Study of Current Measurment in Magnetic Force Microscopy
by
Margaret Hsin-Yi Wang
Submitted to the Department of Electrical Engineering and Computer Science
in partial fulfillment of the requirements for the degree of
Master of Engineering in Electrical Engineering and Computer Science
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
February 2001
@ Margaret Hsin-Yi Wang, MMI. All rights reserved.
The author hereby grants to MIT permission to reproduce and distribute publicly
paper and electronic copies of this thesis document in whole or in part.
A uthor ................
C ertified by ............
. . . . . . . . . . . . . . . . . .Department of Electrical Engineerd aRd ComputerScipCnce
September'14, 2001
....................................
Rajeev J. Ram
Associate Professor
Thesis Supervisor
Accepted by ..................
Arthur C. Smith
Chairman, Department Committee on Graduate Students
MASSACHUJ
JIl2TUIE
OF TECHNOLOGY
JUL 3 1 '1V
LIBRARIES
BARKER
3
Study of Current Measurment in Magnetic Force Microscopy
by
Margaret Hsin-Yi Wang
Submitted to the Department of Electrical Engineering and Computer Science
on September 14, 2001, in partial fulfillment of the
requirements for the degree of
Master of Engineering in Electrical Engineering and Computer Science
Abstract
Current measurement in magnetic force microscopy (MFM) senses the deflection of a magnetized
scanning probe tip caused by the magnetic field from the current flow. Due to the high spatial
resolution of MFM, accurate measurement of current can serve as a powerful tool for failure detection
on small integrated circuits. This thesis presents theoretical analysis and experimental results on
DC and AC current imaging. The movement of the scanning probe and the magnetic tip-sample
interaction were studied and applied to the theory. Sensitivity for the two current imaging methods
were determined to be 1 mA and 15 pA respectively. A general theory for non-linear tip-sample
interaction was developed and improvement to current sensitivity through parametric gain was
proposed and modeled.
Thesis Supervisor: Rajeev J. Ram
Title: Associate Professor
4
5
Acknowledgments
There are many people without whom this thesis would not have been possible. First and foremost,
I would like to thank my professor, Rajeev Ram, for his guidance these last four years. His zeal for
learning is contagious. And even more than the abundance of knowledge he possesses, he is a man
of much wisdom.
I am very fortunate to have had the opportunity to work with such an intelligent, supportive, and
fun-loving research group. I thank Mathew for being my partner-in-crime. The late nights in the
AFM lab would not have been the same without him. I appreciate his attitude and outlook on life.
Harry, a man with much technical expertise, has been truly selfless with his time. His openness and
sincerity are refreshing. I am grateful for the walks to the Student Center with Kevin. He has shown
me the meaning of carpe diem with his life. Peter, my fellow desk partner, has been most patient
these last few months with my taking over our desk. His ability to balance graduate life with other
deserving priorities has been inspiring. And for the rest of the group, Farhan, George, Tom, and
Seung-Ho, I appreciate their friendships. I hope that their graduate experiences are most-fulfilling.
So long guys.
I would also like to thank the National Science Foundation for their support in tuition, equipment,
and fees for using the AFM.
My family and friends have always stood by my side and selflessly given me their love and
encouragement. Papa Wang and Mama Wang have been my role models and I cannot ask more.
Phil and Tif are very dear to me and I look forward to spending more time at home. Stan has been
a loyal friend for the past 10 years. Jarter has been a supportive friend, through many laughters
and good times. Frank has been a dear friend from afar. Bonnie and Ellen have seen the most of
my ups and downs this year and I appreciate their patience and care for me. I am truly blessed with
wonderful people whom I cherish from the bottom of my heart.
6
Contents
1
1.1
1.2
Magnetic Force Microscopy
18
. . . . . . . . . . .
. . . . . . . . . . . . . . . . .
. . . .
18
. . . . .
. . . .
19
1.1.3
M FM Tip . . . . . . . . . . . . . . . . .
. . . .
20
1.1.4
Photodetection System
. . . . . . . . .
. . . .
20
1.1.5
Comparison to Other Magnetic Imaging Techniques
. . . .
21
. . . . . . . . . . . . . . . .
. . . .
22
DC Current Imaging . . . . . . . . . . .
. . . .
25
. . . . . . . . . . . . . .
. . . .
27
. . . . . . . . . . . . . . . . .
. . . .
27
1.1.1
Overview
1.1.2
Movement of Cantilever Beam
Point-Mass Model
1.2.1
2
17
Introduction
1.3
Flexural Beam Model
1.4
Thesis Overview
29
DC Current Imaging
2.1
Cantilever Dynamics . . . . . . . . . . . . . . .
29
2.1.1
Transverse Beam Equation
. . . . . . .
30
2.1.2
Free Cantilever without Damping . . . .
34
2.1.3
Free Cantilever with Damping
. . . . .
36
2.1.4
Comparison to the Point-Mass Model
.
37
2.2
Magnetic Tip-Sample Interaction . . . . . . . .
.. . . . .. . . . .
38
2.3
T heory . . . . . . . . . . . . . . . . . . . . . . .
. .. . . .. . . . .
40
2.3.1
2.4
Comparison to the Point-Mass Model at the First Vibration Mode
41
Experim ent . . . . . . . . . . . . . . . . . . . .
. . .. . . . .. . .
43
. . . . . . . . . . . . . . . . . . .
. . . .. . . . .. .
43
2.4.1
Setup
7
8
CONTENTS
2.5
3
2.4.3
Comparison to Theory at the First Vibration Mode
2.4.4
Second Vibration Mode with Magnetic Tape
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 45
. . . . . . . . . . . . . .
51
. . . . . . . . . . . . . . . . . .
51
Summ ary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
57
3.1
T heory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
3.1.1
Magnetic Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
3.1.2
Boundary Conditions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
3.1.3
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
Experim ent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
3.2.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
R esults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
3.3.1
Decomposition Method
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
3.3.2
Decomposed Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
3.3.3
Response to Varying Lift Heights . . . . . . . . . . . . . . . . . . . . . . . . .
66
Comparison to Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
3.4.1
M FM
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
3.4.2
E FM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
Summ ary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
3.3
3.4
3.5
Setup
Method for Non-Linear Tip-Sample Interaction
83
4.1
T heory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
4.1.1
Limit to DC and AC Current Imaging . . . . . . . . . . . . . . . . . . . . . .
89
Application to Parametric Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
4.2.1
Principles of Parametric Gain . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
4.2.2
Degenerate: wc=..p
92
4.2.3
Nondegenerate: w,
4.2
5
R esults
AC Current Imaging
3.2
4
2.4.2
...............
. .............................
93
4.3
Non-Linearity in AFM System
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
4.4
Sum m ary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
Conclusion
99
5.1
99
Summary
CONTENTS
5.2
5.3
9
Directions for Future Work
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.2.1
M odeling
5.2.2
Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Final W ords . . . . . . . . . . . . . . ..
. . . . . . . . . . . . . . . . . . . . . . . . . 102
105
A Experimental Setup
A.1
Instrument Setup .........
......................................
105
A.1.1
Atomic Force Microscope
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
A.1.2
Lock-in Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
A.2
Sample Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
A.3
Data Acquisition and Processing Software . . . . . . . . . . . . . . . . . . . . . . . . 107
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
A.3.1
LabVIEW Code
A.3.2
Matlab Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
A.4 Recipe for DC and AC Current Imaging . . . . . . . . . . . . . . . . . . . . . . . . . 107
B Magnetic Field Over a Wire
109
C Matlab Code
115
C.1 DC Current Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
115
C.2 AC Current Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
118
C.1.1
DCcurrent.m
C.2.1
ACcurrent.m
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
118
C .2.2
decay.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
C .2.3
decayuse.m
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
C.3 Non-linear Current Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
C.3.1
degenerate.m
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
C.3.2
nondegenerate.m
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
10
CONTENTS
List of Figures
1-1
MFM setup for measuring current. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
1-2
The first four vibration modes of a freely vibratingcantilever . . . . . . . . . . . . . .
20
1-3
Schematic drawing of a MFM tip . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
1-4
Schematic drawings of the position detection and the slope detection systems.
23
1-5
Comparison of magnetic imaging techniques . . . . . . . . . . . . . . . . . . . . . . .
24
1-6
The presence of a magnetic field causes a shift in the resonant f1requency, Awo =
O- wo, affecting both the amplitude and phase responses.....
. . . . . . . . . .
25
1-7
Illustration of force on an electric dipole. . . . . . . . . . . . . . . . . . . . . . . . . .
26
2-1
Bending moment exerted on a small segment of elastic beam.
. . . . . . . . . .
30
2-2
Moment expressed as M =
fA zodA. . . . . . . . . . . . . . .
. . . . . . . . . .
31
2-3
Relationship between beam position and angle of deflection. .
. . . . . . . . . .
32
2-4
Free body diagram of beam element dx.
. . . . . . . . . . . .
. . . . . . . . . .
33
2-5
Vibration amplitude of a free, undamped cantilever tip.
. . .
. . . . . . . . . .
36
2-6
Vibration amplitude of a free, damped cantilever tip. . . . . .
. . . . . . . . . .
37
2-7
Schematic of the extended point probe model. . . . . . . . . .
. . . . . . . . . .
39
2-8
Comparison of cantilever magnitude and phase response at resonance between point
mass model and flexural beam model . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-9
42
Comparison of cantilever magnitude response between point mass model and flexural
beam model slightly off resonance.
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
2-10 MFM setup for DC current imaging. . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
2-11 Experimental phase response to varying DC current. . . . . . . . . . . . . . . . . . .
46
2-12 Comparison between phase responses to currents in opposite direction. . . . . . . . .
47
2-13 Comparison between phase responses at different cantilever drive frequencies.....
48
11
LIST OF FIGURES
12
2-14 Experimental phase response to varying DC current. . . . . . . . . . . . . . . . . . .
49
2-15 Experimental phase response to varying lift height. . . . . . . . . . . . . . . . . . . .
50
2-16 Comparison between the experimental and simulated phase responses using the extended monopole model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
2-17 Comparison between the experimental and simulated phase responses using extended
dipole m odel.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
. . .
55
. . . . . .
56
2-18 Relationship between DC current and maximum to minimum phase difference.
2-19 First and second vibration mode phase imaging of sample magnetic tape.
3-1
Simulated cantilever response to varying AC current . . . .
3-2
Simulated cantilever response to -2 mA AC current . . . . .
3-3
MFM setup for AC current imaging. . . . . . . . . . . . . .
3-4
Circuit diagram for AC current .. . . . . . . . . . . . . . . .
3-5
Experimental magnitude and phase responses to varying AC current..
3-6
Experimental magnitude and phase responses to varying AC current in the opposite
direction.
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-7
Magnitude responses to electrostatic force. . . . . . . . . . .
3-8
Schematic drawing of the decomposition method. . . . . . .
3-9
Method for extracting the odd (MFM) component. . . . . .
3-10 Method for extracting the even (EFM) component. . . . . .
3-11 Total and extracted EFM and MFM magnitude responses due to 2 mA AC current.
3-12 Total and extracted EFM and MFM magnitude responses due to 2 mA AC current
in the reverse direction.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-13 Extracted EFM and MFM magnitude responses to varying AC current.
. . . . . . .
3-14 Extracted EFM and MFM magnitude responses to varying reverse AC current. . . .
3-15 Experimental magnitude and phase responses to varying lift heights. . . . . . . . . .
3-16 Extracted EFM and MFM magnitude responses to varying lift heights. . . . . . . . .
3-17 Comparison between the experimental and simulated magnitude responses using the
extended monopole model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-18 Decay in magnetic field and gradient of magnetic field to tip-sample distances.
. . .
3-19 Maximum H, and dHz/dz as a function of z. . . . . . . . . . . . . . . . . . . . . . .
3-20 Comparison between experimental and simulated MFM responses.
. . . . . . . . . .
LIST OF FIGURES
13
3-21 Simulated EFM response to 2 mA AC current.
......................
80
4-1
Magnitude and phase response at w, = 0. . . . . . . . . . . . . . . . . . . . . . . . .
90
4-2
Cantilever response with no piezo drive. . . . . . . . . . . . . . . . . . . . . . . . . .
91
4-3
Schematic drawing of time-varying parallel-plate capacitor.
. . . . . . . . . . . . . .
92
4-4
Magnitude response at varying frequencies.
. . . . . . . . . . . . . . . . . . . . . . .
93
4-5
Magnitude responses to varying piezo drive amplitudes when W, + WP =
WO. . . . . .
94
4-6
DC current and nondegenerate cantilever response. . . . . . . . . . . . . . . . . . . .
95
4-7
DC current cantilever response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
4-8
FFT spectra of output voltage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
4-9
Non-linearity in input piezo voltage.
. . . . . . . . . . . . . . . . . . . . . . . . . . .
98
4-10 Non-linearity in output voltage of a free cantilever. . . . . . . . . . . . . . . . . . . .
98
5-1
Force calibration plot for voltage to metric conversion. . . . . . . . . . . . . . . . . .
103
B-1 Magnetic field produced by a straight conductor at point P. . . . . . . . . . . . . . .
110
B-2 Magnetic field about a point wire.
112
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
B-3 Magnetic field about a wire with finite width and height . . . . . . . . . . . . . . . . 112
B-4 The vertical components of H, VH, and V 2 H.. . . . . . . . . . . . . . . . . . . . . .
113
14
LIST OF FIGURES
List of Tables
1.1
Table of cantilever parameters.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1
Table of components used for DC current imaging
. . . . . . . . . . . . . . . . . . .
44
3.1
Table of components used for AC current imaging
. . . . . . . . . . . . . . . . . . .
61
A.1
Experimental settings for the lock-in amplifier.
. . . . . . . . . . . . . . . . . . . . .
106
15
21
16
LIST OF TABLES
Chapter 1
Introduction
Magnetic force microscopy (MFM) can image magnetic fields down to 10 nm spatial resolution by
measuring the interaction between a sharp, magnetized probe tip and the magnetic sample. The
technique has become a powerful tool in the magnetic recording head industry, which continually
strives for higher density.
The integrated circuit (IC) industry, similarly, demands smaller circuits for higher density on a
chip, which lowers costs and enables greater portability. However, while there exist highly-developed
techniques for voltage measurement on small ICs, current measurement remains a challenging task[6].
Such a technique provides valuable information for quality assurance and failure detection, and the
absence of one is costly.
From Ampere's Law, charge flow in a conductor results in a "cylindrical" magnetic field about
the conductor[13]. By imaging this magnetic field, current measurements can be made. This noninvasive method of current measurement is particularly advantageous because it minimizes the risk
of damaging the fragile sample. Thus MFM, with its high spatial resolution, proves to be a good
candidate.
This thesis explores DC and AC current imaging in MFM theoretically and experimentally.
First, an overview of MFM, including its setup, a brief history, and a comparison to other magnetic
imaging techniques, is presented in this chapter. A simplified model describing the MFM response
to magnetic field is then introduced to provide physical understanding to the magnetic tip-sample
interactionq. Its limitations are also discussed, thus motivating the employment of a more complete
model. Lastly, an overview of the thesis is provided.
17
Laser
Metal wire carrying current
Figure 1-1: MFM setup for measuring current.
1.1
1.1.1
Magnetic Force Microscopy
Overview
MFM is a special application of atomic force microscopy (AFM), which was invented by Binnig
et al.[4] in 1986. An atomic force microscope images the sample by scanning a sharp probe tip,
attached at the end of a cantilever beam, above the sample. Several operational modes are available
for different imaging purposes. Specifically, MFM uses the tapping mode[41], where the cantilever
is dynamically driven close to or at one of its resonant frequencies.
Interactions between the sample and the tip cause the cantilever to deflect. This deflection is
then measured by a pair of photodiodes, which receive the reflection of a collimated laser beam
directed at the tip. In the case of MFM, the tip is coated with a ferromagnetic thin film and deflects
in the presence of magnetic field. Fig. 1-1 shows the MFM setup of a Digital Instrument Dimension
3000 microscope.
The first MFM images were demonstrated in 1987 by Martin et al.[21], who measured forces
from a microfabricated magnetic recording head. Individual interdomain boundaries have also been
analyzed[10].
Over the years, its high spatial resolution has made it a powerful tool in the data
1.1.
19
MAGNETIC FORCE MICROSCOPY
storage industry. Some applications include measurements of the head performance on hard disks and
the domain behavior of thin-film and magnetoresistive heads. Outside of data storage measurements,
other applications include imaging flux lines in low- and high-T, superconductors[25] and biological
samples such as magnetotactic bacteria[27].
Current imaging in MFM was first performed by Campbell et al.[6].
Subsequently, Babcock
et al.[2] used current strips to determine tip coercivity and magnetic moment, and Kong et al.[18]
similarly quantified magnetic moments using current rings.
Gaining a clear understanding of the MFM cantilever response to current necessitates a more
detailed introduction of three key components: the mechanical movement of the cantilever beam,
the interaction between the MFM tip and a magnetic field, and the setup of the photodetection
system.
1.1.2
Movement of Cantilever Beam
The AFM cantilever is a miniature elastic beam with characteristic vibration modes and frequencies[30],[34].
A complete solution to the cantilever deflection includes the responses from an infinite set of flexural
vibration modes. In general, however, a lock-in amplifier can be used in the measurement process
to isolate the cantilever response at a particular frequency.
Boundary conditions determine the behavior of the cantilever. The fixed end of the beam is
clamped to a bimorph piezoelectric plate, which drives the cantilever. The bimorph piezoelectric
plate consists of two pieces of piezo-ceramic that are bonded together so that the differential changes
in length of the two pieces due to a difference in voltage given can produce relatively large movements
in the cantilever.
The other end is attached to the probe tip. In the absence of any tip-sample interaction, the
tip-end vibrates freely. Fig. 1-2 illustrates the first four vibration modes of the standard thin film
MFM tips from Digital Instruments (type Magnetic Etched Silicon Probe). This model does not
include damping and lateral motion from the cantilever. The first four resonant frequencies occur
at 89.93, 563.89, 1578.82, and 3093.97 kHz, respectively. The typical material parameters for these
cantilevers are shown in Table 1.1, which were obtained from measurements made by Lohau et al.[19]
However, if interaction forces are present, then the boundary conditions would differ.
This
changes the cantilever response. The mathematical model including the presence of DC current is
presented and solved in Chap. 2. Chap. 3 contains the theoretical model and solution to AC current
imaging.
CHAPTER 1.
20
n=1
INTRODUCTION
n=2
30
20
T
20
T
10
10
0
0
-10
-
0
0
50
-20
-30'
10
20
2
100
150
x [um]
200
25 0
200
250
-20
0
50
100
150
x [um]
200
-30
2E50
n=3
n=4
30
30
-
20
T
10
E -10
20
T
10
10
0
0
E -10
E-10
-20
-30
-20
0
50
100
150
x [um]
200
250
-30
0
50
150
100
x [um]
Figure 1-2: The first four vibration modes, n = 1 to 4, of a freely vibrating cantilever.
1.1.3
MFM Tip
An understanding of the MFM tip is required to accurately quantify the magnetic force.
It is
pyramidal in shape and coated with ferromagnetic thin film, as shown in Fig. 1-3. However, its
exact magnetic properties are generally unknown.
A simplification of the probe's magnetic behavior to magnetic field is described in the point probe
approximation[20], and it has proven to yield satisfactory results. In this model, the force acting
on the probe is a function of the effective monopole and dipole moments, which are treated as free
parameters to be fitted to the experimental data.
1.1.4
Photodetection System
Several methods are available for measuring the cantilever deflection. The photodetection system
used in the Dimension 3000 microscope is the slope detection method[23], where a collimated laser
1.1. MAGNETIC FORCE MICROSCOPY
Cantilever parameter
Length of beam, L
Width of beam, a
21
Thickness of beam, b
Value
227
28.9
3.37
Unit
um
um
um
Young's modulus, E
1.69x10"
N/M 2
Density, p
Bimorph vibrational amplitude, zo
2330
10
kg/m 3
nm
Table 1.1: Table of cantilever parameters
beam is focused on the tip and is reflected back into two ajacent photodiodes whose currents are fed
into a differential amplifier. It is also called the optical beam deflection technique.
This system uses the slope at the tip to determine its deflection, assuming a direct relationship between two, while another system, the optical interferometer technique[21],[31], measures the
deflection directly. Fig. 1-4 schematically describes both methods.
The two have essentially the same sensitivity[28]. However, the optical beam deflection force microscope has the added advantage of being able to measure lateral forces as well as normal forces[24].
1.1.5
Comparison to Other Magnetic Imaging Techniques
Other magnetic imaging techniques are available and have also been used to measure current. However, a tradeoff between spatial resolution and magnetic field sensitivity separates the different
methods to specific applications and needs.
Among these techniques, scanning superconducting quantum interference device (SQUID) microscopy[39]
offers the best magnetic field sensitivity. SQUIDs are magnetic field detectors that use the relationship between the quantum mechanical phase difference between two closely-spaced superconductors
and the supercurrent that flows between them to extract information about the magnetic field. The
critical current varies sinusoidally with the integral of the magnetic field through the area of the
SQUID loop and the period of modulation is the superconducting flux quantum, h/2e = 2.07x
10-15 Tm 2. Therefore, its sensitivity is very high and increases linearly with the area. Two main
disadvantages of scanning SQUID microscopy are that its spatial resolution is limited to about 10
pm and that it requires sensor operation at low temperatures[17].
