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Three-dimensional measurement of atomic force microscope cantilever deformation to determine the three-dimensional appli...

Permanent Link: http://ufdc.ufl.edu/UFE0042140/00001

Material Information

Title: Three-dimensional measurement of atomic force microscope cantilever deformation to determine the three-dimensional applied force vector
Physical Description: 1 online resource (76 p.)
Language: english
Creator: Kumanchik, Lee
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: afm, atomic, cantilever, deformation, force, interferometry, light, metrology, microscope, netwon, realization, scanning, standards, stiffness, white
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The atomic force microscope (AFM) is the instrument of choice for measuring nano- to micro-Newton forces (10^-9 to 10^-6 Newtons). However, calibration is required for accurate measurements. AFM calibration has been studied for decades and remains a significant focus within the metrological community, in particular at international standards organizations. While progress has been made, there is still much to accomplish as current force calibration techniques yield relative uncertainties (plus or minus1 standard deviation/mean) of 10%-20%. For example, measuring a force of 500 nN would yield a result between 400-600 nN 68% of the time. The critical issue is the existing AFM metrology, which monitors deformation at a single (spatial) point on a structure that encounters a three-dimensional (3D) force and responds with a 3D deformation. This single-point calibration technique considers only to a limiting set of information, while additional information is available. Similarly, subsequent measurements by the AFM after calibration are restricted to the same limits. As a response, this project aims to improve AFM calibration and use by implementing a new metrological platform and analysis technique. The new platform incorporates a scanning white light interferometer (SWLI) for 3D cantilever deformation measurements. The SWLI introduces two important changes over standard AFM metrology. First, it provides a multi-point measurement of the backside surface of the cantilever rather than a single-point measurement near the free end. Second, it is a direct displacement sensor which does not infer displacement from the measurement of another variable, such as the surface angle in the optical lever technique. In this study, the AFM is first described with a focus on its use as a force sensor. Then, the new platform design and construction, cantilever imaging tests, and the development of a new force model, which takes advantage of the 3D deflection data, are presented. The new force model addresses many of the challenges associated with traditional calibration strategies. Experimental validation is presented for the cases of normal force loading (i.e., perpendicular to the cantilever axis and resulting in bending deformation) and torsional loading.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Lee Kumanchik.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Schmitz, Tony L.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042140:00001

Permanent Link: http://ufdc.ufl.edu/UFE0042140/00001

Material Information

Title: Three-dimensional measurement of atomic force microscope cantilever deformation to determine the three-dimensional applied force vector
Physical Description: 1 online resource (76 p.)
Language: english
Creator: Kumanchik, Lee
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: afm, atomic, cantilever, deformation, force, interferometry, light, metrology, microscope, netwon, realization, scanning, standards, stiffness, white
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The atomic force microscope (AFM) is the instrument of choice for measuring nano- to micro-Newton forces (10^-9 to 10^-6 Newtons). However, calibration is required for accurate measurements. AFM calibration has been studied for decades and remains a significant focus within the metrological community, in particular at international standards organizations. While progress has been made, there is still much to accomplish as current force calibration techniques yield relative uncertainties (plus or minus1 standard deviation/mean) of 10%-20%. For example, measuring a force of 500 nN would yield a result between 400-600 nN 68% of the time. The critical issue is the existing AFM metrology, which monitors deformation at a single (spatial) point on a structure that encounters a three-dimensional (3D) force and responds with a 3D deformation. This single-point calibration technique considers only to a limiting set of information, while additional information is available. Similarly, subsequent measurements by the AFM after calibration are restricted to the same limits. As a response, this project aims to improve AFM calibration and use by implementing a new metrological platform and analysis technique. The new platform incorporates a scanning white light interferometer (SWLI) for 3D cantilever deformation measurements. The SWLI introduces two important changes over standard AFM metrology. First, it provides a multi-point measurement of the backside surface of the cantilever rather than a single-point measurement near the free end. Second, it is a direct displacement sensor which does not infer displacement from the measurement of another variable, such as the surface angle in the optical lever technique. In this study, the AFM is first described with a focus on its use as a force sensor. Then, the new platform design and construction, cantilever imaging tests, and the development of a new force model, which takes advantage of the 3D deflection data, are presented. The new force model addresses many of the challenges associated with traditional calibration strategies. Experimental validation is presented for the cases of normal force loading (i.e., perpendicular to the cantilever axis and resulting in bending deformation) and torsional loading.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Lee Kumanchik.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Schmitz, Tony L.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042140:00001


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THREE-DIMENSIONAL MEASUREMENT OF ATOMIC FORCE MICROSCOPE
CANTILEVER DEFORMATION TO DETERMINE THE THREE-DIMENSIONAL
APPLIED FORCE VECTOR


















By

LEE KUMANCHIK


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2010
































2010 Lee Kumanchik

































To my mom and dad









ACKNOWLEDGMENTS

I thank my parents, Dwight and Bunny Kumanchik, for their support and for

encouraging my dreams. I thank my love, Carolina Cardoza. If I have fire in my eyes it's

because she lit fire under my feet. I thank my three aces, Steve, Alex, and Joel, who

were just good old-fashioned entertainment. I thank the members of my committee and

my advisor, Tony Schmitz, who is the most patient man I know. And I thank my peers in

the Machine Tool Research Center for their shenanigans.









TABLE OF CONTENTS

page

ACKNOWLEDGMENTS.............................................................. .......

LIST O F TA BLES ................................. .......................... ...... ..............7

LIST O F FIG U R ES .................................. .......................... ..... ..............8

LIST O F A BBR EV IA T IO N S ..................................................................................... ............ 10

A B S T R A C T .................................11.......... .................

CHAPTER

1 IN T R O D U C T IO N .......................................................................................... ........ 13

The Atom ic Force M microscope ....................................... .... ...........................13
T h e D e te c to r ............................................................................................................... 1 5
The C antilever..................................................... 18
C antileve r D im e ns io ns ......................................................................................... 2 0
Force Application Location ..... ..................................... ................21
Boundary C conditions ...................................... ........ ............................... 22
T ra c e a b ility ........................................... ............................................... 2 3
T he F force E equation .... ........................................................... ........ ... .... 24

2 LITERATURE REVIEW ..................... ................ ................27

H historical D evelopm ent ................................. ...... ...... .......... .............. 27
Applications of the Atomic Force Microscope .......................... ...............28
F o rc e M od e lin g ......... ............ .... ........... ............................................... 2 9
C a lib ratio n .............. .. ..... ............. ....................................2 9

3 THREE-DIMENSIONAL IMAGING ................. .............. ......... 33

A New Platform ........................... ..................33
Scanning White Light Interferometer.......................................33
Cantilever Holders ................... ....... ................ 35
S ta g e s ......... .. ........ ..................................... .................. ..... 3 6
P platform A ssem bly .......................... .................................. .. ............. 37
Visualization of Q uasi-static Bending....................................................... 38
Residual Stress, "Batwings", and Differencing ............................. 38
S n a p -in .................. ................. ......... ................ ..... ................ 4 0
O their Experim ents ............................... ............. ................ .. 41

4 THREE-DIMENSIONAL FORCE MODELING ............ .... ......... ........ 44









5 VALIDATION OF THE FORCE MODEL.......................................... ..................... 50

Normal Loading ................ ........ ..... ........... 50
Torsion Loading ....................... .................. 58
Uncertainty Analysis ........................... ......... ......... 63

6 CONCLUSIONS ........................................................................... ......... ...................... 68

S u m m a ry ............. .. ................................................................................. ......... 6 9
F future W ork ......... ...... ............................................................................ 70

LIST OF REFERENCES ............. ..... ...... ........................ ..... ...........71

BIO G RA PH ICA L SKETC H .............................................................. ...... ......... 76






































6









LIST OF TABLES

Table page

5-1. C antilever geom entries. .................................... .... ...... .. .... .. .......... ..... 59

5-2 U uncertainty contributions. ......... .............. ............................................................ 64









LIST OF FIGURES

Figure page

1-1. Schematic of atomic force microscope (AFM) operation................ ..... .......... 13

1-2. The optical lever technique.... ...................................... ................ ............... 15

1-3. The small angle approximation .............. ............ ........ ....................... 16

1-4. A perfectly smooth sample slides past the stylus....................................... 17

1-5. Using the optical lever technique to detect twist........................... ........................ 19

1-6. A scanning electron micrograph of a commercial cantilever...............................20

1-7. C antilever stiffness transfer. ........... ........... ...................... .......... ...... 21

1-8. Foreshortening from vertical bending ........................... .. ................. ................ 23

1-9. Cantilever with a friction force proportional to the applied force, F .................... 24

3-1. Prototype platform assembly ...... .... ............. .. .. ............ .... 33

3-2. SW LI schematic. ............... .............. ............... ................... 34

3-3. Example cantilever placement on the aluminum holders (top views)....................36

3-4. Positioning stage scale error. .................................... .............. .... 37

3-5. The Olym pus OMCL-AC240TS cantilever. .............. .......................... ........ ........38

3-6. Olympus BL-RC150VB AB cantilevers. ................................ ............. 39

3-7. Cantilever snap-in demonstration........... ... .............................. ................ 41

3-8. Two Veeco 1930 (1.3 N/m, 35 pm wide) cantilevers in contact............................41

3-9. D ata dropout. ................................... ............................. ..... ..........42

3-10. Tip w ear estim action ................ ................................... .. ..... ................. 43

3-11. Anom alous bending behavior. .................................................... .................. 43

4-1. A cantilever loaded at the free end by two independent forces P and F ...............45

4-2. Force measurement does not depend on the global geometry of the cantilever... 45

4-3. The 3D view of the forces acting on the cantilever.................. ........ .........46









4-4. SWLI measurement coordinates. ................................................................. 48

5-1. SWLI image of the fabricated cantilever. .............................................. 50

5-2. Fabricated masses........................................................................ 51

5-3. Results of Equation 5-1 for the 10 masses ............................................ 53

5-4. Beam loaded under identical conditions, but measurement field of view was
v a rie d ............. ......... .. .. ............. ... ........................................ .. .. 5 4

5-5. The aluminum "tee" bonded to a 1.5 mm wide cantilever. ...................................56

5-6. Results of Equation 5-1 for the 8 masses using the surface fit approach. .............57

5-7. "T ee" attached to cantilever ......... ................. ............................... ................ 58

5-8. Torsion results when mass is loaded at the positive torque arm .......................... 60

5-9. Torsion results when mass is loaded at the negative torque arm........................61

5-10. Torsion loading results for mass loaded at the positive and negative torque
a rm s ......... ...... ............ .................................... ........................... 6 2

5-11. Normal loading results for mass loaded at the positive and negative torque
a rm s ......... ...... ............ .................................... ........................... 6 2









LIST OF ABBREVIATIONS

AFM Atomic force microscope

STM Scanning tunneling microscope

SWLI Scanning white light interferometer

SI Systeme International d'Unites

3D Three-dimensional









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

THREE-DIMENSIONAL DEFORMATION MEASUREMENT OF THE ATOMIC FORCE
MICROSCOPE CANTILEVER SENSOR FOR THE DETERMINATION OF THE
APPLIED FORCE VECTOR

By

Lee Kumanchik

August 2010

Chair: Tony Schmitz
Major: Mechanical Engineering

The atomic force microscope (AFM) is the instrument of choice for measuring

nano- to micro-Newton forces (10-9 to 10-6 Newtons). However, calibration is required

for accurate measurements. AFM calibration has been studied for decades and remains

a significant focus within the metrological community, in particular at international

standards organizations. While progress has been made, there is still much to

accomplish as current force calibration techniques yield relative uncertainties (1

standard deviation/mean) of 10%-20%. For example, measuring a force of 500 nN

would yield a result between 400-600 nN 68% of the time. The critical issue is the

existing AFM metrology, which monitors deformation at a single (spatial) point on a

structure that encounters a three-dimensional (3D) force and responds with a 3D

deformation. This single-point calibration technique considers only to a limiting set of

information, while additional information is available. Similarly, subsequent

measurements by the AFM after calibration are restricted to the same limits. As a

response, this project aims to improve AFM calibration and use by implementing a new

metrological platform and analysis technique.









The new platform incorporates a scanning white light interferometer (SWLI) for 3D

cantilever deformation measurements. The SWLI introduces two important changes

over standard AFM metrology. First, it provides a multi-point measurement of the

backside surface of the cantilever rather than a single-point measurement near the free

end. Second, it is a direct displacement sensor which does not infer displacement from

the measurement of another variable, such as the surface angle in the optical lever

technique. In this study, the AFM is first described with a focus on its use as a force

sensor. Then, the new platform design and construction, cantilever imaging tests, and

the development of a new force model, which takes advantage of the 3D deflection

data, are presented. The new force model addresses many of the challenges

associated with traditional calibration strategies. Experimental validation is presented for

the cases of "normal" force loading (i.e., perpendicular to the cantilever axis and

resulting in bending deformation) and "torsional" loading.









CHAPTER 1
INTRODUCTION

The Atomic Force Microscope

The atomic force microscope (AFM) is a multipurpose instrument used for

interacting with a sample at an atomic scale (including imaging, force interaction, and

manipulating individual atoms). Interaction is performed by an atomically sharp stylus

that is driven by stages to a location of interest on a sample. The small tip radius of the

stylus (<30 nm or approximately the radius of a virus) focuses the interactions onto a

small (nearly point) area enabling extremely fine spatial resolution. The stylus is

attached to a cantilever that is monitored by a detector as it deforms under the

interaction forces (Figure 1-1). Although the entire cantilever deforms, the detector

typically determines the motion at or near the stylus (located at the free end of the

cantilever) only. This is a natural choice since the AFM can perform imaging with the

stylus motion having a 1:1 correlation (ideally) to a sample's surface contours. When

used for force measurement, the stylus motion follows Hooke's law, F=ky, where F is

the vertical force, k is the cantilever bending stiffness, and y is the stylus vertical

displacement. Using an AFM, force resolution as low as 1 pN (10-12 N) can be realized.

atomically detector
sharp stylus observes
i deformation




cantilever
flexure
canti---sample


Figure 1-1. Schematic of atomic force microscope (AFM) operation. Forces at the
stylus from the moving sample cause deformation in the cantilever. A detector
observes the deformation.









Due to the AFM's popularity as a multi-function instrument, it is applied across a

diverse range of disciplines where users often may not possess detailed knowledge of

the underlying mechanics. Therefore, vendors strive for "turn key" system operation.

