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3’96) LIBRARY

Michigar State
University

 

 

 

This is to certify that the
thesis entitled

MODELING AND CONTROL OF ATOMIC FORCE MICROSCOPE
FOR IMAGING USING HIGH ORDER HARMONIC MODES

presented by

Chinwe Pamela Nyenke

has been accepted towards fulfillment
of the requirements for the

Master of degree in Electrical Engineering

 

 

Science

 

Major Professor’s Signature
2/2/0—7

Date

MSU is an affirmative-action, equal-opportunity employer

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DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6/07 p:/ClRC/DateDue.indd-p.1

MODELING AND CONTROL OF ATOMIC FORCE MICROSCOPE FOR IMAGING
USING HIGH ORDER HARMONIC MODES

By

Chinwe Pamela Nyenke

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE
Department of Electrical and Computer Engineering

2007

ABSTRACT

MODELING AND CONTROL OF ATOMIC FORCE MICROSCOPE FOR IMAGING
USING HIGH ORDER HARMONIC MODES

By

Chinwe Pamela Nyenke

This research assesses the imaging of specimen surface features using higher flexural
modes associated with the oscillation of the atomic force microscope (AFM) cantilever.
A computer simulation based on the Euler—Bemoulli Beam Equation is designed to
measure variations in the oscillation amplitude due to variations in the specimen surface.
The results are then compared to those of an experiment employing a real AFM and lock-
in amplifier. Both simulation and experimental results demonstrate that examining
higher modes yields more accurate information about a specimen surface property —
particularly elasticity and t0pography — which is consistent with previous studies. More
importantly, this research achieves improved sensitivity to surface information in lower
modes when exciting the cantilever at a higher resonance frequency. The latter finding
eliminates the need to extract and enhance the amplitude of higher modes, which are

magnitudes smaller and consequently more difficult to measure and analyze.

Copyright by
CHINWE PAMELA NYENKE
2007

ACKNOWLEDGMENTS
The author would like to thank Dr. Ning Xi in the Department of Electrical and Computer
Engineering for his productive discussions and guidance in completing this project. The
author also acknowledges Jiangbo Zhang for his assistance in modeling the cantilever
dynamics and providing AFM image results. This research was supported by Michigan

State University.

iv

TABLE OF CONTENTS

LIST OF FIGURES ........................................................................................................... vi
1. INTRODUCTION .......................................................................................................... l
1.1 BACKGROUND AND MOTIVATION ..................................................................... l
1.2 BRIEF OVERVIEW OF PREVIOUS WORK ............................................................ 2
1.3 BRIEF SUMMARY OF PROPOSED PROJECT ....................................................... 3
1.4 ORGANIZATION OF THE THESIS .......................................................................... 4
2. LITERATURE REVIEW .............................................................................................. 5
3. MODELING OF AFM TAPPING MODE .................................................................... 8
3.1 INTRODUCTION TO TAPPING MODE .................................................................. 8
3.2 INFINITE DIMENSION MODEL .............................................................................. 9
3.3 MODE VERIFICATION ........................................................................................... 14
4. COMPUTER SIMULATION STUDIES .................................................................... 16
4.1 SIMULATION PACKAGE ................................................................................. -. ..... 16
4.2 SIMULATION RESULTS ........................................................................................ 17

4.2.2. Response to Change in Sample Stiffness ....................................................... 18

4.2.2. Response to Change in Sample Topography ................................................. 42

4.2.3. Multiple Changes in Sample Topography ..................................................... 78
4.3 DISCUSSION ............................................................................................................ 88

4.3.1. Response to a Change in Sample Stiffness .................................................... 88

4.3.2. Response to a Change in Sample Topography .............................................. 88

4.3.3. Multiple Changes in Sample Topography ..................................................... 89
5. EXPERIMENT IMPLEMENTATION AND TESTING ............................................ 90
5.1 DESCRIPTION OF EXPERIMENT IMPLEMENTATION .................................... 90
5.2 EXPERIMENTAL RESULTS ................................................................................... 90
5.3 DISCUSSION ............................................................................................................ 92
6. CONCLUSION AND FUTURE WORK .................................................................... 93
6.1 CONCLUSION .......................................................................................................... 93
6.2 FUTURE WORK ....................................................................................................... 93
7. APPENDIX: COMPUTER SIMULATION CODE .................................................... 95
8. REFERENCES .......................................................................................................... 105

 

LIST OF FIGURES

Figure 1. Illustration of cantilever-tip ensemble, photodetector, laser, specimen and

piezotube. ............................................................................................................................ 1
Figure 2. AFM image of an electrode during the tapping mode. ........................................ 8
Figure 3. The cantilever dimensions: A = area = WH. ....................................................... 9
Figure 4. Control loop model of the AFM tapping mode system in Simulink ............... 17
Figure 5. Variations in the Young’s modulus of a specimen ........................................... 19

Figure 6. Point-mass model: tip displacement while scanning a sample of constant
stiffness (7OGPa). .............................................................................................................. 22

Figure 7. Point—mass model: tip displacement while scanning a sample of varying
stiffness (20-70GPa). ........................................................................................................ 23

Figure 8. Point-mass model: tip displacement while scanning a sample of varying
stiffness (2Pa-7OGPa). ...................................................................................................... 23

Figure 9. Elastic-beam model, cantilever driven at the lst mode: tip displacement while
scanning a sample of constant stiffness (7OGPa). ............................................................. 25

Figure 10. Elastic-beam model, cantilever driven at the lst mode: tip displacement while
scanning a sample of varying stiffness (20-70GPa). ........................................................ 28

Figure 11. Elastic-beam model, cantilever driven at the lst mode: tip displacement while
scanning a sample of varying stiffness (2Pa-7OGPa). ...................................................... 32

Figure 12. Elastic-beam model, cantilever driven at the 2nd mode: tip displacement
while scanning a sample of constant stiffness (7OGPa). ................................................... 34

Figure 13. Elastic-beam model, cantilever driven at the 2nd mode: tip displacement while

scanning a sample of varying stiffness (20-70GPa). ........................................................ 37
Figure 14. Elastic-beam model, cantilever driven at the 2nd mode: tip displacement

while scanning a sample of varying stiffness (2Pa-7OGPa) .............................................. 40
Figure 15. Variations in sample topography. ................................................................... 44

Figure 16. Point-mass model: tip displacement while scanning a flat sample surface
(height of -22.7nm). .......................................................................................................... 49

vi

Figure 17. Point-mass model: tip displacement while scanning a varying sample surface
height (-22.7nm to -4.5nm). .............................................................................................. 50

Figure 18. Point-mass model: tip displacement while scanning a varying sample surface
height (-22.7nm to -4nm). ................................................................................................. 51

Figure 19. Point—mass model: tip displacement while scanning a varying sample surface
height (-22.7nm to Onm). .................................................................................................. 52

Figure 20. Elastic-beam model driven at lst mode: tip displacement while scanning a flat
sample surface (-22.7nm) .................................................................................................. 54

Figure 21. Elastic-beam model driven at lst mode: tip displacement while scanning a
varying sample surface height (-22.7nm to -4nm). ........................................................... 57

Figure 22. Elastic-beam model driven at lst mode: tip displacement while scanning a
varying sample surface height (-22.7nm to -4.5nm). ........................................................ 60

Figure 23. Elastic-beam model driven at lst mode: tip displacement while scanning a
varying sample surface height (-22.7nm to Onm). ............................................................ 63

Figure 24. Elastic-beam model driven at 2nd mode: tip displacement while scanning a

flat sample surface (-22.7nm). .......................................................................................... 67
Figure 25. Elastic-beam model driven at 2nd mode: tip displacement while scanning a
varying sample surface height (-22.7nm to -4.5nm). ........................................................ 70
Figure 26. Elastic-beam model driven at 2nd mode: tip displacement while scanning a
varying sample surface height (-22.7nm to -4nm). ........................................................... 73
Figure 27. Elastic-beam model driven at 2nd mode: tip displacement while scanning a
varying sample surface height (-22.7nm to Onm). ............................................................ 76
Figure 28. Point-mass model: bulge and depression in sample surface. ......................... 80

Figure 29. Elastic-beam model driven at lst mode: bulge and depression in sample
surface and 2nd mode amplitude. ..................................................................................... 83

Figure 30. Elastic-beam model driven at 2nd mode: bulge and depression in sample
surface and lst mode amplitude. ....................................................................................... 86

Figure 31. Image results while exciting the cantilever at the lst mode. (a) height
information, (b) first mode amplitude, (c) second mode amplitude. ................................ 91

Figure 32. Image results while exciting the cantilever at the 2nd mode. (a) height
information, (b) first mode amplitude, (c) second mode amplitude. ................................ 91

vii

1. INTRODUCTION

1.1 BACKGROUND AND MOTIVATION

The atomic force microscope (AF M) is an instrument used to characterize a specimen on
the scale of a few nanometers. It consists of a cantilever-tip ensemble which moves
across the specimen surface while responding to attractive and repulsive forces
underneath the tip (see Figure 1). As the cantilever bends in reaction to these forces, its
deflection is measured by means of a laser beam reflected from the top of its free end
onto a photodetector. This measured deflection is then used to adjust the height of the
piezotube scanner and thus move the specimen closer to (or further from) the cantilever
in order to maintain a constant desired force between the tip and surface. In some cases,
piezoresistive cantilevers are used, as opposed to lasers and photodetectors; the

piezoresistive element acts as a strain gage to detect the cantilever deflection.