Scanning Hall probe microscopy (SHPM)[7] consists of a scanning probe attached to a Hall bar.
CHAPTER 1. INTRODUCTION
22
Si ti
Cantilever beam
Thin film
coating
Figure 1-3: Schematic drawing of a MFM tip (not drawn to scale).
It measures the Hall resistance of the bar, which is directly proportional to the magnetic flux through
the bar itself. Its sensitivity depends on the proximity of the Hall probe to the sample and the size
of the bar. The magnetic field resolution is less than that of scanning SQUID microscopy, but the
spatial resolution of the probe is about lpm[26].
Magnetic force microscopy (MFM) has the highest spatial resolution among the magnetic imaging
systems, as well as the added advantage of requiring essentially no special sample preparation[12].
However it lacks the magnetic sensitivity that the other systems can provide. Fig. 1-5 compares the
three imaging techniques in terms of the two criterion.
1.2
Point-Mass Model
An introduction of a simplified model describing the cantilever response is helpful for understanding
the system. The point-mass model approximates the cantilever deflection at the tip as a damped harmonic oscillator[34]. The cantilever beam is represented by an effective point mass, meff, attached
to a spring with stiffness k,.
1.2.
-
a) Displacement detection
b) Slope detection
Figure 1-4: Schematic drawings of the position detection and the slope detection systems.
meff is chosen to match the fundamental vibration frequency of the cantilever beam[30],
k
WO =c
meff
And kc, which is generally about 2 to 5 N/m, is defined by the physical properties of the beam,
which will be explained in Sec. 2.1.4.
In the absence of any tip-sample interaction, the cantilever deflection, z(t), is described by
meff
d 2 z(t)
2
dt
fb~t
+ mefb dt
+ kz(t) = Foejwt
(1.2)
where b is the damping term and Foei't is the driving force of the cantilever from the piezo motion.
By assuming a solution z(t) = Aeidt, the amplitude and the phase responses of the cantilever
can be found:
A(w)
=
(2
Fo/meff
-w 2 ) 2 + (WO
<b(-) = tan-'
where
Q
=
(1.3)
)2
ww
(1.4)
wo/b is the quality factor of the cantilever and is usually about 180.
Tip-sample interaction creates an additional force on the cantilever, which can be Taylor-expanded
to first order,
F(z) = F(zo) +
aF(z0 )
(z - zo)
(1.5)
The zero-order term shifts the equilibrium position of the cantilever tip, but the first-order term
EElectron
Microscopy
10
-I
10
-
10
-
10_
-
-
'kIM F
10 uA
I--
>! 10
C
1 uA
0-7
0.1 uA
10-
10 nA
10.9
1010 -
10
10-1
0.01
0.1
0
110-o
0-I(DO--o C
i
1
10
100
Spatial Resolution (prn)
Figure 1-5: Comparison of magnetic imaging techniques. A tradeoff between magnetic field sensitivity and spatial resolution is shown among scanning superconducting quantum interference device
(SSQUID) microscopy, scanning Hall probe microscopy (SHPM), and magnetic force microscopy
(MFM).
changes the effective spring constant,
k' = k, - F'
(1.6)
where F' =
Consequently, the resonant frequency of the oscillator is shifted by
2ke
assuming that F'
(1.7)
< kc.
Fig. 1-6 describes the resonant frequency shift, Awo =
responses as a result of tip-sample interaction.
w' - wo, in both the amplitude and phase
25
1.2. POINT-MASS MODEL
2
12.
w'
W.0
10-
1
no interaction
with mag. force
W
0o
1.5
-
8-
0.5
AA
/ I
i
-
ID
"0
6-
/WO
0
'D
-4
W,0
-0.5
4
-1
-
2-
95
100
Frequency [kHz]
105
95
100
Frequency [kHz]
105
Figure 1-6: The presence of a magnetic field causes a shift in the resonant frequency, Awo =
affecting both the amplitude and phase responses.
1.2.1
W, - wo,
DC Current Imaging
In the presence of DC current, the magnetized tip experiences an external force F, which can be
quantified by approximating the tip as either a point dipole or a point monopole.
F = V(m -poH)
F = qpoH
and
(1.8)
(1.9)
where m and q are respectively the effective magnetic dipole and monopole moment of the tip, po
is the permeability of free space (47r x 10-1 Wb/Am), and H is the magnetic field produced by the
current[12].
Eqn. 1.8 may be easier to understand intuitively by considering the analogous case of an electric
dipole in the presence of electric field. In Fig. 1-7(a), the electric dipole rests in an uniform electric
field, and the positive and negative ends experience an equal torque. The dipole begins to spin but
its center of mass does not move. In the case of a gradient electric field (Fig. 1-7(b)), the end that is
in the stronger electric field experiences greater torque and the difference causes the center of mass
of the dipole to shift. The concept can similarly apply to a magnetic dipole in a gradient magnetic
Electric dipole
a) Uniform
b) Gradient
+
electric field
electric field
Figure 1-7: Illustration of force on an electric dipole. (a) The electric dipole rests in an uniform
electric field. There is no net movement. (b) The electric dipole in the presence of a gradient electric
field. The end in the stronger electric field experiences greater torque, causing movement.
field by imagining the magnetic dipole having a positive magnetic charge on one end and a negative
magnetic charge on the other.
The monopole approximation, on the other hand, is equivalently an elongated dipole model,
where the length of the dipole is so long that only one end experiences the magnetic force.
Because the current-induced magnetic field is cylindrical, a first-order approximation of F is
sufficient. Thus, Eqn. 1.2 becomes
me!!
d 2 z(t)
dzt
eb dz(t) + kz(t) - F'z(t) = FoejWt
dt 2 + mt+
dt2
f Jdt
(1.10)
where
F' = az
1)z
because only the z-component of the magnetic force affects the cantilever's vertical deflection.
The phase change can be quantified, by taking the partial derivative of the original phase with
respect to w, evaluated at the resonant frequency, and multiplied by the change in the resonant
frequency:
8<>
A~b = 9io x Awo
Q
- VF
(1.12)
If the tip is approximated as a point dipole, the phase change is as follows:
A
=
Q (m.
ioH)
(1.13)
27
1.3. FLEXURAL BEAM MODEL
And the phase change of the monopole model yields
A4) =
(q - VpoH)
Therefore, the phase response varies with the gradient of the magnetic force.
(1.14)
If the tip is
2
approximated as a point dipole, the phase responds to V H, and as a point monopole, the phase
responds to VH. Both models are used in the DC current imaging theory in Chap. 2 and are
compared with experimental results to determine the validity of the two models.
1.3
Flexural Beam Model
The point-mass model is a single-mode approximation and fails to model the cantilever response
beyond the first mode. Its limitation necessitates the employment of the flexural beam model[30
which includes all the vibrational modes of the cantilever. This enables imaging at the higher-order
modes and provides the basis for theory of the non-linear tip-sample interaction between the piezo
movement and AC current. The more accurate model also provides a standard with which the
point-mass model can compare.
The flexural beam model is derived in Chap. 2 and provides the foundation for the current
imaging theories.
1.4
Thesis Overview
This thesis uses both theoretical and experimental results to explain DC and AC current imaging.
Chap. 2 begins by describing the flexural modes of the cantilever beam mathematically. Different
methods of modeling the magnetic tip are discussed. A quantitative simulation of the cantilever
response to DC current follows. Comparisons to experimental data are shown and discussed.
AC current imaging is presented in Chap. 3. Theory is developed using the flexural beam model
and experimental results are shown. The chapter concludes with a discussion of sensitivity.
Chap. 4 includes a method for non-linear tip-sample interactions and an examination of the
intrinsic non-linearity in the AFM system. The non-linear theory is validated by the limit to DC
and AC current imaging and used to excite higher-order frequency components.
A summary of the thesis is presented in Chap. 5. It concludes with suggestions for future work
in the area of current measurement in MFM.
28
CHAPTER 1. INTRODUCTION
Chapter 2
DC Current Imaging
Before approaching current imaging, models used to quantitatively interpret MFM images must be
first presented. In particular, the cantilever beam movement and the tip-sample interaction require
detailed examination.
As mentioned in Chap. 1, the cantilever vibration can be simplified as a mass-spring model,
where the resonant frequency of the system is chosen to match the first vibration frequency of the
beam. However, higher-order vibration modes of the beam are neglected, and this model becomes
inadequate when imaging at the higher-order modes are of interest or when the higher-order modes
are excited. Therefore, a complete model of the cantilever dynamics, the flexural beam model[30],
is introduced.
A brief description of several models describing the magnetic tip-sample interaction is provided.
Reasons for employing the extended monopole model are presented.
The two models are first applied to DC current imaging.
Simulations at the first resonant
frequency are compared with experimental results as well as the point mass model.
2.1
Cantilever Dynamics
The cantilever beam is driven in the transverse direction by a piezoelectric bimorph. In order to
understand how the tip-end of the cantilever behaves in the presence of magnetic force, first the
dynamics of the cantilever movement needs to be discussed. This is described by the transverse
beam equation, also known as the Euler-Bernoulli equation.
29
30
CHAPTER 2. DC CURRENT IMAGING
dx
2111111
Neutral axis
M
(
M
ArAR
d(D
x
z
Figure 2-1: The result of bending moment M exerted on a segment dx of an elastic beam. Tension
occurs above the neutral axis and compression occurs below.
2.1.1
Transverse Beam Equation
The derivation of the beam equation requires first a relationship between beam geometry and bending
moment. Subsequently the two mechanical equilibrium conditions-the sum of the forces and the
sum of the moments are zero-are used to arrive at the final equation.
The bending moment, M, which is also known as torque to people outside the world of mechanical
engineering, is defined as the cross product of force and distance. If it is exerted on both sides of a
small segment dx of the beam as shown in Fig. 2-1, a neutral axis emerges-a line above which the
beam experiences tension and below which the beam experiences compression. Since strain, c, is
defined as elongation over length, it can be described by
(R - z)d1 - Rd4
Rd4
_
z
R
at any distance z from the neutral axis, where R is the radius of bending and d
(2.1)
is the angle of
bending.
Because the material of the beam is isotropic, the stress,
o, and the strain at any given point are
2.1. CANTILEVER DYNAMICS
31
AA
y
os AA
x
z
Figure 2-2: Moment, which is also torque, can be expressed as the stress times the distance away
from the neutral axis, summed over the cross-sectional area.
linearly related:
z
o = EE = -ER
(2.2)
where E is the Young's modulus and is in units of N/m2.
Stress is defined microscopically as the force per unit area (Fig. 2-2). Since moment is the cross
product between force and distance, the total moment on the bending beam can be expressed as the
product of the distance away from the neutral axis, z, and stress in the x-direction, summed over
the cross-sectional area,
M =
fA
zadA =
E z2dA
(2.3)
z2dA is the moment of inertia of the beam cross-section and is represented by the symbol I.
Therefore, moment is
EI
R
(2.4)
In order to directly relate moment to the position of the beam, R is defined[35]. From Fig. 2-3,
several geometric relations can be deduced. First, since R is the bending radius and arc length is
dx
z(x)
ds ds
CD(x)
8
z
Figure 2-3: Relationship between beam position and angle of deflection.
the product of the bending radius and bending angle,
ds = RdO
(2.5)
tan 4 = dz/dx
(2.6)
Also,
If the beam deflection is assumed to be very small compared to the beam length, then 4 is also
very small and justifies the approximation tan
-
4.
By using the above approximation and combining Eqns. 2.5 and 2.6, the relationship is obtained,
1
R
d 2z
dx 2
(2.7)
and results in a new moment equation,
M =
EI
2
Z
dX2
(2.8)
The derivation of the transverse beam equation requires lastly the satisfaction of the mechanical
equilibrium conditions[8]. Fig. 2-4 shows a beam element of length dx experiencing an external force,
f(x). The shear forces (V and V+dV) and bending moments (M and M+dM) on the two faces are
dx
f(x)dx
M
M+dM
V
V+dV
Z
Figure 2-4: Free body diagram of beam element dx. In equilibrium, no translation nor rotation
occurs. Therefore the net force and the net moment are zero
also labelled. The condition of equilibrium at the center of the beam element are
+ fdx =0
E Fz = (V +dV) -V
Y
M = (M +dM)
- M +(V +dV)
dx
dx
2 +V 2 =0
2
2
(2.9)
(2.10)
The sum of the forces are zero in equilibrium because there is no displacement. The net moment,
or the net torque, is also zero to ensure no rotation. All except the second term in Eqn. 2.10 causes
a torque in the clockwise direction. Therefore, only the second term is negative.
As dx approaches zero, the second-order differential terms can be neglected.
Therefore, the
equations are simplified to
dV+ f (x)
=
0
(2.11)
dx
dM
d
dx
+ V= 0
(2.12)
Finally, if the beam is no longer in equilibrium, then the sum of the forces is no longer zero. It
instead equals the inertial force plus the frictional force. The inertial force is defined as the product
of mass and acceleration, where mass can be expressed as the material density, p, times the crosssection area, A, and acceleration is expressed as d.
The frictional force is defined as the product
of the damping term, -y, the mass, pA, and velocity, L. By substituting in Eqn. 2.8 into Eqn. 2.11,
the transverse beam equation is obtained,
dez
dz
d2 z
EIdx4 +ypAz + pA dt2
=
f(x,t)
(2.13)
CHAPTER 2. DC CURRENT IMAGING
34
In summary, the restoring force from the elastic beam, EI4,
works to oppose the external
force, f(x, t). When the beam is not in equilibrium, the beam moves. Its velocity is described by
the frictional force -ypAd,
and its acceleration is given by the inertial force pAp. The sum of the
restoring force, the frictional force, and the inertial force, equals the external force.
2.1.2
Free Cantilever without Damping
The cantilever motion is first examined without the damping term. In the case of a freely-vibrating
cantilever, there is no uniform external force applied on the beam. Therefore, the transverse beam
equation is reduced to,
EIdx4 + pA dt2 = 0
(2.14)
By separating the variables, the solution z(x, t) = Z(x)q(t) can be assumed[9]. Because the piezo
drive at x = 0 is proportional to eiwt, by linearity q(t) = Pei"i. Eqn. 2.14 hence becomes
EId Z(X)
pAZ(x)
d2q(t)
dX4
q(t) dt 2
W2(2.15)
(
This introduces two separate equations:
dt 2 -- + w 2 q(t)
dd4 Z(X)
Substituting the dispersion relation k
Z(x) = Alekx
=
W2
=
0
Z(x) = 0
(2.16)
(2.17)
and solving Eqn. 2.17 gives a solution for Z(x),
+ A 2 e--kx + A 3 eikx + A 4 e jkx
(2.18)
An alternative way of expressing Eqn. 2.18 is
Z(x) =
C1(cos kx + cosh kx) + C 2 (cos kx - cosh kx) +
C 3 (sin kx + sinh kx) + C4 (sin kx - sinh kx)
by redefining the coefficients.
(2.19)
2.1. CANTILEVER DYNAMICS
35
In order to determine the coefficients C 1, C 2 , C 3 , and C4 , boundary conditions need to be applied.
Considering that the cantilever of length L is forced to vibrate from the piezo at the clamped end
at x = 0 with the amplitude z(x = 0, t) = a cos wt, the boundary conditions are thus:
Z(x = 0) = a
dZ(x = 0)
=
dx d z0
d2 Z(x = L)
dx 2 d2 = 00
3
d Z(x = L)
=0
dx3
(2.20)
(2.21)
(2.22)
(2.23)
is 0 because the beam is clamped to the moving piezo. As derived earlier, d2Zx=L) and
rltdt
are related to the moment and the shear force at x = L respectively. Because the beam
dZ(x=O)
3
d Z( x =L)
dx
3
ends at x = L, there cannot be curvature at that point. Also, no shear force is pulling down on the
tip-end, allowing the cantilever to vibrate freely.
The solution after satisfying the boundary conditions is
Z(x)=
((coskx+coshkx)-
(sin
k L sinh kLL
cossincshkL+
kU cosh kU U+ I1 (cos kx - cosh kx) +
(sin k Lcosh kL + sinh kL coskLL\
sic cos
s cosh
+ kL
(sin kx - sinh kx))
kUL cosh
kU + U1I c
(2.24)
Z(x) peaks as cos kL cosh kL + 1 approaches zero. Therefore, the characteristic equation
cos knL coshknL +1
=
0
(2.25)
gives the infinite set of wave numbers k, that define the flexural vibration modes, where
n is
the mode number. Fig. 1-2 in Chap. 1 shows the first four vibration modes of a freely vibrating
MFM cantilever. And as illustrated in Fig. 2-5, the amplitude of vibration peaks at the resonant
frequencies.
CHAPTER 2. DC CURRENT IMAGING
36
160
140
120
100
E 80
80
40
20
0
400
200
600
1000
800
Frequency [kHz]
1200
1400
1600
1800
Figure 2-5: Vibration amplitude of a free, undamped cantilever tip.
2.1.3
Free Cantilever with Damping
Inclusion of the damping term, by definition, diminishes the response of the cantilever. The beam
equation becomes
d4 z
dz
EIdx4 +YPA
d 2z
+PAdt2
(2.26)
0
While the spatial distribution of the beam does not change (Z(x) remains the same), Eqn. 2.16
changes,
d2q(t)
dt 2
dq(t)
2
dt +wq(t)=0
(2.27)
Thus the driving frequency is no longer the same as w, and the dispersion relation changes[9].
By defining the driving frequency as Wdi and the quality factor as
Wdi
Q =
, a relationship between
and the original w can be obtained:
Wdi
2
where w = k
+I
(2.28)
EI
Fig. 2-6 shows the damping effect in terms of vibration amplitude as a function of driving
frequency. The value of
Q needs to be determined experimentally by the equation Q = fo/Afo
where fo is the resonant frequency and Afo is the half-width frequency. According to the cantilever
tuning fork routine on the DI 3000,
Q of the MESP-HM tips at the first resonant frequency was
2.1. CANTILEVER DYNAMICS
37
6-
5-
4-
2-
200
400
600
800
1000
Frequency [kHz]
1200
1400
1600
1800
Figure 2-6: Vibration amplitude of a free, damped cantilever tip.
Q at the second resonant frequency is more varying and is found to be between
450 and 550. The model above assumes that Q is 180 at the first resonant frequency and 500 at the
found to be 180± 10.
second.
2.1.4
Comparison to the Point-Mass Model
As mentioned in Sec. 1.2, the cantilever dynamics can be approximated by a mass-spring model.
The spring constant, kc, is defined as
F
k -I
(2.29)
and can be expressed in terms of the beam parameters[34].
If there is a downward force F pushing on a point a on the beam, the moment at any point x is
M = F - (x - a)
(2.30)
or equivalently, force times distance.
By combining Eqn. 2.8 and Eqn. 2.30, a relationship between F and z can be found,
z =
6EI
x 2 (x - 3a)
(2.31)
The point of interest is at x = L, assuming that a = L as well. Therefore, an expression for k,
CHAPTER 2. DC CURRENT IMAGING
38
is derived:
ke = 3E1
(2.32)
The resonant frequency of the point-mass model is determined by the fundamental vibration
mode of the cantilever beam, and the fundamental mode occurs at kiL = 1.875 according to
Eqn. 2.25. Therefore by using the dispersion relation,
WO
2.2
=
(1.875)2
E
(2.33)
Magnetic Tip-Sample Interaction
The force on a tip due to the magnetic field of the sample can be represented as a convolution between
the probe moment and the sample stray field for a particular tip-sample separation.
However,
while the sample stray field is generally known, the tip magnetization, which originates from a
ferromagnetic thin film coating on a pyramidal silicon tip (see Fig. 1-3), may be irregular in shape
and is much less predictable. Furthermore, tip hysteresis loops and coercivities are usually unknown
and there may be strong interactions between the magnetization of the sample and the tip.
Current models describing the magnetic force on a tip have so far restricted themselves to the
hard magnetic case, where the magnetization of the sample and the tip remains undisturbed during the scanning process, although the field-dependence of MFM probes have been investigated
experimentally[2].
The point dipole approximation, first proposed by Mamin et al.[20] describes the tip as an ideal
dipole, with magnetic dipole moment m.
A slightly more complicated model describes the tip as the sum of an ideal monopole and an
ideal dipole, where the monopole is an approximation of an elongated dipole. The force experienced
by the tip due to magnetic field is,
F = po(q + m - V)H
(2.34)
where q is the magnetic monopole moment of the tip and H is the magnetic stray field from the
sample[14].
Several groups have experimentally determined the effective values of q and m for various MFM
tips[2],[32].
Kong et al.[18] used microfabricated current rings, to take advantage of well-defined
MFM tip
magnetic pole
d- -
sample
Figure 2-7: Schematic of the extended point probe model.
magnetic fields produced by the rings. They presented values for the monopole (2.8x 10-6 emu/cm)
and dipole (3.8 x 10-9 emu) moment of a tip covered with 65 nm cobalt for a current ring with 5pm
diameter and found that the dipole moment value when using a 1pLm diameter ring is about one
order of magnitude smaller.
Lohau et al.[19] explained the difference between the two rings by relating a characteristic decay
length of the magnetic field to an effective volume of the magnetic tip that is actually interacting
with the field. They concluded that the usage of the point probe approximation needs to be adjusted
by an effective distance,
6, above the tip apex, because the location of the point probe within the
real physical tip depends on the magnetic stray field.
