit-cell of 2-D mushroom structure. The dimension of designed
structure is: T=254 pm, P=145 pm, g=32 pm, C=222 pm, H=85 pm, h=4.5 pm, d=35 pm, h+H=89.5
pm (b) A schematic of 5-section 1-D mushroom structure, (c) An equivalent circuit model for a unit
cell.
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135
capacitance between the adjacent caps and can be used to tune the response. W e designed the
param eters of the mushroom structure by scaling down the dim ensions o f the mushroom
structure constructed by Sanada et al. [1], in order to make it operate in a higher frequency
ranges. W e perform ed a full-wave sim ulation for the unit cell (shown in figure 4.1) using
A nsoft HFSS com mercial software (Eign-m ode) to calculated the dispersion curve. In figure
C .l(a), we applied periodic boundary conditions for each tw o walls facing each other, and an
air box above the structure which term inated with a PM L (perfect matched layer). W e
optim ized the unit cell dimension values to achieve an approximate 37 GHz cut-off frequency,
which was chosen based on the capability o f our m easurem ent system. In figure C .l (a), the
material between the metal cap and the metal patches was SU -8 2005 (£r =5.8), whereas the
material used between the metal patches and the ground plan on the bottom o f the unit cell was
SU -8 2050 (Er =3.3). The patches, cap, and the bottom of the unit cell (ground plane) were
gold conductors (the gold layer thickness was bigger than skin depth at 37 GHz, thus it can be
assum ed as a perfect conductor) [7]. Figure C .l (b) diagrams a 1-D schematic o f the
mushroom structure. The ground-signal-ground (GSG) probe pads were added at each end of
the structure for the measurem ent purposes. The fabrication process was previously reported in
[7]. Figure C .l (c) diagrams the equivalent circuit model for a unit cell o f this structure. In this
figure, C l presents the mutual capacitance between the adjacent caps and patches. Shunt
inductance, Ll , is provided by the metal post. The inductance associated with caps and
patches, introduces LR, and the capacitance between caps or patches and the ground plane
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136
(a)
CD
"O
co
»*o—
D
■<
3a
C
o>
m easu rem en t
- - H FSS (Driven m ode)
■- - - circuit model
ca
30
35
40
45
50
55
60
65
F requency (GHz)
100-
(b)
o 50-
cr
- - circuit model
— HFSS (Eigen mode)
air mode
u.
O.Ojt
0.5^
pd (rad)
I Fi«ltf[A/a3
2 8 2 S ? « -» 3
x .« 5 f * - c e s
■ 77
i. «u-«e9
I. 6*t95<-083
1.52490-883
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m ku d 1. 2757«-88t
i.isiu-aes
: 1 82650-863
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<* 0 3 8 3 * -» 4
i.
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l.S « ts6 0 -« 4
(c)
2.9«aee-ws
■i*
Fig. C.2: (a) S2i (Transmission coefficient): 1-D 9-section mushroom type transmission line Sparameters measurement, full-wave simulation using HFSS (Driven-mode), and circuit simulation using
ADS. (b) Dispersion curve: full wave simulation using HFSS (Eign-mode), and a circuit model based by
using equation 4.1 [7] (c) Electric and magnetic field pattern for LH mode (without patches).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
137
introduces the shunt capacitance
C r.
Thus, the 5-section m ushroom structure transm ission line
(TL) in figure C .l (b), represents a 1-D com posite right/left handed (CRLH) transm ission line.
This structure can behave like a LH-TL or LH m edia in its lower frequency band when
specific relation ship between various param eters is fulfilled [ 1].
C.3 Simulation results in comparison to the experimental results
Figure C.2 (a) shows the transm ission coefficient (S 21) measurem ent results from a m icro­
fabricated 1-D 9 section m ushroom structure [7]. The S 21 from a full-wave simulation (Driven
mode, HFSS from A nsoft Co.), and S 21 calculated using circuit model are also shown on the
same figure for the com parison. The S 21 m easurem ent on 1-D m ushroom structure was
perform ed using an A gilent 85 IOC vector net w ork analyzer, a Cascade probe station, and 150
pm pitch probes. Figure C.2 (a) shows an excellent agreem ent between sim ulated structure and
m easurem ent data. Circuit param eters in figure C .l (c) w ere calculated using Agilent ADS
circuit sim ulator by com parison o f circuit model to the m easurem ent results. The values were
found to be: CR=0.26 pF, LR=0.062 nH, C L=0.12 pF, Ll =0.017 nH, R=2.88 ohm. Figure C.2
(b) shows the dispersion curve calculated from full-wave simulation (Eign-mode, HFSS from
Ansoft Co.) that was used for the design of the structure. Also, the dispersion curve calculated
from circuits values (determ ined earlier) showed at the same figure. The circuit model
dispersion curve is in a good agreem ent with the low er m ode dispersion curve calculated from
the full-wave simulation except in the air mode. The dispersion curve predicted by the circuit
m odel can be found from following equation [7]:
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138
p.d = 2 arcsin
{(o2CRLL -\)(co2LR CL - \ )
C1
4 co2L l C l
The low er band o f the dispersion curve in figure C. 2(b) shows anti-parallel group velocity
(Vg
dco
co
and phase velocity (Vp = — ) which proves the existence o f a back-w ard wave
[6 ], thus, the structure exhibits itself as a LH-TL.
C.4 Conclusion and future work
I have presented a design for achieving a m ushroom type LH -TL structure by
dispersion curve engineering. The m easurem ent results of the m icro-fabricated 1-D m ushroom
type transm ission line showed a very good agreement with the simulation. Future w ork will
include exploration o f two dim ensional m edia based on this unit cell, which are o f interest due
to their N R I properties.
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139
C.5 References
[1] A. Sanada, C. Caloz, and T. Itoh, ’’Plannar distributed structures with negative refractive
index,” IEEE Transaction on microwave theory and techniques, 52, 2702 (2004).
[2] V. G. Veselago, ‘T h e electrodynam ics o f substances with simultaneously negative values
of
E
and ft,” Soviet physics Usp., 10, pp. 509(1968).
[3] F. Elek, G. V. Eleftheriades,” D ispersion analysis o f the shielded Sievenpiper structure
using m ulti-conductor transm ission-line theory,” IEEE Microwave and wireless
components letters, 9, 434 (2004).
[4] J. B. Pendry ’’N egative refraction m akes a perfect lens” Physical review letters, 85,
3966 (2000).
[5] D. Sievenpiper, L. Zhang, R. F. J. Broas, N. G. A lexopolous, and E. Yablonovitch, “Highim pedance electrom agnetic surfaces with a forbidden frequency band,” IEEE Transaction
on microwave theory and techniques, 47, 2059 (1999).
[6] S. Ramo, J. R. W hinnery, and T. van Duzer, Fields and waves in communication
electronics, Third edition, John W iley & sons, 1993.
[7] C. Qin, A. B. Kozyrev, A. Karbassi, V. Joshkin, and D. W . van der W eide,”
M icro-fabricated left-handed transm ission line operating at 50 GHz, “IEEE International
Microwave Symposium (IMS), 3-8 June 2007.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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