However, achieving the required accuracy under this paradigm can be challenging. For

example, cantilever stiffness studies have found that the manufacturer-specified

stiffness can vary by as much as 300% from the calibrated stiffness (0.2 N/m nominal

stiffness vs. 0.067 N/m calibrated stiffness) [1]. One study combined all manufacturer

specifications required for force measurement (including stiffness) and found the

combined uncertainty was greater than 1500% [2]. While this level of divergence may

not always be present, calibrating the AFM is clearly required to obtain meaningful

results. However, after decades of calibration research, the best stiffness calibrations,

performed by international standards agencies, are accurate to only 5% [3]. Using

methods available to the average user, the accuracy of stiffness calibration is limited to

10%-20%, in general [2,4]. Because stiffness is one of the parameters required to

determine force, the accuracy of the force measurement will be no better than the

stiffness accuracy and, most likely, will be worse.

Since the force is computed with Hooke's law (F=ky), there are two components to

a force calibration: 1) stylus displacement calibration; and 2) stiffness calibration, with

the majority of research focused on stiffness calibration. This force relation, however, is

too simplistic. Cantilever mechanics studies have shown how cross-axial sensitivity to

forces perpendicular to the measurement direction lead to a systematic error between

10%-20% [5]. This error is independent of calibration. While compensatory post-

processing techniques are available [6], they are largely unused for several reasons.









First, their complexity requires expertise in mechanics to understand and programming

knowledge to implement. Second, they require the geometrical parameters of the AFM

setup (which are often difficult to obtain) and knowledge of the stylus-sample

interactions (which are typically unknown a priori. Third, they continually evolve and

must be updated every couple of years [5-8]. Therefore, the simple force equation,

F=ky, remains in use. In a comprehensive review (1254 cited references) of AFM

calibration and implementation in numerous fields, there was no mention of any

alternative force relation to F=ky [9]. In the following sections, force measurement by

AFM is discussed in three parts: the detector, the cantilever, and the force equation.

The Detector


laser source segmented position 1 position 2




< position 2
S1 S1
cantilever position 1 segmented photo-detector


Figure 1-2. The optical lever technique. A laser is reflected off a cantilever onto a
photo-detector. The light level on each segment of the photo-detector (S1 and
S2) represents the amount of cantilever deflection.

The stylus displacement (perpendicular to the cantilever axis in the "normal" force

direction) is monitored by a detector (represented by y in F=ky). The most popular

detection system in use today is the optical lever technique (Figure 1-2). Here, a laser is

reflected from the backside of a cantilever onto a two-segment photo-detector. As the

cantilever deforms, the angle of incidence changes and the reflected laser beam moves








along the photo-detector segments. The light levels on the segments are directly related

to the angle of incidence of the cantilever (after compensating for photo-detector non-

linearity). The longer the optical path, the more pronounced the effect; this amplification

follows the simple lever rule. Though detection is directly related to angle, it is difficult to

calibrate for angular motion. Additionally, in imaging mode and when applying Hooke's

law, the vertical displacement (not cantilever angle) is required. Therefore, a rigid

surface calibration is typically performed. Here, the stylus is brought into contact with a

rigid sample and then the sample stage is displaced vertically. Using the known

commanded displacement, the small angle approximation can be used to determine a

rate of displacement directly proportional to the angular change of the AFM cantilever

(Figure 1-3). In this way, the stylus vertical displacement accuracy is approximately

equal to the stage vertical displacement accuracy.


ZAO c Ax


Ax
I IT


Figure 1-3. The small angle approximation. Small deformations enable the cantilever
angular change to be related to the stage vertical motion.

While this detection system is simple in practice, it suffers from a critical drawback.

Since it indirectly measures displacement through angular change, any force which

causes the cantilever to deform without displacing the stylus vertically will still register

as stylus motion. For example, a sample moving parallel to the beam axis while in

contact with the stylus will generate a kinetic friction force that will tend to deform the

cantilever flexure (Figure 1-4). Though the sample may be perfectly smooth (no









topography), the lateral friction will register as vertical motion due to the change in

reflected beam angle. This phenomenon has been exploited in tribology since bi-

directional motion registers as complementary positive and negative detector output that

relates to friction [10]. As an alternative to motion parallel to the beam axis,

perpendicular motion can also be used to measure friction. This motion causes beam

twist (torsional loading) and can be measured using a quadrant photo-detector (Figure

1-5). The arrangement of four total segments isolates twist from stylus vertical

displacement so that topography is not registered as friction in the twist direction. The

challenge is




sliding friction
causes
cantilever -
deformation
moving sample



Figure 1-4. A perfectly smooth sample slides past the stylus. The resulting friction force
deforms the cantilever which registers as motion by the detector.

that twist is a measure of angle and therefore requires a more complex calibration

compared to the rigid surface calibration used for displacement. Additionally, twist angle

is proportional to torsion (i.e., the product of the friction force and stylus length), not just

the frictional force. Therefore, the stylus length (or height) must be measured to

calculate the frictional force.

Calibrating the detector for every measurement is essential for obtaining good

force accuracy. In practice, however, this may not be practical. In the absence of

comprehensive calibration, misalignment between the detector and the cantilever may









go undiagnosed. For the optical lever technique, the angle of incidence determines the

displacement, which imparts some robustness to misalignment. The angular deflection

equation for a rectangular cantilever with an end force and its sensitivity to

measurement location are,

3F oa 2Lx-2x2 ax
(x) = (2Lx 2) & ao 2 2 (1-1
2L0k 0 2Lx-x2 x

where L is the cantilever length, x is the location along the beam axis (0 x x 5 L), and e

is the angle of the cantilever. Misalignment of the laser from the calibrated position

gives an error rate in displacement measurement, x:y, between 1:1 and 1:0 from the

base to free end, respectively. For example, a system calibrated for x = L, but

misaligned by a few micrometers, would introduce virtually no error in corresponding

displacement measurements due to this misalignment. Since this is the most common

position to point the laser, the optical lever technique is fairly robust to misalignment.

For AFM systems with direct displacement sensors such as displacement measuring

interferometers, alignment errors are not mitigated but instead amplified. The deflection

equation and sensitivity to measurement location for displacement are,

y(x) = (3Lx2 3) & 3 2Lx2 -x3 (1-2)
(3L LY = 3 (1-2)
2L3k y 3Lx2-x3 x

Misalignment of the detector from the calibrated position gives an error rate, x:y,

between 1:3 and 2:3 from base to free end with the free end being the best position.

The Cantilever

The cantilever is an interchangeable component in the AFM system. Cantilevers

are made with different geometries and tips (in addition to sharp styli, spheres are also

often used, for example) to suit many applications with the same AFM system.

Regardless of the cantilever manufacturer (often different from the AFM system









manufacturer), the bending stiffness is always specified for use in the force equation (k

in F=ky). However, the stiffness is extremely sensitive to cantilever geometry and,

therefore, must be calibrated on an individual basis. Potentially the most confounding

issue faced by stiffness calibration is that the stiffness of a generalized beam depends

on the boundary conditions. For example, a fully constrained (fixed/fixed) beam has four

times the stiffness of a cantilever (fixed/free) beam. For the AFM cantilever, the stiffness

estimation is naturally based on the cantilever model; the complexities that arise from

the stylus contact mechanics, which alter the system stiffness, are left for the user to

handle.


Front view position
laser laser
beam beam


photo-
V detector



cantilever N
axis
position

Figure 1-5. Using the optical lever technique to detect twist. The left-to-right motion of
the laser registers as twist. The top-to-bottom motion registers as vertical
deflection, as shown previously.

There are numerous stiffness calibration techniques and each has benefits and

drawbacks. Non-contact techniques avoid stiffness altering contact mechanics, but do

not take into consideration the load application location. Contact techniques are often

simpler conceptually and more convenient for achieving traceability to international









standards, but must compensate for contact behavior. Rather than list the techniques

here (see Chapter 2), the largest sensitivities to stiffness are described.

Cantilever Dimensions

A cantilever's dimensions define its stiffness. From solid mechanics, a rectangular

cantilever beam can be modeled using Euler-Bernoulli beam theory. At the stylus, the

deflection equation is,

k
4F rearrange Ew )3 (1-3)
Ew 4

where E is Young's modulus for the beam material, w is the beam width, L is the beam

length, and t is the beam thickness. Rearranging yields an expression for the stiffness

according to Hooke's law (analytical models exist for other beam geometries and can

similarly be used to estimate stiffness). Direct measurement of each term provides the

best result so that the stiffness uncertainty is a combination of the measurement



















Figure 1-6. A scanning electron micrograph of a commercial cantilever. Extra material
is cut away during manufacturing. This undercut is not visible in a top-down
view of the cantilever.









uncertainties in each input and the model accuracy (i.e., considering the limitations

imposed by the model assumptions). In this example, the most sensitive parameters are

thickness and length. Thickness of silicon substrates (the typical AFM cantilever

material) is difficult to predict from manufacturing and should be measured directly.

Length can vary from nominal due to undercut at the base when using silicon-based

manufacturing techniques (Figure 1-6). Assuming all other quantities are perfectly

known, there is approximately a 1:3 relationship between the error in length or thickness

to the error in stiffness. For example, a 1% error in thickness or length leads to an error

in stiffness of ~3% (30 nm thickness error or 2 pm length error on a 3 pm x 200 pm

cantilever with a nominal stiffness of 6 N/m gives a stiffness error of 0.18 N/m).

Force Application Location


pre-calibrated
V cantilever


A c v e effective
--- L -----_V



Figure 1-7. Cantilever stiffness transfer. The AFM cantilever is pushed against the pre-
calibrated cantilever. The force application point on the AFM cantilever is
precisely located at the stylus. However, the force application point on the
pre-calibrated cantilever is subject to the position accuracy of the AFM. The
effective length of the pre-calibrated cantilever determines the measured
stiffness.

A more direct technique to determine stiffness is to apply a known force to the

cantilever and measure the displacement so that k=F/y. For relatively stiff cantilevers, a

mass can be attached to the free end to apply a known (gravity-based) force. Here, the

force application point requires precise alignment to the stylus location, where the

cantilever will experience all subsequent forces. The sensitivity to alignment errors is









identical to the length sensitivity with the same 1:3 relationship. For mass loading, there

is a fundamental limitation imposed by the availability of traceable mass artifacts smaller

than 1 mg. Therefore, a variation of the force loading scheme, referred to as a stiffness

transfer, is used where the unknown stiffness of a cantilever is related to the pre-

calibrated stiffness of a second cantilever. The result is similar to a system of two

springs in series such that kI=k2(xt/x-1) where k, is the unknown stiffness, k2 is the

known stiffness, xt is the total motion as one cantilever is pressed against the other, and

x, is the response motion of the unknown cantilever. Here, the force application point on

the unknown cantilever is determined precisely since the stylus makes direct contact

with the cantilever of known stiffness. However, the force location on the second

cantilever is again subject to alignment errors and results in an error in stiffness at a

rate of 1:3 (Figure 1-7).

Boundary Conditions

For contact-style stiffness calibrations, as well as any use of an AFM cantilever

after calibration, boundary conditions can contribute considerable measurement

uncertainty. Boundary conditions define the deflection equation (the equations

presented so far have been for an ideal cantilever with a vertical load at the stylus). This

simplification fails when the cantilever comes into contact with a sample, either during

contact style calibrations or when used to perform measurements. An example contact

phenomenon is friction, as discussed previously. Foreshortening occurs whenever the

cantilever deforms (Figure 1-8). The foreshortening is resisted by friction parallel to the

sample surface. As an example, the stylus vertical motion for a cantilever experiencing

sliding friction proportional to the vertical force is,










y = F (1-4)

system compliance

where p is the coefficient of kinetic friction and r is the stylus height. Assuming this to be

the only boundary condition effect, the ideal stiffness k can be obtained if the coefficient

of sliding friction is known as well. However, this is not typically the case. Boundary

conditions are an important concept for this research and are discussed in Chapter 4.










Figure 1-8. Foreshortening from vertical bending.

Traceability

Proper calibration of stiffness should follow a clear path of traceability to the

System International d'Unites (SI). Traceability is the chain of calibrations that link any

sensor/instrument through a direct path, with defensible uncertainty statements for each

measurement, to a primary standard. This primary standard, which reflects a physical

realization of the unit of measure, is defined and maintained (if necessary) by national

measurement institutes (NMIs). Traceability enables measurement uncertainty to

propagate through the chain of calibrations and thus identifies the uncertainty for

measurements performed by a given sensor or instrument. This notion of standardized

units and measures allows quantitative data to be compared between organizations,

manufacturers, and laboratories, facilitating collaboration and reproducibility of results.

Traceability is sometimes difficult to obtain depending on the calibration technique.









Therefore, the approach adopted by the National Institute of Standards and Technology

(the national measuring institute for the US) has been stiffness transfer by cantilever

artifacts. This technique has been traceably compared to the geometrical model as well

as the popular thermal calibration technique [11] and agreement within ~5% has been

obtained [4].

The Force Equation



fixed-free cantilever




J F



Figure 1-9. Cantilever with a friction force proportional to the applied force, F.

As noted, the most popular approach for AFM force measurement is to apply

Hooke's law, F=ky The choice of this equation stems from the single point detection

scheme used to monitor the cantilever deformation. With only a single point of

displacement, the force transduction constant must be stiffness (N/m). The general

Euler-Bernoulli beam equation relating a beam's deflection to the applied load is,

a2 a 2
-a l E )= u (1-5)
8X2 ax2

where x is the distance along the beam axis, y is the deformation perpendicular to the

beam axis, I is the second moment of area, and u is a distributed load. Four integration

are necessary to obtain the force-displacement equation, which requires four boundary

conditions. Because the moment boundary condition is impossible to know without









detailed knowledge of the contact mechanics, the challenge of force measurement by

AFM can be summarized as attempting to solve an underdetermined equation.