Similar observations have been made by
Mamin et al.[20] for the dipole model and Belliard et al.[3] for the monopole model. Fig. 2-7 shows
the schematic of the extended point probe model. The lift height, d, is a fixed distance above the
sample where the tip scans and is manually controlled by the user, and the point probe is effectively
at a height z = d + 6 above the sample.
Lohau et al. also determined that "an unambiguous MFM-image analysis can only be performed
when using either the monopole or the dipole contribution of the magnetic tip."
Similar conclusions were made by van Schendel et al.[16],[38], who used transfer functions in the
Fourier domain to quantify the magnetic force on the tip. Calculations are simplified in Fourier
space because transfer functions are multiplied instead of convoluted.
When compared to the point-pole tip models, the authors found the extended monopole model
to agree best with their simulations. While the dipole and monopole models only match the ex-
CHAPTER 2. DC CURRENT IMAGING
40
perimentally determined tip surface charge distribution at specific spatial wavelengths of sample
magnetization, the extended monopole model matches over a greater range of wavelengths.
2.3
Theory
In the presence of a device carrying DC current, the tip-end of the cantilever experiences an attraction
or a repulsion force in the z-direction. This force can be expanded into a Taylor series,
F(z) = F(zo) +
OF
1 a2 F
1. (z - zo) + 2z 2 Iz (z - zo) 2 +...
The first term shifts the equilibrium position of the cantilever beam at x
(2.35)
L. The excition force
around the equilibrium can be approximated by the second term, which can be represented as an
effective spring with stiffness k*[29], where:
k* = 9F(z)|
(2.36)
Oz
Therefore, the boundary condition at x = L changes while the cantilever, clamped to the piezo,
is still driven at x = 0. The moment remains zero at x = L because nothing is attached beyond
that end to create a curvature. However, the shear force now equals the magnetic force[30],
El
dx 3
= k*Z(x
=
(2.37)
L)
Z(x) can again be found by using the four boundary conditions to solve for the coefficients of the
general solution (Eqn. 2.19):
Z(x) = 2 ((coskx+coshkx)M sin kL sinh k L-cosh kL sin kL-cos kL sinh kL
cos kL sinh k L-cosh kL sin kL-M(cos kL cosh kL+1)
'
2 cos kL cosh kL-M(cos kL sinh kL+cosh kL sin kL)
cos kL sinh k L-cosh kL sin kL-M(cos kL cosh kL+1)
where M =
(cos kx
(sin kx
sk
- cosh
-
kkx) +
sinh kx(23
3
Ek.
The characteristic equation that defines the resonant frequencies is calculated in the same way
2.3. THEORY
41
as for a free cantilever. Z(x) approaches infinity as the terms in the denominator approach zero:
cos k, L sinh k, L - cosh k, L sin k, L =
E Ik3
k*,3 (cos k, L cosh k, L + 1)
(2.39)
As k* approaches zero, in the limit that the spring is infinitely soft, the characteristic equation
reduces down to the same for a free cantilever. And if the spring is infinitely hard, k* = 00 and the
characteristic equation becomes the same as in the case of a pinned end[30].
The concept of the force on the tip shifting the resonant frequency of the cantilever, as described
in the point mass model, is shown here mathematically. Althernatively, E
can be expressed as
kc(k1) . Depending on the effective stiffness of the external force k* compared to the effective
spring
3
constant of the beam k,, the resonant frequency ranges between the limits of a free cantilever and a
pinned cantilever. At the first vibrational mode, kiL of a free cantilever is 1.8751, kiL of a pinned
cantilever is 3.9266. If L! = 0.1, kiL = 1.91891, and if
j- =
10, kiL = 3.1677[30].
By using the extended monopole model, the magnetic force is
F = qpoH
(2.40)
where H of DC current flowing through an infinitely long wire with finite height and width is derived
in App. B and the effective distance between the sample and the monopole is z = d + 6.
Because the tip is specifically magnetized in the z direction, the z component of H dominates
the magnetic tip-sample interaction. Therefore,
Fz = qpoHz
(2.41)
This means that the magnetic force is proportional to the magnetic field. From the point mass
model, it was shown that the magnitude response is proportional to the magnetic force and that
the phase response is proportional to the gradient of the force in the z-direction, providing a better intuitive understanding of the effects of the magnetic force on the tip behavior. Therefore, a
comparison of the cantilever's responses between the two models is necessary.
2.3.1
Comparison to the Point-Mass Model at the First Vibration Mode
The cantilever response at the first vibration mode, using both the point mass model (PMM) and
the flexural beam model (FBM), is shown in Fig. 2-8. The cantilever, using the parameters given in
CHAPTER 2. DC CURRENT IMAGING
42
Table 1.1, is driven at its fundamental frequency (89.9 kHz) where the maximum phase response is
expected. The wire used in the simulation is 2pm wide and 200 nm tall, and q and z are assumed
to be 10-6 Am and 400 nm respectively. The two models yielded very similar results. Furthermore,
the phase of the deflection response, Z(L), and the slope response dZ(L)/dx of the flexural beam
model were found to be identical.
2820
2800-C2780 2760 ' 2740 2272027002680
-10
-8
-6
-4
-2
I
0
x [um]
2
I
20
4
6
I
8
10
I
10-
0
-10
-10
-8
-6
-4
-2
0
x [um]
2
4
6
8
10
Figure 2-8: Comparison of cantilever magnitude and phase response, at resonance, between point
mass model (PMM) and flexural beam model (FBM) along a 2pm-wide wire centered at x = 0,
carrying 20 mA DC current.
For the most sensitive magnitude response, the cantilever is not driven at the first resonant
frequency[34]. Instead, it is driven at the frequency that has the highest magnitude slope, which is
slightly off the resonant frequency. The magnitude response at this frequency is shown in Fig. 2-9.
In these simulations, this frequency is determined to be at 89.71 kHz.
It is evident that the two models also behave very similarly at the fundamental mode. The
magnitude response of dZ(L)/dx is plotted separately because it yields a larger response.
2.4. EXPERIMENT
43
2.6
2.5
2.3
'E2.2
22.1
2
10
1.55
-8
-6
-4
-2
-8
-6
-4
-2
0
x [um]
2
4
6
8
10
0
2
4
6
8
10
X 104
1.5
1.45
1.4
1.35
2
1.3
1.25
1.2'
-1c 0
x[um]
Figure 2-9: (Top) Comparison of cantilever magnitude response, slightly off resonance, between
point mass model (PMM) and flexural beam model (FBM) along a 2pm-wide wire centered at x =
0, carrying 20 mA DC current. (Bottom) Magnitude slope response of the cantilever.
2.4
2.4.1
Experiment
Setup
The experimental setup for DC current imaging is shown in Fig. 2-10 and the components are listed
in Table 2.1.
The experiments were performed at room temperature. At the beginning of each session, the
tips are calibrated by first imaging a magnetic tape sample, which was from Digital Instruments.
To optimize scanning sensitivity, the tip is scanned very slowly across the sample, at a rate of 0.05
Hz.
Data acquisition was performed external to the AFM, using a lock-in amplifier along with LabVIEW and Matlab code. The data sampling rate of the lock-in was set at 32 Hz and the time
constant was at 30 ms. Two output signal from the AFM's signal access module are available: RMS
amplitude (IN 0) and vertical deflection (AUX A). The vertical deflection output was unpredictable
during DC current imaging sessions, sometimes showing strange slants in the response. Thus, the
Laser
E
f
'\
Lock-in
Amplifier
S
A
Figure 2-10: MFM setup for DC current imaging. The function generator drives the cantilever at
frequency f and provides the reference signal for the lock-in amplifier. The input signal to the
lock-in is taken from the photodetector signal of the AFM.
Component
Function Generator
DC Power Supply
Lock-in Amplifier
Atomic Force Microscope
MFM tip
Vendor
Agilent
Agilent
Stanford Research Systems
Digital Instruments
Digital Instruments
Part Number
33120A
E3633A
SR844
Dimension 3000
MESP-HM
Table 2.1: Table of components used for DC current imaging
signal was taken from the RMS amplitude output, which is also the feedback signal. Because the
feedback system is amplitude-modulated, the magnitude of the feedback signal is not believed to
represent the direct magnitude of tip deflection. The phase response, on the other hand, should
accurately represent the phase of the cantilever vibration.
The wire sample was fabricated by Mathew Abraham. The fabrication process as well as the
data acquistion steps are explained in App. A.
2.4. EXPERIMENT
2.4.2
45
Results
By driving the cantilever at the first resonant frequency of the tip, the phase response to varying
current was obtained. Fig. 2-11 shows the raw data acquired by the lock-in, the normalized data,
and a plot of the difference between the maximum and minimum phase response as a function of
current. The normalized data is centered at its mean and provides a better comparison among the
responses at different current levels. It is the change in the phase response that indicates the change
in the effective spring constant by DC magneitc field.
The increased phase response due to greater current is evident. A line-fit through the third plots
yields a slope of 0.118 deg/mA. The resonant frequency of the tip was 59.44 kHz. The wire used in
the experiment is 2pim wide and 200 nm tall. The lift height between the sample and the tip was 200
nm. This means that the tip is first scanned right above the sample, obtaining good topographical
data. As a second pass, the tip is lifted by an amount specified by the user and by incorporating
the topographical data, the tip-sample distance is maintained throughout second scan.
One disadvantage to driving the cantilever externally is a tradeoff between topographically accuracy and magnetic field sensitivity.
The setup of the AFM's internal piezo driver allows the
topographical and lift scans to be performed at different tip drive frequencies as well as tip drive
amplitudes. Therefore, the topographical scan can occur at the frequency of optimum magnitude
response, which is usually 200-300 Hz off the resonant frequency, and the lift scan can occur on the
resonance, giving maximum phase response. However, by applying the cantilever drive externally,
we are limited to one frequency. The topographic effects on phase response are shown in Fig. 2-12.
Sync outputs generated by the DI instrument were available inside the controller machine. However, the signal had noticeable delay which prevented accurate synchronization to the start of each
scan.
As the tip scans across the sample in the lift mode, it is raised by 200 nm (the height of the wire)
upon reaching the wire and again lowered by the same amount once it passes the wire. Because
the maximum phase response also occurs near the wire edges, the signals can be contaminated if
topography wasn't well-tracked and the tip does not lift and lower right at the edge of the wire.
In the case where the magnetic field is in the counter-clockwise direction, the tip lowered before
it hit the edge of the wire and thus the phase response was amplified. Similarly, when the magnetic
field is clockwise about the wire, the tip lifted before reaching the wire and therefore diminished the
response to magnetic field.
The tip appears to be more susceptible to topographical contamination in the presence of a
CHAPTER 2. DC CURRENT IMAGING
46
b)
a) 82.5 r
82
1
0.5
(5;
4)81.5
0
C)
Ca
IL
IL -0.5
81
80.5
-1
80
-10
0
-5
-5
0
10
5
-10
-5
x [um]
C)
0
x [um]
5
10
2
-8 1.5
C)
U .
c15 0.5
(L
0
10
5
15
I [mA]
Figure 2-11: Experimental phase response to varying DC current flowing through a 2pim-wide wire
centered at x = 0. The cantilever is driven on resonance. Current levels are 0, 0.5, 1, 2, 3, 4, 5, 7, 9,
10, 12, and 15 mA. The arrows are drawn in the direction of increasing current. Tip-sample distance
was 200 nm. a) Raw data from the lock-in. b) Data normalized by the mean. c) The difference
between maximum and minimum phase response.
2.4. EXPERIMENT
0.3
47
0.6
3 mA
7 mA
0.2
0.4
0.1
'53
0)
a)
Cz
0
-0.3
0
U)
-0.1
-0.2
0.2
(D
_r_ -0.2Cz
CL
CW
magnetic field
-0.4
-1 0
-5
0
x[um]
-0.4-
CCW
magnetic field
5
-0.6-0.8
-1 0
10
0.6
0.4
5;
a)
Ca
1
9 mA
-5
0
x[um]
5
10
0
x[um]
5
10
12 mA
0.5
0.2
0
0
-0.2
C
.C
-
a_
-0.4
-0.5
-0.6
-0.8'
-1 0
-5
0
x[um]
'
5
'-1
10
1
-10
-5
Figure 2-12: Comparison between phase responses to currents in opposite direction.
distance is 200 nm
Tip-sample
CHAPTER 2. DC CURRENT IMAGING
48
repulsive magnetic field. The phase responses of the two current directions in attractive magnetic
fields are very similar. To compensate, the tip was driven slightly off its resonance, at 59.41 kHz, to
obtain reasonable topographical information as well as good sensitivity. Fig. 2-13 shows the results
compared to driving the tip on its resonance. The response is improved in the repulsive magnetic
force regime and confirmed in the attractive magnetic force regime.
Fig. 2-14 shows the results of driving the cantilever off resonance. A line-fit of the phase difference
response to current gives a slope of 1.06 deg/mA. As expected, the phase sensitivity is slightly less
than if the cantilever were driven on resonance. However, the results are more reliable.
Some topographical artifacts are believed to have affected the phase reponse. As seen in the
case where there is no current flowing through the wire, some phase response is still detected and
it corresponds to the topography of the wire. This effect can be reduced by sensing at a higher lift
height.
0.4
0.6
5 mA
7 mA
-
0.4
Detuned
-- Resonant
0.2
0.2
Cu
a.CO
0
CCu3
(L
(L
0
-0.2
-0.2
-0.4
-0.41 0
-5
0
x [um]
5
-10
10
0.6-
0.4-
1
9 mA
-5
0
x [um]
5
10
-5
0
x [um]
5
10
12 mA
0.5
0.2Cu
-o
PA0
Cu
.C
Cu C-0.2
0
Cu
a-0.4
-0.5-
-0.6
-0.8
-0
-5
0
x [um]
5
10
-10
Figure 2-13: Comparison between phase responses to cantilever driven on (59.44 kHz) and off
resonance (59.41 kHz). Tip-sample distance is 200 nm.
2.4. EXPERIMENT
a)
49
b)
92
91.5
1r
0.51
91
0
[L
90.5
-C
IL
-0.5
90
89.5-10
-5
'
5
0
x [um]
-1'
-1 0
'0
10
-5
0
x [um]
5
10
2
C)
-~1.
5
C
6
a)
*~0.5
00L
10
5
15
I[mA]
Figure 2-14: Experimental phases response to varying DC current flowing through a 2pm-wide wire
centered at x = 0. The cantilever is driven off resonances, at 59.41 kHz. Current levels are 0, 0.5,
1, 2, 3, 4, 5, 7, 9, 10, 12, and 15 mA. The arrows are drawn in the direction of increasing current.
Tip-sample distance was 200 nm. a) Raw data from the lock-in. b) Data normalized by the mean.
c) The difference between maximum and minimum phase response.
CHAPTER 2. DC CURRENT IMAGING
50
a) 8 2 .5 r
b)
82
1
0.5
81 .5
.D
0
IL
Ca
81
-0.5
80.51
80'-10
-5
5
0
x [um]
C)
-1
-1
10
0
-5
0
x [um]
5
10
2
1.8
a,
2.1.6
1.4
1.2
6\
0.8
200
400
O
600
800
Lift Height [nm]
0
1000
Figure 2-15: Experimental phases response to varying lift height. Current level is 15 mA. The
cantilever is driven on resonances. Lift heights are 200, 250, 300, 350, 400, 450, 500, 600, 700, 800,
and 900 nm. The arrows are drawn in the direction of increasing lift height. a) Raw data from the
lock-in. b) Data normalized by the mean. c) The difference between maximum and minimum phase
response.
2.4. EXPERIMENT
2.4.3
51
Comparison to Theory at the First Vibration Mode
The extended monopole (Fig.2-16) and dipole (Fig. 2-17) models are both compared to the detuned
experimental results with 15 mA DC current. For the extended monopole model, q and
6 are fitted
to match the experimental results. The values obtained for q and J are 1.9x 107 Am and 650 nm,
given that the lift height is 200 nm. The experimental and simulated results are closely-matched,
particularly within the the width of the wire, which is between x = -1p1m and x =
1pm. Further
away from the wire, the experimental results fall off slower than the simulated. This is possible due
to the physical volume of the tip, which is not simulated in the monopole model. As the tip moves
away from the wire, the outer rim of the pyramidal tip may still interact with the magnetic field,
causing some phase response.
The extended dipole model simulation agrees less-well with the experimental result. With 15
mA current, m, and 6 are found to be 2.5x10-
Am 2 and 1000 nm, respectively. However, away
from the edge of the wire, the phase response of the extended dipole model falls off faster and even
overshoots a little before approaching zero. Therefore, it appears that the extended monopole model
better represents the magnetic tip-sampe interaction.
In order to approximate a reasonable 6 for the extended monopole model, the maximum to
minimum phase response is plotted along a linear increase in current (see Fig. 2-18). Different lift
height values are examined. They range from 200 nm to 1000 nm, incrementing by 100 nm.
According to (c) in Fig. 2-14, the phase difference varies linearly with current. In Fig. 2-18, a
linear dependence begins to emerge around lift height z = 700 nm, justifying the need to include an
effective distance, 6, to the physical tip-sample distance.
2.4.4
Second Vibration Mode with Magnetic Tape
In addition to modeling the cantilever response at the fundamental frequency, the flexural beam
model offers the flexibility of examining the responses at higher-order cantilever vibration modes.
The vibration modes of a free cantilever are determined by Eq. 2.25 and the first five kaLs are 1.875,
4.694, 7.855, 10.996, and 14.137, respectively.
To image at the second vibration mode, the second resonant frequency is calculated. Using the
same cantilever parameters given in Table 1.1, the calculated value for k2L, and Eq. 2.28, the second
resonant frequency, f2, is found to be 563 kHz. It is about 6.26 times larger than the first resonant
frequency, fi.
CHAPTER 2. DC CURRENT IMAGING
52
With this information, as well as the experimentally determined values for
Q at
the first and the
second resonant frequencies, the point mass model can be used to find the phase response at the
second vibration mode the same way it calculated the response at the first. Assuming the effective
mass, meff, of the tip remains constant, a factor of 6.26 increase in the frequency corresponds to a
factor of 6.262 increase in the effective spring constant.
Q, from
experimental data, increases by a
factor of 2.37 to 3.24.
Therefore, according to Eq. 1.13, the phase response at the second vibration mode should be 12
to 16 times smaller than the phase response at the first vibration mode. This decrease in sensitivity
prevents accurate DC current imaging at the second mode.
However, we did image magnetic tape with the second mode, because the tape has a stronger
magnetic field. Fig. 2-19 shows the results. The data was taken straight from the DI instrument.
6
The numeric values obtained from the ASCII export function require data conversion: x/21 x 180
where x represents the phase values.
The measured first and second resonant frequencies of the tip are 62.86 and 399.96 kHz, respectively. The
increase in
Q at the first is 184.89, and it is 533.28 at the second.
the Q and the frequency from the first to the second
There is a factor of 2.88 and 6.36
vibration mode. Therefore, from
the theory, a factor of 14.03 decrease in the phase response is expected. The experimental phase
response at the second mode was found to be 15.68 times less than the first. The flexural beam
model is needed to model imaging at the high-order vibration modes. However, when the magnetic
field from the sample is non-time-varying, the point mass model suffices once the high-order resonant
frequency is calculated.
When driving at the first and second modes, the drive amplitudes should be tuned accordingly to
ensure the free cantilever vibrational amplitudes are equivalent. It serves as an experimental control
for a more precise comparison between the responses at the two modes. The drive amplitudes vary
with the tip as well as the placement of the tip on the cantilever holder and can only be determined
experimentally.
2.5
Summary
This chapter covers the derivation of the flexural beam model, the different proposals for the modeling
magnetic force on the tip, the theoretical and experimental results of DC current imaging at the
first vibrational mode, and imaging of the magnetic tape at the second mode.
2.5. SUMMARY
53
Three key points are summarized.
First, by matching the experimental results to both the
extended monopole and dipole model, we found the extended monopole model to better represent
the magnetic tip-sample interaction. The effective monopole moment and displacement for 15 mA
DC current were found to be 1.9x 10-7 Am and 850 nm respectively. According to the point mass
model, the phase of the cantilever vibration responds to the gradient of the magnetic field created
by the DC current.
Secondly, the overall magnetic imaging sensitivity at the second harmonic is worst than the first,
even thought the
Q value
at the second harmonic is higher. The phase response is proportional to
the ratio of Q over the effective spring constant. While
spring constant increases even more, by a factor of
Q increases
(W2/w1)
2
.
at the second mode, the effective
Therefore, the overall phase response
at the second mode is less.
Third, sensitivity in DC current imaging was determined to be at 1 mA. One possible limiting
factor is thermal noise. In Chap. 3, AC current imaging is discussed. The time-varying current acts
as a driving source and the piezo no longer needs to drive the cantilever.
CHAPTER 2. DC CURRENT IMAGING
54
0.80.60.40.2 0 ----------------------0.2
-0.4q = 1.9xo10- Am
LH =850 nm
-0.6-0.8-1111
-10
-8
-4
-6
-2
0
x [um]
2
4
8
6
10
Figure 2-16: Comparison between the experimental and simulated results of the cantilever phase
response to a 2 pm-wide wire carrying 15 mA DC current. The simulation used the extended
850
monopole model to describe the magnetic tip-sample interaction. q = 1.9xiO- 7 Am and z
nm give the best fit result.
per
--
-
1
nal
0.80.60.4 -
0.2 0 ----
- - -/---
- - -
~
-0.2
0.4 -
--
m =2.5x104 Am2
LH = 1200 nm
-0.6-0.8-
-10
-8
-6
-4
-2
0
x [UM]
2
4
,
8
10
Figure 2-17: Comparison between the experimental and simulated results of the cantilever phase
response to a 2 pm-wide carrying 15 mA DC current. The simulation used the extended dipole
model to describe the magnetic tip-sample interaction. mz = 2.5 x 10-13 Am 2 and z = 1200 nm give
the best fit result.