  
    
   

. photodetector

cantilever

specimen

piezotube "
scanner

Figure 1. Illustration of cantilever-tip ensemble, photodetector, laser, specimen and
piezotube.

The AFM is operated in one of three modes — contact, non-contact, and tapping —
depending on the type of specimen being examined. In the contact mode, the cantilever
tip is dragged across the surface such that only repulsive forces are present between the
tip and specimen. The non-contact mode operation requires the cantilever to be oscillated
at or near its resonance frequency while the oscillation amplitude is modified by
attractive forces. (In this latter operation, a soft specimen can be scanned without
dragging the tip across the surface and potentially damaging the specimen or tip.) The
tapping mode combines the benefits of both contact and non-contact by allowing for
high-resolution scanning and manipulation of soft specimen with minimal damage,
respectively. Because of its advantages, this last operation is commonly used when

investigating material properties.

The cantilever’s oscillation during the tapping mode has typically been modeled as a
point-mass system, which only considers the first harmonic. However, because it is
actually a superposition of multiple flexural modes, researchers have questioned whether
ignoring the contribution of higher harmonics results in a loss of valuable information
about a specimen surface [1]-[2]. One of the current research areas in atomic force
microscopy is to obtain more accurate information about the surface properties of a

specimen using these higher modes, which may in turn improve imaging.

1.2 BRIEF OVERVIEW OF PREVIOUS WORK

Several studies have already demonstrated advantages to analyzing the higher harmonics,

including increased image contrast and improved sensitivity to certain surface features,

such as the elastic modulus and height variation [1]-[8]. One common setback among
these studies is that the amplitudes of these harmonics are magnitudes smaller than that of
the first mode and, thus, require enhancement [1], [3] or implementing another method to
improve measurability. Little has been done with exciting these higher modes such that

the lower modes, which have larger magnitudes, can be used for analysis instead [4].

1.3 BRIEF SUMMARY OF PROPOSED PROJECT

Like previous studies, this one assesses the effect of using higher harmonics to image a
specimen surface. More importantly, this study investigates the effect of exciting these
higher modes such that lower ones may be used for detection of these properties as well.
Any significant improvements in imaging that may be achieved when examining the
higher modes can have serious implications for the analysis of DNA strands, protein
peptides, polymer films, carbon nanotubes, and other structures in nanotechnology-
related fields. Should the lower harmonics present comparable improvements, the
finding would be vital in reducing the need to extract and enhance the small amplitudes

of higher harmonics.

The objective of this project is to use a computer simulation to measure and compare the
sensitivity of multiple modes to specimen surface information. An actual AFM is then
employed to verify the results of the computer simulation study. The approach is to
model the cantilever as an elastic beam, record changes in the oscillation amplitude using
various harmonics, and assess the level of sensitivity to specimen properties via these

changes.

1.4 ORGANIZATION OF THE THESIS

This paper presents a review of literature on previous research in Section 2; an
explanation of the tapping mode and infinite dimensions of the cantilever motion in
Section 3; a description of the AFM computer simulation, including results and
discussion, in Section 4; a description of the experiment performed with an actual AFM,
including results and discussion, in Section 5; a conclusion of the project and ideas for
future work in Section 6; an appendix of simulation code and further results in Section 7;

and a list of references used for this research in Section 8.

2. LITERATURE REVIEW

There has been speculation that modeling the cantilever as a point mass, or a system with
only one vibration mode, potentially limits the accuracy of information obtained about a
given specimen surface. Thus, researchers are beginning to consider the higher
harmonics that contribute to the cantilever motion during the tapping mode. By
understanding that this motion is a ‘superposition of several eigenmode vibrations’ [1]-
[2], such properties as surface stiffness and topography may be detected more accurately.
Recent work by several researchers demonstrates that analyzing the higher modes of the
cantilever motion may improve sensitivity to certain specimen properties that might

otherwise go unnoticed when analyzing the first mode alone.

Hillenbrand, M. Stark, R. W. Stark, and Heck] have all investigated the use of higher
modes to increase image contrast. Hillenbrand and M. Stark collaborated to evaluate the
detection of tip-specimen interaction variations on an etched silicon wafer. Their results
demonstrated that higher harmonics, particularly the 13th, are more sensitive to
deviations in the time and strength of tip-specimen interaction and, thus, to deviations on
the wafer surface [5]. A few years later, R. W. Stark refined the findings of this previous
study by determining that variations in specimen topography and elasticity are accurately
detected by the 8th and 13th mode, respectively [6]. One minor shortcoming in both
references is that the experimental data is mainly qualitative; visual comparisons of
distinctions in image results are used to assess the sensitivity of each harmonic to
specimen properties. R. W. Stark attributes the lack of quantitative data in his work to

the low bandwidth of the lock-in amplifier equipment which limited the ability to

measure the small amplitudes of the higher harmonics. Nevertheless, both references

show increased image contrast when looking beyond the first vibration mode.

Higher harmonics have yielded improvements in areas beyond image contrast as
demonstrated in studies by Sahin, Rodriguez, and Garcia. Sabin reveals that hard and
soft samples can be investigated more accurately with the 24th and 8th harmonic,
respectively [1]. Unlike R.W. Stark, he obtains more quantitative results and examines
the detection of elasticity (rather than the detection of variations in elasticity) for a given
specimen. However, in order to accomplish this task, Sahin implements a method of
using specially-designed cantilevers to enhance the small amplitude signals of higher
modes. Meanwhile, Rodriguez and Garcia show that the second harmonic can detect
very small force variations such that they are then able to sense changes in the atom and
molecule composition of the specimen [7]. This work seems inconsistent with that of
R.W. Stark, Hillenbrand, and Sahin, who all suggest that modes below the 8th omit
valuable information about material properties. However, Rodriguez and Garcia present
thorough quantitative data demonstrating that the second mode is sufficient, at least in
detecting the atom and molecule composition by determining the Hamaker constant.
Additionally, the two researchers avoid implementing a method to measure higher mode

amplitudes since they only examine the first two harmonics.

The general results of the literature demonstrate advantages in exploiting the higher
modes but also reveal a challenge in measuring the small amplitudes (or high frequency)

of these modes. For this reason, this research involves the investigation of using lower

modes to detect specimen properties when exciting the cantilever at a higher resonance
frequency. It is believed that these lower modes, which have larger amplitudes, can then
be used for analysis. The advantages of examining higher modes are also investigated
further in this research. Similar to Rodriguez and Garcia, few harmonics are used and
shown to be sufficient. Unlike previous works, this project provides more experimental

results.

3. MODELING OF AFM TAPPING MODE

3.1 INTRODUCTION TO TAPPING MODE

 

Figure 2. AFM image of an electrode during the tapping mode.

During the tapping mode, the cantilever is oscillated at or close to its fundamental
resonance frequency, causing the tip to touch the specimen surface intermittently. The
oscillation amplitude is used as feedback and compared to a desired constant amplitude
set by the operator. Any error between the actual and desired amplitudes results in an
adjustment of the piezotube scanner height and thus an adjustment in the tip-specimen
separation. The error may be induced by a bulge or depression in the specimen surface
which causes the oscillation amplitude of the cantilever to change in size. Figure 2 shows

a tapping mode image obtained while scanning an electrode in this project.

During the tapping mode, the tip-specimen interaction force enters two attractive and
repulsive regimes. In the attractive regime, the tip is far enough from the specimen
surface that van der Waals’ forces dominate while, in the repulsive regime, the tip is
close enough to the surface such that Pauli and ionic repulsive forces dominate. On the
nanometer scale, a small layer of condensed water vapor is present on the specimen

surface and allows for capillary forces to also exist.

3.2 INFINITE DIMENSION MODEL

 

 

 

 

 

\

 

Figure 3. The cantilever dimensions: A = area = WH.