As an exercise, the beam equation is solved assuming friction proportional to the

applied force at the tip in order to illustrate how F=ky can be replaced with an improved

force-displacement relationship (Figure 1-9). Assuming y=y(x), u=O, and El is constant,

integration is straightforward (El d4y/dx4=0) and yields four equations,

-F
d3y
El = C (1-6)
d2y
El = Cx + C2 = M (1-7)

dy X2
El = C + C2+ C (1-8)
dx 2
=0
X3 X2
Ely = C + C2 + Cx + C (1-9)

where C3-4 are zero due to the fixed base boundary condition (zero motion and slope at

the base), C, is equal to the perpendicular force component (by force balance with

shear force), and Equation 1-7 is equal to the bending moment M along the beam. The

bending moment equation captures the friction contact mechanics and is,

M = F(L x) + uFr (1-10)

where p is the coefficient of sliding friction and r is the tip (stylus) height. The resulting

deflection equations for any point on the beam and for the tip location (x=L) are,

y(x) = (3Lx2 x3) + IFrX2
6El 2El
FL3 3 (1-11)
y(L) = 1+ ) F (1-11 F

system compliance

The addition of friction makes the force equation more complex than F=ky. In practice,

this friction analysis is too simple. Under the assumption that macroscopic friction









applies to the atomically sharp stylus, there must be static friction as well. Unlike kinetic

friction, static friction is not a proportional relationship to normal force except at its

maximum. Therefore, measurements performed where there is no sliding (such as part

of stick-slip events) include an unknown quantity in the final displacement equation,

Gmax=IstaticF

y(x) = (3Lx2 x3) + -GmaxGmax 2 (1-12)
6EI 2EI

where G is the friction resisting sliding. Simplifying to the tip location (x=L) does not give

a clear relationship between the applied force F and the tip motion y(L). When the

macroscopic friction model is insufficiently accurate or when other forces contribute,

such as electrostatic, osmotic, and/or magnetic forces, this entire analysis is invalid.

Ambiguities like this will always exist in the force equation if a force relationship is

unknown a priori, which is true in virtually every measurement situation. This makes

exploratory force measurements particularly difficult since the environment has yet to be

characterized. Therefore, a new force measurement paradigm is necessary to move

toward more accurate force metrology.









CHAPTER 2
LITERATURE REVIEW

Historical Development

The atomic force microscope was introduced by Binnig, et al., in 1986 to measure

micro-scale forces [12]. Just prior to the invention of the AFM, the scanning tunneling

microscope (STM) [13] was developed. This was a surface profilometer which used the

strong variation in electron tunneling current with separation distance to map a sample

surface. It only worked on conductive samples. The AFM was developed to expand the

types of surfaces that could be probed. The idea was to use a compliant structure with a

sharp tip that, when probing a surface, would deform from forces applied to the tip.

Since all materials generate reaction forces to the probe's applied pressure, the AFM

was not limited to imaging conductive materials. More importantly, force could be

inferred from the deformation of the compliant structure. Initially, a simple rectangular

cantilever was utilized as the compliant structure and a scanning tunneling microscope

was used as the cantilever displacement sensor.

Since 1986, the AFM cantilever has been adapted to suit a wide range of

applications. For example, two-arm triangular cantilevers increase the torsional rigidity

and reduce the sensitivity to torsional loads. Coating the cantilever surface with thin

films has become a standard method to increase detector sensitivity, enable biological

imaging, and/or make the cantilever conductive. Additionally, various tip geometries

such as spheres and carbon nanotubes are applied. Self-actuating piezoelectric

cantilevers are used in cantilever arrays which enable a large sample area to be imaged

rapidly. Since cantilevers are interchangeable, they are specialized to suit the specific

environment. Therefore, no one design is used exclusively of the others.









The displacement detector has also taken several forms. A capacitive sensor over

a double leaf cantilever was developed [14]. Heterodyne and homodyne interferometry

have been used with different laser sources such as He-Ne lasers and laser diodes

[15,16,17,18]. Additionally, heterodyne interferometry applied over a small spot on the

cantilever enabled the 3D surface under the spot to be visualized [19]. Torsion of the

cantilever was detected with this approach. A laser was used in the optical lever

technique [20] and torsion could be detected depending on the photodiode

arrangement. Finally, the cantilever itself has been used as the detector through the

piezoresistive effect [21]. Each detector scheme has situational advantages, but the

most broadly adopted scheme used today is the optical lever technique. Since the

detector is not interchangeable, most commercially-available systems rely solely on this

technique.

Applications of the Atomic Force Microscope

AFM has become ubiquitous in the research setting. Since the AFM can image

non-conducting samples, it is commonly used in ultramicroscopy (imaging at atomic

resolutions). For example, the AFM is used in biological [22], crystalline growth [23], and

pharmaceutical drug [24] imaging applications. However, the AFM is not restricted to

imaging. It is also used in atomic scale manipulation [25], such as nanolithography [26],

and data storage and retrieval, such as IBM's Millipede project [27]. The AFM has also

become a potential tool for quality control in microelectromechanical systems (MEMS)

as a parallel to electronic quality control in microchip manufacturing [28]. This is due to

the AFM's force measuring capabilities, which enable material and mechanical testing.

Many fields can take advantage of the AFM's ability to measure micro-scale forces. For

example, the structure of DNA is known, so current efforts are directed at measuring the









bonding forces between complementary nucleotides [29]. The mechanical stability of

proteins has also been tested [30]. In surface science, the AFM is applied to friction

measurements, such as between microspheres [31] and in capillary force analysis [32].

Force Modeling

Force is computed through its relationship to the cantilever's deformation. This

relationship, called the force model, is based on Hooke's Law which is an integral part

of linear elastic theory. Hooke's law provides a constitutive equation that relates stress

with strain (force, F, with displacement, y). Since the detector in AFM measures

displacement, Hooke's law is applied directly as F=ky. This displacement relationship is

the most common force model for AFM [9]. However, it has been recognized that this

model is too simplistic. Other force models have been proposed which take into account

forces that arise from contact mechanics at the cantilever's tip. These have attempted to

capture tip geometry [33], elastic deformation of the sample [34], cantilever inclination

angle [5], adhesion [35], and friction [36,37]. Many times the model is situational. For

example, if the sample is very hard, plastic deformation is unlikely to occur or, if the

cantilever and sample are submerged in water, capillary force is mitigated. Therefore, it

is necessary to understand the test environment in order to choose a model and

determine its parameters.

Calibration

The AFM system must be calibrated to obtain reasonable measurement accuracy.

The literature focuses heavily on stiffness calibrations because stiffness is the most

difficult parameter in the force equation to calibrate. Calibration may be divided into four

categories: dimensional, intrinsic, dynamic, and static.









Dimensional methods use estimations of the geometrical parameters of the

cantilever and beam theory to predict stiffness. Computations may be carried out

analytically or via finite element analysis. Relevant parameters include length,

thickness, width, film thickness, modulus of elasticity, and second moment of area.

Depending on the method, compensation can be made for V-shaped beams (double

arm cantilevers) [38] and the trapezoidal cross-section seen in some commercial

cantilevers [39]. Accuracy is limited by the combined standard uncertainty from each

measurement; thickness, length, and Young's modulus are typically the largest error

contributors.

Intrinsic methods attempt to apply naturally occurring phenomena. An example is

force measurement on DNA [40]. Since the bonding energy of DNA is specific and

highly reproducible, the force to pull DNA apart is reproducible. Such artifacts are nearly

identical in nature and can be mass-produced by the millions. A critical parameter for

such a calibration is temperature since, in general, increasing temperature reduces the

additional energy required to rupture bonds.

Dynamic methods use cantilever vibration to estimate stiffness from frequency

shift or phase change. One of the first techniques that applied this dynamic approach

was developed by Cleveland et al. [41]. In this technique, a known mass was attached

to the end of the cantilever and the corresponding reduction in natural frequency was

measured. The location of the mass on the beam is a critical parameter and removing

the mass after calibration is not trivial. Mass has also been added using different

materials such as thin gold films [42], water droplets dispensed from an inkjet [43], and

even other cantilevers [44]. Other researchers altered the method so that mass addition









was not required. A well-known method proposed by Sader et al. [45] uses the resonant

frequency together with the Q (damping) factor in a fluid, usually air. The use of fluid

dynamics has become more common, especially in colloidal probe microscopy,

because the cantilever has a sphere for a tip and spheres are convenient to model in

fluid dynamics [46]. Another approach uses the equipartition theorem and thermal

oscillations of the cantilever to determine cantilever stiffness [47]. With corrections

made for the laser spot size [48] (a laser is used to determine cantilever deflection), this

method has become popular due to its application ease.

Static methods involve applying known forces directly to the cantilever and

observing the resulting deflection. They are the most direct measurement of cantilever

stiffness. Many devices have been used to apply a direct force to the cantilever. These

include macro-sized ("relatively large") cantilevers [49], piezoresistive levers [50, 51],

and nanoindentation machines [52]. The measurement uncertainty in these experiments

has been as high as 20%. The most sensitive parameter is claimed to be the load

application point. Since the stiffness is related to length by an inverse cube law, small

changes to load application point result in large errors in stiffness. To address this

issue, Cumpson et al. [53-55] at the National Physics Laboratory, Teddington

Middlesex, UK, have developed a series of micromachined artifacts for cantilever

calibration. These artifacts have fiducials for locating the load application point and are

available in a variety of shapes to accommodate a wide stiffness range. Some even

include a built-in mechanism for calibrating themselves [56]. Gates et al. developed

arrays of reference cantilevers with sufficient fabrication repeatability that a calibration

performed on a single cantilever is representative of the entire array [57].









There are also additional calibration challenges. The stiffness has been shown to

change in response to wear at the tip [33] and over time even if unused [2]. Given the

difficulties associated with stiffness calibration and force model development, this

research introduces a new measurement paradigm. Instead of a single point

measurement of the tip displacement, the full-field displacement of the 3D surface of the

cantilever is obtained.









CHAPTER 3
THREE-DIMENSIONAL IMAGING

A New Platform

Three-dimensional imaging requires a platform different from traditional AFM. This

chapter discusses the design of the prototype platform used in this study and presents

its capabilities and restrictions. The platform consists of a scanning white light

interferometer (SWLI), holders for the cantilevers, a translation stage, a rotation (tip-tilt)

stage, and an adapter plate to connect these units to the SWLI table (Figure 3-1).

SWLI objective

S-.Tip-tilt alignment
Three-axis translation Probes stage
stage











SWLI tip-tilt table


Figure 3-1. Prototype platform assembly. It is composed of a scanning white light
interferometer (SWLI), a translational stage, and a tip-tilt stage.

Scanning White Light Interferometer

A key platform component is the SWLI, an optical 3D surface profiler that uses

interference of a broad spectrum light source, or "white light", to measure surface

topography (Figure 3-2). Light reflected from the sample interferes with light reflected

from a reference surface, but unlike coherent source interference, the white light

interference only occurs over a small optical path difference. By translating the objective










(which carries the reference surface) relative to the sample, a plot of the interference

intensity for this path difference range can be captured on a pixel-by-pixel basis by the

SWLI detector. The location of the modulated intensity region (due to the alternating

constructive/destructive interference) indicates the relative height of the sample at that

pixel. Each pixel on the detector corresponds to a lateral position on the sample; the

field of view for the corresponding height map depends on the system magnification. A

Zygo NewView 7200 was used in this research. The selected system included a

motorized translation/rotation (or X/Y/tip/tilt) table for sample alignment to the optical

axis. As shown in Figure 3-1, the prototype platform is mounted to this table.

Detector


Light
source



Translator

1 Interferometric
S objective

2

Sample 10 20 30 40 50
Translation (pm)

Figure 3-2. SWLI schematic. A Mirau objective is represented, although other types are
available. The left-to-right offset between the interference intensity from
positions 1 and 2 indicates their relative height difference (positive translation
in the upward direction).

There are constraints associated with SWLI measurements. First, a full height map

takes several seconds to acquire, in general, since the objective must be translated.

During this time the sample dynamics must be interrupted (this pseudo-static approach

is applied in this study) or synchronized with the measurement (this strobing approach









is not used here). Second, the lateral resolution is typically orders of magnitude coarser

than the vertical resolution along the optical axis. For the experiments reported here,

two objectives were used: a 5x Michelson with a lateral resolution of 2.2 pm/pixel and a

20x Mirau with a 0.55 pm/pixel lateral resolution (lx zoom and 640x480 pixel detector).

In comparison, the NewView 7200 literature specifies a vertical resolution of 0.1 nm,

although this value is dependent on the noise floor imposed by the measurement

environment. Therefore, tests were performed to determine the repeatability for the

experimental setup. A smooth silicon surface was placed on the stage at the cantilever

loading location and 130 scans were completed using a 40 pm vertical scan range. The

height repeatability for each pixel was then assessed. It was found that, on average,

each pixel reported the same position within a standard deviation of 2.7 nm. Therefore,

a resolution of 2.7 nm was assumed. Third, the SWLI cannot detect large changes

between adjacent points, so there is a maximum slope that can be detected. This slope

varies depending on the selected objective/zoom; it was 4 deg for the 5x objective and

18 deg for the 20x objective (each at 1x zoom).

Cantilever Holders

Aluminum plates with dimensions of 60 mm x 60 mm x 3 mm were fixed to the

stages (Figure 3-3). Cantilevers and test surfaces were adhered to the plates at the

midpoint of three sides (a fourth side with a specialized geometry was also available to

enable cantilever alignment in future work). This design allowed the holder to be

unscrewed and rotated to select the next cantilever for experimentation while

approximately maintaining the same position in the SWLI field of view. A similar

configuration was used for tests with a cantilever contacting a rigid surface. Cantilevers









and rigid surfaces were bonded to the holders with adhesive to enable the required top-

down view for the SWLI measurements.

a) b)
Q ''--'- =


-I-
.








Figure 3-3. Example cantilever placement on the aluminum holders (top views). The
individual cantilevers are too small to be seen at this scale, but the monolithic
base chips are labeled A1-C2 (a) A-C (b), and a rigid sample is labeled S
(b). If the holders were unscrewed and rotated in the indicated direction, the
next experiment would be C1 against C2 (a) and C against S (b).

Stages

The positioning stage (Thorlabs MAX301) was a three-axis, parallel kinematics,

flexure based design with 4 mm of coarse motion (thumbscrew actuation) and 20 pm of

fine motion driven by piezoelectric actuators with strain gauge feedback. The motion

was controlled by a Thorlabs BPC103 controller. The second stage was a manual tip-tilt

platform (Thorlabs ATP002) positioned on a base assembly (Thorlabs AMA501) which

enabled equal height, side-by-side use with the positioning stage. The tip-tilt platform

provided 4 deg of roll and pitch and acted as the fixed stage in the experiments. Any

cantilever on this stage could be tilted into alignment with the positioning stage and then

held fixed for the duration of the experiment.