2.5. SUMMARY
55
3-
2.5-
2
S1.5-
_r-
Ca
0.5 -
0
5
10
15
I [mA]
Figure 2-18: Relationship between DC current and maximum to minimum phase difference at varying
liftheights. The extended monopole model is used. Lift heights range from 200 nm to 1000 nm,
incrementing by 100 nm. The arrow is in the direction of increasing lift heights.
CHAPTER 2. DC CURRENT IMAGING
56
a)
b)
15
0.8
0.6
10
0.4
5
o 0.2
0
a
a.
U)
0
0
-0.2
-5
-0.4
II
-0.6
)
0
3
2
c)
2
1
0
3
x [um]
x [um]
15
First mode
10
-(Second
mode)
x 15.68
D5
"a
.c 0
-5
-10
0
2
1
3
x [um]
Figure 2-19: First (a) and second (b) vibration mode phase imaging of sample magnetic tape (3
ym). (c) Best fit of second mode imaging to first mode imaging.
Chapter 3
AC Current Imaging
The theory and experimental results for AC current imaging are presented in this chapter. Although
the tip-sample force is now time-varying, the system remains linear by reducing the piezo drive to
zero and limiting the driving force to the magnetic field created by AC current.
Magnetic force from a time-varying current is first derived and applied to the boundary conditions. Simulations of the cantilever response are presented.
Secondly, experimental results are shown and compared to the theory.
3.1
Theory
The flow of AC current instead of DC current through the wire sample introduces a new frequency
component in the beam vibration. The AC magnetic field produced by the current acts as a new
driving force on the cantilever. As long as the current frequency is different from the piezo drive
frequency, the linear cantilever response at the current frequency can be filtered out by a lock-in
amplifier.
3.1.1
Magnetic Force
The derivation of the magnetic force is similar to the DC case. Referring to App. B, the current
flowing through the conductor is now a product of an amplitude, I, and a time-varying term, e3'.
The new term is linearly carried into the expression of the magnetic field. The z-component of the
57
CHAPTER 3. AC CURRENT IMAGING
58
magnetic field with AC current flowing through a wire with finite width and height is thus,
Ieswct
HZ = 27rW
I
H
f
0
W/2
W2
w/2
0zO
- H (X-
_
(3.1)
)dzdx
XO p2 + (Z -~zo)2
Since the approximate magnetic force from the extended monopole model is linearly related to
the magnetic field, it also can be expressed as the product of the force from the DC current and
eJwt.
3.1.2
Boundary Conditions
When using Taylor series expansion to approximate the magnetic force, the first term, F(zo), represents the new driving force. Therefore, it can no longer be neglected. Furthermore, it is orders of
magnitude larger than the higher-order terms. Therefore, only the first term in the Taylor series is
included.
At
x = 0,
the cantilever beam is clamped. Its position and first derivative are both zero. Still
no moment is applied at x = L. However, the shear force there equals the magnetic force.
The new boundary conditions are as follows:
z(X = 0,t) = 0
(3.2)
&z(x=,t) = 0
(3.3)
ElEI
2z(x,t)=
OX2Lt-
(.4
(3.4)
El az(x=L,t) = Fz(zo)eJowt
(3.5)
where w, is the current frequency, F(zo) is the same Taylor series component as in the DC case
since the time-varying term is added externally, and zo is the effective tip-sample distance.
3.1.3
Solution
The solution is assumed to take the form of z(x, t) = Z(x)ewct, where a general solution for Z(x) is
given in Eqn. 2.19. After applying the boundary conditions, Z(x) is solved:
Z(x) =
1
( sin kL+sinhkL
M os kLcosh kL+1
(COS k
(co ,(
- cosh kx) -
k)~~
cos kL+coshL
coskLcoshkL+i
(sin k
snx
- sinh k)
ihkx
(3.6)
36
3.1.
THEORY
59
and wc = k2
where M =
( 1
-
+
2).
Fig. 3-1 shows the simulated cantilever response to varying AC current flowing through a 2 umwide wire. The current values are 0.5, 1, 1.5, and 2 mA, with the arrow pointing in the direction
of increasing current.
The current frequency matches the first vibration mode of the beam. The
effective monopole moment and tip-sample distance are: q = 8x10-6 Am and z = 1 jm. The
discontinuities at x = -1 jm and x = 1 jm are a result of the constant lift height between the tip
and the top of the sample.
70
6050 -40 -
'30 -
220 100
-30
-20
-10
-20
-10
0
x [um]
10
20
30
100
50-
0
Co
-50-
-100'
-30
I
0I.___________
10
x [um]
__L_______
20
30
Figure 3-1: Simulated cantilever magnitude and phase response to AC current at the first beam
vibration mode. The current values range from 0.5 to 2 mA, in increments of 0.5 mA. The arrow
points in the direction of increasing current. The wire is 2 pm-wide and centered at x = 0. q
8x10- 6 Am, z = 1 pm.
The magnitude response increases linearly with current. It is symmetric about the center of the
wire and gradually falls off to zero as the tip travels further away from the wire.
The phase response, on the other hand, does not vary with current. But it shifts 180 degrees as
the tip crosses the middle of the wire. On one side of the wire, if the tip experiences a "push-pull"
force from the AC magnetic field, then it should experience instead a "pull-push" force when it is
on the other side, because the z-component of the magnetic field switches sign at the middle of the
wire.
CHAPTER 3. AC CURRENT IMAGING
60
When the current is in the reverse direction, the magnitude response remains the same since it is
symmetric, but the phase is flipped (see Fig. 3-2). Because the current is now going in the opposite
direction, the "push-pull" effect on the tip reverses.
70
60-50 40 "E30--
220-10 --30
-20
-10
0
10
20
30
10
20
30
x [um]
100
50-
0.
-50-
-1n
II
-30
-20
-10
0
x[um]
Figure 3-2: Simulated cantilever magnitude and phase response to -2 mA AC current at the first
beam vibration mode. The wire is 2 pm-wide and centered at x = 0. q = 8x10- 6 Am, z = 1 pm.
Contrary to DC current imaging, the cantilever response to AC current is mainly contained in
the magnitude data. Therefore, the cantilever vibrates in direct response to the magnetic force, and
not its gradient. When using the extended monopole model, we can conclude that the cantilever
responds directly to AC magnetic field while it responds to the gradient (in z) of DC magnetic field.
3.2
Experiment
Experimental results of AC current imaging are shown in this section. The experimental setup is first
explained. The magnitude and phase responses to varying current levels and directions are presented.
The asymmetric magnitude responses lead to the hypothesis of electrostatic force interference. A
method for decomposing the EFM and MFM response from the magnitude data is developed and
the hypothesis is verified.
Laser
;
Lock-!i.....n
Amp IIffier:
A Function generator
Figure 3-3: MFM setup for AC current imaging. The current is driven by an external function
generator at frequency we, which becomes the only driving force on the cantilever. The lock-in
amplifier receives signal from the AFM photodetector.
3.2.1
Setup
The components used in the AC current imaging experiments are listed in Table 3.1 and the setup
is shown in Fig. 3-3.
Component
Vendor
Part Number
Function Generator
Agilent
33120A
Lock-in Amplifier
Stanford Research Systems
SR844
Atomic Force Microscope
Digital Instruments
Dimension 3000
MFM tip
Digital Instruments
MESP-HM
Table 3.1: Table of components used for AC current imaging
As mentioned earlier, the DI instrument uses the LiftMode technology to sense long-range forces,
such as electrostatic and magnetic forces. LiftMode allows the tip to first perform a topographical
scan across the sample, in which the tip is positioned close to the sample surface and amplitude
feedback is used to extract the topographical makeup of the sample. Once topography is known and
Function Generator
50Q
I I
+ I1 II
II II
|i
I
I
I
I_
_
_
_
_
_
_
Wire Sample
Figure 3-4: Circuit diagram for AC current. The output impedance of the function generator is 50
Q. Two wires on the sample are bonded to the contact pads, each has about 100 Q of resistance.
stored into memory, the tip rescans across the sample at some fixed distance above the topography to
ensure that the tip is responding to the long-range forces and not contaminated by the topography.
In the AC current experiments, the tip is first driven at a frequency where the magnitude response
is most sensitive, in order to acquire good topographical data. During the lift scan, the piezo drive
amplitude and frequency are both set to zero, leaving the AC magnetic field as the only driving
force.
The current is driven externally by the function generator at frequency w,. The sync output from
the function generator provides the reference signal for the lock-in amplifier. The AFM photodetector
signal is taken from the vertical deflection output (AUX A) of the Signal Access Module (SAM) and
fed to the lock-in as the input signal.
Fig. 3-4 shows the circuit diagram between the voltage from the function generator and the wire
sample. The output impedance of the function generator is 50 Q. Only two wires are bonded to the
contact pads and each has about 100 Q of resistance. Therefore, the amplitude of current passing
through one of the wires is vi, x 100.
With the 2 pm-wide wire in the middle, the tip scans a distance of 50 Pm, across the wire, at
a rate of 0.1 Hz (5 pm/s). By the use of LabVIEW, the lock-in samples the input signal at 32 Hz.
The data is stored into an internal buffer and subsequently downloaded to a computer and read via
Matlab code. The details of the data acquisition process is given in App. A.
In order to minimize the effects of the 12-degree incline between the cantilever and the sample
63
3.3. RESULTS
surface, the tip should scan symmetrically across the wire. This is accomplished by positioning the
long-dimension of the wire sample parallel to the long-dimension of the cantilever and setting the
scan angle to 90 degrees.
3.3
Results
Figs. 3-5 and 3-6 show the experimental results to varying AC current in both directions. The lift
height was fixed at 200 nm and the wire is 2 1am-wide, centered at x = 0.
A 180-degree shift in the phase response is observed, and it agrees with theory. The offset in the
actual experimental phase values could be due to the AFM's internal circuitry or delay between the
photodetector output signal to the lock-in.
0.06
0.05
'0.04
0.030.02 0.01
-25
-20
-15
-10
-5
-20
-15
-10
-5
0
5
10
15
20
25
0
5
10
15
20
25
200
100-
0-
- 100 -
-25
x
lum]
Figure 3-5: Experimental magnitude and phase responses to varying AC current through a 2pmwide wire centered at x = 0. The cantilever is driven on resonance. Current levels are 5, 25, 50,
100, 250, 500 pA and 1, 2, 3, 4 mA. The arrows are drawn in the direction of increasing current.
Tip-sample distance was 200 nm.
The magnitude response, however, does not follow the simple AC current imaging theory. While
the theory predicted a symmetric magnitude response about the center of the wire, the experimental
results were asymmetric.
CHAPTER 3. AC CURRENT IMAGING
64
OX 8
0.061
57
0.041
CO
0.02
0 -25
-20
-15
-10
-5
0
5
10
15
20
25
-20
-15
-10
-5
0
x [um]
5
10
15
20
25
200
100-(D
50--
0-50-25
Figure 3-6: Experimental magnitude and phase responses to varying AC current in the opposite
direction. The cantilever is driven on resonance. Current levels are 5, 25, 50, 100, 250, 500 pA and
1, 2, 3, 4 mA. The arrows are drawn in the direction of increasing current. Tip-sample distance was
200 nm.
3.3. RESULTS
65
The difference in the degree of asymmetry between the two directions of current indicated possible
concurrent sensing of electrostatic force in addition to magnetic force. The Cr on the CoCr MFM
tips is conductive and the tips are sensitive to electrostatic force. In fact, the same tips are also
widely-used for electrostatic force microscopy (EFM), which measures the voltage difference between
the tip and the sample. In our experiments, the tip is grounded while there is a voltage drop through
the wire.
The EFM magnitude responses for the two current directions are expected to be different. The
direction of the voltage drop switches and the potential at the same location along the wire may be
different.
Fig. 3-7 shows the magnitude responses to electrostatic force alone for the two current directions.
Specialized conductive probes from DI were used (Model SCM-PIT). They are Platinum/Iridium
coated and are non-magnetic. The phase responses are not shown but they are nearly constant, with
only about 2 to 3 degrees of variation.
3.4
x 10
3.2
3
2.4
-2 5
6
-20
-15
-10
-5
-20
-15
-10
-5
0
x {um]
5
10
15
20
25
0
[um]
5
10
15
20
25
x10
5.8-
5.2
5
4-2 5
x
Figure 3-7: Magnitude responses to electrostatic force caused by 2 mA of current in both directions.
A non-magnetic tip (Model SCM-PIT) was used.
CHAPTER 3. AC CURRENT IMAGING
66
3.3.1
Decomposition Method
Due to the symmetry of the total EFM response and the asymmetry of the total MFM response,
a decomposition method is proposed to separate the total cantilever response into even and odd
components, of which the even would represent the EFM response and the odd would represent the
MFM response. Fig. 3-8 graphically illustrates the decomposition method.
The decomposition involves two operations. The first flips the total response about the z-axis and
subtracts it from itself. From Fig. 3-9, it is evident that the odd component is reconstructed by this
operation, while the even component is simultaneously cancelled out. The second operation extracts
the even component by halving the sum of the flipped response and the original (see Fig. 3-10).
3.3.2
Decomposed Results
Figs. 3-11 and 3-12 compare the extracted EFM and MFM response to the experimental magnitude
responses. Note that the total MFM response includes both the magnitude and the phase.
The MFM responses from the two current directions are identical outside of the 7r phase difference.
The EFM responses, however, have different offsets, which agree with previous analysis. The height
of the EFM responses also appear to be smaller than that of the MFM.
Furthermore, the results from the seven lowest current values from Figs. 3-5 and 3-6 are taken
and decomposed to its EFM and MFM responses in Figs. 3-13 and 3-14. Clear difference in the
MFM response is seen down to 15 pA, suggesting that the sensitivity in AC current imaging is about
15 pA.
3.3.3
Response to Varying Lift Heights
The cantilever response to varying lift heights are shown in Fig. 3-15 and the decomposed EFM and
MFM results are showns in Fig. 3-16. As lift height increases, the MFM response broadens and
slightly diminishes. The EFM response also decreases.
3.4
Comparison to Theory
Having decomposed the cantilever response into an EFM and an MFM component, it is necessary
to compare the MFM response to theory. EFM simulations are also compared to the EFM results
3.4. COMPARISON TO THEORY
67
Z
Odd
Component
a
-C
C
X
+
Z
Even
Component
C
-c
z
(a+b)/2
Total
Response
a
x
-c
L
V
(a-b)/2-
Figure 3-8: Schematic drawing of the decomposition method. The total response can be represented
by the addition of an even and an odd component.
68
CHAPTER 3. AC CURRENT IMAGING
Z
x
z
z
b/4
_ b/4
C
x
-c
Xx
-b/4
z
z
b/4
b/4_|_
x
-c-b/4.
-c
X
.
Figure 3-9: Method for extracting the odd (MFM) component. The odd component can be reconstructed by subtracting half of the flipped response from half of itself, while the even component
vanishes during the same operation. Therefore, by doing the same to the total response, only the
odd component remains.
3.4. COMPARISON TO THEORY
69
z
a
-C
C
X
Z
-C
*z
C
x
+
a/2
-C
z
z
a/2
-C
C
C
X
+
a/2
-C
C
Figure 3-10: Method for extracting the even (EFM) component. The even component can be reconstructed by adding half of the original response and half of the flipped response. The odd component
is cancelled out during the same operation. By applying this method to the total response, the even
component is extracted.
CHAPTER 3. AC CURRENT IMAGING
70
0.035
0.03
0.025
0.02
0.015
0.01
0.005
-20
-10
10
0
20
x [um]
x
10'
0.03
15
0.02
5710
0.01
2
.~5
CO
00
-0.01
0
-0.02
-0.03
-20
-10
0
x [um]
10
20
-20
-10
0
10
20
x [um]
Figure 3-11: (Top) Total experimental magnitude response due to 2 mA AC current. (Bottom
Left) Extracted EFM magnitude response from total. (Bottom Right) Extracted MFM magnitude
response from total.
3.4. COMPARISON TO THEORY
71
0.04
0.035
5
003
0.025
0.02
z 0.015
0.01
0.005
-20
-10
0
x[um]
10
20
0.03
0.02
-0.01
CD
C:
CD
co
2
0.01
-0.015
03
0
-0.01
-0.02
-0.02
-0.025
-0.03
-20
-10
0
x [um]
10
20
-20
-10
0
10
20
x [um]
Figure 3-12: (Top) Total experimental magnitude response due to 2 mA AC current in the reverse
direction. (Bottom Left) Extracted EFM magnitude response from total. (Bottom Right) Extracted
MFM magnitude response from total.
CHAPTER 3. AC CURRENT IMAGING
72
x
x 10
10-
3.5
2.51
_0
:3
'E
0)
as
2
'0
1.5
0
0)
Ca
Z
-2
0.5
-4
0
-6
-0.5
-20
-10
0
x [um)
10
20
-20
-10
0
10
20
x [um]
Figure 3-13: Experimental EFM and MFM magnitude responses to varying AC current. EFM data
is the even component of the total response and MFM data is the odd component. The current
levels are 5, 15, 25, 50, 100, 250, and 500 pA. The responses increase with current. Tip-sample
distance was 200 nm.
3.4. COMPARISON TO THEORY
73
X 10,
x 10
-0.5-1
-1.5-2
:3
0)
ca
i
-2.5 F
'E
0)
Cz
2
-3
0
-2
-3.5 F
-4
-4
-4.5 F
-6
-5
-20
-10
0
x [um]
10
20
-20
-10
0
x [um]
10
20
Figure 3-14: Experimental EFM and MFM magnitude responses to varying reverse AC current.
EFM data is the even component of the total response and MFM data is the odd component.
The current levels are 5, 15, 25, 50, 100, 250, and 500 pA. The responses increase with current.
Tip-sample distance was 200 nm.
CHAPTER 3. AC CURRENT IMAGING
74
0.05-
0.02
-
S)0.015 --
0.01
0.005
-
-25
-20
-15
-10
-5
0
x [um]
5
10
15
20
25
-20
-15
-10
-5
0
x [um]
5
10
15
20
25
100-
50-
0
-25
Figure 3-15: Experimental magnitude and phase responses to varying lift heights. Current level was
at 2 mA, flowing in the reverse direction. Lift height ranges between 200 and 2000 nm and increases
in increments of 600 nm. The arrow is drawn in the direction of increasing lift height.
75
3.4. COMPARISON TO THEORY
x 10'
0.025-1
0.020.015-
-2
0.01
-
-3
0.005
a)
C-,
--4
0
-0.005
-5
-0.01
-6
-0.015
-7
-0.02
-0.025
-8
-20
-10
0
x [um]
10
20
-20
-10
0
10
20
x [um]
Figure 3-16: Extracted EFM and MFM magnitude responses to varying lift heights. Current level
was at 2 mA, flowing in the reverse direction. Lift height ranges between 200 and 2000 nm and
increases in increments of 600 nm. The arrows are drawn in the direction of increasing lift height.
CHAPTER 3. AC CURRENT IMAGING
76
0.04
0.030.02
2
.Simulated
-
0.01
r
q =3.2x1 0~6 Am
o -
LH =10.5
.
um
-0.01
q 2.8x10-5 Am
LH = 6.5 um
-0.02-
-0.03-
-0.04
-25
-20
-15L
-10L
-5
0
x [um]
5
10
15
20
25
Figure 3-17: Comparison between the experimental and simulated results of the cantilever magnitude
response to a 2 pm-wide wire carrying 2 mA AC current. The current frequency matches the first
vibrational resonance of the cantilever beam. The simulation used the extended monopole model
to describe the magnetic tip-sample interaction. Two fits were used to match the experiment. The
first yields q = 3.2x 10-5 Am and z = 10.5 pm and the second has q = 2.8x 10-5 Am and z = 6.5
pm.
The experimental data, however, is in the units of volts and requires conversion into meters,
representing the amount of physical tip deflection. This conversion varies with each measurement
session because the tip placement and laser alignment differ from session to session. Typically, a
series of steps is performed at the end of each session to determine the relationship between the
voltage signal and the physical tip deflection. Unfortunately, these steps were not performed when
the measurements were made. An estimate of 1 mV to 1 nm is assumed. The process of finding the
voltage-to-meter relationship is explained in Chap. 5.
3.4.1
MFM
Fig. 3-17 shows the experimental and simulated magnitude responses to 2 mA AC current. The
simulations assume the extended monopole model. The first uses the parameter values q = 3.2 x 10-5
Am and z = 10.5 pm and the second uses q = 2.8x10-
5
Am and z = 6.5 pm.
The first matches well at further distances away from the wire while the second fits well near
the wire. The need for two fits may be because the magnetic field decays faster as a function of
3.4. COMPARISON TO THEORY
77
tip-sample distance near the wire. Fig. 3-18(a) shows the magnetic field strength up to 10 pm away
from the center of a 2 pm-wide wire at various tip-sample distances.