The cantilever was modeled as a long, slender linearly elastic beam with a uniform cross-
section and small deformations (see Figure 3). Because of these and other modeling
assumptions, the Euler-Bemoulli Beam Equation was appropriately applied to calculate

the deflection w of the cantilever. The equation which describes the deflection is

2 2
9—[51—6 WOW] = p, (1)

6x2 6x2

where E is the Young’s modulus of the cantilever, I is the area moment of inertia, and p is

the'end load. Assuming that E and I are constant along the length of the cantilever, the

above equation simplifies to

4
516 wow) z p

a 4 (2)
x

Since the end load p (e.g., tip-specimen interaction force) causes the cantilever to vibrate,

it can be expressed using the Newton-Euler equation

2
p =—pAa—W(—:’-Q. (3)

6x

where the area A and linear mass density p, are assumed to be constant along the length

of the cantilever. Combining Equations (2) and (3) produces the following

4 2
E16 w(x,t)+pAd w(x,t) :

0 (4)
6x4 6x2

The initial conditions for the cantilever are

10

6w(x,t)

M/(x,t):0|t:0, Tlt=0=0 (5)

Additionally, the fixed end of the cantilever must have zero deflection and velocity at all

times. The free end should not have a bending moment or shearing force. Thus, the

boundary conditions are

_ 3W(x,t) _
w(xat)Ix:O-' 0’ Tlx=0_ 0: (6a)
62w(x t) 531905 3)
EI——2—’—| _L=0, EI—3—’——| _L=0 (6b)
6x x— 6x x—

To solve Equation (4), the separation of variables method is used. We first guess that

w(x,t) can be separated into two independent variables as follows

W(x, t) = ¢(x)q(t), (7)

where one depends on space and the other on time. Plugging this solution into Equation

(4) yields

4 2
E, a (¢(x)q(t» + m a woman z 0
6x4 0x2

(8)

9

Consequently, the general solution for ¢(x) is (6):

11

¢(x) = A sin(kx) + B cos(kx) + C sinh(kx) + D cosh(kx), (9)

where k = pA /(E1)a)2 . While using the boundary conditions to find the coefficients A,

B, C, and D, the following frequency equation is obtained
cos(kL) cosh(kL) = —1, (10)

There are an infinite number of k values that satisfy Equation (10) such that

a). = k.2 IE, 1': l,2,3,...,oo, (11)
I l pA

where (0,- is the eigenfrequency of the ith mode. For each value ki, there exists a (15,-, and

thus a characteristic vibration w,-

wl.(x. t) = ¢,.(x)q,.(t). (12)

The general solution to Equation (4) is a sum of the infinite number of characteristic

vibrations

awkzgmma (m

i=1

12

Thus, by Euler-Bernoulli Beam Equation it is shown that the cantilever vibrates at infinite
resonant frequencies simultaneously, rather than at a single frequency. During the
tapping mode, the forces (driving and tip-specimen interaction) that act on the cantilever

tip require a modification of Equation (4) to

62w(x,t) _

64w(x,t)
+ 2 ‘ drivin
ax g

E]
6x4

cos (or + F (t)

pA (14)

tip — specimen”

The tip-specimen forces are primarily attractive and repulsive, as described in Section

3.1. When the tip-specimen instantaneous separation d is larger than the intermolecular
distance a0, attractive forces are present and modeled as

F(O,d)ts = ——HR (‘5)

6(d2)’

where H is the Hamaker constant and R is the radius of the tip. When d is smaller than
a0, or when the tip is relatively close to the specimen surface, the attractive forces are

now modeled as

HR
6(a02)

 

F(O,d)ts = - (16)

13

using the Derjaguin—Mfiller-Toporov (DMT) model [9]. In this region, repulsive forces

are also present and represented with the following equation

HR
6( 2)

 

_ 4 . _ 3/2
F(0,d)tS—- +§EJR(aO d) (17)

“0
using DMT. The repulsive forces become dominant when d is smaller than a0

The solution to Equation (14) is still a superposition of infinite vibrations but the forces
are now factors of q( t). The cantilever dimensions in this experiment are: H = 4um, W =
35um, L = 120 um, and E = l3OGPa. The tip radius R is 20nm and its height is

negligible.

3.3 MODE VERIFICATION

An accurate measurement of the cantilever’s fundamental resonance frequency is

necessary for correct determination of the specimen properties during the tapping mode

operation. The manufacturer typically attaches a nominal fundamental frequency f1 to

each AFM cantilever which may be a few percentages off the actual value. As a result, f1

must be calibrated.

l4

For this project, the Nanoscope 111 Tuning Software was used to calibrate f I (SOkHz) for
the cantilever in the AFM experiment. The cantilever was driven at a range of
frequencies close to f I and the frequency response was recorded with the software. The

frequency that yielded a response with the largest amplitude was then assumed to be the
actual fundamental resonance frequency. For this project, that value was 51 .6kHz. The
same procedure was applied to the nominal second resonance frequency, which was

210kHz; the actual value was approximately 208.8kHz.

15

4. COMPUTER SIMULATION STUDIES

4.1 SIMULATION PACKAGE

A simulation package (Veeco Company) of the atomic force microscope during the
tapping mode was used. The simulation code, which originally modeled the cantilever
motion as a point-mass, was modified to account for infinite flexural modes. Tests were
performed in Simulink, a platform developed by Mathworks for the simulation of

dynamic systems. All tests were run on a personal computer.

Figure 4 shows a diagram of the AFM control-loop model that was run in Simulink; the
cantilever block contains the software that calculates the multiple modes based on the
Infinite Dimension Model in Section 3.2. (See Appendix for code.) The setpoint
amplitude is a constant amplitude set by the user during a closed loop simulation of the
tapping mode. The error signal between this setpoint amplitude and the actual oscillation
amplitude of the cantilever is amplified by a gain and ultimately used to adjust the height
of the holder position. The results in this project were obtained while running open-loop
simulations of the tapping mode; the actual cantilever oscillation amplitude was

monitored while the holder position remained unadjusted.

l6

gain

trigger
3%,.

    
       
 

   
        
 
      
  
     
    

_ deflection controller ' ' j ' "
setpornt gain amplifier and
amplitude piezo actuator holder
oscillation
amplitude
_, Closed or

 

driving
amplitude

oscillation ‘
amplitudes of ‘ holder
modes 1, 2, 3 cantilever sample position

(elastic-beam) surface

Figure 4. Control loop model of the AFM tapping mode system in Simulink

4.2 SIMULATION RESULTS

Results were obtained and compared for a point-mass and elastic-beam simulation of the
cantilever motion. For the point-mass model, the simulated cantilever was driven at the
first resonance frequency and only the first harmonic was analyzed. During the elastic-
bearn simulation, the first three harmonics were analyzed while the cantilever was first
driven at the first resonance frequency and then at the second resonance frequency.
Thus, three total simulations were actually performed: point-mass, elastic-beam while
excited at the first resonance frequency, and elastic-beam while excited at the second
resonance frequency. The accuracy of the point-mass and two elastic-beam models was

investigated by running three simulation scenarios: 1) a variation in sample stiffness, 2) a

17

variation in sample topography, and 3) multiple variations in sample topography. One
variable, the cantilever oscillation amplitude, was measured during each simulation. The
resulting measurements were used to evaluate sensitivity to changes in specimen stiffness

and topography as well as determine imaging accuracy.

4.2.2. Response to Change in Sample Stiffness

A 2-millisecond single-line scan simulation of the cantilever was performed on a
computer-generated specimen having a Young’s modulus of 7OGPa, or constant stiffness.
The same simulation was then performed as the Young’s modulus was adjusted to 20GPa
after 0.8 milliseconds of scanning and reset to 7OGPa after 1.2 milliseconds to replicate a
momentary change in sample stiffness. The same procedure was lastly performed for an
adjustment to 2Pa to replicate a more significant change in sample stiffness. Figure 5
illustrates these changes in Young’s modulus. In all cases, the fixed end of the cantilever
(holder position) was set to Onm while the specimen was simulated as flat surface with a

constant height of -4nm, or 4nm below the holder position.

18

Figure 5. Variations in the Young’s modulus of a specimen

l9

 

 

 

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1.6

1.4

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(a)

 

 

 

 

 

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1.6

1.4

1.2

0.8

0.6

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am 1 I 1 I. l 1 I 1 I

7 ............................. . .........
16‘ 5... ............... .................. ................... ..................................... ................ .................................... ................ _

5 3 5 i f i
am, 5_ .............. .................. ............. .................. S- ............... ............ . a

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E 4.. ................. ................. ..... l .................. .................. .................. ..................................... ................ _

-"’ 1 i é f 3 §
21 3_ ................. .................. .................. ................... .................. ................. _

3 2 i 2 2 2 a
>- 2__ ............ £__ ............. ._
1-. ................ .................. .................. ................... ................... .................. ................ ................. _

0 I I i I . . i i
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
time (5) x10’3

(C)

Figure 5 continued.

During the point-mass simulation, the sinusoidal driving force amplitude was set to 95nm

which yielded a maximum oscillation amplitude of 8nm such that the cantilever would

barely tap the flat specimen surface. This was done so that the height of the specimen

would not interfere with the results of this simulation. The results of the point-mass

simulation in Figure 6 demonstrate that there is no change in the oscillation amplitude as

the Young’s modulus remains constant, which is expected.

In Figures 7 and 8, the

cantilever’s oscillation amplitude again remains unchanged while the stiffness decreases

in the center of the specimen. Thus, the variations in sample stiffness were not detected

using the first mode alone.

21

 

I I III I I I II II

amplitude (m)
O

 

 

III .
I III |IIIIIl I I W111

_5 I l l
0 0.2 0.4 0.6 0.8 I 1.2 2

time(s ) x103

 

Figure 6. Point-mass model: tip displacement while scanning a sample of constant
stiffness (7OGPa).