The positioning stage's vertical linearity was tested from 7 pm to 20 pm within the

0-20 pm range of the piezoactuators. A silicon surface was adhered to the holder on the

positioning stage. A second silicon surface was adhered to the holder on the tip/tilt










stage. The SWLI was used to measure the height difference between the two surfaces

as the positioning stage was translated along the optical axis using 1 pm increments.

The result (Figure 3-4) reveals a scale error of 79 nm/pm. Though this quantity was

available, the experimental results performed in this study did not require motion of the

positioning stage, i.e., only features within the SWLI scan range were used, so this

scale factor was not required in the data analysis.


1.2
Data
1 Fit

0.8

-. 0.6
y = 0.079x
2 0.4

0.2



-0.2
0 2 4 6 8 10 12
Commanded displacement [iLm]

Figure 3-4. Positioning stage scale error. This was determined by comparing stage
vertical position to displacement determined from SWLI measurement.

Platform Assembly

The platform assembly rested on the SWLI motorized table. During use, the

platform positioning stage was first aligned to the SWLI optical axis using the table.

Then, the tip-tilt stage was aligned to the positioning stage. Since the SWLI angular

detection limit can be low (depending on the objective and effective magnification),

proper alignment is important to maximize the available measurement range.

Additionally, aligning to the optical axis ensures that the vertical motion from the

positioning stage produces no lateral motion in the field of view.









Visualization of Quasi-static Bending

The prototype platform enables the direct visualization of cantilever deformations

without the need for interpretation via a single point measurement of the cantilever tip

motion used in conjunction with beam models. In this section, cantilever deflection

measurements are provided. Various loading conditions are represented including no

loading, rigid surface loading, and cantilever-on-cantilever loading. The results are

discussed and the primary imaging limitations noted [58].

Residual Stress, "Batwings", and Differencing







U -






Figure 3-5. The Olympus OMCL-AC240TS cantilever. The deformed shape has
approximately 0.5 pm of initial tip deflection.

The cantilevers presented here had a non-planar shape even in the absence of an

external load. Cantilevers are typically designed to be flat, but residual stresses from the

surface coating often lead to unwanted deformation. This deformation is typically small,

but is observable in the SWLI measurements. Figure 3-5 shows the reflective backside

of an Olympus OMCL-AC240TS (1.8 N/m stiffness) cantilever (i.e., the side the laser

would normally reflect from in an AFM). It is seen that the cantilever deformation is

approximately 0.5 pm at the tip with no external load applied. An imaging artifact
approximately~~ ~ ~ ~ 0. ma h i ihn etra odapid nimain atiac









sometimes referred to as "batwings" is also observed. These spikes along the periphery

of the cantilever are false height readings and tend to occur at steep height transitions.

The speck near the tip (also visible in the section view) is surface contamination and

also causes the bat wing effect. Additional cantilever measurements are shown in

Figure 3-6. The data were collected using a 20x objective at 1x zoom in the 320x240

pixel mode.












Figure 3-6. Olympus BL-RC150VB AB cantilevers. Stiffness is 0.03 N/m (a) and 0.006
N/m (b). The deformation levels are ~ 0.25 pm (a) and ~1 pm (b) in the
absence of external loads.

When studying deformation of beams it is necessary to separate the contribution

due to the external forces) from those caused by other sources such as residual stress.

For linear elastic beam models, deformation follows the rule of superposition. Therefore,

in each of the analysis results presented here, the cantilever was measured before

applying external loads. This reference image was then subtracted from all subsequent

images to isolate the relative deformation caused by the applied forces. This

differencing does not remove the bat wings, however, as these locations tend to report

random erroneous heights from image to image.









Snap-in

When an AFM cantilever is placed in close proximity to a surface, an attractive force

develops which pulls the tip closer to the surface. At a critical gap size, the attractive

force overcomes the cantilever's restorative elastic force and the tip snaps into contact

with the surface. This attractive force represents one of the many forces an AFM

cantilever experiences. Snap-in behavior was measured for an Olympus OMCL-

AC240TS (1.8 N/m) cantilever positioned near a smooth silicon surface (Figure 3-7).

For these measurements, the silicon surface was mounted on the three-axis positioning

stage and moved toward the cantilever in 50 nm steps. The sequence of three plots

included in Figure 3-7 displays the cantilever deformation profile for three stage

locations. In position 1, the silicon surface is sufficiently far from the cantilever so that

no appreciable attractive force is present. In position 2, snap-in has occurred. The

cantilever deflection is approximately 0.1 pm at the tip. In position 3, the stage has

continued to move vertically past the cantilever's undeformed position so that it is

deflected upwards. In Figure 3-7b, the centerline cantilever profiles are provided for the

same three stage locations. The tip deflection versus stage motion is provided in Figure

3-7c; measurement points are identified by the small circles in 50 nm increments and

the three locations from Figures 3-7a and 3-7b are presented by the large circles. Note

that the information in Figure 3-7c is all that is available from standard AFM single point

deflection metrology. All images were captured using a 20x objected at 1x zoom in

320x240 pixel mode. The SWLI data was difference to isolate the deformation caused

by snap-in and it was filtered to remove the bat wing artifacts.




















b) 0.3-. i I
3
0.2


1 1
Q 0

-0.1
100 150 200 250 300
Length (pm)


0.21 pm








-0.41 pm


0.2 0.4
Stage motion (pm)


Figure 3-7. Cantilever snap-in demonstration. a) Height map of the cantilever at various
stage locations, 1-3; b) corresponding two-dimensional profiles, 1-3; and c)
traditional plot of stage motion versus tip deflection with 1-3 labeled. The
arrow indicates the sequence of commanded stage motions.

Other Experiments


Figure 3-8. Two Veeco 1930 (1.3 N/m, 35 pm wide) cantilevers in contact. The left
cantilever lever tip is positioned above the right cantilever lever tip.

One of the benefits of full-field imaging is the ability to capture height maps of the


cantilever and sample simultaneously. For example, in cantilever-on-cantilever

experiments using an AFM, measurements can only be made for the instrumented









cantilever leaving the deflection of the other cantilever to be inferred. Using the

prototype platform, both cantilevers can be viewed simultaneously. Figure 3-8 shows

two Veeco 1930-00 (1.3 N/m, 35 pm wide) cantilevers with one cantilever pushed

against the tip of the other. There is approximately 9 pm of vertical offset between their

bases and both cantilevers exhibit approximately 4.5 pm of end deflection. The

objective used was 20x, 0.8x zoom, in 640x480 pixel mode.











Figure 3-9. Data dropout. Due to SWLI measurement limitations this is commonly
observed on the cantilever periphery (area 1 left image). This results in errors
when determining the plan view dimensions. Including the sample in the
measurement reveals lost data (area 2 right image).

The sample can also be used as an aid in SWLI measurements of a cantilever's

plan view dimensions. Due to SWLI measurement limitations, data dropout is common

at the cantilever periphery. However, by placing a smooth sample underneath the

cantilever as a backdrop, the lost pixels can be revealed (Figure 3-9). Additionally, the

sample can aid in tip wear analysis. Though not demonstrated here, it is feasible to

measure the distance from the sample to the surface of the cantilever to estimate

progressive tip wear as depicted in Figure 3-10.

Finally, an example of the platform's capability to identify anomalous bending

behavior is presented in Figure 3-11. Here, an Olympus OMCL-AC160TS (42 N/m)

cantilever was pressed against a smooth silicon surface. The deflection profile was









linear, rather than the expected cubic behavior. It is assumed that a crack at the

cantilever base caused this hinge-like bending behavior. The cantilever broke off the

base chip before this could be confirmed, but the fracture location supports the crack

assumption. The imaging parameters included a 20x objective, 1x zoom, and 320x240

pixel mode.


cantilever h1 | 2


sample h,- h2 = tip wear


Figure 3-10. Tip wear estimation. Using SWLI measurements that include the sample
surface in the view, an initial measurement, hl, can be compared to a current
measurement, h2, to find the tip wear, hi-h2.


Figure 3-11. Anomalous bending behavior. This Olympus OMCL-AC160TS (42 N/m)
cantilever shows a linear bending profile rather than the expected cubic
behavior.









CHAPTER 4
THREE-DIMENSIONAL FORCE MODELING

The 3D deflection data provided by SWLI significantly increases the amount of

information available to determine force. To illustrate the mathematical method, the

"normal" force component is determined first. A horizontal rectangular beam is loaded

through contact at the stylus (Figure 4-1). Forces both perpendicular, F, and parallel, P,

to the beam axis develop where the relationship between the two forces is unknown (for

example, a proportional friction model is not assumed). The deflection equation is

determined from the Euler-Bernoulli beam model. Assuming y=y(x), u=O, and El is

constant, four integration of the beam equation (El d4y/dx4=u=O) yields,
-F
y(x) = x + C2X2 C3 4 (4-1)
6EI

where C, is determined by force balance with the shear force and C2-4 are normally

determined by applying boundary conditions. Rather than applying boundary conditions

and condensing the deflection equation to y(x=L) as in the single point measurement

paradigm, the SWLI data provides y(x) for Ox
utilized. Equation 4-1 is re-written as a general third-order polynomial,

y(x) = B3x3 + B2X2 + BIx + B0 (4-2)

where B0-3 are the polynomial coefficients. These coefficients can be estimated by a

least-squares fit to the 3D data. Therefore, the normal force component F is directly

related to the third order polynomial coefficient B3,

F = -6EIB3 (4-3)

The transduction coefficient (-6E/) has many advantages over stiffness. First, it is

independent of the beam length, while stiffness has a sensitive relationship to beam









length (1:3 length:stiffness error for a rectangular cantilever). Second, the equation

holds regardless of load application point (this avoids another 1:3 error rate). Similarly,

the measurement location does not affect the estimate of B3 so there is no error for

detector misalignment. Third, the normal force is completely decoupled from other

forces (immune to stylus contact effects).


P


Figure 4-1. A cantilever loaded at the free end by two independent forces P and F.

top view


similar
rectangular cantilever, measurement
locations









double arm triangular

Figure 4-2. Force measurement does not depend on the global geometry of the
cantilever. Both cantilevers shown utilize the same force equation except that
the bottom cantilever should be measured on both arms (ideally each arm
carries half the load).









The result is a direct application of the shear force equation for beams (-El

d3y/dx3=V). The shear force is constant along the length of the beam and is directly

related to the third derivative of the beam deflection. This explains why the technique is

independent of location along the beam and is independent of boundary conditions.

Also, the entire beam does not have to be measured (though it improves the signal to

noise ratio). Instead, a small section of the beam can be measured and, if the section is

small enough (ideally, d3y/dx3 is infinitesimal), the shape change across the length of

the beam is unimportant. This is in contrast to stiffness, which is dependent on the

global shape of the beam to be solved analytically (shape is often restricted to prismatic

beams). Therefore, many beam geometries are easily adapted to the 3D force

technique, such as double arm triangular beams (Figure 4-2).

Front view
Y
y
z

Z T
T







F


Figure 4-3. The 3D view of the forces acting on the cantilever.

A similar development yields expressions for the parallel force component, P, and

torsional force component, T, to identify the full 3D force vector applied to the cantilever









V(F,P, T), (Figure 4-3). First, the parallel force component, P, is derived. Using the same

generic loading case in Figure 4-1, the moment equation is,

EI = M =F(L -x) +Pr (4-4)
dX2

where r is the tip height of the stylus. The deflection equation is then,

C2
==0 =0
-F 3 FL Pr (4-5)
y(x) = x + 2 3 X + C4 (4-5)
( 6EI \2EI 2El)

where C3-4 are zero for a fixed base boundary condition (motion and slope are zero) and

C2 is determined from the moment equation. Unlike shear, the moment depends on

measurement location (not constant over the length of the beam). This is important

when the location of the base and free end are not well known, which occurs when

completing a SWLI measurement with a field of view smaller than the cantilever length.

Here, the coordinate system of the measurement will not correspond to the coordinate

system of the deflection equation which is centered at the base of the cantilever (Figure

4-4). Therefore, a polynomial fit to the deflection data cannot be used directly. Instead,

the deflection equation is re-written for coordinate system independence so that a

mapping can be performed between the polynomial fit of the data and the deflection

equation. For example, both the deflection equation and polynomial fit can be mapped

to,

Y Yo = A(x Xo)3 + A2(x Xo) (4-6)

where A1-2 are coordinate independent coefficients, and yo and xo translate the

polynomial from the origin to an arbitrary coordinate system. Expanding the transform

polynomial to relate to the deflection equation coefficients and polynomial fit coefficients

yields,









-F FL Pr
6El \2EI 2El)
y = Ax3 + (-3Axo)x2 + (A2 + 3Ax02) x + (Yo Axo03 A20) (4-7)
B3 B2 B1 B0

-F
A, = B3 = (4-8)
6El

2 (FL Pr 2
A2 = B2 = -2EI 2E 2E) (4-9)
3B3 F

Note that the A, coefficient does not vary with coordinate system, providing a second

check on the invariance of measurement location for F. Solving for P in terms of the

polynomial fit coefficients yields the measurement-independent result,


P = [l B22 3B3B1- 3B3L (4-10)


This result fully decouples Pfrom the normal force F. However, determining P does

require additional knowledge about the cantilever geometry, including tip height, r, and

beam length, L. Note that beam length is not cubed and is therefore less sensitive to

error than the single point approach.


side view

SWLI


Y- -R SWLI

YOi XEB
xswxo



Figure 4-4. SWLI measurement coordinates. The SWLI has its own coordinate system
(xswu,yswu) which is offset from the global coordinate system defined in the
Euler-Bernoulli deflection equation (XEB,YEB) by an unknown amount (xo,yo)

The final force component, T, applies torsion to the beam causing twist. The twist

equation can be derived using St. Venant's torsion theory which assumes the rate of










change of angular twist over the length of the beam, da/dx, is constant. The resulting

equation for twist is,

a= x (4-11)
G]j

where a is the angular twist, R is the applied torque, x is the position along the length of

the beam, G is the material's shear modulus, and J is the torsion constant. Converting

to deflection yields,


y(x,z) = -xz (4-12)
Gj

where z is the location across the width of the beam. This equation assumes a

particular coordinate system (origin at the base) so it can be generalized as,


y = (x xo)(z zo) + yo (4-13)
Gj

R -R
S= XZ + -X Z +. (4-14)
Gj Gj
D H

A surface fit to the 3D data yields a coordinate-independent estimate of the coefficient D

which is directly related to the applied torque,

Tr
R = GJD (4-15)

T =GD (4-16)
where T is the final force component of the 3D force vector applied to the tip.
where T is the final force component of the 3D force vector applied to the tip.