According to the extended monopole model, the effective displacement of the point pole is dependent on the gradient of the magnetic force in z. If the magnetic field decays at a faster rate
along z, then effectively less of the total tip volume interacts with the field; and if the magnetic
field decays slowly, then more of the tip interacts with the field and the effective monopole should
be modeled as if it were at a higher position along the tip. The large difference in the effective lift
height between the two fits appear to agree with the above reasoning.
32-
0 0--1-2
-3-10
-8
-6
-4
-2
0
x [um]
2
4
6
8
10
-8
-6
-4
-2
0
x [um]
2
4
6
8
10
6
420
T-2 -
-4-6-10
Figure 3-18: (a) Magnetic field strength and (b) gradient of magnetic field from 2 mA AC current
at tip-sample distances ranging from 200 nm to 3200 nm, incrementing every 500 nm. The arrow
indicates the direction of increasing lift heights.
Furthermore, the effective lift height values used to fit the AC experimental results are significantly larger than those used in the DC case. In AC current imaging, critical information lies in
the magnitude response, which is proportional to the magnetic force. This means that AC current
imaging is a direct response to the magnetic field because the monopole model asserts a linear relationship between the magnetic force and the magnetic field. On the other hand, information in DC
current imaging is given by the phase response, which is a response to the gradient of the magnitude
CHAPTER 3. AC CURRENT IMAGING
78
field in z. Fig. 3-18 shows the decay in the magnetic field and gradient of magnetic field as a function
of tip-sample distance.
Fig. 3-19 shows the maximum Hz and dHz/dz value as a function of tip-sample distance. The
height at which Hz and dHz/dz at z = 200 nm decays by l/e are z = 1.2 pm and z = 621 nm
respectively. The effective lift heights between the AC and DC simulations differ by an order of
magnitude, whereas the placement of 1/e decay between Hz and dHz/dz only differ by a factor of
two. Other reasons causing larger effective lift heights in AC current imaging are not understood
and remain a question to be answered in future work.
6-%I
dH /dz
5-'
E
o
1/e decay of dH /dz
4-
z = 621 nm
1/e decay of Hz
z = 1.20 um
3
XI
.................................
...............
........
2-
H
01
0.2
0.4
0.6
0.8
1.2
1
1.4
1.6
1.8
2
z[um]
Figure 3-19: Maximum Hz and dHz/dz as a function of z. Hz at z
pm, and dHz/dz decays 1/e at z = 621 nm.
=
200 nm decays l/e at z
=
1.2
Fig. 3-20 shows comparison between the experimental and simulated MFM responses to 2 mA
AC current at various tip-sample distances. q is 3.2x10-- Am. The physical lift heights d range
from 200 nm to 2000 nm. The fitted effective lift heights values, z, increase at a slower rate and
range from 10.5 pm to 11.6 pm. As the physical tip-sample distance increases, the 1/e decay rate
decreases. Therefore, the effective lift heights do not increase linearly with the physcial lift heights.
3.4. COMPARISON TO THEORY
0.02
79
d = 200 nm
z=10.5um
0.021
0.01
0.01
-E
)
0
/-
)
-0.01
-0.02
-20
5;.
-a)
0
-0.01
/-
-0.02
0.02
d = 800 nm
z=10.8um
-10
0
x [um]
d = 1400 nm
z = 11.2 um
10
20
-20
0.02
/
0.01
57
0
-0.01
0
x [um]
10
20
20
L
. =2
/-
-0.02
-10
10
/
/-
-20
20
'
0)
-0.02
10
0.01
-a)
-0.01
0
x [um]
d = 2000 nm
z=11.6 um
0
/-
ca)
-10
-20
-10
0
x [um]
Figure 3-20: Comparisons between experimental and simulated MFM responses at various lift
heights. The solid lines represent experimental results and the dashed lines represent simulated
results.
CHAPTER 3. AC CURRENT IMAGING
80
3.4.2
EFM
EFM simulations are also needed to check the experimental results. Electrostatic force is caused by
a difference in voltage between the tip and the sample. And since force is the derivative of energy,
electrostatic force can be expressed as a function of the difference voltage, V, and its capacitance,
C[1],[11]:
V2 C
Fz
(3.7)
2 Oz
A parallel-plate capacitor is assumed, with an effective surface area of V/R where
R
=
40 nm[11].
The tip-sample distance represents the distance between the parallel plates. Furthermore, fringing
effects are neglected and F, is only experienced when the tip is above the wire.
Fig. 3-21 shows the simulated EFM response to 2 mA AC current.
Because the simulation
assumes no electrostatic force outside the region of the wire, no magnitude nor phase response is
detected. The symmetry of the EFM response agrees with the experimental results. However, the
magnitude response is nearly three orders of magnitude smaller than that from the experiments.
Presently, the reason is still unknown.
x10,
E
1.510.5-
-25
-20
-15
-10
-5
0
x [um]
5
10
15
20
2! 5
-20
-15
-10
-5
0
x [um]
5
10
15
20
25
1D0
80604020-25
Figure 3-21: Simulated EFM response to 2 mA AC current. V = 100 mV.
3.5. SUMMA RY
3.5
81
Summary
The chapter establishes theory for AC current imaging by using new boundary conditions to solve
the flexural beam model. Experimental results are shown and confirm the theory.
Four key points are concluded from the discussion of AC current imaging. First, from the theory
and simulations, the response to AC current is contained in the cantilever magnitude response.
Therefore, AC current imaging is a direct response to the magnetic field and not its gradient.
However, the magnetic field as a function of tip-sample distance decays slower than the gradient
of the field and requires the use of a larger effective lift height. The effective lift height used to fit
the experimental response to 2 mA AC current at a physical lift height of 200 nm is 10.5 Pm, with
an effective monopole moment of 3.2x 10-5 Am.
Therefore, even though the sensitivity of AC current imaging is around 15 pA, the large effective
lift height will reduce the original spatial resolution of MFM.
Lastly, we provided a decomposition method which separates the MFM and EFM responses from
the experimental data, upon recognizing that the MFM response corresponds to the odd component
of the total response and the EFM response correscopnd to the even.
82
CHAPTER 3. AC CURRENT IMAGING
Chapter 4
Method for Non-Linear
Tip-Sample Interaction
The presence of both a piezo and an AC current drive causes non-linear tip-sample interactions and
requires a new method to describe the cantilever response.
Several authors have examined the tip-sample interaction caused by the periodic contact made on
the sample surface by the tip in tapping mode and reported higher-harmonic generation by non-linear
interaction[15],[36]. Stark et al. also used Fourier transform principles to mathematically reconstruct
the tip-sample interaction from the spectral response[37]. However, no general mathematical model
is currently available.
This chapter presents the non-linear theory. The limiting case where the current frequency
approaches zero is examined and used to validate the theory by comparing the results to DC current
imaging responses. By reducing the piezo drive amplitude to zero, the theory is also compared to
AC current imaging.
One direct application of the non-linear theory is in the area of parametric gain. The AC current
source at frequency w, transfers the power applied to the piezo at frequency wp to a new frequency,
W, + WP. By matching the sum of the two frequencies to a cantilever vibration mode, the response
is enhanced by the high
Q of the
cantilever mode. This enables the piezo to be driven much harder
because it can now be driven off resonance and rely on the current frequency to transfer power to
the resonance.
At the end of the chapter, the intrinsic non-linearity in the AFM system is examined.
83
CHAPTER 4. METHOD FOR NON-LINEAR TIP-SAMPLE INTERACTION
84
4.1
Theory
If the piezo is driven at frequency wp and with amplitude a and the current has frequency wc, then
the new boundary conditions are as follows:
(4.1)
z(x = 0, t) = a cos wpt
z(x-0,t)
= 0
(4.2)
El 0 z(x=L,t) = 0(4.3)
z~x=L-Lt) EI
EIz
where Fz (zo) and
-
x-L,t) = Fz(zo) coswct + (
Ijzo coswct)z(x = L, t)
(4.4)
Izo are the first two Taylor series components of a DC magnetic force since the
time-varying terms are included externally, and zo is the effective tip-sample distance.
From Eqn. 4.4 we see that the first Taylor series term drives the cantilever at the current frequency and that the second term contributes to the generation of higher-order harmonics and intermodulation products. Consequently, a general solution for z(x, t) can be guessed, assuming the
higher-orders are sufficiently smaller:
z(x,t)
=
Zdc(x) +
Zca(x)eJwct
+ Zcb(x)e-3wct + ZCc(x)eJ2wct + Zcd(x)e- 2_2wc
t
+
Zpa(x)e31pt + Zpb(x)e-wPt +
Zcpa(x)e3(w±+wP)t + Zepb (x)e3(w -w )t + Zcpc(x)eJ(-wc+wp)t + Zcpd(x)e-3(wc+wp)t
(4.5)
4.1. THEORY
85
where
(4.6)
Zdc(x)
=
dc1 +dc2 x+dc3 x 2 +dc 4x 3
Zca (x)
=
ca 1 (cos kcax + cosh kcax) + ca 2 (cos kcaX - cosh kcax) +
ca3(sin kcax + sinh kcaX) + ca4(sin kcaX - sinh kcaX)
Zeb (x)
Zcpd(x)
=
cb1 (cos kebx + cosh
kCbx)
+ cb2 (cos kebx - cosh kcbx) +
cb 3 (sin kcbx + sinh
kebx)
+ cb4 (sin kcbx - sinh kcbx)
(4.7)
cpd1(cos kcpdx + cosh kcpdx) + cpd2 (cos kepdx - cosh kepdx) +
cpd3 (sin kcpdx + sinh kcpdx) + cpd4 (sin
kcpdx -
sinh kcpdx)
The spatial distribution of each frequency component contains four coefficients that need to be
determined by the boundary conditions.
must satisfy
Zdc(x),
because it is not related to a time-dependent term,
= 0 (see Eqn. 2.26), of which Eqn. 4.6 is a general solution. Each of the ten
d 4d"
other spatial distributions
(Zca(X), Zcb(x), etc.)
require a dispersion relation relating its frequency
component to its wave number.
The method is identical to the situation where only one frequency component is present (Sec. 2.1.3),
because each frequency component is independent from another. For example, to find the dispersion
relation between wc and kca, Za(x)elt is plugged into the damped beam equation (Eqn. 2.26),
and the relationship is found:
ca=
(2 4Q
2J
)
(4.8)
Or for Zcpa(x)e3(oc+WP)t,
(PC+ W)
where
Wca =
k2ca'a
=W
(
4
1-
+1
)
(4.9)
E
= k2
L and wcpa
pa
cpa VpA~
pA
The same relationship applies for the terms that have positive frequency components. In the case
of a negative frequency component, such as Zc6 (x)e3wc , the dispersion relation is slightly different,
WC
where
Wcb =
k
V
.
(
4 -
-
)
(4.10)
CHAPTER 4. METHOD FOR NON-LINEAR TIP-SAMPLE INTERACTION
86
It is important to note that if w, > wp, then (-we + wp) is negative and (w, - wp) is positive.
Likewise, if wp > we, then the signs are flipped.
Once the dispersion relations are established, the coefficients can be determined using the boundary conditions. The first three boundary conditions do not generate any new frequencies and the 11
spatial distributions remain independent from each other.
By applying Eqn. 4.1 to the general equation, the first coefficient of each spatial distribution is
zero except for pal and pbi, which equal a/2. And from Eqn. 4.2, dc 2 and the third coefficients of
the others (i.e. ca 3 , cb 3) are also set to zero. Eqn. 4.3 relates dc3 and dc4 and the second and fourth
coefficients of the others.
The last boundary condition relates the spatial distributions to each other. By collecting the
terms with the same frequency dependence together, 11 new equations emerge:
3
DC
wc
-wc
2w,
d Zd (x
=L) =
3
dx
d3 Ze. x=zL) =3
3
3
3
d Z dx(x=L) -=
d Z,dx(x=L)
M1Zeb(x = L)
+
M1Zca(x = L)
+± M1Zdc(x = L)
+
M 1 Zee(x
MlZcd(x = L)
+
MlZdC(x = L)
+
-
L)
M1Zca(x = L)
3
-2wc
wp
-
wp
(w + wp)
(w - wp)
Swp)
(-we
d Zcd(x=L)
3
d33Z, dX(x=L) =
d Z ,(x=L)
3
dx
3
d Z b(x=L) =
3
dx
d3 Z~~p.(x=L)-
M1Zep
(x
MlZepd(x
MlZeb(x
L)
L)
= L)
+
M 1 Zcpa (x
= L)
+
M1Zcpb(x = L)
MlZpa(x = L)
3
dx
d3 Z,,b(x=L)-
MlZpb(x = L)
M1Zpa(x = L)
3
dx
d Z~pd(x=L)3
dx
3
-
(we
where Mo =
Swp)
F
(zo) and M, =
1
MlZpb(x = L)
K
Together, the 22 equations from the third and the fourth boundary conditions are enough to
solve the 22 unknown coefficients, two from each spatial distribution. A 22x22 matrix scheme is
4.1.
THEORY
87
used and each row represents one equation,
A1,1
A, 2 A1,3
A
2 ,1
A 2 ,2
A 3 ,1
A 3 ,2
A 1 ,4
A 2 ,3
...
...
...
...
...
...
...
A 2 2 ,1
...
...
A
4 ,1
A
...
1 , 22
dc3
B1
...
dc4
B
. ..
ca 2
...
ca4
...
cb2
B5
...
cpd 4
B22
2
B 3
=
B
4
(4.11)
CHAPTER 4. METHOD FOR NON-LINEAR TIP-SAMPLE INTERACTION
88
where
A
A
A
A
A
A ,
1 2
6L
T(kca)
A ,
2 4
U(kca)
,5
T(kcb)
A 3,
6
U(kcb)
4
,7
T(kcC)
A
5
,9
T(kcd)
A
T(kpa)
A
1
,1
2
,3
3
A 6,
11
A
A
A
A
A
A
:
7
,1 3
T(kpb)
A
8
,1 5
T(kcpa)
A
9
,1
T(kcpb)
A
7
4
,8:
U(kCC)
5
,1 0
U(kcd)
6
,1 2
U(kpa)
7
,1 4
U(kpb)
8
,1 6
U(kcpa)
9
,1 8
U(kcpb)
10
,1 9
T(kcpc)
A
1 0 ,2 0
U(kcpc)
11
,2
T(kcpd)
A
1 1 ,2 2
U(kepd)
12
,2
-6
A
1 2 ,3
MiX(kca)
A
1 3 ,2
Ml L
A
1 3 ,8
1
A1 ,
2 6
A
13
,1
13
,7
14
,1
MI L
M
1
X(kcc)
A
A
A
,4 :
MiY(kca)
A
12
,5 :
M1X(kcb)
A
13
,3 :
kaV
cca)
A
13
,4 :
kgcW(kca)
A
14
,5 :
k3bV(kcb)
A
14
,6
kb W(kc
k3cV(kcc)
A
15
,8
k cW(kcc)
IcdV(kcd)
A
16
,1 0
cdW(kcd)
17
,1 6
MI Y(kcpa)
18
,1 8
MY(kcpb)
19
,1 6
WpaW(kcpa)
20
,1 8
k cpb W(kcpb)
21
,2 0
22
,2 2
MY(kcc)
2
A
12
3
2
A
A
MI Y(kcb)
M
1
L
3
A
A
14
,9
MX(kcd)
15
,3
MiX(kca)
A
16
,5
Ml X(kcb)
A
A
MI L
14
,2
14
,1 0
MY(kcd)
15
,4
MiY(kca)
A
16
,6
M 1 Y(kcb)
A
17
,1 2
k3aW(kpa)
A
18
,1 4
IebW(kpb)
A
,1 2
MaY(kpa)
19
A
20
,1 4
M
A
1 7 ,1 1
A
1 8, 13
k3bV(kpb)
A
1 9 ,1 1
M 1 X(kpa)
A
A
2 0, 13
M1X(kpb)
A
A
2 1, 11
M 1 X(kpa)
A
2 1 ,1 2
A
2 2, 13
M
A
2 2 ,1 4
1
X(kpb)
A
B
B
B
B
Y(kpb)
A
MiY(kpa)
A
M 1 Y(kpb)
A
1
-*X(kpa)
6
16
,9
17
,1 5
M 1 X(kcpa)
A
18
,1 7
MiX(kcpb)
A
19
,1 5
k
paV(kcpa)
A
20
,1 7
k
PbV(kcpb)
A
21
,1 9
22
,2
B
B
13
4kpaS(kpa)
17
19
B21
1 5 ,7
:
-M
M 1 W(kpa)
1 W(kpa)
B
B
B
A
kpcV(kcpc)
k3
1
V(kcpd)
pb)
7
14
a k3bS(kpb)
18
20
22
:
-
TM1 W(kpb)
-
M1W(kpb)
A
k
6
)
W(kp)
W
kp
cpd W(kcpd)
4.2. APPLICATION TO PARAMETRIC GAIN
89
and all the other elements are zero. A key to the abstractions are as follows:
S(k) = sin kL + sinh kL
T(k) = -cos kL - cosh kL
U(k)
-
sin kL - sinh kL
V(k) = -sinkL + sinhkL
W(k) = cos kL + coshkL
X(k) = cos kL - cosh kL
Y(k) = sin kL - sinh kL
The coefficients are solved numerically in Matlab by multiplying the inverse of matrix A by
matrix B. Once the coefficients are found, the response at difference frequencies can be determined.
For example, the complete solution at w, is Z,(x) where IZc(x)I = 21Za(x)I = 2IZeb(x)I and LZ
LZa(X)
4.1.1
=
=
-LZZb(x).
Limit to DC and AC Current Imaging
In order to validate the non-linear theory, the cantilever response at the limit where the current
frequency approaches zero is compared to the DC current response shown in Chap. 2. The piezo is
driven on resonance. The result is shown is Fig. 4-1, and it is identical to Fig. 2-8.
At the limit where the piezo drive amplitude is zero, the cantilever is again purely responsive to
the AC current and the result is identical to AC current imaging (compare Figs. 4-2 and 3-1).
4.2
Application to Parametric Gain
The non-linear system, including both piezo and AC current drives, provides new possibilities for
cantilever excitation. One is to examine the w, + w frequency component. By matching w, + wp to
the cantilever's first resonant frequency, the power applied to the piezo drive can be tranferred to
the resonance by the current drive.
These principles of parametric gain are first discussed. The degenerate case where W, = WP is
presented next, followed by the nondegenerate case where the two frequencies are different.
CHAPTER 4. METHOD FOR NON-LINEAR TIP-SAMPLE INTERACTION
90
2820
28002780 R 276002740272027
-10
-8
-6
-4
-2
0
x [um]
2
4
6
8
10
-8
-6
-4
-2
0
x [um)
2
4
6
8
10
10
0
.1
- 10
-10
Figure 4-1: Magnitude and phase response to 20 mA current at the limit where w, approaches zero.
WP is driven on resonance . q = 10-6Am. z = 400 nm.
4.2.1
Principles of Parametric Gain
Parametric gain can be illustrated by a damped harmonic oscillator in which an energy storage
parameter is modulated at some frequency wi [40]. An example of a degenerate parametric oscillator
is an RLC circuit in which the capacitor is time-varying.
Consider a normal parallel RLC circuit. The voltage across the elements as a function of time is
d2 v(t)
1 dv(t)
dt 2 d2RC
+
dt
1
LC v)
t
=
=0
(4.12)
However, if the capacitor has an added time-varying term, where
C = Co + ACsinwit
(4.13)
a new expression for the voltage would be
d2 v(t)
1 dv(t)
1
AC
dt2 + RC0 dt + LC0 (1± C0 sinwitv(t))=
(4.14)
4.2. APPLICATION TO PARAMETRIC GAIN
91
70
605040 0
'E20 -20 --
10
-20
-15
-10
-5
0
x [um]
5
10
20
15
-0)
100
50-
0--
-50-
-100
-20
-15
-10
0
-5
5
10
15
20
Figure 4-2: Cantilever response with no piezo drive to 2 mA AC current flowing through a 2pm-wide
wire, centered at x = 0. q = 8 x 10-6Am. z = 1 pm.
assuming that AC < Co.
By assuming a solution v(t)
=
a cos(wt +
#)
and neglecting the (wi + w) term, Eqn. 4.14 can be
solved:
(P
_
edot+0)
+ RC
RCo
(2)e3(-t++)
where wo = LCO
Therefore, if w1/2 = w = wo,
#
=
0 or -r, and RAC
2Co
=
0
=-e3[W)t*]
(4.15)
2/wo, then steady-state oscillation can be
achieved.
Physically, the time-varying capacitance can be thought of as a parallel-plate capacitor whose
distance between the two plates varies with time, since capacitance is inversely proportional to
distance. Fig. 4-3 schematically shows the voltage gain achieved by the modulated capacitance.
Because the capacitance changes at twice the resonant frequency, if the plates are pulled apart at
the maxima and minima of the charge function, q(t) and are pushed together at the zeros of the
charge function, parametric gain in the voltage function can be achieved. At the instance right
before the plates are pulled apart, a certain amount of charge accumulates on the plates (Fig. 43(a)). Once pulled apart, the capacitance drops but the charge cannot change instantaneously and
a) t = ti
b)t= t2
d,
d2 = d1 +8
Figure 4-3: Schematic drawing of time-varying parallel-plate capacitor. (a) At t = ti, charge accumulates on the parallel-plates. q(ti) = C(ti)v(ti). (b) At t = t 2 , the parallel plates are pulled apart
some distance 6. Instantaneously, the charge remains the same, q(t 2 ) = q(ti), while the capacitance
drops, C(t 2 ) < C(ti). Therefore, because q(t 2 ) = C(t 2 )v(t 2 ), v(t 2 ) > v(t)1.
remains the same. Therefore, to satisfy q(t 2 )
= C(t 2 )v(t 2 ),
the voltage increases instantaneously. In
other words, the work done to pull the parallel plates apart was given back as voltage gain. On the
other hand, no work is needed to push the plates back since the timing coincides with the zeros of
the charge function.