22

 

III III III I III III III III I II III

amplitude (m)
O

 

 

I
I' III IIIIIII I II III III II I III IIIII

0 0.2 0.4 0.6 0.8 1.2 1.6 1.8 2
time (5) x103

 

Figure 7. Point-mass model: tip displacement while scanning a sample of varying
stiffness g(20-70GPa).

nx10g

 

III I I. I II I II

amplitude (m)
.0

 

‘I
II II IIIII II’IIIIII I III II I' I ‘ I II

_5 I l I I l I I I 1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
time (5) x103

 

 

Figure 8. Point-mass model: tip displacement while scanning a sample of varying
stiffiiess (2Pa-7OGPa).

23

In order to evaluate the elastic-beam model, the first three flexural modes were extracted.
The tip response was first recorded while driving the cantilever at the first resonance
frequency for each of the three Young’s modulus values. The driving force amplitude
remained set at 95mm. The results in Figure 9 show that the amplitudes of each of the
modes remain steady as the stiffness of the specimen remains constant, which is
expected. In Figures 10 and 11, while the stiffness decreases in the center of the
specimen, the mode amplitudes again remain unchanged. Despite the fact that three
modes are now analyzed instead of one, the variations in surface stiffness continue to go
undetected. (Note how in Figures 9-11, the first mode amplitudes for each of the three

Young’s modulus values are identical to those of the point-mass model, as expected.)

24

Figure 9. Elastic-beam model, cantilever driven at the lst mode: tip displacement while
scanning a sample of constant stiffness (7OGPa).

25

x10'

 

1111 1'11" 1111' 1'11

   
 

amplitude (m)
53'

11111 111111111111111111111111111I

 

-5 I 4
0 0.2 0.4 0.6 0.8 1 1 2 1.4 1.6 1.8
time (5) x103

(a) first mode

 

luf 1 1 1 1 1 1 1 1

 

 

‘ i ‘ 1'11 1 1 1 1'» '11‘11 1
E ' 1 11'111111 111111111111'1'111'
%
E- 11111 11" 11 .111
1 M11111 1111111 l"11 1111' 1 ~ 1111111111
'1'50 0.12 014 016 018 .1 112 114 116 1,18 2
tIme(s) x10'3

0)) second mode

26

"1'1111111111 111111111111;1111111 11111'

     
 

 

2 111 12111 1'1 11111 ~11:
I1 1 1111111 1 1 1 1
1111? 1 1111' 11 1111 11 1111111111 1111111111111- 1' I1
‘30 012 014 016 018 1 1.12 1.14 116 118 2
time (5) x10'3
(c) third mode

Figure 9 continued.

27

Figure 10. Elastic-beam model, cantilever driven at the lst mode: tip displacement while
scanning a sample of varying stiffness (20-7OGPa).

28

 

:x10'

11111111I

 

  

amplitude (m)
C)

 

 

111111111111111111111111111111111'11111111111111111111

1 1

-5
0 0.2 0.4 0.6 0.8 1 1. 2 1.4 1.6 1.8 2
t1me( ) x103

(a) first mode

 

0.5 111111131111121111111111

1'11 111 11
111 1 1111 111111111111111111111111111111111111111

amplitude (m)
9

 

 

1 1. I .
-0.5 11 1111111 11111111111111111111111111 1111111111 111 1 1.11 : 11111111111111111
, . I : 1 11
_1 . 1 . 1 1 =
'1‘50 012 014 016 010 1 112 114 116 118 2
time(s) x103

(b) second mode

29

.12
4x10

3

1 121111 1 1 11111111111111111 11111111 1111111111

30

 
 

ampliu Itu de

_2 11111.11 11 111111111111111111111111111111 1111 11 111111 11111 1111 11111111

 

| 11 1
_3 1 1 1 1 1 1 1 J 1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
time (5) x10'3
(0) third mode

Figure 10 continued.

30

Figure 11. Elastic-beam model, cantilever driven at the lst mode: tip displacement while
scanning a sample of varying stiffness (2Pa-7OGPa).

31

 

x10"9

5 l j l j l r ‘ l I 1

 

l
l
l
l

0

    

amplitude (m)

r4 gggggggll

I
Ch

0
o
M
o
A
o
0')
Q
00
_\.
._.L
[\J
_L
A
_x
07
_L
Co

mN-

(a) first mode

 

p, “5‘ l MM
u M lll “lliml‘
! ‘l l j

l‘,

amplitude (m)
O

  

 

1 5 1 l l 1 r l l
‘ 0 0.2 0 4 06 0.8 1 1.2 1.4 1.6 1.8 2

(b) second mode

32

 

O

amplitude (m)

 

 

-2 ‘
1 1
_31 1 l l l 1 1 1
0 02 0 4 06 0.8 1 1.2 1 4 1 6 1 8 2
time (5) x103
(c) third mode

Figure l 1 continued.

In the second simulation for evaluating the elastic-beam model, the cantilever was driven
at the second resonance frequency for each of the three Young’s modulus values. The
driving force amplitude was increased to 660nm to achieve the same oscillation
amplitude of 8nm for the driven mode as in the previous simulations. While the Young’s
modulus remains a constant 7OGPa during scanning, all three mode amplitudes in Figure
12 remain steady. However, when the specimen stiffness decreases momentarily to
ZOGPa, there is a visible change in the first mode amplitude (see Figure 13). When the
Young’s modulus decreases to 2 Pa, this amplitude change becomes more significant and

a slight amplitude variation begins to appear in third mode (see Figure 14).

33

Figure 12. Elastic-beam model, cantilever driven at the 2nd mode: tip displacement
while scanning a sample of constant stiffness (7OGPa).

34

 

amplitude (m)

amplitude (m)

 

11 111111 11111 1111111 111 11111111 1111111

1
11111.

11111121111111‘11111 111' 11111 I11 1111 11111111 1 1:11:

11 1111

1 1 1 1 1 1 1 1 1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
time (s) x 1041

 

 

(a) first mode

  
 

C)___

Ln
1

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
time (5) x103

(b) second mode

35

 

amplitude (m)

 

Figure 12 continued.

time (s)

(0) third mode

36

 

Figure 13. Elastic-beam model, cantilever driven at the 2nd mode: tip displacement while
scanning a sample of varying stiffness (20-7OGPa).

37

 

 

amplitude (m)

coamplitude (m)

 

 

 

 

 

 

 

III III , III
III IIIII II’ IIIIIII IIIII
I‘III IIII ; IIII II
4 -2IIIII IIIIKII- IIII .... III-I
I II 1 ‘ ‘ ' II II
III I I II .I
....... _8IIII III IIIIII III IIIIII II I - IIIIIIIIIIII, I:IIIII IIIII
II 1
1 i I i 1 1 I 1 l
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
time(s) x10'3
(a) firstmode
Kx10‘g
J l I I T I l I I
0
I 1 l I I I I l 1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
time(s) x103

(b) second mode

38

amplitude (m)

 

 

25L J I I I I 1 I I I
I 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
time (5) x10'
(0) third mode

Figure 13 continued.

39

Figure 14. Elastic-beam model, cantilever driven at the 2nd mode: tip displacement
while scanning a sample of varying stiffness (2Pa-7OGPa).

40

 

-H
2x10

I III
0 » III

 

 

”I221MIH II

 

II II! II ‘Iltr

 

 

 

g-l I II ii I: i
.3- ....... II III “I I I III I, i“ I!" ill! ‘ .............. ...... It III “I .. l‘ M W Ii! ! II‘II IiIi
-10—- ' ' '
0 0i2 0i4 0i6 0L8 11 12 1.14 1i6 1.:8 2
time (5) x103

amplitude (m)

(a) first mode

C)

0 0.2

0.4

 

time (s) x 10.3

(b) second mode

41

 

amplitude (m)

 

 

-1.5
-2
-2.5 l ‘I I l l l l l l
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
time (5) x103
(c) third mode

Figure 14 continued.

4.2.2. Response to Change in Sample Topography

For this scenario, a 2-millisecond single-line scan simulation was performed on a
computer-generated flat specimen surface with a constant height of -22.7nm (i.e., 22.7nm
below the holder position). The height was chosen to give the cantilever ample room to
move and reach its maximum oscillation amplitude of 8nm. A simulation was then
performed on a specimen surface that was adjusted to -4.5nm afler 0.7 milliseconds of
scanning and reset to -22.7nm after 1.4 milliseconds to replicate a momentary change in

specimen height. The procedure was repeated for surface height adjustments to -4nm and

42

Onm. Figure 15 illustrates these four computer-generated specimen surfaces. All
simulations were run using a sample of constant stiffiiess (7OGPa) and a cantilever holder

fixed at Onm.

43

Figure 15. Variations in sample topography.

44

 

 

 

 

1.8

1.6

1.4

1.2

0.8

0.6

0.4

0.2

 

 

 

a fi _ A
TE .......................
T ............. .........................

m _ . _

5.

nw

Eb Ego: 03.556

3

x10‘

time (s)

x 10'8

 

 

 

 

 

 

.............................................................................

 

1.8

1.6

1.4

1.2

0.6 0.8

0.4

0.2

 

 

9:5 £90: vifiwm

-2.50

3

x10'

time (s)

(b)

45

Figure 15 continued.