CHAPTER 5
VALIDATION OF THE FORCE MODEL

Normal Loading

Equation 4-3 was evaluated for the case of a meso-scale horizontal beam (no tip)

with a "normal" mass-based force applied (i.e., F was ideally the only force component).

The beam was fabricated from a 50.8 mm diameter, double-sided polished <111>

silicon wafer by diamond cutting at the flat. The design dimensions for the beam were 1

mm wide, 5 mm long, and 0.3 mm thick. After fabrication, the dimensions were

measured using the SWLI. The average width was 0.941 mm and the length was 4.820

mm (the SWLI lateral resolution was 4.4 pm for a 5x objective at 0.5x zoom with a

640x480 detector). A thickness measurement was completed by placing a glass surface

beneath the cantilever and using the SWLI to measure the distance from the top of the

glass surface to the top of the cantilever. The thickness value, t, was 0.305 mm but

varied depending on location by 9 pm. A stitched SWLI image of the entire cantilever is

shown in Figure 5-1.















Figure 5-1. SWLI image of the fabricated cantilever. The enlarged field of view was
enabled by stitching multiple images together. It is seen that the width of the
beam varies slightly along its length. The wafer edge roll-off is observed at
the free end (left end) of the beam.









The normal forces for the beam were produced using a series of 10 masses

(Figure 5-2a); the individual masses were each suspended from a tether which was

looped over the beam (Figure 5-2b). The mass values, mmeas, were measured using a

Mettler Toledo AB265-S/FACT precision balance (0.1 mg resolution). The beam was

fixed to a holder using CrystalbondTM heat-activated adhesive and the holder was bolted

to the tip-tilt stage.









a) _, 2 24.650 b)

Figure 5-2. Fabricated masses. a) Masses used for vertical loading experiments, b)
cantilever attached to the tip-tilt stage and loaded near the free end.
Measurements are valid at any section between the base and the load point.

The measurement procedure was initiated by aligning the cantilever base to the

SWLI optical axis (the optical axis was treated as parallel to the gravity vector, although

the degree of alignment was not determined). To perform this alignment, a 5x objective

with 0.5x zoom was applied to give a 2.82 mm by 2.12 mm field of view and 4.4

[tm/pixel lateral resolution. The cantilever was then laterally positioned under the

objective using the motorized stage. The cantilever was longer than the field of view so

only a section of the cantilever was measured; stitching was not applied for the force

measurements. The following steps were completed for each mass.

Attach the mass to the cantilever.
Complete a first SWLI measurement.
Remove the mass.
Complete a second SWLI measurement.









The two SWLI measurements were difference to isolate the deflection caused by

loading. As described before, differencing isolates the deformation caused by the

applied force from other sources. However, differencing requires lateral alignment

between images. Therefore, the mass was applied first in the measurement sequence

since it required the longest setup time. This reduced the time between steps 2 and 4 to

a few seconds, which mitigated the effect of lateral drift due to stage settling and

thermal effects. In addition to drift-related lateral motion, vertical deflection of the

cantilever from an applied force also leads to a small shift (foreshortening) of the

cantilever along the beam axis. Because the beam deflections were small in this study,

this shear effect was neglected.

A deflection profile was obtained for each applied force by taking a section view

through the beam center in the difference map (similar to the approach used for the

AFM cantilevers shown in Figure 3-7). The section view data was then fit in a least-

squares sense using a cubic polynomial to obtain the coefficient B3. Equation 4-3 was

adjusted for F=mestg and divided by the measured mass mmeas to assist in the data

analysis,

mest 6EIB3/g -1 (5-1)
mmeas mmeas

where mmeas is the mass measured by the precision balance in kg, mest is the mass

estimated by the cantilever force equation in kg, E<11o>=168.9 GPa [59], I=wt3/12 for the

rectangular cross-section, t=0.305 mm, and g=9.81 m/s2. There was a noticeable taper

in the width so the average value within the measurement field of view was used

(w=0.926 mm). Equation 5-1 should equal 1 within the measurement uncertainty. The

uncertainty was estimated from the uncertainties of each variable in Equation 5-1 and is









discussed later in the chapter. The results are displayed in Figure 5-3 with error bars

based on the uncertainty analysis and multiplied by a coverage factor of 2. As seen in

the figure, the estimated mass under predicts the measured mass by a mean of 9.6%

but is within the error bars.




a) ''---- ---- --- ----:-
U) E --i------0 --^--.) ---( -------------- t-)--( -----(
uo 0.8

0.6 o est/meas
C-- nominal
u) 0.4
Co ..4 --mean

2 0.2

5 10 15 20 25
Mass (g)

Figure 5-3. Results of Equation 5-1 for the 10 masses. The error bars are based on the
full uncertainty analysis multiplied by a coverage factor of 2.

The variability of B3 with profile location in a single measurement was also

investigated. Eleven deflection profiles near the beam centerline were selected from a

single image and used to individually calculate B3. The standard deviation was found to

be 0.5% of the mean value. This deviation is an indicator of the model limits since the

Euler-Bernoulli equations describe a two-dimensional beam. Noise in the SWLI height

map from the single measurement result would also contribute.

Next, the cantilever was loaded and 11 separate measurements were sequentially

completed before removing the load. Only the centerline profile was extracted from

each measurement and was used to calculate B3. The standard deviation was found to









be 1.5% of the mean. This provides an indication of the limit imposed by the

environmental noise over the required measurement time of approximately 4 minutes.












Figure 5-4. Beam loaded under identical conditions, but measurement field of view was
varied. The boxes represent the partially overlapping measurement regions
along the beam's axis. For identical loading, the shear should be identical
regardless of location.

Ideally, the force calculated using Equation 5-1 is independent of the

measurement location for constant loading conditions. To test this behavior, a

measurement (denoted 1) was completed for a field of view near the free end and

compared to a measurement (denoted 2) completed near the base; see Figure 5-4. The

values of B3 differed, however, due to the tapering beam width (see Figure 5-1). To

enable a direct comparison of the two results, it was necessary to consider the non-

constant beam width and corresponding variation in the moment of inertia along the

beam's axis. This was accomplished using the average widths, Wavg, of the beam for the

two selected fields of view. The result was evaluated by considering the ratio of the two

measurement results 1 and 2 (same mass, but different location). After simplification,

the ratio is,



\ 12 1 2
( avB
var~ied.B: The boxes represen t (5-2)









which is nominally equal to 1. Experimentally, the ratio was determined to be 1.03

based on a mass load of m=8.617 g and the following values: (B3)1=-37.02 m2,

(Wavg)1=0.88 mm, (B3)2=-33.89 m-2, and (wavg)2=0.93 mm. This deviation from unity is

reasonable given the 1.5% standard deviation obtained from the repeated B3 tests for a

single load with the profile extracted along the beam's centerline.

In a second study of the polynomial fit-based force determination approach, a

horizontal Olympus OMCL-AC240TS probe (rectangular cross-section, 30 |tm x 2.8

|tm, 240 |tm length, k=1.8 N/m, approximately 14 |tm tip height, less than 10 nm tip

radius) was deflected vertically against a rigid, smooth silicon surface. The experimental

B3 value was -2.11 xl 04 m-2 from the centerline deflection profile. If the manufacturer-

specified El value of 8.29x10-12 N-m2 is applied, the resulting vertical force is F=-

6B3E/=-6(-2x1 1x1 04)(8.31x1 0-12)=1.05x10-6=1.05 [N. Based on the measured deflection

of 0.59 |jm at the probe's free end (from the SWLI height map), the force obtained from

the manufacturer's spring constant is F=1.8(0.59x10-6)=1.06x10-6=1.06 [N. This gives a

1% agreement and provides a preliminary validation of the method for a typical AFM

probe. Data was collected using a 20x objective, 1x zoom, 320x240 pixel detector.

In a final study, the full surface data of a cantilever was used to calculate the

polynomial coefficients rather than taking a section view down the center of the

cantilever axis. This was done in preparation for the torsion experiments which required

the three-dimensional deformation data (see Equation 4-14). To accommodate a full

surface fit, least-squares curve fitting was used again. The equation for the surface is a

linear combination of each deformation,









y(x,z) = B3x3 + B2x2 + C1x + +Dxz + Hz (5-3)
bending bending torsion
and
torsion

Since the generic equations for bending and torsion contain both x and constant terms,

these components are added together in Equation 5-3 and are represented by new

coefficients C1 and Co. For normal loading, the only component of interest was B3.

Given a number of (x,z,y) points, Equation 5-3 can be written in matrix form,

(B3-)
X13 1X2 X1 1 1Z1 Z1 B2

Xn Xn2 Xn 1 nZm Z (nxm)x6 Ynxm(nxm)xl
H 6x1

where m is the number of measurements across the width of the cantilever, n is the

number of measurements down the length of the cantilever, and mx n is the total

number of measurements. This can be compactly written as [XZ]{B}={Y} and solved for

in the least squares sense using the pseudo-inverse,

{B} = [XZ]+{Y} (5-5)











A B

Figure 5-5. The aluminum "tee" bonded to a 1.5 mm wide cantilever. A) a microscope
image of the tee, B) a SWLI height map of the tee. The images are stitched to
enlarge the field of view. The groove near the cantilever is for alignment
purposes.









A 1.46 mm wide by 0.311 mm thick by ~5 mm long cantilever was mass loaded to

test this approach. Additionally, a removable "tee" was bonded to the cantilever to be

used for the torsion experiments described later. The "tee", shown in Figure 5-5, had

three sockets to accept a stainless steel sphere which, in turn, held the masses. This

configuration enabled repeatable positioning of the masses into the sockets. For normal

loading, the center hole was used. Eight masses (3.260 g, 5.426 g, 7.552 g, 9.734 g,

11.877 g, 14.059 g, 16.122 g, and 18.285 g) were applied to the cantilever, with five

trials per mass. The coefficients were determined using Equation 5-5 and the masses

were computed using Equation 5-1. The results are shown in Figure 5-6 where the error

bars are based on the uncertainty analysis and multiplied by a coverage factor of 2. The

estimated mass under predicts the measured mass by a mean of 6.2% but is within the

error bars.




U) I



0 0.6 o est/meas
U) -nominal
) 0.4 ------- mean

0.2


5 10 15
Mass (g)

Figure 5-6. Results of Equation 5-1 for the 8 masses using the surface fit approach.
The error bars are based on the full uncertainty analysis multiplied by a
coverage factor of 2.









Torsion Loading


















using the "tee".

Equation 4-16 was evaluated on multiple cantilevers using offset loading as shown
R=rmg Xmas

A -B


Figure 5-7. "Tee" attached to cantilever. A) Attaching a "tee" to the cantilever enabled
offset loading, B) an array of cantilevers with one cantilever being loaded
using the "tee".

Equation 4-16 was evaluated on multiple cantilevers using offset loading as shown

in Figure 5-7a. An aluminum "tee" was fabricated and bonded to the cantilevers (Figure

5-5). Three holes were drilled through the tee as load zones for a stainless steel sphere.

The resulting ball and socket arrangement enabled precise placement of the sphere on

the tee. A hook inserted through a hole in the sphere enabled it to support the masses.

The tee was bonded to the free-end of the cantilevers using CrystalbondTM heat-

activated adhesive. The assembly shown in Figure 5-7b enabled offset loading with a

constant torque arm.

New cantilevers were fabricated with dimensions and torque arms shown in Table

5-1. Positive arms generated counter-clockwise twist and negative arms generated

clockwise twist. The torque arm was defined as the perpendicular distance from the

center of the hole to the cantilever beam axis. Since the tee was bonded manually,

placement of the tee onto the cantilever was not tightly controlled resulting in cantilever-









to-cantilever variability in the torque arm alignment. Additionally, the field of view of the

microscope was too small to simultaneously capture the holes and the cantilever in the

same measurement and stitching was time intensive. Therefore, the torque arm was

estimated based on the orientation of the alignment grooves to the cantilever axis and

the known dimensions of the tee.

Table 5-1. Cantilever geometries. The thickness, t, was approximately 311 pm.
Cantilever Lenth Width Width Torsion Positive Negative
antler L h nominal measured, multiplier, torque torque arm
number (mm) (mm) w(mm) 3 [60] arm (mm) (mm)
1 5 1 0.959 0.263 5.339 NA
2 5 1.5 1.454 0.286 5.371 5.605
3 5 2.5 2.462 0.306 5.528 5.448
4 5 3 2.962 0.310 5.355 5.622
5 5 0.7 0.656 0.233 5.221 5.756
6 13 1.5 1.455 0.288 5.378 5.598
7 13 3 2.963 0.310 5.347 5.630
8 13 0.7 0.657 0.233 5.590 5.387


The cantilevers were each loaded with at least three different masses five times

per mass in the same manner as for the normal loading experiments. The first mass

was the smallest in the set (2.166 g). The third was the largest mass the cantilever

could accept while still reflecting light back to the SWLI (note the SWLI has a maximum

surface angle that it can measure which depends on the magnification). The other

masses were selected to be between the first two. The coefficients were estimated

using Equation 5-5 with the coefficient of interest being D. Equation 4-16 was adjusted

to assist in data analysis similar to Equation 5-1,

mest GD/r_ 1 (5-6)
mmeas mmeas

where mmeas is the mass measured by the precision balance in kg, mest is the mass

estimated from the cantilever force equation in kg, 54.0 5 G 5 64.7 GPa depending on









width and thickness [61], J=p/wt3 (P/ and w are given in Table 5-1), t=311 pm, and r is the

torque arm given in Table 5-1. The results for positive torque and negative torque are

shown in Figures 5-8 and 5-9, respectively. The estimated mass under predicts the

measured mass for positive torque loading by 7.4% and under predicts the measured

mass for negative torque loading by 11.2%. The mean of the two results is 9.3%.