Another example of a damped harmonic oscillator is the mass-spring system, from which the
point-mass model is derived. By relating the elements of a mass-spring system to the RLC circuit,
intuition about the non-linear cantilever response can be gained. The effective spring constant k,
of the cantilever beam in the point-mass model is analogous to the capacitor. Since magnetic force
on the tip shifts the effective spring constant, k, is modulated at the AC current frequency w.
The force on the effective spring and its displacement is related by F(t) = k(t)z(t) where F(t) is
analogous to q(t) and z(t) is analogous to v(t). If the maxima and minima of the force function
coincides with the "loosening" of the spring, then an instantaneous gain in displacement must occur
to satisfy the force equation.
4.2.2
Degenerate: w, = wp
Parametric gain in the degenerate case can be shown in the response of 2W frequency component.
With w, = wp = W, the response of the eJ2wt term is maximized when 2w matches the resonant
frequency of the cantilever. Fig. 4-4 shows the magnitude responses of both the e-wt and the e
2
wt
terms as a function of drive frequency. The frequencies at which the peaks occur are evident: for
eswt, it happens when w = wo; for e 2"t, it happens when w/2 = wo.
4.2. APPLICATION TO PARAMETRIC GAIN
93
3000
250020001500100050040
50
60
70
80
Frequency [kHz]
90
100
110
50
60
70
80
Frequency [kHz]
90
100
110
0.014
0.012
0.01
0.008
8 0.006
2 0.004
0.002
40
Figure 4-4: Magnitude response to 2 mA AC current through a 2pum-wide wire, centered at x = 0
across sweep of frequencies. w, = w=
,. q = 5x10-7Am. z = 800nm. (Top) Response of e3"
term. (Bottom) Response of eJ2wt term.
4.2.3
Nondegenerate: we
wp
Fig. 4-5 shows the magnitude responses to 1 mA AC current of various combinations of W, and wp,
with the criteria that w, + wp equals the first vibrational frequency, wo. The piezo is not driven in
the top figure, and w, =
wo = 89.9 kHz. This is identical to AC current imaging and the maximum
response is 42 nm. In the middle figure, w, = 34.9 kHz and wp = 55 kHz. The piezo drive amplitude
required to achieve the same cantilever response as the top figure is 12 pm. The piezo frequency in
the bottom figure is 100 Hz less than the resonant frequency and the drive amplitude needed is only
8 nm.
By driving the piezo near the resonant frequency, the response is enhanced drastically by the
of the cantilever. Fig. 4-6 compares the cantilever response at wp
=
wo
Q
- 187 Hz and wp = wo - 100
Hz. w, = 0 and 100 Hz respectively. The first corresponds to DC current imaging at the frequency
of optimal magnitude response.
The slope of cantilever response versus piezo drive amplitude is the same for the two frequency
settings as well as for different current levels. However, the DC current response is more than two
CHAPTER 4. METHOD FOR NON-LINEAR TIP-SAMPLE INTERACTION
94
40-
E
£30. 20 10 -15
-10
-5
0
x [um]
5
40 -
a
E 0)
20 CU
-15
10
15
10
15
1.5 um
a =1.2 um
-10
-5
0
x [um]
5
60-
-
= 12 nm
4-a
S40 o20 01
-15
a=8 nm
-10
-5
0
x [um]
5
10
15
10-5
89.9 kHz. q
=
Figure 4-5: Magnitude responses to 1 mA AC current when w, + w,
Am. z = 1 pm. (Top) w, = w0 and w, = 0. (Middle) w, = 34.9 kHz and wp = 55 kHz. The piezo
drive amplitude, a, needed to be about 12 pm to match the magnitude response of the top figure.
(Bottom) w, = 100 Hz and w, = 89.8 kHz. The piezo drive amplitude only needed to be 8 nm to
match the magnitude.
orders of magnitude larger. The simulation uses fitting parameter values similar to those determined
in DC current imaging (q = 10-
Am and z = 1 pm), and they are assumed to be the same for
both frequency settings. But without nondegenerate experimental data with which to compare, this
assumption cannot be verified. Therefore, focus should be on the sensitivity to current level within
the two frequency settings.
When wp = wo - 100 Hz, the change in cantilever response to current level is very drastic. Fig. 4-7
zooms in on the DC current response to examine its change to current level. This concludes that the
nondegenerate case is more sensitive to current level. However, two practical issues remain. First, q
and z must be fitted to experimental results in order to accurately simulate the physical cantilever
response. Second, the noise floor must be examined to determine a feasible sensitivity level.
95
4.2. APPLICATION TO PARAMETRIC GAIN
106
........
lo
102
1 nA, 1, 2, 10 mA
S100
10 mA
2
mA
-
Nondegenerate
.....
DC current
-
1 mA
S10~
-)
1 nA
10-8
10'
10
Piezo drive amplitude [nm]
10
102
Figure 4-6: DC current and nondegenerate cantilever response to varying piezo drive amplitude. DC
current: wp = wo - 187 Hz and w, = 0. Nondegenerate: w, = wo - 100 Hz and w, = 100 Hz. q =
10- 7 Am and z = 1 ptm.
x010
454.I
7
4.5654.56
4.555
4.55
mA
-10
.
4.545
4.54
2 mA
0 4.535
-
4.53
1 mA
4.525
I nA
4.52
200
200.2
200.4
200.6
200.8
201
201.2
Piezo drive amplitude [nm]
201.4
201.6
201.8
Figure 4-7: DC current cantilever response at w, = wo - 187 Hz and w,
amplitude. q = 10-7 Am and z = 1 pm.
=
202
0 to varying piezo drive
CHAPTER 4. METHOD FOR NON-LINEAR TIP-SAMPLE INTERACTION
96
4.3
Non-Linearity in AFM System
The nonlinear response of the AFM system is examined to ascertain that the non-linearities observed
experimentally are mainly from the non-linear tip-sample interaction and not from the system itself.
The test for non-linearity in the AFM system was performed with a regular tapping-mode tip.
It was driven by the piezo and kept far above the sample to ensure no interaction. The piezo drive
amplitude was increased slowly and both the input piezo voltage and the output voltage from the
photodetector are recorded by the oscilloscope and Fourier transformed into the frequency domain.
One raw FFT spectra of output voltage is shown in Fig. 4-8.
-20-
-40-
E
-60
C.
-80
-100-
-120-
-140'
0
500
1000
1500
2500
2000
3000
Frequency [kHz]
3500
400
4500
5000
Figure 4-8: FFT spectra of output voltage to 0.205 input voltage.
The input and output voltages at the drive frequency as well as two and three times the frequency
are plotted against the input voltage at the first harmonic. The figures are shown in Figs. 4-9 and
4-10.
The plot of vi, at the first three harmonics versus vi, at the first harmonic is used for reference.
The slopes of the higher harmonics are all much smaller, suggesting minimal generation of nonlinearities in the system. The slope of the output voltage at the first harmonic is very close to one
and confirms its response to the input voltage.
By examining the difference among the harmonics at vi,
=
0.257 V, non-linear generation can be
determined by comparing the results of the input and the output voltages. The difference between
4.4. SUMMARY
97
the first and second harmonic input voltages is 0.233 V, whereas the same comparison of the output
voltages yields a difference of 0.232 V. This shows that there is no generation of the second harmonic
between the piezo drive and the photodetector output. The same comparison between the first and
third harmonics yield 0.248 V for the input voltages and 0.238 for the output voltages. Therefore,
no third-harmonic-generation is observed either.
This concludes that the non-linearity in the AFM system is minimal.
4.4
Summary
In this chapter, a general method for non-linear tip-sample interaction is presented and shown to be
consistent with DC and AC current imaging theories in Chaps. 3 and 4. By using the sum of the
current and piezo frequencies to excite the first vibrational mode, parametric gain is achieved. The
degenerate case shows a large cantilever response at wo/2 in the 2w frequency component when w,
=P
= W=
wo/2.
The nondegenerate cantilever response is enhanced when the piezo is driven near the resonant
frequency. The results are compared to DC and AC current imaging. The difference in the nondegenerate responses to varying current level is much more drastic than the DC current responses.
And by an 8 nm piezo drive amplitude at wp =
wo - 100 Hz, nondegenerate response achieves the
same cantilever oscillation amplitude as the pure AC current response.
The comparisons to DC and AC current imaging assumed the same effective monopole moment and displacement values respectively. However, in order to accurately determine these values,
experimental results are needed.
Lastly, the intrinsic non-linearity in the AFM system is determined to be negligible.
98
CHAPTER 4. METHOD FOR NON-LINEAR TIP-SAMPLE INTERACTION
0.4
1
1
1
Fundamental mode
0.35-
slope = 1
0.3
-
0.25-
-
-
0.2 -
0.15Second harmonic
0.05
-
Third harmonic
slope = 0.028
slope = 0.122
0.1-
-
-
0
0.2
0.22
0.24
0.26
0.28
0.3
Vi [V]
0.32
0.34
0.36
0.38
0.4
Figure 4-9: Non-linearity in input piezo voltage at the first three harmonics.
0.4
Fundamental mode
0.35-
slope = 0.957
0.3 -
0.25 -
3
-
0.2-
Third harmonic
slope = 0.068
0.15Second harmonic
slope = 0.105
0.1
0.05 -
00.2
0.22
0.24
0.26
0.28
0.3
Vin IV
0.32
0.34
0.36
0.38
0.4
Figure 4-10: Non-linearity in output voltage of a free cantilever at the first three harmonics.
Chapter 5
Conclusion
5.1
Summary
The goal of this thesis is to explain DC and AC current imaging in magnetic force microscopy as
well as understand the effects and applications of non-linear tip-sample interactions.
Chap. 2 derives the flexural beam model and presents various methods to describe the magnetic
tip-sample interaction. DC current imaging theory is subsequently presented and compared to the
experimental data. The extended monopole model matched the experimental results better than the
extended dipole model. The effective monopole moment and displacement for 15 mA DC current
were found to be 1.9 X10-7 Am and 850 nm respectively. Magnetic imaging at the second cantilever
vibration mode was also examined with a standard magnetic tape. Even though the
Q
is higher
at the second mode, the sensitivity decreases because of a faster increase in the effective spring
constant. The overall sensitivity to DC current imaging was determined to be at 1 mA.
AC current imaging is discussed in Chap. 3. By filtering the response at the current frequency,
the results were independent of the piezo drive frequency. Simulations show that the critical magnetic
field information is contained in the cantilever's magnitude response, suggesting that the tip responds
directly to the magnetic field in AC current imaging. Electrostatic force contamination in the overall
cantilever response necessitated a decomposition method which separates the total response to its
even and odd components, of which the even represented the tip's response to electrostatic force and
the odd represented its response to magnetic force. Comparison with the experimental MFM results
suggested a need to employ a much higher effective lift height in the AC simulations (q = 3.2x 10-5
99
CHAPTER 5. CONCLUSION
100
Am and z = 10.5 pm for 2 mA AC current). Even though the current sensitivity was determined to
be around 15 pA, the high effective tip-sample distance lowers the spatial resolution of this method.
Chap. 4 presents a general theory for non-linear tip-sample interaction, which is not limited to
current measurement nor MFM. Comparisons to DC and AC current imaging validate the theory.
Parametric gain with the non-linear theory is shown in the degenerate and nondegenerate cases. The
nondegenerate cantilever response is enhanced when the piezo is driven near the resonant frequency.
With the sum of the piezo and current frequencies set to the first vibrational frequency and L
= 100
Hz, the nondegenerate responses show greater current sensitivity than DC current imaging. Lastly,
the non-linearity in the AFM system was experimentally determined to be negligible.
5.2
Directions for Future Work
Suggestive directions in modeling and experimental work for current measurement research in MFM
are presented below. The list focuses on possible next steps in the research.
5.2.1
Modeling
Magnetic Tip-Sample Interaction
Theoretical development in this thesis focused heavily on the transverse vibration of the cantilever
beam due to interest in the higher-order vibrational modes.
Less emphasis was placed on the
modeling of the magnetic tip-sample interaction. While the use of the extended monopole model is
verified experimentally, it remains an approximation.
To model the magnetic tip-sample interaction more precisely, the Fourier transform method can
be employed[16]. It uses transfer functions in the Fourier domain to simplify the convolution of the
magnetic field from the sample and the magnetic surface area of the tip. According to McVitie et
al.[22], who evaluated the point charge and Fourier transform models against measured magnetic
field distributions from MFM tips, the latter provided the best match.
Large effective lift height in AC current imaging
The need for large effective lift heights ( 10 pm) in AC current imaging remains a mystery. The
original hypothesis attributed this need to the slower decay rate of the magnetic field versus the
gradient of the field. However, upon comparing the two l/e decay length at a tip-sample distance
5.2. DIRECTIONS FOR FUTURE WORK
101
of 200 nm, the result for the magnetic field was only a factor of two larger than that of the gradient.
According experimental results obtained by Lohau et al.[19], the effective displacement appear to
relate linearly with the decay length. Therefore, it does not fully explain the near factor of 10
increase in the effective lift heights between DC and AC current imaging.
EFM
More detailed EFM simulations are also needed. The current model approximates the capacitance
between the MFM tip and the sample as a parallel-plate capacitor, with an effective area of a circle
with radius R = 40 nm[11]. However, the magnitude of the simulated EFM response is nearly three
orders of magnitude smaller than the experimental EFM results.
Noise analysis
A more comprehensive discussion of current sensitivity would require detailed analysis of the possible
sources of noise. Noise floors for the different imaging methods should be determined. Butt et al.[5]
examined thermal noise in the first six vibrational modes. Salapaka et al.[33] presented an overall
analysis on multi-mode noise.
5.2.2
Experimental
MFM with shielded sample
To ensure pure MFM response without EFM contamination, the wire sample can be shielded with a
layer of PMMA, which serves as insulation. Therefore, the voltage across the wire sample is shielded
from the tip and electrostatic force should be eliminated.
EFM with demagnetized MFM tips
If AC current imaging is performed without a shield sample, more experimental EFM results are
needed. In Chap. 3, the decomposed EFM response from a MFM tip is compared to the response of a
non-magnetic EFM tip. The two results are similar qualitatively. However, quantitative comparisons
could not be made because the physical dimensions and makeup of the tips are different.
In order to ascertain the decomposed EFM response quantitatively, one possibility is to perform
EFM with a demagnetized MFM tip. Demagnetization of MFM tips can be achieved by heating the
tip for several hours or by allowing the tip to undergo AC demagnetization.
CHAPTER 5. CONCLUSION
102
AC current imaging at second harmonic
Second harmonic imaging of AC current should also be explored. The magnitude response does not
have the same dependence on
Q and k,
as the phase response does in DC current imaging and better
sensitivity may be achieved at the second vibration mode in AC current imaging.
Parametric gain
Chap. 4 shows the theoretical potential for parametric gain with piezo and current drive frequencies.
Experimental verification is needed. Furthermore, q and z must be fitted to experimental data. The
phase between the piezo and current drive frequencies must remain linearly related and the lock-in
should reference the sum frequency.
Conversion from electrical amplitude to mechanical displacement
The cantilever oscillation is measured by the photodetection system, which outputs a voltage signal.
In order to determine the cantilever oscillation in nanometers, an added step is needed at the end
of each experimental session.
The method to measure cantilever oscillation uses the TappingMode force curve function in the
DI instrument menu. The force curve graphs cantilever oscillation amplitude against vertical piezo
scanner movement (see Fig. 5-1).
The scanner begins on the right side of the graph, where the
cantilever vibrates in free air. As the scanner lowers and the tip begins to tap on the sample surface,
the feedback system reduces cantilever oscillation amplitude, which is represented by the downward
slope on the force curve. Eventually, as the scanner continues its downward movement, the tip hits
the surface and its amplitude reduces to zero. Therefore, the horizontal length of the downward
slope represents the peak cantilever amplitude and can be compared to the peak voltage signal.
5.3
Final Words
The tradeoff between current sensitivity and spatial resolution between DC and AC current imaging
became apparent. While MFM is able to sense AC current down to 15 pA, the large effective lift
height suggests diminished spatial resolution. Its performance appears to be more comparable to
scanning hall probe microscopy, which has about 1 pA current sensitivity and 1 pm spatial resolution.
The non-linear approach to current measurement is promising. Parametric gain can be achieved
5.3.
Cantilever amplitude
Cantilever
oscillation
amplitude
[nm]
0
Vertical scanner movement [nm]
Figure 5-1: Force calibration plot for voltage to metric conversion.
and sensitivity to current is enhanced. However, experimental work is needed to support the theoretical analysis and to determine its spatial resolution.
The development of current measurement in MFM can serve as a powerful tool for various research.
It may provide an simple and accurate way of quality assurance on integrated circuits.
Moreover, it allows imaging of current flow through various materials and insights on electron transport can be gained.
104
CHAPTER 5. CONCLUSION
Appendix A
Experimental Setup
This appendix covers the parts of the experimental setup that is common in both DC and AC current
imaging. The basic setup of the instruments and the fabrication processes of the wire sample are
first presented, followed by details of the data acquisition and process software.
A set of experimental steps involved in DC and AC current imaging are also provided. Familiarity
with MFM is assumed.
A.1
A.1.1
Instrument Setup
Atomic Force Microscope
The usage of the AFM follows standard D13000 MFM procedures, in terms of the tip calibration,
laser alignment, and sample loading processes. A scan rate of 0.1 Hz is used to ensure detailed
imaging as well as clean data acquisition by the lock-in. Typically, the lift height during LiftMode
is set at 200 nm because the sample has a height of 200 nm.
A.1.2
Lock-in Amplifier
The values used for the lock-in amplifier are listed in Table A.1.
asterisk (*) are controllable through LabVIEW user interface.
105
The controls marked with an
APPENDIX A. EXPERIMENTAL SETUP
106
Instrument Control
Value
GPIB address*
8
Sensitivity*
300 mV (DC)
100 mV (AC)
Time Constant*
30 ms
Filter
24 dB/Octave
Channel 1
R [V]
Channel 2
Reference in
External, 50 Q
Signal in
1MQ, 30 pF
Table A.1: Experimental settings for the lock-in amplifier.
A.2. SAMPLE FABRICATION
A.2
107
Sample Fabrication
The wire sample was prepared by Mathew Abraham. A brief description of the process is included
here. First, a 100 nm layer of oxide was grown on a 4-inch Si wafe in an oxidation tube, in order to
electrically isolate the metal wires and the Si substrate. Al wires were then defined on the oxide layer
by a liftoff process. AZ5214 was used as the image reversal resist and 200 nm of Al was deposited
using an E-beam evaporator.
A.3
Data Acquisition and Processing Software
Data acquisition of the lock-in amplifier is controlled through LabVIEW. Once the values are stored
into a computer file, Matlab code is used to process and plot the data.
A.3.1
LabVIEW Code
Due to the graphical nature of LabVIEW programming, it is difficult to include the actual code.
Therefore, only the basic functions of Controller.vi is described.
First, the data buffer of the lock-in is cleared. Then the instrument parameters are set, including
time constant, sensitivity, and sampling rate. Once data sampling is triggered, the buffer begins to
store data. The program waits long enough to scan in a specified number of scan lines and queries
the stored data, which is subsequently placed in files named by the user.
A.3.2
Matlab Code
lvread.m reads the stored values from the file specified by LabVIEW and translates them into a
matrix in Matlab.
A.4
Recipe for DC and AC Current Imaging
The important steps in DC and AC current imaging are listed. General knowledge of MFM imaging
with the Dimension 3000 is assumed.
First, the SAM needs to be connected to the AFM, between the extender electronics module
and the microscope. The controller should be turned off and the user should exit the DI program
prior to adding the SAM. The tip is then loaded onto the holder and magnetized in the z-direction.
APPENDIX A. EXPERIMENTAL SETUP
108
Standard laser alignment and frequency sweep routine are performed and the interleave parameters
are set.
The tip is calibrated with a magnetic tape and its magnetization is checked by phase imaging in
the interleave mode. Typical parameters are as follows: the scan size is 3 to 5 pm; the lift height is
40 nm; the phase z-range is about 30±10 degrees; the height z-range is 100 nm. Note that for better
tracking purposes, the tip should engage with the scan size set to 1 prm and increased to the desired
scan size after it is engaged.
Once the tip's topographic and magnetic force imaging abilities are proven, the user should
disengage, remove the magnetic tip and load the wire sample. Specifically, the wire sample should
be loaded with the horizontal wires parallel to the user. The new surface is focused upon and the
tip engages for the second time. Common settings for the wire is: scan size = 20 to 50 pm; the lift
height is 200 nm; the phase z-range is 2 to 5 degrees; the height z-range is 500 nm.
For DC current imaging, the piezo is driven externally by a function generator and the current
comes from a DC power supply. The piezo drive input is ANA 1. The lock-in references the sync
output of the function generator and takes in the photodetector output from the SAM.