46

 

 

 

 

 

 

 

x10'8

.

f
l
1.8

1.6

1.4

L
l
1.2

0.8

0.6

0.4

0.2

 

 

_ _ _, m . a _
0 0 4| 1 1 a.l 2

E: urgor OEEmm

x10

time (s)

(C)

 

 

 

 

 

 

1.8

1.6

1.4

1.2

0.8

0.6

0.4

0.2

 

 

Act 590... Barnum

3

x10'

time (s)

(d)

47

In order to evaluate the point-mass model, a simulation was performed on the four
samples shown in Figure 15 while extracting only the first mode of vibration. The results
in Figure 16 show a steady oscillation amplitude as the specimen surface remains flat,
which is expected. In Figure 17, the amplitude remains fairly constant despite the
presence of a slight bulge on the specimen surface. It is only when the bulge is large
enough to restrict the tip displacement that the oscillation amplitude decreases (see Figure
18); this occurs when the separation between the bulge and the cantilever’s rest position
is 4nm. As the bulge on the specimen surface increases in height, the first mode
oscillation amplitude changes more significantly. When the bulge height is Onm, or at the
same height as the cantilever’s rest position, the amplitude is approximately zero because

the cantilever is entirely obstructed and unable to vibrate (see Figure 18).

48

 

II III

III 'IIII III-III' IIIIIIIII
III I I I I I I

amplitude (m)

 

II II III III III I II III. I IIIIIIIIII I III II I I

-5
0 0.2 08 1 1.2 1.4 1.62

 

Figure 16. Point-mass model: tip displacement while scanning a flat sample surface
(height of -22.7nm).

49

9

III III II III II IIII III II

:x10

amplitude (m)
0

[III

 

 

I'IIIHH'IIIH iIiii II iIIiI

-5
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
fime(s)

 

x10a

Figure 17. Point-mass model: tip displacement while scanning a varying sample surface
height (-22.7nm to -4.5nm).

50

x1o‘a
5f f 1

 

11111111 1111

1111 111111111111"
111111111 1111111111 111

amplitude (m)
0

 

 

111111111111111.11111.11111111111111111111111111111111111111111111 1111111111111

0 0.12 0.4 0.60.8 11 1.2 1.4 11.6 1.8 2
time(s ) x10'3

 

Figure 18. Point-mass model: tip displacement while scanning a varying sample surface
height (-22.7nm to -4nm).

51

x 10'9

H111111‘1

Ln

191 5.13.1
i 11 1i? J i‘ i l i ii ‘1

. ll
'1

*1
l

   
 

      

l

amplitude (m)
C)

l
i ’5 ii I ii i i

will

’

w 1' 1 ,
i i ii i i 1i i1 Ii i .1

E 1 1 1 1 1 1 1 1 1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

time (s) x 10-3

1‘1

 

Figure 19. Point-mass model: tip displacement while scanning a varying sample surface
height (-22.7nm to Onm).

In order to evaluate the elastic-beam model, the first three flexural modes were extracted.
In the first simulation, the cantilever was driven at the first resonance frequency and the
oscillation amplitude was recorded for each mode during each variation in surface height.
In Figure 20, the results show steady mode amplitudes when scanning a flat sample
surface as expected. However, the second and third mode amplitudes become
increasingly sensitive to changes in surface height starting at -4.5nm (see Figures 20-23).
The first mode amplitude is only affected when the tip starts to impede the surface at 4nm
below the holder position, as in the case of the point-mass simulation. Amplitude

changes in all three modes are most significant when the surface is at the same height as

52

the cantilever holder. (Note that the amplitudes of the undriven modes — second and

third — are magnitudes smaller than that of the driven, or first, mode).

53

Figure 20. Elastic-beam model driven at lst mode: tip displacement while scanning a flat
sample surface (-22.7nm).

54

 

11 111

amplitude (m)

 

 

111111111111 11111111111111111111111111111 1111:11111

0 0.2 0.4 0.6 0.8 1 1.2 1.4
time (s)

 

x 10'3

(a) first mode

1"" 1 1 1 1 1 1 1

1111111 '111111'111111'1'11111 1111111111111 1111111111111 111 111111111

 
 

 

g

:10

3105-11111111111111111111111111111111111111111111111111 1111111
1

-1

-15 1 1 1 1 1 1 1 1 1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

time(s) X1041

(b) second mode

55

O

amplitude (m)N

1 11 11111111111111111111111 11111111111111111 1111111

  

11111111111111111.111111111111111111 11111 111111111 111111 111111111
-2 ,1 . , .

 

_3 1 1 1 L 1 1 1
0 0.2 0.8 1 1.2 1.4 1.6 1.8 2
time (5) x103
(c) third mode
Figure 20 continued.

56

Figure 21. Elastic-beam model driven at lst mode: tip displacement while scanning a
varying sample surface height (-22.7nm to -4nm).

57

 

x10-

 

 

l

 

 

.w l l ll l“ l llllll l l [l llll'
l‘lll‘llllll .lll‘l'lll ll l
g l" ‘
l . . Ill
llll‘llllWM W lll ll ll
.50 of2 014 05 0.3 timgw 1,2 1.4 1.6 8x103
(a)firstmode

 

 
 
   

 

"0 l‘fl H: l ‘:l " Swill" ‘llnl lml‘fll [l -1_‘m§“fl‘ MW”
@053 l l M' ill I‘ ’ l l H h l H ll-l l Ill ll lll‘l MM '2 ”lull l'llllll l l ll“ llnill l‘ W l l J l ' l l l ll l l
a)
3 0
Elol H! l
«1-0 W My ll ll. lllllll'l "lll‘l’ll‘ll'l W'lll .l'Hl ll WW
-1
15 l l l I l l l l l
' 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
time(s) x10'3

(b) second mode

58

x1o'12

 

4‘ I l I I I 1 1

l

3 1
l l l 1111.111 l l l 1 1 1 'll 1‘ l 1 l 1 1' ll 11111111
:20

 

2 ‘ 1| 11 111111111111 1 11 ll, 11.111111 11111.11 11 1.1 1| M 11 1111111111111 5,1 1'1 ”111111111

 

_3 1 1 1 1 L 1 1 1 1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
time (S) X 10'3
(c) third mode

Figure 21 continued.

59

Figure 22. Elastic-beam model driven at lst mode: tip displacement while scanning a
varying sample surface height (-22.7nm to -4.5nm).

60

 

amplitude (m)

 

x 10'9
5

j

11111111W 111111

O

1.11

 

 

: 111111111 111111 111111111111,111111111111

 

- 0 0.2 0.40.6 0.8 1 1.2 1.4 1.6 1.8 2

 

 

 

 

 

. amplitude (m)

 

 

 

time (s) x 10-3
(a) first mode
x 10‘“
" 1 1 1 1 1 l- 1 1 1
2- 1
0
2 __
1 l 1 1 1 1 1 l 1
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
time (s) x 10-3

(b) second mode

61

x 10"2

 

‘ 1
111

1111111111111 11 1‘ 11 11 1 111 11 '1 1111111111111‘ 11‘ “1‘11‘ 1‘ 11|1 11 1111 111111111111 11 11111111111111

    

amplitude (m)
0

‘1 l 1‘ 1~ ‘1 ‘ 1 11 ‘ ‘1 1‘ l1 l1 11‘ 11 1 1; ; 1 1 l 11 11111111 121111

 

l l l l l 1 l l l

-4

 

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
(c) third mode

Figure 22 continued.

62

Figure 23. Elastic-beam model driven at lst mode: tip displacement while scanning a
varying sample surface height (-22.7nm to Onm).

63

 

amplitude (m)

amplitude (m)

xwfi

 

Ln

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(b) second mode

64

111111111'111‘1111‘1‘1 : 1 1 “1111111
1 ‘1‘ 11111 .1‘ ‘1 1 1 11 1 1
11'11 1‘1‘1‘ 111111
‘111; 1111
11111111 1 1 “‘11 1 11111111111
1 ‘1111; ‘1‘ 111
'% 02 04 05 03 1 11 15 1A 16 18 2
ime 5 x10'3
(a)firstmode
:MW
" 1 1 1 1 1 1 fi fi j
r _
z _
L .1
c
-3 1 1 1 1 1 1 1 1 1
0 02 04 05 03 1ma$ 12 1A 15 13 3
x10"

 

[\J

—L

 

 

amplitude (m)

 

 

 

 

 

 

 

(c) third mode

Figure 23 continued.

In a second simulation for evaluating the elastic-beam model, the cantilever was driven at
the second resonance frequency during each of the four sample height variations. When
the sample surface is flat, the amplitude of each the three modes remains steady as
expected (see Figure 24). However, the first and third mode amplitudes show an
increasing sensitivity to changes in surface height (see Figures 24-27). In Figure 25, the
first mode is able to detect the surface height adjustment of 4.5nm below the holder
position. Amplitude changes are significant in both the first and third modes and barely
visible in the second mode when the surface height is adjusted closer to the cantilever at 4

mm below the holder position (see Figure 26). In Figure 27, amplitude changes in all

65

three modes are most significant when the surface is at the same height as the cantilever
holder. During each sample height variation, the amplitudes of the undriven modes (first

and third) are again magnitudes smaller than that of the driven mode (second).