1.4

1.2

8 ]

U 0.8-
St- 0 +cant2
S* +cant3
0 0.6 +cant4
S+cant5
S04 +cant6
+cant7
A +cant8
0.2- nominal
---+mean

5 10 15 20 25
Mass (g)

Figure 5-8. Torsion results when mass is loaded at the positive torque arm. The error
bars are based on the full uncertainty analysis multiplied by a coverage factor
of 2.

A second study was performed to determine if the normal and torsional

components could be determined simultaneously with the same measurement. Figure

5-7a shows how an offset load produces both twist and bending deformation. Because

the tee increased the effective length of the cantilevers, which subsequently increased

the measurement sensitivity to external noise, two alterations to the setup were required

to lower the noise levels. First, foam insulation was placed around the work area to










block air currents. Second, the tethers for the masses were replaced by a rigid,

stainless steel link. Motion at the test mass was rigidly linked to the sphere which

caused the sphere to rock in its socket and quickly damped vibrations.


1.4

1.2-


a)
a ---- m ------------------------------------------------- -

Uo 0.8 o -cant2
06 -cant3
-cant5
0.4-

S-cant6
S0.4- -cant7
-cant8
0.2- nominal
---- mean

5 10 15 20 25
Mass (g)

Figure 5-9. Torsion results when mass is loaded at the negative torque arm. The error
bars are based on the full uncertainty analysis multiplied by a coverage factor
of 2.

A 1.46 mm wide by 0.311 mm thick by ~5 mm long cantilever was used along with

5 masses (3.260 g, 5.426 g, 7.552 g, 16.122 g, and 18.285 g) applied at both positive

and negative torque arms (5.464 mm and 5.515 mm respectively) with five trials per

mass. The coefficients were determined using Equation 5-5, the mass estimated, and

the results analyzed by Equation 5-6. Using the normal force coefficient yields the

results shown in Figure 5-10 and using the torsion force coefficient yields the results

shown in Figure 5-11 with the error bars based on the uncertainty analysis and

multiplied by a coverage factor of 2. The mean estimated mass using the normal force










coefficient under predicts the measured mass by 8.0% and the mean estimated mass

using the torsional force coefficient under predicts the measured mass by 11.9% (mean

positive torque arm of 12.5% and mean negative torque arm of 11.2%).


1.2



--------------------
E 0.8

2 0.6
Us o positive
04 negative
-- nominal
Mean
0.2


5 10 15 20
Mass (g)

Figure 5-10. Torsion loading results for mass loaded at the positive and negative torque
arms. The error bars are based on the full uncertainty analysis multiplied by a
coverage factor of 2.


1.2

1
(I

E 0.8
(/)
U,
(0
S0.6

u 0.4

0.2


5 10 15 20
Mass (g)

Figure 5-11. Normal loading results for mass loaded at the positive and negative torque
arms. The error bars are based on the full uncertainty analysis multiplied by a
coverage factor of 2.


-- -- -----------------^- -





o positive
D negative
nominal
mean








Uncertainty Analysis

The measurement uncertainty reported in the figures was determined by the

propagation of errors of Equations 5-1 and 5-6,

( m.mesjme os 17E<110> t 2 1 mm 2
/ normal E 2 2 B2. + + mmas (5-7)
/~mmeas normal ( means


('mest/Immeas )s )( 2 + (j2 +,)2 + 2 )2 + ( me-s)2 (5-8)
\ hmmeas )torsion ~ 9 r ) (Mm)

where terms with asterisks can be expanded further. The terms I and J expand to,

= = j=2/ (32 (5-9)

= ,7t2 + )2 (5-10)
t '. k t resolution t standard dev

where w is the beam width, t is the beam thickness, "resolution" is associated with the

SWLI height resolution, and "standard dev" is based on measurements of thickness at

multiple locations on the beam. The terms B3 and D expand to,


B3 =( 2 + (17B3\)2 (5-11)
B3 B3 least-squares B3 standard dev


D = (-D 2 + ()2 (5-12)
D D least-squares D D standard dev

where the "least-squares" term is associated with the covariance matrix obtained during

fitting and the "standard dev" term is based on the repeated trials of each mass load.

The term r, the torque arm, expands to,


= -- o ( )2 + ()2ign (5-13)
r r resolution vr alignment









where "resolution" is associated with the SWLI lateral resolution and "alignment" is

associated with the extrapolation of the hole location using the alignment grooves

instead of a stitched image of the entire assembly. Representative values are provided

in Table 5-2 where the largest uncertainty contributor, thickness (standard dev), is also

the term that gets multiplied by three in Equation 5-9.

Table 5-2. Uncertainty contributions.

Name Variable Value
name

Young's modulus E/E<1l0> 0.005

Shear modulus GI/G 0.01


Cantilever width

Cantilever thickness
(resolution)
Cantilever thickness
(standard dev)
Normal force coefficient
(least-squares)
Normal force coefficient
(standard dev)
Torsional force coefficient
(least-squares)
Torsional force coefficient
(standard dev)

Gravity

Torque arm (resolution)

Torque arm (alignment)

Mass measurement


aw/w

(Ut/t)resolution

(Ot/t)standard dev

(UB3 /B3)least-squares

(o3 s/B3)standard dev

(UD /D) least-squares

(cD/D) standard dev

o9/g

(Orr/r)resolution

(Or/r) alignment

cmmeas /mmeas


0.003-0.013

1.7x10-5

0.029

0.005

0.02

0.0002

0.02


0.004

0.002

0.02

1.1x10-5 -6.1 x10-5









Young's modulus for single crystal silicon depends on crystal orientation. For (111)

silicon with the cantilever beam axis on the (111) plane, Young's modulus is

independent of orientation with a value of 168.9 GPa. However, manufacturing limits

prevent perfect alignment to the (111) plane. The manufacturer specified alignment

tolerance to the (111) plane was 0.5 deg. Using the stiffness tensor for silicon and

applying tensor rotations, the maximum error in modulus for wafer misalignment was

determined to be 0.5% of 168.9 GPa.

A similar process was used to determine the uncertainty for shear modulus due to

misalignment to the (111) plane. However, shear modulus does depend on the

orientation of the beam axis on the (111) plane. The beam axis was nominally oriented

along the <-1-12> direction by cutting perpendicular to the wafer flat (the wafer flat was

along the <1-10> direction). The flat had a tolerance of 1 deg to the <1-10> direction

and the cutting process had a tolerance of 2 deg. Finally, the shear modulus depended

on the width to thickness ratio of the cantilever. The combined uncertainty for shear

modulus was less than 1% of the nominal shear modulus (545 G < 64.7).

The cantilever width uncertainty was specified as two times the SWLI lateral

resolution (2 x 4.4 pm). Dividing by the cantilever width gives a range from 0.3% to

1.3%. Similarly, cantilever thickness uncertainty was specified as two times the SWLI

vertical resolution (2 x 2.7 nm) which was negligible for thicknesses of 0.305 and 0.311

mm. However, thickness had a much greater variability when measuring at multiple

locations on the same cantilever. Thickness measurements were made by placing a

glass slide under the cantilever and measuring from the top of the glass to the top of the

cantilever surface. This distance was affected by contaminants between the glass and









cantilever and by variability in wafer thickness. Also, measuring the thickness risked

damaging the cantilevers. Therefore, a maximum variability of 9 pm was assumed

based on multiple measurements of a few cantilevers at different locations on each

cantilever. For a 0.311 mm thick cantilever the uncertainty is 2.9%.

The coefficient matrix used in the force equations was obtained from a least-

squares fit to the three-dimensional deformation data, i.e. a series of (x,y,z) points. The

uncertainty in the deformation data is related to the uncertainty in the coefficient matrix

by the covariant matrix. Rewriting Equation 5-3 more generally,

y(x,z) = B3fl(x,z) + B2f2(x, z) + C1f3(x,z) + Cof4(x,z) + Df5(x,z) + Hf6(x, z) (5-14)
bending bending torsion
and
torsion

a new symmetric matrix, a, can be defined with elements,

lk =e l 1 [(1 x i, kxi,)] (5-15)

where I and k are the row and column numbers of a respectively and vary from 1 to 6, m

is the number of measurements across the width of the cantilever, n is the number of

measurements down the length of the cantilever, and mx n is the total number of

measurements. Since measurements are obtained by the SWLI camera, each (x,z)

location is a different pixel, where i and j are the pixel indices. The term oy is the y-

coordinate measurement uncertainty at a given pixel (ij) and was assumed to be the

same for all pixels. This enabled o to be estimated by,

a2 =- =Yj (B3x + B 22 + C x, + Co + Dxiz + Hzj)]2 (5-16)

where N=n x m and 6 corresponds to the number of free parameters (the coefficients).

The covariant matrix is the inverse of a. The elements of the covariance matrix are the

variances and covariances of the fitted coefficients,









UB2 B3 B2 UB3UC1 UB3 Co B3 UD B3 UH
2
UB2 U B2UCI UB2UCo UB2UD UB2 UH
a-=1 ac, 2 UC, 'C,0 UCUD OUCUH (5-17)
aCo 2 UCo D UCoUH
D 2 D UH
2
UH

A typical result of the covariance matrix using torsion loading of 7.552 g on a 1.46 mm

wide by 0.311 mm thick by ~5 mm long cantilever on the positive torque arm is,

-816.1 -3.456 0.004 0.000 0.001 0.000-
0.015 0.000 0.000 0.000 0.000
-1_ 0.000 0.000 0.000 0.000 x 1-5 (5-18)
S 0.000 0.000 0.000
0.001 0.000
0.000-

with values for IB31 and IDI of 18.2556 and 0.4874 respectively. Taking (a01)11/B3 and

I(ac1)55/D gives the uncertainty in these terms of 0.5% and 0.02% respectively.

The coefficients also had measurement-to-measurement deviation when loaded

with the same mass multiple times. Therefore, the standard deviation was computed for

every measurement sequence and found to be around 2% of the mean for both B3 and

D. Gravity was assumed to be 9.81 m/s2 but alignment to gravity was not checked so a

conservative 5 deg deviation was assumed leading to an uncertainty of 0.4% of 9.81

m/s2. The torque arms had a minimum uncertainty due to the SWLI lateral resolution (2

x 4.4 pm was used). Additionally, the alignment grooves were used to extrapolate the

torque arm distance. Using the extrapolation procedure and comparing to a direct

measurement of the torque arm by stitching multiple images together, it was found the

torque arm could be off by as much as 100 pm so the uncertainty was set to this limit.

Finally, all masses were weighed by a precision balance with a resolution of 0.1 mg so

the uncertainty was set to two times this value (2 x 0.1 mg).









CHAPTER 6
CONCLUSIONS

All results under predicted the measured mass (6% to 12%). The thickness was

the largest uncertainty contributor at 3% (Table 5-2) providing a 9% uncertainty in the

estimated mass compared to the measured mass. This uncertainty alone nearly closes

the gap in the under predictions. Thickness measurement is a significant challenge that

is not unique to this research. Not only is it difficult to measure, but its uncertainty is

amplified by being cubed in the force equation.

For the first torsion study (parameters in Table 5-1), the estimated mass was

systematically higher for the positive torque arm than for the negative torque arm

(Figures 5-8 and 5-9). This is contrasted with the second torsion study where the

opposite trend was observed (Figure 5-11). The reason for the difference is the method

for computing the torque arms. In the first torsion study, because many cantilevers were

tested, the torque arms were extrapolated based on the alignment grooves. In the

second torsion study, the torque arms were directly measured by stitching multiple

images together to obtain both the tee and cantilever in a single view. Note that in the

first study, the gap between the mean positive torque arm and the mean negative

torque arm was 3.8% while in the second study, the gap was 1.3% and in the opposite

direction (total difference of 5.1%). The torque arm concept used in this research is

analogous to AFM tip height. Determining these values is a significant challenge in

cantilever force research.

In the study where the normal force coefficient simultaneously with the torsion

force coefficient (Figure 5-10 and Figure 5-11) were computed from a single test, the

estimated mass using the normal force could be directly compared to the estimated









mass from the torsional force. The two results both under estimated the measured mass

but by different amounts. The normal force under estimated by 8% and the torsion force

by 11.9% (difference of 3.9%). Common parameters in the force equations contribute

equally and in the same direction to both results and, therefore, the difference must be

because of uncommon parameters. For example, shear modulus uncertainty does not

contribute to the normal force result but does contribute to the torsion force result.

Summary

Measuring the deflection of an AFM cantilever at a single spatial point requires the

use of a scalar stiffness as the force transduction constant. Stiffness, however, is a

poorly defined quantity for a cantilever sensing an unknown 3D force vector. If, instead,

deflection of the cantilever is measured at multiple points simultaneously, the force

vector can be determined unambiguously. Measurements made this way are immune to

many of the problems associated with traditional single point measuring schemes.

There is no dependency on where the load is applied or what portion of the beam is

measured and there is no cubic dependency of measurement location on length. Each

force component can be independently determined, whereas, in single point techniques,

the total force vector must be decomposed using prior knowledge about the

relationships between the force components.

A prototype platform was designed to measure the full-field deflection profile of

cantilevers under load. The primary platform component was a scanning white light

interferometer (SWLI), which measures surface contours through the low coherence

interference of white light. This platform was used to test a new 3D force model. Forces

were applied to single crystal silicon cantilevers to cause bending and torsion. From the

3D deformation, the forces were determined unambiguously.









Future Work

This study identifies many follow-on research topics. A natural extension is to

validate the model for the third force component, P. Then, the complete force model

could be verified using AFM-scale cantilevers. The obvious challenge for micrometer-

scale cantilevers is identifying sources of suitably small reference forces and

determining the material properties and the thickness of multi-layered cantilevers.

Finally, the prototype platform could be improved. For example, the SWLI measurement

rate is on the order of seconds. Adapting the SWLI for shorter measurement times or

selecting an alternative measurement approach could reduce dynamic disturbances

such as vibrations. With a new high frequency measurement transducer, a new dynamic

3D force model could be investigated.









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BIOGRAPHICAL SKETCH

The author was born in Miami, FL, in 1982. In 2001, he started his bachelor's

degree at the University of Florida (UF). Before graduation, as part of an undergraduate

research fellowship, he worked at the National Institute of Standards and Technology

(NIST), the United States standards agency. There, he became interested in small force

metrology and afterward worked in this field for his master's degree project. In 2007, he

earned his master's degree from UF and then interned at NIST. Later, he started his

doctoral degree which extended his master's work to the full three-dimensional force

model presented in this study.