For AC current imaging, the piezo is driven internally during the topographic scans and is not
driven during the lift scans. The function generator provides the AC current to the wires and the
reference for the lock-in.
Data acquisition is enabled by Controller.vi.
Appendix B
Magnetic Field Over a Wire
The cantilever response depends critically on the magnetic field produced by the straight conductor.
This magnetic field is derived by first considering the case of a long wire with negligible width and
height, which will be called a point wire. Secondly, the magnetic field of a wire with finite width
and height, a bar wire, is presented.
The law of Biot and Savart expresses the total magnetic field H at any point in space due to the
current as
H =
-
dx
(B. 1)
r2
47r
where dl is a current element and r is the distance from the element to the observation point[13].
Applying it to the case of a straight conductor with current, I, and length, 2a (Fig. B-1), the
magnetic field at point P can be solved, with r =
H=
I
4r
I
/ 2dya(X2
'Y=
afX
+
-f
v/x 2 + y2 , dl =
47r
(2+y)/
Eqn. B.2 is simplified if the conductor is very long, a
H =
dy, and sinG = X/
2a
x a
xv'z2 + a2
X2 + y 2:
(B.2)
> x,
-(B.3)
27rx
Furthermore, as long as the observation point is at the same distance away from the wire, it expe109
x
P
r
0
-a
dl
a
Figure B-1: Magnetic field produced by a straight conductor at point P.
riences the same magnetic field. Hence Eqn. B.3 can be rewritten as
H=
(B.4)
27rr
where r is the distance between the observation point and the wire.
Because the MFM tip is usually magnetized in one specific direction, separating the x- and zcomponents of the magnetic field is necessary. As illustrated in Fig B-2, the direction of the magnetic
field is related to the 0-component,
H= = - (sin Oi - cos 0-)
27rr
27rr
(B.5)
It can also be re-expressed in cartesian-coordinates,
H= 27r
(
2
(X2 + Z2)
-2
(B.6)
(X2 + Z2)
The tip is most often magnetized in the vertical direction (z) because it reacts more sensitively to
the vertical force. Therefore it is helpful to isolate the z-component of the magnetic field:
Hz = -- 2
27r (x + z 2)
Physcially, Hz is asymmetric about the z-axis and is maximized along the x-axis.
(B.7)
111
Fig. B-3 illustrates a wire with finite width, W, and height, H. The total magnetic field can
be expressed as the sum of the magnetic fields produced by each infinitissimal units over the crosssectional area, and the z-component is hence
Hz =
I
I
2,rWH
fW/2
r0
J-W/2 f-H
xx
(x
- XO) 2 +
(z
dzdx
(B.8)
- ZO)2
Fig. B-4 shows the vertical components of H, VH, and V 2H as a function of lateral (x) and
vertical (z) distance from a 2 pm-wide wire carrying 50 mA current. V 2H is directly related to the
change in the cantilever's phase response as explained in Sec. 1.2.
APPENDIX B. MAGNETIC FIELD OVER A WIRE
112
(x, z)
Magnetic field,H
-
0
/
r
X
o
4 x
Wire c arrying current, I,
in +y direction
Figure B-2: Magnetic field about a point wire with current, I, in the +y direction.
z
(x, z)
(X zo)
LulW
Figure B-3: Magnetic field about a wire with width, W, and height, H. It is equivalent to the
summation of the magnetic fields of infinitissimal pieces at (xo, zo) with width dx and height dz over
the cross-sectional area.
113
500
a)
450
400
E
0
,s350
N
300
50
250
-100
200
-2
500
-1.5
-1
0
-0.5
0.5
1
1.5
2
H in[Oe]
b)
450
400
E
'S350
N
300
250
200
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-1.5
-1
-0.5
0
x [um]
0.5
1
1.5
2
500
450
400
E
, 350
N
300
250
200
-2
Figure B-4: The vertical components of a)H, b)VH, and c)V 2 H as a function of lateral (x) and
vertical (z) distance from a 2 pm-wide wire carrying 50 mA current
114
APPENDIX B. MAGNETIC FIELD OVER A WIRE
Appendix C
Matlab Code
C.1
DC Current Imaging
C.1.1
DCcurrent.m
% DCcurrent.mM
% calculates magnitude and phase response to DC currentM
% y represents cantilever deflectionM
clear allM
syms q yo M x-cant k M
syms x z x
zO IcM
% transfer function U M
10
matrixA = [-cos(q)-cosh(q) -sin(q)-sinh(q);
...
.
M*(sin(q)-sinh(q))-cos(q)+cosh(q) M*(-cos(q)-cosh(q))-sin(q)+sinh(q)];M
matrixB = [yo/2*(cos(q)-cosh(q)); yo/2*(cos(q)+cosh(q)-M*(sin(q)+sinh(q)))];
U
= simple(matrixA -1*matrixB);M
y = yo/2*(cos(k*x-cant)+cosh(k*x-cant))
U(1)*(cos(k*x-cant)-cosh(k*x-cant))
+ ... M
+ U(2)*(sin(k*x-cant)-sinh(k*x.cant));I
del-y = diff(y,x-cant,1);M
M4
115
20
APPENDIX C. MATLAB CODE
116
E = 1.69e11;
rho = 2330;
a = 28.9e-6;
b = 3.37e-6;
L = 227.le-6;
% Young's modulus [N/m^2]M
x-cant
% setting to y(x-cant=L) and dy(x-cant=L)/dx-cantM
% amplitude of bimorph driveM
yo
= L;
10e-9;
A = a*b;
I = a*b^3/12;
% density [kg/rn3]M
% width of beam [m]M
% thickness of cantilever [m]A
% length of cantilever [m]k
% cross section area of cantileverM
% moment of inertiaM
%##################################################################
% solve for H fieldA
M
total current sent through wireM
Ic = .001;
width of wire sampleM
W = 2e-6;
height of wire sampleM
H = 200e-9;
magnetic dipole moment of tip in x M
m-x = 0;
m-z =
30
0;
40
magnetic dipole moment of tip in z [A*m^2] A!
m-q = le-8;
% magnetic monopole moment [Am]M
u-0 = 4*pi*le-7;
Q = 180;
kc = 3*E*I/L-3
%permeability of free spaceM
quality factorM
effective spring constant of cantilever [N/m];M
% function for z component of H field at one singular pointM
func = Ic/(2*pi*W*H) * -(x-x0)/((x-x0)^2 + (z-z0)^2);M
50
% H field summed over width and heightA!
Hz = int( int(func, x, -W/2, W/2), z, -H, 0);M
% first derivative of Hz to find F-mag = m-z * dHz/dzM
dHz = diff(Hz, z, 1); M
ddHz
=
u-0*(m-z*dHz+m-q*Hz); M
Fmag
kmag
diff(Hz, z, 2);M
=
diff(Fmag, z, 1);
% kmag
60
=
dFmag/dz evaluated at z-eqM
points = 5001;M
startx = -5*W;M
endx = -startx;M
x = [startx: (endx-startx)/points:endx];M
C.1.
DC CURRENT IMAGING
heightabove = 800e-9;
z = zeros(1,points);M
117
M
70
counter = 1;M
for dummy = startx:(endx-startx)/points:-W/2M
z(counter) = heightabove-H;M
counter = counter + 1; M
endM
for dummy = -W/2:(endx-startx)/points:W/2K4
z(counter) = heightabove;M
counter = counter + 1;M
endM
80
for dummy = W/2: (endx-startx)/points:endxM
z(counter) = heightabove-H;M
counter = counter + 1;M
endM
z = z(1:length(x));M
num-Hz = eval(vectorize(Hz));M
num-kmag = eval(vectorize(kmag));M
90
% for Point Mass ModelM
num-ddHz = eval(vectorize(ddHz));M
phi = Q/kc*m-z.*num-ddHz;M
%##############################################################A
wO = (1.8751041)^2/L^2*srt(E*b^2/12/rho)*.5*(sqrt(4-1/Q^2));M
% natural resonance of fundamental mode M
fO = wO/2/pi M
% [Hz]M
100
w = wO;m
% driving at first resonant freq.M
w-k = 2*w/(sqrt(4-1/Q^2)+j/Q);M
damping = w-k/Q;M
'(4
k = sqrt(w-k*sqrt(rho*A/E/I));4
M
q
110
= k*L;M
M = kc*q^3/3./num-kmag;M
APPENDIX C. MATLAB CODE
118
num-y = eval(vectorize(y));M
num-deLy = eval(vectorize(de_y));
mass = kc/wO^2;M
PM = 4.9960e+003./ (w0^2-w^2+num_kmag/mass + j*w*wO/Q);M
120
% Plotting of FiguresM
figureM
superplot(x*1e6,180/pi*(angle(numy) -angle(num-y(1))))M
ylabel('Phase [deg]')M
xlabel('x [um]')M
AC Current Imaging
C.2
ACcurrent.m
C.2.1
% ACcurrent.mM
% calculates magnitude and phase response to AC currentM
%
z-ca represents cantilever deflectionM
4
clear all
% transfer function UM
K4
syms k-ca L dcF E I x-cantM
10
K4
-sin(kca*L)-sinh(kca*L); ... M
matrixA = [-cos(kca*L)-cosh(kca*L)
(sin(kca*L)-sinh(kca*L)) (-cos(k_ca*L)-cosh(k-ca*L))];M
matrix-B = [0; dcF/E/I/k_ca^3];M
U = (matrixA^-1*matrixB);M
4
z-ca =
U(1)*(cos(k-ca*x-cant)-cosh(k-ca*x-cant))
+ ...
4
U(2)*(sin(k-ca*x-cant)-sinh(k-ca*x-cant));M
M
20
% cantilever parametersM
E = 1.69e11;
rho = 2330;
% Young's modulus [N/m2]M
% density [kg/m^3]M
C.2. AC CURRENT IMAGING
a
b
=
L
=
28.9e-6;
3.37e-6;
227.le-6;
119
% width of beam [m]M
% thickness of cantilever [m]k
% length of cantilever [m]M
x-cant = L;
K4
30
A = a*b;
I = a*b^3/12;
% cross section area of cantileverM
% moment of inertiaM
4
% wire parameters and magnetic fieldM
4
syms x z x
zA M
Ic
= .002;
W = 2e-6;
H
200e-9;
m.x = 0;
m-z = 0;
q = 2.8e-5;
% total current sent through wireM
% width of wire sampleM
% height of wire sampleM
40
% magnetic moment of tip in x M
% magnetic moment of tip in z M
% magnetic monopole momentM
u-0 = pi*4e-7;M
Q = 180;
kc
3*E*I/L^3;
% quality factorM
% effective spring constant of cantilever [N/m];M
M4
% function for z component of H field at one singular pointM
func = Ic/(2*pi*W*H) * -(x-x0)/((x-x0)^2 + (z-z0)^2);M4
50
% H field summed over width and heightM
Hz = int( int(func, x,
-W/2, W/2), zO, -H,0);M
% first derivative of Hz to find F-mag
dHz = diff(Hz, z, 1); M
=
m-z
*
dHz/dzM
4
Fmag
4
kmag
u_0*(m-z*dHz+q*Hz); M
=
diff(Fmag, z, 1);K4
60
K4
points = 2002;M
startx
-12.5*W;M
endx = -startx;m
K4
x = [startx: (endx-startx) /points:endx] ;4
4
heightabove = 8400e-9;
z = zeros(1,points);-
M
APPENDIX C. MATLAB CODE
120
70
counter = 1;M
for dummy = startx: (endx-startx)/points: -W/2M
z(counter) = heightabove-H;M
counter = counter + 1;
M
endM
for dummy = -W/2:(endx-startx)/points:W/2M
z(counter) = heightabove;M
counter = counter + 1;K
endM
M
80
for dummy = W/2:(endx-startx)/points:endxM
z(counter) = heightabove-H;M
counter = counter + 1;K
endM
z = z(1:length(x));M
numHz = eval(vectorize(Hz));M
% case of AC currentM
90
dcF = eval(vectorize(Fmag));M
%##################################################################$t
% drive frequencyM
t =
0;M
wO = (1.8751041)^2/L^2*srt(E*b^2/12/rho)*.5*(sqrt(4-1/Q^2));M
% natural resonance of fundamental mode M
wc
=
wO;M
100
fc = wc/2/piM
w-ca
=
2*wc/(sqrt(4-1/Q^2)+j/Q);M
k-ca = sqrt(w-ca*sqrt(rho*A/E/I));M
num-z-ca = eval(vectorize(z-ca));K
%##################################################################$
% plotsM
M
figureno = 2M
if figureno == 1M
figureM
110
C.2. AC CURRENT IMAGING
121
subplot(2,1,1)M
splot(x*1e6, abs(num-z-ca)*1e9);M
xlabel('x [um]')M
ylabel('Magnitude [nm]');M
subplot(2,1,2)M
splot(x*1e6, angle(num-z-ca)*180/pi);M
xlabel('x [um]')M
ylabel('Phase
120
[deg]');M
M
elseif figureno == 2M
newzca(1:points/2) = -abs(num-z-ca(:points/2))*e9-Q
newzca(points/2+1:points+1) = abs(num-z ca(points/2+1:points+1))*1e9;M
splot(x*1e6,newzca/1000,
'--')M
axis tightM
endM
C.2.2
decay.m
% decay.mM
% shows decay of Hz and dHzM
% called by decay-use.mM
function [numHz, num-dHz] = decay(num-z)M
E = 1.69e11;
rho = 2330;
a = 28.9e-6;
b = 3.37e-6;
L = 227.le-6;
A = a*b;
I = a*b^3/12;
I
.002;
W = 2e-6;
H = 200e-9;
Q = 180;
kc = 3*E*I/L^3;
%
%
%
%
Young's modulus [N/m^2]M
density [kg/rn ^3]M
width of beam [m]M
thickness of cantilever [m]M
10
% length of cantilever [m][
% cross section area of cantileverM
% moment of inertiaM
% total current sent through wireM
% width of wire sampleM
% height of wire sampleM
% quality factorM
% effective spring constant of cantilever [N/m];M
20
x
= sym('x');M
z = sym('z');4
x = sym('xO');M
zA = sym('zO');M
4
% function for z component of H field at one singular pointM
APPENDIX C. MATLAB CODE
122
func = I/(2*pi*W*H) * -(x-xO)/((x-xO)^2
+ (z-z0)^2);M
% H field summed over width and heightM
Hz = int( int(func, x,
-W/2, W/2), zA, -H, 0);M
30
% second derivative of Hz to find F-mag = m-z*ddHzM
ddHz = diff(Hz, 'z', 2);M
dHz = diff(Hz, 'z', 1);K
startx = -5*W;M
endx = -startx;M
points = 599;M
40
x = [startx:(endx-startx)/points:endx];M
z = num_z;M
numHz = eval(vectorize(Hz));M
num-dHz = eval(vectorize(dHz));M
num-ddHz = eval(vectorize(ddHz));M
C.2.3
decayuse.m
% decay.mM
% shows decay of Hz and dHzM
% called by decay-use.mM
function [num-Hz, num-dHz] = decay(num-z)M
E = 1.69e11;
rho = 2330;
a = 28.9e-6;
b = 3.37e-6;
L = 227.le-6;
A = a*b;
I = a*b^3/12;
Young's modulus [N/m^2]M
density [kg/m^3]1
width of beam [m]M
thickness of cantilever [m]M
length of cantilever [m]k
I
total current sent through wireM
width of wire sampleM
= .002;
W = 2e-6;
H = 200e-9;
Q = 180;
kc = 3*E*I/L^3;
10
cross section area of cantileverM
moment of inertiaM
height of wire sampleM
quality factorM
effective spring constant of cantilever [N/m];M
20
x
=
sym('x');M
C.3. NON-LINEAR CURRENT IMAGING
123
z = sym('z');M
x = sym('xO');M
zO = sym('zO');M
% function for z component of H field at one singular pointM
func = I/(2*pi*W*H) * -(x-xO)/((x-xO)^2 + (z-zO)2);M
% H field summed over width and heightM
Hz = int( int(func, x, -W/2, W/2), zO, -H, 0);M
I4
30
% second derivative of Hz to find F-mag = m-z*ddHzM
ddHz = diff(Hz,
dHz = diff(Hz,
'z', 2);M
'z', 1);M
I4
startx = -5*W;M
endx = -startx;m
points = 599;K
M
40
x = [startx:(endx-startx)/points:endx];
z = num-z;M
numHz = eval(vectorize(Hz));M
numdHz = eval(vectorize(dHz));M
num-ddHz = eval(vectorize(ddHz));M
use.tex
C.3
Non-linear Current Imaging
C.3.1
degenerate.m
% degenerate.mM
% calculates magnitude and phase response to degenerate non-linearM
clear allM
% cantilever parametersM
K4
E = 1.69e11;
% Young's modulus [N/m-2]M
rho = 2330;
% density [kg/m3]M
a = 28.9e-6;
% width of beam [m]M
b = 3.37e-6;
% thickness of cantilever [m]M
L = 227.le-6;
% length of cantilever [m]m
10
APPENDIX C. MATLAB CODE
124
x-cant = L;
% setting to y(x-cant=L) and dy(x- cant=L)/dx-cantM
yo
% amplitude of bimorph driveM
Oe-9;
A = a*b;
I
a*b^3/12;
M~
% cross section area of cantileverM
% moment of inertiaM
20
% wire parameters and magnetic fieldM
syms x z x
zA M
Ic = 0.002;
W = 2e-6;
H = 200e-9;
% total current sent through wireM
% width of wire sampleM
mx
% magnetic dipole moment of tip in x M
% magnetic dipole moment of tip in z M
= 0;
m-z = 0;
% height of wire sampleM
% magnetic monopole moment M
q = 8e-6;
u-0 = pi*4e-7;M
% quality factorM
Q = 180;
kc = 3*E*I/L-3;
% effective spring constant of cantilever [N/m];M
K4I
% function for z component of H field at one singular pointM
func = Ic/(2*pi*W*H) * -(x-x0)/((x-x0)^2 + (z-z0)^2);M
% H field summed over width and heightM
Hz = int( int(func, x, -W/2, W/2), zA, -H,0);M
M
% first derivative of Hz to find Fmag = m-z * dHz/dzM
dHz = diff(Hz, z, 1); M
30
40
u0*(m-z*dHz+q*Hz); M
Fmag
=
kmag
= diff(Fmag, z, 1);
% kmag = dFmag/dz evaluated at z-eqM
points = 5001;K
endx = 10*W;M
startx = -endx;M
50
x = [startx: (endx-startx)/points:endx];M
heightabove = 1000e-9;
z = zeros(1,length(x));M
counter = 1;M
M
C.3. NON-LINEAR CURRENT IMAGING
125
for dummy = startx:(endx-startx)/points:-W/2M
z(counter) = heightabove-H;M
counter = counter + 1;
endM
M
60
for dummy = -W/2:(endx-startx)/points:W/2M
z(counter) = heightabove;M
counter = counter + 1;K
endM
for dummy = W/2: (endx-startx)/points:endxM
z(counter) = heightabove-H;M
counter = counter + 1;M
70
endM
z = z(1:length(x));M
numHz = eval(vectorize(Hz));M
A
t
ACcurrent = 1;M
if ACcurrent ==IM
% case of AC currentM
sO = zeros(1,length(x));M
80
si = eval(vectorize(kmag));M
dcF = eval(vectorize(Fmag));M
elseM
% case of DC currentM
sO = eval(vectorize(kmag));M
si = zeros(1,length(x));K
dcF = zeros(1,length(x));M
endM
90
phi = O;M
MO
Ml
cMl
sO/(E*I);M
sl*exp(j*phi)/(2*E*I);M
sl*exp(-j*phi)/(2*E*I);M
%##################################################################A
% drive
frequencyM
t = 0;M
100
%offset = 186.44*2*pi;
% for magnitudeM
offset
% for phaseM
=
0;
APPENDIX C. MATLAB CODE
126
wO = (1.8751041)^2/L^2*sqrt(E*b^2/12/rho)*.5*(sqrt(4-1/Q^2));M
% natural resonance of fundamental mode M
w = wO-offset;M
wpl = 2*w/(sqrt(4-1/Q^2)+j/Q);M
w-nl = 2*w/(sqrt(4-1/Q^2)-j/Q);M
w-p2 = 4*w/(sqrt(4-1/Q^2)+j/Q);M
w-n2 = 4*w/(sqrt(4-1/Q^2)-j/Q);M
k-pl
k-nl
k-p2
k-n2
110
= sqrt(w-pl*sqrt(rho*A/E/I));M
= sqrt(w-nl*sqrt(rho*A/E/I));M
= sqrt(w-p2*sqrt(rho*A/E/I));M
= sqrt(w-n2*sqrt(rho*A/E/I));M
y0 = zeros(1,length(x)); M
y-pl = zeros(1,length(x));M
y-nl = zeros(1,length(x));M
y-p2
zeros(1,length(x));M
y-n2 = zeros(1,length(x));M
dr
g
zeros(1,length(x));M
dely-p =
delypl
zeros(1,length(x));M
delynl = zeros(1,length(x));M
dely-p2 = zeros(1,length(x));M
dely-n2 = zeros (1,lengt h(x));M
for i = 1:length(x)M
a3 a4 b2 b4 c2 c4 d2 d4 f2 f4A
%
0
0;K
0
0
0
0
0
matrixA = [2 6*L 0
0 ... K
-sin(k-nl*L)-sinh(k-nl*L)
0 -cos(k-nl*L)-cosh(k-n1*L)
0
0
0
0
0
0;K
0 -cos(k-p1*L)-cosh(k.p1*L) -sin(k.p1*L)-sinh(k-pl*L) ... M
0
0
0
0
0
0
0;K
0 ... M
0
0
0
0
0
-cos(k-n2*L) -cosh(k-n2*L) -sin(k-n2*L)-sinh(k-n2*L) 0 0;M
0 ... M
0
0
0 0
0
0
0
-cos(k-p2*L)-cosh(k-p2*L) -sin(k-p2*L)-sinh(k-p2*L);M
M0(i)*L^2
M0(i)*L^3-6 ... M
M1(i)*(cos(kLnl*L)-cosh(k-nl*L))...M
M1(i)*(sin(k-nl*L)-sinh(k-nl*L)) ...A
cMl(i)*(cos(k-pl*L)-cosh(k-pl*L)) ...M
cMl(i)*(sin(kLpl*L)-sinh(k-pl*L)) ...