66

Figure 24. Elastic-beam model driven at 2nd mode: tip displacement while scanning a
flat sample surface (-22.7nm).

67

amplitude (m)

amplitude (m)

 

 

 

 

- 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

time (s) .3

(b) second mode

68

 

amplitude (m)

 

 

-2
2 5| 4 l l l 1 1 l l l
' 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
(c) third mode
Figure 24 continued.

69

Figure 25. Elastic-beam model driven at 2nd mode: tip displacement while scanning a
varying sample surface height (-22.7nm to -4.5nm).

70

 

Iamplitude (m)

 

 

"”0 02 04 06 0.8 1 1.2 14 16 13 2
time(S) X103
(a) firstmode

 

amplitude (m)

 

 

time (S) X 103
(b) second mode

71

_C)

Q

amplitude (m)

 

 

1,5
-2 ‘
_25 I I I I I I I I I
' 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
time (5) x10'
(c) third mode

Figure 25 continued.

72

Figure 26. Elastic-beam model driven at 2nd mode: tip displacement while scanning a
varying sample surface height (-22.7nm to -4nm).

73

 

 

amplitude (m)
IL)
L

I
A
l

l

 

 

 

 

_ I l I I I I I I I
80 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
time (s) x 10-3

(a) first mode

amplitude (m)
C)

 

l l l l I I I l I
0 0.2 0.4 0.6 0.8 I 1.2 1.4 1.6 1.8 2
time (5) x103

(b) second mode

74

 

_L

. amplitude (m)
O O
"on o '01

I
A

 

l
[\J

_21: I I I I I I I L I
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

time (s) x 10-3

A!

(c) third mode

Figure 26 continued.

75

Figure 27. Elastic-beam model driven at 2nd mode: tip displacement while scanning a
varying sample surface height (-22.7nm to Onm).

76

 

amplitude (m)

C)

-0.5_

 

 

 

 

time (s) x 10.3

(a) first mode

 

amplitude (m)

 

 

time (s) x 10.3

(b) second mode

77

 

amplitude (m)

 

 

 

 

time (s) x 10-3

(c) third mode

Figure 27 continued.

4.2.3. Multiple Changes in Sample Topography

For the following simulation, the mode amplitude that was most sensitive to variations in
specimen topography for each model was recorded. A 2-millisecond single-line scan
simulation was performed on a computer-generated sample surface consisting of a bulge
and depression in the topography. The specimen height was adjusted to -4nm midway
through scanning, which was the point at which the tip tapped the surface in previous
simulations. The sample was then heightened to ~2nm to replicate a bulge and lowered to

-6nm to replicate a depression in the surface (see Figure 28).

78

 

For the point—mass simulation, the cantilever was driven at the first resonance frequency
while the first mode amplitude was recorded. The results in Figure 28 illustrate that the
depression at -6nm is barely detectable. The amplitude change at the bulge is

approximately -57% while that at the depression is approximately 29%.

79

 

Figure 28. Point-mass model: bulge and depression in sample surface.

80

sample height

amplitude (m)

 

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

 

time(s) x10'3
(a)samp1e
I‘ II III III
IIIIIII HIII III ‘II‘IIIII III III
I II'I'II

   

 

_5 I I I I I I
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

times (5 ) X 10'3

(b) first mode

81

The cantilever was then driven at the first resonance frequency while the second mode
amplitude was recorded. As seen in Figure 29, the second mode is more sensitive to
variations in sample topography. The amplitude during the bulge is more than 20 times
the original amplitude; however, it quickly decays to a negligible size. Meanwhile, the
amplitude during the depression is half the size of the original amplitude but is barely

detectable.

82

Figure 29. Elastic-beam model driven at lst mode: bulge and depression in sample
surface and 2nd mode amplitude.

83

sample height

amplitude (m)

x10'

 

 

 

 

 

 

 

....................................

 

 

 

 

 

 

 

 

 

0 0.2 0.4 0.6

0.8 1
time (s)

(a) sample

1.4

1.6

1.8
x10'

 

 

 

 

time (s)

(b) second mode

84

 

 

 

A second elastic-beam simulation was performed in which the cantilever was driven at
the second resonance frequency while the first mode amplitude was recorded. The results
in Figure 30 show that the first mode is even more sensitive to multiple variations in
sample topography. The amplitude change during the bulge is well more than 75 times
the original amplitude initially; it then quickly decays to a steady 30 times. The
amplitude change during the depression is barely noticeable since the cantilever is still

recovering after having left the bulge.

85

Figure 30. Elastic-beam model driven at 2nd mode: bulge and depression in sample
surface and lst mode amplitude.

86

sample height

amplitude (m)

C)

A

N

 

 

 

 

 

 

 

 

 

 

(a) sample

 

 

 

 

 

 

 

I I I I I I

 

0.2

0.4

0.6 0.8 1 1.2 1.4 1.6 1.8

time (s)

(b) first mode

87

 

 

4.3 DISCUSSION

4.3.]. Response to a Change in Sample Stiffiiess

The point-mass simulation, which considers the first mode alone, is unable to detect the
changes in surface stiffness. For the elastic-beam simulation in which the cantilever is
driven at the first resonance frequency, the first three modes are all also unable to detect
such changes. Both simulation results are consistent with studies in which the 8th

harmonic, at a minimum, is necessary to sense surface stiffiiess variations.

The elastic-beam simulation in which the cantilever is driven at the second resonance
frequency, on the other hand, detects the variations in surface stiffness. The first mode is
significantly more sensitive to these variations than the third mode. This latter result may
be due to the fact that the cantilever is more flexible, and thus easier for surface forces to
excite, at lower harmonics. The second mode, however, is unable to detect any of the
changes in surface stiffness. Since the cantilever is driven at the second resonance
frequency, the amplitude of the second mode is dominated by the driving force

amplitude.

4.3.2. Response to a Change in Sample Topography

When the cantilever is driven at the first resonance frequency, the second and third
modes of the elastic-beam simulation are more able to detect changes in sample
topography than the point-mass model. This result is consistent with theoretical and

experimental studies demonstrating that certain sample surface information may be

88

omitted when examining the first mode alone. Additionally, the second mode of the
elastic-beam simulation is more sensitive than the third. This observation is again
consistent with the cantilever being more flexible and, thus, easier to excite at lower

harmonics.

Meanwhile, for the elastic-beam simulation in which the cantilever is driven at the second
resonance frequency, the second harmonic is least sensitive to sample topography
variation. This result is due to the fact that the oscillation amplitude of the second mode
is dominated by the driving force amplitude, as was the case in sample stiffness variation.
The first and third modes are more able to identify the surface topography changes. The
first mode, however, is more sensitive to these changes since the cantilever is more

flexible at lower harmonics.

4.3.3. Multiple Changes in Sample Topography

The point-mass simulation is again least sensitive to the variations in sample topography.
Though the elastic-beam simulation was more sensitive to these variations, when the
cantilever was driven at the fundamental resonance frequency, the amplitude change of
the higher (second) mode was difficult to distinguish over the depression in the specimen
surface. When the cantilever was driven at the second resonance frequency, the first
mode sensitivity was magnitudes greater than the second mode in the previous run. This
result is consistent with the theory that lower modes are more responsive to changes in
the sample surface since the cantilever is more flexible, and thus easier to excite, at these

lower modes.

89

5. EXPERIMENT IMPLEMENTATION AND TESTING

5.1 DESCRIPTION OF EXPERIMENT IMPLEMENTATION

An atomic force microscope with a DMASP active probe (Veeco Instruments) was used
to verify the sample topography results obtained in the computer simulations. The
specimen in this experiment was an electrode on a silicon substrate and its height was
approximately 30nm. The first mode amplitude of the cantilever was measured by a DSP
lock-in amplifier (Stanford Research Systems, l02 kHz bandwidth). Nanoscope (V eeco),
which is a program that records images of the sample height and second mode amplitude,
was modified to also generate an image of the first mode amplitude. The cantilever

(probe) holder was fixed at a height of approximately 4nm above the specimen surface.

5.2 EXPERIMENTAL RESULTS

The cantilever was driven at the first and second resonance frequencies. Images of the
specimen height, first mode amplitude, and second mode amplitude were generated while
the second mode was used as a feedback signal. (The AFM was designed by Veeco to
use the second mode vibration as feedback.) The results show that when the cantilever is
driven at the first resonance frequency, the first and second mode amplitudes are almost
indistinguishable (see Figure 31). However, when the cantilever is driven at the second
resonance frequency (see Figure 32), the specimen surface features are more visible in
the first mode amplitude than the second mode. Moreover, a comparison of the first

mode amplitudes in Figure 31b and in Figure 32b reveals that the first mode displays

90

more features via the dark spots when driving the cantilever at the second resonance

frequency (see Figure 32b).

 

(a) (b) (C)

Figure 31. Image results while exciting the cantilever at the 1st mode. (a) height
information, (b) first mode amplitude, (0) second mode amplitude.