PAGE 1

1 THREE -DIMENSIONAL MEASUREMENT OF ATOMIC FORCE MICROSCOPE CANTILEVER DEFORMATION TO DETERMIN E T HE THREE -DIMENSIONAL APPLIED FORCE VECTOR By LEE KUMANCHIK A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2010

PAGE 2

2 2010 Lee Kumanchik

PAGE 3

3 To my mom and dad

PAGE 4

4 ACKNOWLEDGMENTS I thank my parents, Dwight and Bunny Kumanchik, for their support and for encouraging m y dream s I thank my love, Carolina Cardoza. If I have fire in my eyes its because she lit fire under my feet. I thank my three aces Steve, Alex, and Joel, who were just good old -fashioned entertainment. I thank the members of my committee and my advisor Tony Schmitz, who is the most patient man I know. And I thank my peers in the Machine Tool Research Center for their shenanigans

PAGE 5

5 TABLE OF CONTENTS ACKNOWLEDGMENTS ...................................................................................................... 4 page LIST OF TABLES ................................................................................................................ 7 LIST OF FIGURES .............................................................................................................. 8 LIST OF ABBREVIATIONS .............................................................................................. 10 ABSTRACT ........................................................................................................................ 11 CHAPTER 1 INT RODUCTION ........................................................................................................ 13 The Atomic Force Microscope ................................................................................... 13 The Detector ............................................................................................................... 15 The Cantilever ............................................................................................................. 18 Cantilever Dimensions ......................................................................................... 20 Force Application Location .................................................................................. 21 Boundary Conditions ............................................................................................ 22 Traceability ........................................................................................................... 23 The Force Equation ............................................................................................. 24 2 LITERATURE REVIEW .............................................................................................. 27 Historical Development ............................................................................................... 27 Applications of the Atomic Forc e Microscope ..................................................... 28 Force Modeling .................................................................................................... 29 Calibration ................................................................................................................... 29 3 THREE -DIMENSIONAL IMAGING ............................................................................ 33 A New Platform ........................................................................................................... 33 Scanning White Light Interferometer ................................................................... 33 Cantilever Holders ................................................................................................ 35 Stages .................................................................................................................. 36 Platform Assembly ............................................................................................... 37 Visualization of Quasi -static Bending ......................................................................... 38 Residual Stress, Batwings, and Differencing ................................................... 38 Snap-in ................................................................................................................. 40 Other Experiments ............................................................................................... 41 4 THREE -DIMENSIONAL FORCE MODELING ........................................................... 44

PAGE 6

6 5 VALIDATION OF THE FORCE MODEL .................................................................... 50 Normal Loading ........................................................................................................... 50 Torsion Loading .......................................................................................................... 58 Uncertainty Analysis ................................................................................................... 63 6 CONCLUSIONS .......................................................................................................... 68 Summary ..................................................................................................................... 69 Future Work ................................................................................................................ 70 LIST OF REFERENCES ................................................................................................... 71 BIOGRAPHICAL SKETCH ................................................................................................ 76

PAGE 7

7 LIST OF TABLES Table page 5 -1. Cantilever geometries. .............................................................................................. 59 5 -2. Uncertainty contributions. ......................................................................................... 64

PAGE 8

8 LIST OF FIGURES Figure page 1 -1. Schematic of atomic force microscope (AFM) operation. ........................................ 13 1 -2. The optical lever technique. ...................................................................................... 15 1 -3. The small angle approximation. ............................................................................... 16 1 -4. A perfectly smooth sample slides past the stylus .................................................... 17 1 -5. Using the optical lever technique to detect twist. ..................................................... 19 1 -6. A scanning electron micrograph of a commerci al cantilever .................................. 20 1 -7. Cantilever stiffness transfer. ..................................................................................... 21 1 -8. Foreshortening from vertical bending. ...................................................................... 23 1 -9. Cantilever with a friction force proportional to the applied force, F. ........................ 24 3 -1. Prototype platform assembly. ................................................................................... 33 3 -2. SWLI schematic. ....................................................................................................... 34 3 -3. Example cantilever placement on the aluminum holders (top views). .................... 36 3 -4. Positioning stage scale error. ................................................................................... 37 3 -5. The Olympus OMCL-AC240TS cantilever. .............................................................. 38 3 -6. Olympus BL -RC150VB AB cantilevers. ................................................................... 39 3 -7. Cantilever snap -in demonstration. ............................................................................ 41 3 -8. Two Veeco 1930 (1.3 N/m, 35 m wide) cantilevers in contact .............................. 41 3 -9. Data dropout. ............................................................................................................ 42 3 -10. Tip wear estimation. ................................................................................................ 43 3 -11. Anomalous bending behavior. ................................................................................ 43 4 -1. A cantilever loaded at the free end by two independent forces P and F ............... 45 4 -2. Force measurement does not depend on the global geometry of the cantilever. .. 45 4 -3. The 3D view of the forces acting on the cantilever. ................................................. 46

PAGE 9

9 4 -4. SWLI measurement coordinates. ............................................................................. 48 5 -1. SWLI image of the fabricated cantilever. ................................................................. 50 5 -2. Fabricated masses. ................................................................................................... 51 5 -3. Results of Equation 5-1 for the 10 masses. ............................................................. 53 5 -4. B eam loaded under identical conditions, but measurement field of view was varied. ..................................................................................................................... 54 5 -5. The aluminum tee bonded to a 1.5 mm wide cantilever. ...................................... 56 5 -6. Result s of Equation 5 -1 for the 8 masses using the surface fit approach. ............. 57 5 -7. Tee attached to cantilever. ..................................................................................... 58 5 -8. Torsion results when mass is loaded at the positive torque arm. ........................... 60 5 -9. Torsion results when mass is loaded at the negative torque arm. .......................... 61 5 -10. Torsion loading results for mass loaded at the positive and negative torque arms. ....................................................................................................................... 62 5 -11. Normal loading results for mass loaded at the p ositive and negative torque arms. ....................................................................................................................... 62

PAGE 10

10 LIST OF ABBREVIATIONS AFM Atomic force microscope STM Scanning tunneling microscope SWLI Scanning white light interferometer SI Systme International dUnits 3D Three dimensional

PAGE 11

11 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THREE -DIMENSIONAL DEFORMATION MEASUREMENT OF THE ATOMIC FORCE MICROSCOP E CANTILEVER SENSOR FOR THE DETERMIN ATION OF THE APPLIED FORCE VECTOR By Lee Kumanchik August 2010 Chair: Tony Schmitz Major: Mechanical Engineering The atomic force microscope (AFM) is the instrument of choice for measuring nano to micro -Newton forces (109 to 106 Newtons ). However calibration is required for accurate measurements. AFM calibration has been studied for decades and remains a significant focus within the metrological community, in particular at international standards organizations. Whi le progress has been made, there is still much to accomplish as current force calibration techniques yield relative uncertainties (1 standard deviation/mean) of 10% 20% For example, measuring a force of 500 nN would yield a result between 400-600 nN 68% of the time. T he critical issue is the existing AFM metrology which monitors deformation at a single (spatial) point on a structure that encounters a three-dimensional (3D) force and responds with a 3D deformation. This sing le -point calibration technique considers only to a limiting set of information, while additional information is available. Similarly, subsequent measurements by the AFM after calibration are restricted to the same limits. As a response, this project aims to improve AFM calibration and use by implementing a new metrological platform and analysis technique.

PAGE 12

12 The new platform incorporates a scanning white light interferometer (SWLI) for 3D cantilever deformation measurements. The SWLI introduces two important changes over standard AFM metrol ogy. First, it provides a multi point measurement of the backside surface of the cantilever rather than a single point measurement near the free end. Second, it is a direct displacement sensor which does not infer displacement from the measurement of anoth er variable, such as the surface angle in the optical lever technique. In this study, the AFM is first described with a focus on its use as a force sensor. Then, the new platform design and construction, cantilever imaging tests, and the development of a n ew force model which takes advantage of the 3D deflection data are presented. The new force model addresses many of the challenges associated with traditional calibration strategies E xperimental validation is presented for the case s of normal force lo ading (i.e., perpendicular to the cantilever axis and resulting in bending deformation) and torsional loading.

PAGE 13

13 CHAPTER 1 INTRODUCTION The Atomic Force Microscope The atomic force microscope (AFM) is a multipurpose instrument used for interacting with a sample at an atomic scale (including imaging, force interaction, and manipulating individual atoms). Interaction is performed by an atomicall y sharp stylus that is driven by stages to a location of interest on a sample. The small tip radius of the stylus (<30 nm or approximately the radius of a virus) focuses the interactions onto a small (nearly point) area enabling extremely fine spatial resolution. The stylus is attached to a cantilever that is monitored by a detector as it deforms under the interaction forces (Figure 1 -1). Although the entire cantilever deforms, the detector typically determines the motion at or near the stylus (located at t he free end of the cantilever) only. This is a natural choice since the AFM can perform imaging with the stylus motion having a 1:1 correlation (ideally) to a samples surface contours. When used for force measurement, the stylus motion follows Hookes law F = ky, where F is the vertical force, k is the cantilever bending stiffness, and y is the stylus vertical displacement. Using an AFM, force resolution as low as 1 pN (1012 N) can be realized. Figure 11. Schematic of atomic force microscope (AFM) operation. Forces at the stylus from the moving sample cause deformation in the cantilever. A detector observes the deformation.

PAGE 14

14 Due to the AFMs popularity as a multi -function instrument, it is applied across a diverse range of disciplines where users often may not possess detailed knowledge of the underlying mechanics. Therefore, vendors strive for turn key system operation. However, achieving the required accuracy under this paradigm can be challenging. For example, cantilever stiffness studies have found that the manufacturer -specified stiffness can vary by as much as 300% from the calibrated stiffness (0.2 N/m nominal stiffness vs. 0.067 N/m calibrated stiffness) [1]. One study combined all manufacturer specifications required for force measurement (including stiffness) and found the combined uncertainty was greater than 1500% [2]. While this level of divergence may not always be pr esent, calibrating the AFM is clearly required to obtain meaningful results. However, after decades of calibration research, the best stiffness calibrations, performed by international standards agencies, are accurate to only 5% [3]. Using methods availabl e to the average user, the accuracy of stiffness calibration is limited to 10% -20% in general [2,4]. Because stiffness is one of the parameters required to determine force, the accuracy of the force measurement will be no better than the stiffness accurac y and, most likely, will be worse. Since the force is computed with Hookes law (F = ky), there are two components to a force calibration: 1) stylus displacement calibration; and 2) stiffness calibration, with the majority of research focused on stiffness calibration. This force relation, however, is too simplistic. Cantilever mechanics studies have shown how cross axial sensitivity to forces perpendicular to the measurement direction lead to a systematic error between 10% -20% [5]. This error is independent of calibration. While compensatory post processing techniques are available [6], they are largely unused for several reasons.

PAGE 15

15 First, their complexity requires expertise in mechanics to understand and programming knowledge to implement. Second, they require the geometrical parameters of the AFM setup (which are often difficult to obtain) and knowledge of the stylus sample interactions (which are typically unknown a priori ). Third, they continually evolve and must be updated every couple of ye ars [58]. Therefore, the simple force equation, F = ky, remains in use. In a comprehensive review (1254 cited references) of AFM calibration and implementation in numerous fields, there was no mention of any alternative force relation to F= ky [9]. In the following sections, force measurement by AFM is discussed in three parts: the detector, the cantilever, and the force equation. The Detector Figure 12. The optical lever technique. A laser is reflected off a cantilever onto a photo-dete ctor. The light level on each segment of the photo detector (S1 and S2) represents the amount of cantilever deflection. The stylus displacement (perpendicular to the cantilever axis in the normal force direction) is monitored by a detector (represented b y y in F=ky ). The most popular detection system in use today is the optical lever technique (Figure 1 2). Here, a laser is reflected from the backside of a cantilever onto a two-segment photodetector. As the cantilever deforms, the angle of incidence changes and the reflected laser beam moves

PAGE 16

16 along the photo-detector segments. The light levels on the segments are directly related to the angle of incidence of the cantilever (after compensating for photo -detector non linearity). The longer t he optical path, the more pronounced the effect; this amplification follows the simple lever rule. Though detection is directly related to angle, it is difficult to calibrate for angular motion. Additionally, in imaging mode and when applying Hookes law, the vertical displacement (not cantilever angle) is required. Therefore, a rigid surface calibration is typically performed. Here, the stylus is brought into contact with a rigid sample and then the sample stage is displaced vertically. Using the known com manded displacement, the small angle approximation can be used to determine a rate of displacement directly proportional to the angular change of the AFM cantilever (Figure 13 ). In this way, the stylus vertical displacement accuracy is approximately equal to the stage vertical displacement accuracy. Figure 13. The small angle approximation Small deformations enable the cantilever angular change to be related to the stage vertical motion. While this detection system is simple in practice, it suffers from a critical drawback. Since it indirectly measures displacement through angular change, any force which causes the cantilever to deform without displacing the stylus vertically will still register as stylus motion. For example, a sample moving parallel to the beam axis while in contact with the stylus will generate a kinetic friction force that will tend to deform the cantilever flexure ( Figure 1-4 ). Though the sample may be perfectly smooth (no

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17 topography), the lateral friction will register as vertica l motion due to the change in reflected beam angle. This phenomenon has been exploited in tribology since bi directional motion registers as complementary positive and negative detector output that relates to friction [10]. As an alternative to motion parallel to the beam axis, perpendicular motion can also be used to measure friction. This motion causes beam twist (torsional loading) and can be measured using a quadrant photo -detector ( Figure 1 -5 ). The arrangement of four total segments isolates twist from stylus vertical displacement so that topography is not registered as friction in the twist direction. The challenge is Figure 14. A perfectly smooth sample slides past the stylus. The resulting friction force deforms the cantilever which registers a s motion by the detector. that twist is a measure of angle and therefore requires a more complex calibration compared to the rigid surface calibration used for displacement. Additionally, twist angle is proportional to torsion (i.e., the product of the fri ction force and stylus length) not just the frictional force. Therefore, the stylus length (or height ) must be measured to calculate the frictional force. Calibrating the detector for every measurement is essential for obtaining good force accuracy. In pr actice, however, this may not be practical. In the absence of comprehensive calibration, misalignment between the detector and the cantilever may