0
0
0
0;K
120
130
140
C.3. NON-LINEAR CURRENT IMAGING
cM(i)/k-n1^3*L^2 cMl(i)/k-n1^3*L^3...M
MO(i)/k-n1^3*(cos(k-nl*L)-cosh(k-nl*L))-sin(k-n1*L)+sinh(k-nl*L).. .M
MO(i)/k-n1^3*(sin(k-nl*L)-sinh(k-nl*L))+cos(k-nl*L)+cosh(k-nl*L)...M
o o ... M
M1(i)/k-n1^3*(cos(k-n2*L)-cosh(k-n2*L))...M
M1(i)/kn1^3*(sin(k-n2*L)-sinh(k-n2*L)).. .M
o
0
0
127
150
0;m
cMl(i)/k-n2^3*(cos(k-nl*L)-cosh(k-nl*L)) ... M
cMl(i)/k-n2^3*(sin(k-nl*L)-sinh(k-n1*L)) ... M
o
0
... M
MO(i)/k-n2^3*(cos(k-n2*L)--cosh(k-n2*L))-sin(k-n2*L)+sinh(k-n2*L)...M
MO(i)/k-n2^3*(sin(k-n2*L) -sinh(k-n2*L))+cos(k-n2*L)+cosh(k-n2*L). ..
0 0;K
M1(i)/k-pl^3*L^2 M1(i)/k-p1^3*L^3...D
0
160
... m
0
MO(i)/k-p1^3*(cos(k-pl*L)-cosh(k-pl*L))-sin(k-pl*L)+sinh(k.pl*L).. .m
MO(i)/k-p1^3*(sin(k-p1*L)-sinh(k-p1*L))+cos(k-p1*L)+cosh(k-p1*L)...M
0
0
0
0
...m
cMl(i)/k-p1^3*(cos(k-p2*L)-cosh(k-p2*L))...M
cMl(i)/k-pl^3*(sin(k-p2*L)-sinh(k-p2*L));M
0
0 M1(i)/k-p2^3*(cos(k-p1*L)-cosh(k-p1*L)).. .M
M1(i)/k-p2^3*(sin(k-pl*L)-sinh(k-pl*L))
0 0 ... M
MO(i)/k-p2^3*(cos(k-p2*L)-cosh(k-p2*L))-sin(k-p2*L)+sinh(k-p2*L)...M
MO(i)/k-p2^3*(sin(k-p2*L)-sinh(k-p2*L))+cos(k-p2*L)+cosh(k-p2*L)M
170
];m
matrix-B = [O;M
yo/4*(cos(k-nl*L)-cosh(k-n1*L));M
yo/4*(cos(k-pl*L)-cosh(k-pl*L));M
o;M
0; K
(-Ml(i)*yo/4*(cos(k-nl*L)+cosh(k-nl*L)) - ... M
cMl(i)*yo/4*(cos(k-p1*L)+cosh(k-pl*L)));M
yo/4*(sin(knl*L)+sinh(k-nl*L) - .M.S
MO(i)/k-nl^3*(cos(k-nl*L)+cosh(k-nl*L)))-...4
dcF(i)/2/E/I/k-n1^3*exp(-j*phi);
M
-cMi(i)/k-n2^3*yo/4*(cos(k-nl*L)+cosh(k-nl*L));M
yo/4*(sin(k-pl*L)+sinh(k-p1*L) - ... n
MO(i)/k-p1^3*(cos(k-pl*L)+cosh(k-pl*L)))-...M
dcF(i)/2/E/I/k-nl^3*exp(j*phi);M
-Ml(i)/k-p2^3*yo/4*(cos(k-pl*L)+cosh(k-pl*L))M
];m
M
U = (matrix-A-1*matrix-B);M
M4
180
190
APPENDIX C. MATLAB CODE
128
y_0(i)
U(1)*x-cant^2 + U(2)*xcant^3;M
=
y-pl(i) = (yo/4*(cos(k-pl*x-cant)+cosh(k-pl*x-cant)) +
U(5)*(cos(k-pl*x-cant)-cosh(k-pl*x-cant))
+
.
.M
U(6)*(sin(k-pl*x-cant)-sinh(k-pl*x-cant)));M
y-nl(i) = (yo/4*(cos(k-nl*x-cant)+cosh(k-nl*x-cant)) +
U(3)*(cos(k.nl*x-cant)-cosh(k-nl*x-cant))
.
+ .M..
U(4)*(sin(k-nl*x-cant)-sinh(k-nl*x-cant))) ;M
yp2(i) = (U(9)*(cos(k-p2*x-cant)-cosh(k-p2*x-cant)) + ... M
U(10)*(sin(k-p2*x-cant)--sinh(k-p2*x..cant)));M
y-n2(i) = (U(7)*(cos(k-n2*x-cant)-cosh(k-n2*x-cant)) + .M
U(8)*(sin(k-n2*x-cant) -sinh(kn2*xcant)));M
200
delyO(i)
dely-pl(i)
=
U(1)*2*x-cant + U(2)*3*x cant^2;M
k-pl*(yo/4*(-sin(k-pl*x-cant)+sinh(kpl*x-cant)) + .M
U(5) *(-sin(k-p1*x.cant)-sinh(k-pl*x.cant)) + ... M
dely.nl(i)
=
knl*(yo/4*(-sin(k-nl*x-cant)+sinh(k-nl*x-cant)) + ... 1
U(3) *(-sin(k-nl*x.cant)-sinh(k-nl*x-cant)) + .. .M
U(6)*(cos(k-pl*x-cant)-cosh(k-p1*x-cant)));M
210
U(4)*(cos(k-nl*x-cant)-cosh(k-nl*x-cant)));M
dely-p2(i)
=
k-p2*(U(9)*(-sin(k-p2*x-cant)-sinh(k-p2*x-cant))
+ .M.S
U(10)*(cos(k-p2*x-cant)-cosh(k-p2*x-cant)));M
dely-n2(i)
=
k-n2*(U(7)*(-sin(k-n2*x-cant)-sinh(k-n2*x-cant))
+ .M.S
U(8) *(cos(k-n2*xcant)-cosh(k-n2*x-cant)));M
endM
%##################################################################$
% plotsM
220
if ACcurrent == 1M
figureM
subplot(2,1,1)M
plot(x*1e6, 2*abs(yp1)*1e9);M
xlabel('x [um]')M
ylabel('Magnitude [nm]');M
subplot(2,1,2)M
plot(x*1e6, angle(y-pl)*180/pi);M
xlabel('x [um]')M
ylabel('Phase [deg] ');M
figureM
subplot(2,1,1)M
plot(x*1e6, 2*abs(yp2)*1e9);M
xlabel('x [um]')N
ylabel('Magnitude [nm]');M
subplot(2,1,2)M
230
C.3. NON-LINEAR CURRENT IMAGING
129
plot (x*1e6, angle(y _p2)*180/pi);M
xlabel('x [umx]')M
ylabel('Phase [deg] ');M
240
elseM
figureM
subplot(2,1,1)M
plot(x*1e6, 2*abs(y-pl)*1e9);M
xlabel('x [um]')M
ylabel('Magnitude [nim] ');M
subplot(2,1,2)K
plot(x*1e6, 180/pi*(angle(y-pl)-angle(y-pl(1))));M
xlabel('x [um]')M
250
ylabel('Phase [deg] ');M
endM
C.3.2
nondegenerate.m
% nondegenerate.mM
% calculates cantilever response for wp \= wcM
clear allM
M
E = 1.69e11;
rho = 2330;
a = 28.9e-6;
b = 3.37e-6;
L = 227.le-6;
A
I
12e-9;
=
a*b;
a*b^3/12;
syms x z x
% density [kg/m 3]M
% width of beam [m]M
% thickness of cantilever [m]?[
% length of cantilever [m]M
10
'(
% amplitude of bimorph driveM
x-cant = L;
yo
% Young's modulus [N/rm2]M
cross section area of cantileverM
% moment of inertiaM
%
zO M
20
Ic
= .001;
W = 2e-6;
H = 200e-9;
m-x =
0;
% total current sent through wireM
% width of wire sampleM
% height of wire sampleM
% magnetic moment of tip in x M
APPENDIX C. MATLAB CODE
130
m-z = 0;
q = le-5;
u-0 = pi*4e-7;M
Q 180;
% magnetic moment of tip in z M
kc = 3*E*I/L^3;
% effective spring constant of cantilever [N/m];M
% magnetic monopole moment(I
% quality factorM
M
% function for z component of H field at one singular pointM
func = Ic/(2*pi*W*H) * -(x-x0)/((x-x0)^2 + (z-z0)^2);M
30
% H field summed over width and heightM
Hz = int( int(func, x,
-W/2, W/2), z,
-H,0);M
% first derivative of Hz to find F-mag = m-z * dHz/dzM
dHz = diff(Hz, z, 1); M
40
Fmag = u-0*(m-z*dHz+q*Hz); M
kmag = diff(Fmag, z, 1);
% kmag
= dFmag/dz evaluated at
z-eqM
points = 2001;M
startx = -12.5*W;M
endx = -startx;M
x = [startx: (endx-startx)/points:endx];M
50
heightabove = 10OOe-9; M
z = zeros(1,points);M
counter = 1;M
for dummy = startx:(endx-startx)/points:-W/2M
z(counter) = heightabove-H;M
counter = counter + 1;
M
endM
for dummy = -W/2:(endx-startx)/points:W/2M
z(counter) = heightabove;M
counter = counter + 1;M
endM
for dummy = W/2:(endx-startx)/points:endxM
z(counter) = heightabove-H;M
counter = counter + 1;M
endM
z = z(1:length(x));M
60
C.3. NON-LINEAR CURRENT IMAGING
num-Hz = eval(vectorize(Hz));M
131
70
phi = 0;
si = eval(vectorize(kmag));M
dcF = eval(vectorize(Fmag));M
M1
M1
cMl
=
sl*exp(j*phi)/(2*E*I);M
sl*exp(-j*phi)/(2*E*I);M4
%##################################################################A$
% drive frequencyM
t = 0;K
offset = 186.44*2*pi;
% for magnitudeM
%offset = 0;
% for phaseM
80
wO = (1.8751041)^2/L^2*sqrt(E*b^2/12/rho)*.5*(sqrt(4-1/Q^2));M
% natural resonance of fundamental mode M
fO = wO/2/pi M
% [Hz]M
90
w = wO;M
choice = 2;M
if choice == 1M
wp = 50000*2*pi;M
wc = w-wp;M
elseif choice == 2M
100
wc = 100*2*pi;M
wp = w-wc;M
elseM
wp = 1;M
wc = w;M
endM
fp = wp/2/pi;
fc = wc/2/pi;M
M
w-ca
w-cb
w.cc
w-cd
110
= 2*wc/(sqrt(4-1/Q^2)+j/Q);M
= 2*wc/(sqrt(4-1/Q^2)-j/Q);M
= 4*wc/(sqrt(4-1/Q^2)+j/Q);M
= 4*wc/(sqrt(4-1/Q^2)-j/Q);M
APPENDIX C. MATLAB CODE
132
w/Q
w-pa = 2*wp/(sqrt(4-1/Q^2)-j/Q);M
w-pb = 2*wp/(sqrt(4-1/Q2)-j/Q);M
w-cpa = 2*(wc~wp)/(sqrt(4-1/Q-2)+j/Q);MI
w-cpd = 2*(wc+wp)/(sqrt(4-1/Q^2)-j/Q);M
120
if wp > wcM
w-cpb = 2*abs(wc-wp)/(sqrt(4-1/Q^2)-j/Q);M
w-cpc = 2*abs(-wc+wp)/(sqrt(4-1/Q^2)+j/Q); M
elseM
w-cpb = 2*abs(wc-wp)/(sqrt(4-1/Q^2)+j/Q);M
w-cpc = 2*abs(-wc+wp)/(sqrt(4-1/Q^2)-j/Q); M
endM
k-ca = sqrt(w-ca*sqrt(rho*A/E/I));M4
130
k-cb = sqrt(w-cb*sqrt(rho*A/E/I));M
k-cc = sqrt(w-cc*sqrt(rho*A/E/I));M
k-cd = sqrt(w-cd*sqrt(rho*A/E/I));M
k-pa = sqrt(w-pa*sqrt(rho*A/E/I));M
k-pb = sqrt(w-pb*sqrt(rho*A/E/I));M
kcpa = sqrt(wcpa*sqrt(rho*A/E/I));M
k-cpb = sqrt(w-cpb*sqrt(rho*A/E/I));M
k-cpc = sqrt(w-cpc*sqrt(rho*A/E/I));M
k-cpd = sqrt(w-cpd*sqrt(rho*A/E/I));M
z-dc
z-ca
= zeros(1,length(x));
140
M
zeros(1,length(x));K
z-cb = zeros(1,length(x));K
z-cc = zeros(1,length(x));M
z-cd
zeros(1,length(x));M
z-pa = zeros(1,length(x));M
z-pb = zeros(1,length(x));M
z
cpa
=
zeros(1,length(x));M
z-cpb = zeros(1,length(x));M
z-cpc = zeros(1,length(x));M
z-cpd = zeros(1,length(x));M
for i = 1:length(x)M
m
150
C.3. NON-LINEAR CURRENT IMAGING
133
M
160
matrix-A = [ see Eqn. 4.11
];M
matrix.B = [O;M
O;M
Q;M
O;M
0;M
yo/4*(cos(k pa*L)-cosh(kpa*L));M
yo/4*(cos(k-pb*L)-cosh(k-pb*L));M
0;m
170
0;m
0;M
0;M
0;m
-dcF(i)/2/E/I*exp(j*phi);M
-dcF(i)/2/E/I*exp(-j*phi);M
; M
; M
k -pa^3*yo/4*(sin(k-pa*L)+sinh(k -pa*L));M
k-pb^3*yo/4*(sin(k-pb*L)+sinh(k-pb*L)) ;M
-M1(i)*yo/4*(cos(kpa*L)+cosh(k-pa*L));M
-Ml(i)*yo/4*(cos(k-pb*L)+cosh(k-pb*L));M
-cMI(i)*yo/4*(cos(k-pa*L)+cosh(k.pa*L));M
-cMi(i)*yo/4*(cos(k-pb*L)+cosh(k-pb*L))I
180
];'
U = (matrixA^-1*matrix-B);M
z-dc(i)
=
U(1)*x-cant^2 + U(2)*x-cant^3;M
+
... M
z-cb(i) = U(5)*(cos(k-cb*x-cant)-cosh(k-cb*x-cant)) +
U(6)*(sin(k-cb*x-cant) -sinh(k-cb*x-cant));M
... K
z-cc(i) =
+
... M
+
...
z-ca(i) =
U(3)*(cos(k-ca*x-cant)-cosh(k-ca*x-cant))
190
U(4)*(sin(k-ca*x-cant) -sinh(k-ca*x-cant));M
U(7)*(cos(k-cc*x-cant)-cosh(k-cc*x-cant))
U(8)*(sin(k-cc*x-cant) -sinh(k -cc*x-cant));M
z-cd(i) =
U(9)*(cos(k-cd*x-cant)-cosh(k-cd*x-cant))
U(10)*(sin(k-cd*x-cant) -sinh(k
z-pa(i) =
cd*x-cant));M
yo/4*(cos(k-pa*x-cant)+cosh(k-pa*x-cant))
U(11)*(cos(k-pa*x-cant)-cosh(k-pa*x-cant))
+
+ ... K
... M
200
U(12)*(sin(k-pa*x-cant) -sinh(k.pa*x-cant));M
z-pb(i) =
yo/4*(cos(k-pb*x-cant)+cosh(k-pb*x-cant))
U(13)*(cos(k-pb*x-cant)-cosh(k-pb*x-cant))
+
U(14)*(sin(k-pb*x-cant)-sinh(k-pb*x-cant));M
..
+ ...
n
APPENDIX C. MATLAB CODE
134
z-cpa(i) = U(15)*(cos(k-cpa*x-cant)-cosh(k-cpa*x-cant))
U(16)*(sin(k-cpa*x-cant) -sinh(k-cpa*x-cant));M
z-cpb(i) = U(17)*(cos(k-cpb*x-cant)-cosh(k-cpb*x-cant))
U(18)*(sin(k-cpb*x-cant) -sinh(k-cpb*x-cant));M
z-cpc(i) = U(19)*(cos(k-cpc*x-cant)-cosh(k-cpc*x-cant))
U(20)*(sin(kLcpc*x-cant) -sinh(k-cpc*x-cant));M
z-cpd(i) = U(21)*(cos(k-cpd*x-cant)-cosh(k-cpd*x-cant))
U(22)*(sin(k-cpd*x-cant) -sinh(k-cpd*x-cant));M
+
+
+
+
..
...
...
...
M
M
210
M
endM
%##################################################################$
% plotsM
figureno = 1;M
220
if figureno == IM
%figureM
if choice == 1M
subplot(312)K
splot(x*1e6, 2*abs(z-cpa)*1e9);M
xlabel('x [um]')M
ylabel('Magnitude [nm] ');M
axis([-15, 15, 0, max(2*abs(z-cpa)*1e9)])M
m
230
elseif choice == 2M
subplot(313)M
splot(x*1e6, 2*abs(z-cpa)*1e9);M
xlabel('x [um]')M
ylabel('Magnitude [nm] ');M
axis([-15, 15, 0, max(2*abs(z-cpa)*1e9)])M
elseM
subplot(311)M
splot(x*1e6, 2*abs(z.ca)*1e9);M
xlabel('x [um]')M
ylabel('Magnitude [run] ');M
axis([-15, 15, 0, max(2*abs(z-ca)*1e9)]))
endM
elseif figureno == 3M
figureM
subplot(2,3,1)M
plot(x*1e6, abs(z.dc)*1e9);M
240
C.3. NON-LINEAR CURRENT IMAGING
xlabel('x [uml')N
ylabel('Magnitude [rnm]');M
135
250
subplot(2,3,4)M
plot(x*1e6, angle(z-dc)*180/pi);M
xlabel('x [um]')M
ylabel('Phase
[deg] ');K
subplot(2,3,2)M
plot(x*1e6, 2*abs(z-ca)*1e9);M
xlabel('x [um]')M
ylabel('Magnitude [run] ');M
subplot(2,3,5)M
plot(x*1e6, angle(z-ca)*180/pi);M
xlabel('x [um]')N
ylabel('Phase
260
[deg] '-);4
subplot(2,3,3)M
plot(x*1e6, 2*abs(z-cc)*1e9);P
xlabel('x [um]')N
ylabel('Magnitude [un]');M
subplot(2,3,6)K
plot(x*1e6, angle(z-cc)*180/pi);M
xlabel('x [um]')M
270
ylabel('Phase [deg] ');K
elseif figureno == 4M
I4
figureM
subplot(4,4,1)K
plot(x*1e6, abs(z-dc));M
subplot (4,4,5)M
plot(x*1e6, abs(z-ca));M
280
subplot (4,4,6)M
plot(x*1e6, abs(z-cb));M
subplot(4,4,7)K
plot(x*1e6, abs(z-cc));KI
subplot(4,4,8)K
plot(x*1e6, abs(z-cd));M
subplot(4,4,9)M
plot(x*1e6, abs(z-pa));1
subplot(4,4,10)M
plot(x*1e6, abs(z-pb));M
290
APPENDIX C. MATLAB CODE
136
subplot(4,4,13)K
plot(x*1e6, abs(z-cpa));M
subplot(4,4,14)M
plot(x*1e6, abs(z-cpb));M
subplot(4,4,15)K
plot(x*1e6, abs(z-cpc));M
300
subplot(4,4,16)K
plot(x*1e6, abs(z-cpd));M
figureM
subplot(4,4,1)K
plot(x*1e6, angle(z-dc));M
subplot(4,4,5)M
plot(x*1e6, angle(z-ca))M
subplot(4,4,6)K
plot(x*1e6, angle(z-cb));M
310
subplot(4,4,7)M
plot(x*1e6, angle(z-cc));M
subplot(4,4,8)K
plot(x*1e6, angle(z-cd));M
subplot(4,4,9)K
plot(x*1e6, angle(z-pa));M
subplot(4,4,10)M
320
plot(x*1e6, angle(z-pb));M
subplot(4,4,13)M
plot(x*1e6, angle(z-cpa));M
subplot(4,4,14)K
plot(x*1e6, angle(z-cpb));M
subplot(4,4,15)M
plot(x*1e6, angle(z-cpc));subplot(4,4,16)M
plot(x*1e6, angle(z-cpd));M
e4
end
330
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