 

(a) (b) (C)

Figure 32. Image results while exciting the cantilever at the 2nd mode. (a) height
information, (b) first mode amplitude, (c) second mode amplitude.

91

5.3 DISCUSSION

The first harmonic is more sensitive to specimen features when the cantilever is driven at
the second frequency as opposed to the first. This result confirms the computer
simulation study results in Section 4.2.2; when the cantilever is driven at a higher mode
resonance frequency, the lower modes (in this case, the first mode) become sensitive to
changes in the surface topography. However, it is difficult to tell whether driving the
cantilever at the second resonance yields significantly better results as determined in

Section 4.2.3 given the nature of the image results.

92

6. CONCLUSION AND FUTURE WORK

6.] CONCLUSION

Simulation results demonstrate that examining higher modes yields more accurate
information about a specimen surface — particularly elasticity and topography — which is
consistent with previous studies. More importantly, this research achieves improved
sensitivity to surface information in lower modes when exciting the cantilever at a higher

resonance frequency.

The latter finding presents a potential alternative to extracting and enhancing the
amplitudes of higher modes, which are magnitudes smaller and consequently more
difficult to measure and analyze. By investigating lower modes instead, accurate
specimen surface information can still be obtained while avoiding the implementation of
amplitude enhancement. The findings of this research suggest improved imaging of
specimen with varying surfaces, which is beneficial in the analysis of polymer films,
DNA strands, semiconductor materials, and other samples in nanotechnology-related

research.

6.2 FUTURE WORK

Future studies may investigate a wide range of topographically varying specimen while
driving the cantilever at the first and second resonance frequencies. Image comparisons
may then reveal more concrete experimental evidence as to the benefit of examining

lower mode amplitudes while driving the cantilever at a higher mode frequency. Also,

93

development of a high bandwidth lock-in amplifier may be explored to better detect

higher mode amplitudes for imaging.

Additionally, studies may investigate changes in the harmonic phase signals while
exciting the cantilever at a higher resonance frequency. It is already common practice in
atomic force microscopy to examine the phase (in addition to the amplitude) for sample
surface information when driving the cantilever at the fundamental resonance frequency

alone [3].

94

7. APPENDIX: COMPUTER SIMULATION CODE

The following two pieces of code were written to implement the elastic-beam model into
the Veeco AFM tapping mode simulation package. The following code calls numerous

other modules which are not included in this Appendix.

 

RunMeZSimulate.m — Code run before each simulation.

Declares and initializes cantilever and tip-specimen force parameters; Euler-

Bernoulli Beam Equation variables; other important parameters.

clear;
clear global;

global idx;

global index;

global tspan;

tspan = zeros(l,40002);

global w_L2

global w_L2dot
global Tip_Pos_Min
global Tip_Pos_Max
global Amplitude

///////// //////

global a0

o\°

the DMT model

constant,
global R % the tip radius,

global gamma % in DMT model

used in the DMT model

used in the DMT model
interatomic distance,
global Estar % the effective contact stiffness,

used in the DMT model
used in

global L % Length of the cantilever

global m
global rho
global A
global E
global I

o\° o\° o\°

o\°

o\°

global omega
global L1
global L2
acting point

o\°

o\°

95

density of the cantilever

area of the cross section of the cantilever
Young's modules of the cantilever

Inertia of the cantilever

Eigenfreguency of the cantilver
Position of the drive force acting point
Position of the tip—sample interaction

global
global

global
global

omega(i)

global
global
global
global
global

global
global
global
global
global
global

global

global
global
global

global
global
global
global
global
global
global
global

Pgain
Dgain

Number_of_Modes
Inderive

2004)

%
%

Some parameters are from Physical Review B 69,

O

/

Number_of_Modes 0 Number of the modes

TipPos %Tip position

zeta % the damping coefficient of each mode

KL % Used to calculate the eigenfrequencies,
= KL(i)“2/(L“2) * sgrt( E*I/(rho*A) );

phi_Ll % modeshapes of the cantilever at Ll

phi_L2 % modeshapes of the cantilever at L2

CurrentSamplePos % current Sample Position

Drive_omega

%

Drive_Force_Amp Amplitude of the driving force
IntegralOfModeShapes

d_on

d_off

gamma_l

h_l

IsMeniscusPresent

ResidueMode

Fcap
Fts
HolderPos

Inderive
Et

nuet

Es

nues

V_l

r_l

W

l;
l;

—
.n—

%2;

1;

085412
by Robert W Stark

Use non-normalized mode shapes to avoid numerical

difficulty.

0

/

0

Since normalized mode shapes would be very large and they

have unit

96

% of 1/sgrt(kg) , and modal coordinates would be very small
and they have

0

6 unit of m*sgrt(kg).

R = 208-009; %m % nm

Et = (1.3ell); %*(1e—15)*(1000); % mN/(nm)“2 % the
elastic moduli of tip, in DMT
nuet = 0.3; % no unit % the poisson rations

of tip, in DMT

L = (125e-6); %/(1e- 9); % nm

W= (35e—6); %/(le- 9); % nm

rho = (2.33e3); % *(1e- 9) 3; % kg/(nm)“3 density of
silicon

A = (4e— 6)*(35e-6); %/( (1e—9)“2 ); % (nm)A %4e—6 * 250e-
6;

m = rho*A; % kg/nm

E = (1.31ell); 6*(le-15)*(1OOO); % mN/(nm)“2

I = (1/12) * (35e-6) * ( (4e—6)“3 ); % / ((1e-9)“4) %
(nm) %(1/12)*(250e-6)*(4e-6)“3;

%Sample parameters

Es = 7e10; %*(1e-15)*(lOOO); % mN/(nm)“2 % the
elastic moduli of sample, in DMT
nues = 0.3; % no unit % the poisson rations

of sample, in DMT

gamma = (31e-3); %* (1e- 6); % (pJ)/(nm)“2
H = (6.4e—20); %/(l e- 2); % pJ
a0 = (1.65474e-10); %/(le-9); % nm % sgrt(

H/(24*pi*gamma) );

Estar = 1/( (1-nuet“2)/Et + (l-nues“2)/Es ); %*(1000); %
mN/(nm)“2

%Capillary force parameters

h_l=2e-lO; % / (le-9); %[nm] %Water film thickness
gamma_l=0.072; %*(1e-6); %[pJ/nmAZ] %liquid-vapor
interfaccial energy for water
r_l:(3*pi*gamma*R“2/Estar)A(1/3);

V_l=4*pi*R*h_l“2 + 4/3*pi*h_l“3 + 2*pi*r_l“2*h_l;

d_on=2*h_l; %Tip-sample distance to form the water
meniscus
d_off=V_l“(1/3)—1/(5*R)*V_l“(2/3); %Tip—sample distance

to break the water meniscus

97

IsMeniscusPresent 0;

L1 = (1/10)*L; % nm
L2 = L;% (0.999999)*L; % nm
[omega,KL] =

GetEigenFrequencies(Number_of_Modes,L,E,I,rho,A);

IntegralOfModeShapes = testInteg(KL,L);

Drive_omega = omega(Inderive);

phi_Ll = getModeShape(L1,KL,L);
phi_L2 getModeShape(L2,KL,L);

zeta(1:Number_of_Modes+1)=1/200;

//’////////’////’/’/ //

%totalSteps = 300000;

totalSteps = 40002; %100000;
%%%%%%%%%%%%

%Phase_of Tip = zeros(1, totalSteps);

Amplitude = zeros(1, totalSteps);
Tip_Pos_Max = zeros(1, totalSteps);
Tip_Pos_Min = zeros(1, totalSteps);
w_L2 = zeros(1, totalSteps);
w_L2dot = zeros(1, totalSteps);

98

o
9

Section 4).

sfcn_AFM(t,x,u,flag)

— S-function (“cantilever” block

[sys,x0,str,ts]

Simulates the cantilever as it oscillates in response to driving and tip-sample forces

and scans a simulated sample surface.

Sfcn AFM.m
switch flag

function

__ S __

__ e _.

__ m __

__ i __

__ t __

__ __

__ e __

= l __

__ p __

.r __ m =

\I __ a __

] _. S =

.r ) : __

.r g __ d __

\I a __ n __

l\ l __ a __

S f __ __

e (x __ r __

Z r __ S __

.l t _. n __

S S __ O __

e 2 __ i __

2 m __ t __

.l u __ i __

l n __ d __

a r __ n __

i v __ O __

t __ C __

i .r __ __ __

n .r ) __ l __

I \I U Q g __ a __

l u r n a __ .l __

d .. x i l __ t __

m X I h f __ .l __

I t t = n __

__ t I\ 0 d __ i __

0/0 0/0 0,0 /\ S n e = S __
06 06 .I. e t .1. __ __ e t . __ __
06 n 90 S t U 0 d __ __ Z S n __ __
0,0 O 06 t a p d n __ __ .1 e O __ __
% i 06 r d t a __ __ S 2 .l = __
0,0 t 0/0 r p u 0,0 90 0’0 0’0 h __ __ e .l t __ :
so a 06 t U O 06 06 n __ __ Z S C __ __
0,0 Z 06 S l l 0,0 e 0,0 .r U __ __ .l n __ __
0,0 .l 90 I 0,0 0’0 0,0 d 0/0 0’0 0,0 d 90 t 0/0 ] | __ __ l e U = __
o6 l o6 O o6 o6 m o6 o6 m o6 a % rL e r... __ __ a h f __ __
90 a 06 X 0,0 e 90 06 t % 06 n 06 S /\ __ __ i t . __ __
0/0 .l 0,0 .... 0/0 t 0’0 I __ 0’0 U 0,0 __ 0/0 .l 0,0 __ .l r __ = t S __ :
06 t 0,0 O S 90 a 90 2 06 p 06 3 90 m 06 9 w O __ __ .l n __ __
0/0 .l 0’0 V1 0,0 d 0,0 S 0’0 t 0’0 S 0/0 r 0,0 S r r = __ n r e __ __
90 n 06 e S 06 p 06 e y 06 U 0,0 e y 0,0 e 0,0 e y e r __ = I U h __ __
90 I % S .I. 06 U 06 S S % O % S S 06 T 06 S S h e __ __ l t t __ __
0,0 so a 06 06 a 06 06 a % o6 a t __ __ d e __ __
06 90 06 C 06 06 90 C o6 % 90 C % 06 06 C O .m "n z m R W. = n"

e 0’0 0’0 = 06 0,0 f 0,0 __ 0/0

99

function [sys,x0,str,ts]=mdlInitializeSizes();

O

6 New globals
global w_L2;
global w_L2dot;

global wl_L2;
global wl_L2dot;

global Tip_Pos_Min;
global Tip_Pos_Max;
global HolderPos;

global Drive_omega

global Amplitude;
global tspan

global Number_of_Modes;
global idx;

idx = 1;
tspan(l) = 0;
q = 0*ones(1,Number_of_Modes);

qdot = 0*ones(1, Number_of_Modes);

Tip position

w_L2(1) = 0' %
0; %Tip velocity

w_L2dot(1) =

wl_L2(l) = 0;
wl_L2dot(l) = O;

Tip_Pos_Max(l) = 0;

Tip_Pos_Min(1) = 0;

HolderPos = 0; %22.7e-9; %0; %5e-9;
Amplitude(l) = 0; %2e-11;

fai = O;

%%% Calculate the phase angle

fai = atan2(Drive_omega*(w_L2(l)-
(

%lOe—9;

%22.7e-9;

Tip_Pos_Max(1)+Tip_Pos_Min(1))/2), w_L2dot(1))—

Drive_omega*tspan(l);

100

fai = fai- floor(fai/(2*pi))*2*pi;
if (fai>pi)
fai = fai-2*pi;

end
fai = fai- pi/2.0;
Phase_of_Tip(l) = fai;
sizes = simsizes; %l
sizes.NumContStates = O;
sizes.NumDiscStates = Number_of_Modes*2+5+3;
sizes.NumOutputs - 8; %4; %4; %Number_of_Modes*2+2+l;
sizes.NumInputs = 3;
sizes.DirFeedthrough = 0;
sizes.NumSampleTimes = l;
sys = simsizes(sizes); %1x7
for k = 1:5
w_L(k) = 0;
end
x0 = [transpose([q, qdot]); transpose(w_L); w_L2dot(1);
Phase_of_Tip(l); Amplitude(1)];
str = l];
ts = [Se-8 O];

o\°

end mdlInitializeSizes

’_—————’——_————————————__-——_———_—————————m

_——¢—_————#———_—_———
———_————¢_—_———_——_——

% mdlUpdate
% Handle discrete state updates, sample time hits, and
major time step

0

6 requirements.

/

I-h o\°

unction sys = mdlUpdate(t,x,u);

global TipPos %Tip position

global phi_L2 % modeshapes of the cantilever at L2
global CurrentSamplePos % current Sample Position

global Drive_omega

global Drive_Force_Amp % Amplitude of the driving force

101

global IntegralOfModeShapes
global HolderPos

% New globals

global w_L2

global w_L2dot

global wl_L2
global wl_L2dot

global Tip_Pos_Min
global Tip_Pos_Max
global Amplitude

global idx
global tspan

global Number_of_Modes

Drive_Force_Amp = u(l);
HolderPos = u(2);
CurrentSamplePos = u(3);

q = transpose(x(1:Number_of_ModeS))i % 1x2
qdot = transpose(x(Number_of_Modes+l:Number_of_Modes*2));
% 1x2

idx = idx+1;
tspan(idx) = t;

[q, qdot] = rk4(2, tspan(idx-l),
tspan(idx),q,qdot,Number_of_Modes,0); %sys =
transpose(RK4(1,idx-1,idx+l,q,qdot,Number_of_Modes,O)); %
DOESN'T WORK

%InteractionForce = Fts;
%CapillaryForce = Fcap;

TipPos = O;
w_L2dot(idx) = 0;

for j = l:Number_of_Modes
TipPos = TipPos + phi_L2(j)*q(j);
w_L2(idx) = TipPos;
w_L2dot(idx) = w_L2dot(idx) + phi_L2(j)*qdot(j);

102

6%666 Calculate the contour of the vibration

if w_L2dot(idx-1)<=0 && w_L2dot(idx)>0
Tip_Pos_Min(idx—l) = w_L2(idx—1);

end

if w_L2dot(idx—1)>=0 && w_L2dot(idx)<0

Tip_Pos_Max(idx-1) = w_L2(idx-1);
end
Tip_Pos_Min(idx) = Tip_Pos_Min(idx-l);
Tip_Pos_Max(idx) = Tip_Pos_Max(idx-1);
Amplitude(idx) = Tip_Pos_Max(idx) - Tip_Pos_Min(idx);
%1.15e—8;

%%% Calculate the phase angle

fai = atan2(Drive_omega*(w_L2(idx)-
(Tip_Pos_Max(idx)+Tip_Pos_Min(idx)1/2), w_L2dot(idx))—
Drive_omega*tspan(idx);

fai = fai- floor(fai/(2*pi))*2*pi;

if (fai>pi)

fai = fai-2*pi;
end
fai = fai- pi/2.0;
Phase_of_Tip(idx) = fai;

o\°

%

o\°

for k = 1:2 %1:Number_of_Modes
w_L(k) = phi_L2(k+5)*q(k+5);

end
for k = 3:5
w_L(k) = 0,
end
sys = [transpose([q, qdot]); transpose(w_L); w_L2dot(idx);

Phase_of_Tip(idx); Amplitude(idx)];

I

—————o—————————————_——-
__.—_—————-—o-—o_—._——_————_—.

9 mleutputs

103

% Return the output vector for the S—function

function sys = mleutputs(t,x,u)

global Number_of_Modes;
global idx;
global HolderPos;

M1 = zeros(8, 2*Number_of_Modes);
M2 eye(8,8); %[l 0 O O; 0 1 0 O; 0 O 1 O; O 0 0 1];
sys = [M1, M2]*x;

% end mleutputs

104

8. REFERENCES

[1] Sahin, 0., Quate, C. F., Solgaard, 0., and Atalar, A., “Resonant harmonic response in
tapping-mode atomic force microscopy,” Physical Review B, vol. 69, 2004.

[2] Zypman, F. R. and Eppell, S. 1., “Analysis of scanning force microscope force-
distance data beyond the Hookian approximation,” Journal of Vacuum Science &
Technology B: Microelectronics and Nanometer Structures, vol. 16, no. 4, pp.2099-2101,
July 1998.

[3] Balantekin, M. and Atalar, A., “Enhancing higher harmonics of a tapping cantilever
by excitation at a submultiple of its resonance frequency,” Physical Review B, vol. 71,
no.12,2005.

[4] Zhang, J., Xi, N., Li, G., and Su C., “Atomic Force Microscopy Sensing Using
Multiple Modes,”.

[5] Hillenbrand, R., Stark, M., and Guckenberger, R., “Higher harmonics imaging in
tapping-mode atomic-force microscopy: Insights into the tip-sample interaction,” Applied
Physics Letters, vol. 76, no. 23, June 2000.

[6] Stark, R. W. and Heckl, W. A., “Higher harmonics imaging in tapping-mode atomic-
force microscopy,” Review of Scientific Instruments, vol. 74, no. 12, Dec. 2003.

[7] Rodriguez, T. R. and Garcia, R., “Compositional mapping of surfaces in atomic force
microscopy by excitation of the second normal mode of the microcantilever,” Applied
Physics Letters, vol. 84, no. 3, Jan. 2004.

[8] Rabe, U., J anser, K., and Arnold, W., “Vibrations of free and surface-coupled atomic
force microscope cantilevers: Theory and experiment,” Review of Scientific Instruments,
vol. 67, no. 9, pp. 3281-3293, Sept. 1996.

[9] Garcia, R. and San Paulo, A., “Attractive and repulsive tip-sample interaction
regimes in tapping-mode atomic force microscopy,” Physical Review B, vol. 60, no. 7,
Feb. 1999.

105

   

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