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18 go undiagnosed. For the optical lever technique, the angle of incidence determines the displacement, which imp arts some robustness to misalignment. The angular deflection equation for a rectangular cantilever with an end force and its sensitivity to measurement location are, ( ) = ( 2 ) & = (1 -1) where L is the cantilev er length, x is the location along the beam axis (0 x L ), and is the angle of the cantilever. Misalignment of the laser from the calibrated position gives an error rate in displacement measurement x:y between 1:1 and 1:0 from the base to free end, respectively For example, a system calibrated for x = L but misaligned by a few micrometers would introduce virtually no error in corresponding displacement measurements due to this misalignment Since thi s is the most common position to point the laser, the optical lever technique is fairly robust to misalignment. For AFM systems with direct displacement sensors such as displacement measuring interferometers, alignment errors are not mitigated but instead amplified. The deflection equation and sensitivity to measurement location for displacement are, ( ) = ( 3 ) & = 3 (1 -2) Misalignment of the detector from the calibrated position gives an error rate, x:y between 1:3 and 2:3 from base to free end with the free end being the best position. The Cantilever The cantilever is an interchangeable component in the AFM system. Cantilevers are made with different geometries and tips (in addition to sharp st yli, spheres are also often used, for example) to suit many applications with the same AFM system. Regardless of the cantilever manufacturer (often different from the AFM system

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19 manufacturer), the bending stiffness is always specified for use in the force equation ( k in F=ky ). However, the stiffness is extremely sensitive to cantilever geometry and, therefore, must be calibrated on an individual basis. Potentially the most confounding issue faced by stiffness calibration is that the stiffne ss of a generalized beam depends on the boundary conditions. For example, a fully constrained (fixed/fixed) beam has four times the stiffness of a cantilever (fixed/free) beam. For the AFM cantilever, the stiffness estimation is naturally based on the cant ilever model; the complexities that arise from the stylus contact mechanics, which alter the system stiffness, are left for the user to handle. Figure 15. Using the optical lever technique to detect twist. The left to -right motion of the laser registers as twist. The top -to -bottom motion registers as verti cal deflection, as shown previously. There are numerous stiffness calibration techniques and each has benefits and drawbacks. Non-contact techniques avoid stiffness altering contact mechanics but do not take into consideration the load application location. Contact techniques are often simpler conceptually and more convenient for achieving traceability to international position cantilever axis laser beam Front view laser beam position photo detector

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20 standards, but must compensate for contact behavior. Rather than list the techniques here (see Chapter 2), the largest sensitivities to sti ffness are described. Cantilever Dimensions A cantilevers dimensions define its stiffness. From solid mechanics, a rectangular cantilever beam can be modeled using Euler -Bernoulli beam theory. At the stylus, the deflection equation is, = = (1 -3) where E is Youngs modulus for the beam material, w is the beam width, L is the beam length, and t is the beam thickness. Rearranging yields an expression for the stiffness according to Hookes law (analytical models exist for other beam geometries and can similarly be used to estimate stiffness). Direct measurement of each term provides the best resul t so that the stiffness uncertainty is a combination of the measurement Figure 16. A scanning electron micrograph of a commercial cantilever. Extra material is cut away during manufacturing. This undercut is not visible in a top -down view of the cantilever.

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21 uncertainties in each input and the model accuracy (i.e., considering the limitations imposed by the model assumptions). In this example, t he most sensitive parameters are thickness and length. Thickness of silicon substrates (the typical AFM cantilever material) is difficult to predict from manufacturing and should be measured directly. Length can vary from nominal due to undercut at the bas e when using silicon -based manufacturing techniques ( Figure 16 ). Assuming all other quantities are perfectly known, there is approximately a 1:3 relationship between the error in length or thickness to the error in stiffness. For example, a 1 % error in thickness or length leads to an error in stiffness of ~ 3 % (30 nm thickness error or 2 m length error on a 3 m x 200 m cantilever with a nominal stiffness of 6 N/m gives a stiffness error of 0.18 N/m ). Force Application Location Figure 17. Cantilev er stiffness transfer. The AFM cantilever is pushed against the precalibrated cantilever. The force application point on the AFM cantilever is precisely located at the stylus. However, the force application point on the pre -calibrated cantilever is subjec t to the position accuracy of the AFM. The effective length of the pre -calibrated cantilever determines the measured stiffness. A more direct technique to determine stiffness is to apply a known force to the cantilever and measure the displacement so that k=F/y For relatively stiff cantilevers, a mass can be attached to the free end to apply a known (gravity based) force. Here, the force application point requires precise alignment to the stylus location, where the cantilever will experience all subsequent forces. The sensitivity to alignment errors is

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22 identical to the length sensitivity with the same 1:3 relationship. For mass loading, there is a fundamental limitation imposed by the availability of traceable mass artifacts smaller than 1 mg. Therefore, a variation of the force loading scheme, referred to as a stiffness transfer, is used where the unknown stiffness of a cantilever is related to the precalibrated stiffness of a second cantilever. The result is similar to a system of two springs in series su ch that k1=k2(xt/x11) where k1 is the unknown stiffness, k2 is the known stiffness, xt is the total motion as one cantilever is pressed against the other, and x1 is the response motion of the unknown cantilever. Here, the force application point on the unknown cantilever is determined precisely since the stylus makes direct contact with the cantilever of known stiffness. However, the force location on the second c antilever is again subject to alignment errors and results in an error in stiffness at a rate of 1:3 (Figure 1-7). Boundary Conditions For contact -style stiffness calibrations, as well as any use of an AFM cantilever after calibration, boundary conditions can contribute considerable measurement uncertainty. Boundary conditions define the deflection equation (the equations presented so far have been for an ideal cantilever with a vertical load at the stylus). This simplification fails when the cantilever com es into contact with a sample, either during contact style calibrations or when used to perform measurements. An example contact phenomenon is friction, as discussed previously. Foreshortening occurs whenever the cantilever deforms ( Figure 1 8 ). The foresh ortening is resisted by friction parallel to the sample surface. As an example, the stylus vertical motion for a cantilever experiencing sliding friction proportional to the vertical force is,

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23 = (1 -4 ) where is the coefficient of kinetic friction and r is the stylus height. Assuming this to be the only boundary condition effect, the ideal stiffness k can be obtained if the coefficient of sliding friction is known as well. However, this is not typically the case. Boundary conditions are an important concept for this research and are discussed in Chapter 4 Figure 18. Foreshortening from vertical bending. Traceability Proper calibration of stiffness should follow a clear path of traceability to the Systme International dUnits (SI). Traceability is the chain of calibrations that link any sensor/instrument through a direct path, with defensible uncertainty statements f or each measurement, to a primary standard. This primary standard, which reflects a physical realization of the unit of measure, is defined and maintained (if necessary) by national measurement institutes (NMIs). Traceability enables measurement uncertaint y to propagate through the chain of calibrations and thus identifies the uncertainty for measurements performed by a given sensor or instrument. This notion of standardized units and measures allows quantitative data to be compared between organizations, m anufacturers, and laboratories, facilitating collaboration and reproducibility of results. Traceability is sometimes difficult to obtain depending on the calibration technique.

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24 Therefore, the approach adopted by the National Institute of Standards and Tech nology (the national measuring institute for the US) has been stiffness transfer by cantilever artifacts. This technique has been traceably compared to the geometrical model as well as the popular thermal calibration technique [11] and agreement within ~5% has been obtained [4]. The Force Equation Figure 19. Cantilever with a friction force proportional to the applied force, F. As noted, the most popular approach for AFM force measurement is to apply Hookes law, F=ky The choice of this equatio n stems from the single point detection scheme used to monitor the cantilever deformation. With only a single point of displacement, the force transduction constant must be stiffness (N/m). The general Euler -Bernoulli beam equation relating a beams deflec tion to the applied load is, = (1 -5) where x is the distance along the beam axis, y is the deformation perpendicular to the beam axis, I is the second moment of area, and u is a distributed load. Four integrations are necessary to obtain the force displacement equation, which requires four boundary conditions. Because the moment boundary condition is impossible to know without fixed free cantilever

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25 detailed knowledge of the contact mechanics, the challenge of force measurement by AFM can be summarized as attempting to solve an underdetermined equation. As an exercise, the beam equation is solved assuming friction proportional to the applied force at the tip in order to illustrate how F=ky can be replaced with an improved forcedisplacement relationship ( Figure 1 9 ). Assuming y=y(x), u=0 and EI is constant, integration is straightforward ( EI d4y/dx4=0) and yields four equations, = (1 -6) = + = (1 -7) = + + (1 -8) = + + + (1 -9) where C3 4 are zero due to the fixed base boundary condition (zero motion and slope at the base), C1 is equal to the perpendicular force c omponent (by force balance with shear force), and E quation 1 -7 is equal to the bending moment M along the beam. The bending moment equation captures the friction contact mechanics and is, = ( ) + (1 -10) where is the coefficient of sliding friction and r is the tip (stylus) height. The resulting deflection equations for any point on the beam and for the tip location ( x=L ) are, ( ) = ( 3 ) + ( ) = 1 + = (1 -11) The addition of friction makes the force equation more complex than F=ky In practice, this friction analysis is too simple. Under the assumption that macroscopic friction

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26 applies to the at omically sharp stylus, there must be static friction as well. Unlike kinetic friction, static friction is not a proportional relationship to normal force except at its maximum. Therefore, measurements performed where there is no sliding (such as part of st ick -slip events) include an unknown quantity in the final displacement equation, ( ) = ( 3 ) + (1 -12) where G is the friction resisting sliding. Simplifying to the tip location ( x=L ) does not give a clear relationship between the applied force F and the tip motion y(L) When the macroscopic friction model is insufficiently accurate or when other forces contribute, such as electrostatic, osmotic, and/or magnetic forces, this entire analysis is invalid. Ambiguities like this will always exist in the force equation if a force relationship is unknown a priori which is true in virtually every measurement situation. This makes exploratory force measurements particularly diff icult since the environment has yet to be characterized. Therefore, a new force measurement paradigm is necessary to move toward more accurate force metrology.

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27 CHAPTER 2 LITERATURE REVIEW Historical Development The atomic force microsc ope was introduced by Binnig, et al in 1986 to measure micro -scale forces [12]. Just prior to the invention of the AFM, the scanning tunneling microscope (STM) [13] was developed. This was a surface profilometer which used the strong variation in electron tunneling current with separation distance to map a sample surface. It only worked on conductive samples. The AFM was developed to expand the types of surfaces that could be probed. The idea was to use a compliant structure with a sharp tip that, when probing a surface, would deform from forces applied to the tip. Since all materials generate reaction forces to the probes applied pressure, the AFM was not limited to imaging conductive materials. More importantly, force could be inferred from the d eformation of the compliant structure. Initially, a simple rectangular cantilever was utilized as the compliant structure and a scanning tunneling microscope was used as the cantilever displacement sensor. Since 1986, the AFM cantilever has been adapted to suit a wide range of applications. For example, two arm triangular cantilevers increase the torsional rigidity and reduce the sensitivity to torsional loads. Coating the cantilever surface with thin films has become a standard method to increase detector sensitivity, enable biological imaging, and/or make the cantilever conductive. Additionally, various tip geometries such as spheres and carbon nanotubes are applied. Self -actuating piezoelectric cantilevers are used in cantilever arrays which enable a larg e sample area to be imaged rapidly. Since cantilevers are interchangeable, they are specialized to suit the specific environment. Therefore, no one design is used exclusively of the others.

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28 The displacement detector has also taken several forms. A capaciti ve sensor over a double leaf cantilever was developed [14]. Heterodyne and homodyne interferometry have been used with different laser sources such as He-Ne lasers and laser diodes [15,16,17,18]. Additionally, heterodyne interferometry applied over a small spot on the cantilever enabled the 3D surface under the spot to be visualized [19]. Torsion of the cantilever was detected with this approach. A laser was used in the optical lever technique [20] and torsion could be detected depending on the photodiode a rrangement. Finally, the cantilever itself has been used as the detector through the piezoresistive effect [21]. Each detector scheme has situational advantages, but the most broadly adopted scheme used today is the optical lever technique. Since the detec tor is not interchangeable, most commercially available systems rely solely on this technique. Applications of the A tomic Force M icroscope AFM has become ubiquitous in the research setting. Since the AFM can image non-conducting samples, it is commonly use d in ultramicroscopy (imaging at atomic resolutions). For example, the AFM is used in biological [22], crystalline growth [23], and pharmaceutical drug [24] imaging applications. However, the AFM is not restricted to imaging. It is also used in atomic scal e manipulation [25], such as nanolithography [26], and data storage and retrieval, such as IBMs Millipede project [27]. The AFM has also become a potential tool for quality control in microelectromechanical systems (MEMS) as a parallel to electronic quali ty control in microchip manufacturing [28]. This is due to the AFMs force measuring capabilities, which enable material and mechanical testing. Many fields can take advantage of the AFMs ability to measure micro-scale forces. For example, the structure o f DNA is known, so current efforts are directed at measuring the

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29 bonding forces between complementary nucleotides [29]. The mechanical stability of proteins has also been tested [30]. In surface science, the AFM is applied to friction measurements, such as between microspheres [31] and in capillary force analysis [32]. Force Modeling Force is computed through its relationship to the cantilevers deformation. This relationship, called the force model, is based on Hookes Law which is an integral part of line ar elastic theory. Hookes law provides a constitutive equation that relates stress with strain (force, F with displacement y ). Since the detector in AFM measures displacement, Hookes law is applied directly as F=ky This displacement r elationship is the most common force model for AFM [9]. However, it has been recognized that this model is too simpl istic Other force models have been proposed which take into account forces that arise from contact mechanics at the cantilevers tip. These have attempted to capture tip geometry [33], elastic deformation of the sample [34], cantilever inclination angle [5], adhesion [35], and friction [36,37]. Many times the model is situational F or example, if the sample is very hard, plastic deformation i s unlikely to occur or if the cantilever and sample are submerged in water, capillary force is mitigated. Therefore, it is necessary to understand the test environment in order to choose a model and determine its parameters. Calibration The AFM system mus t be calibrated to obtain reasonable measurement accuracy. The literature focuses heavily on stiffness calibrations because stiffness is the most difficult parameter in the force equation to calibrate. Calibration may be divided into four categories: dimensional, intrinsic, dynamic, and static.

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30 Dimensional methods use estimations of the geometrical parameters of the cantilever and beam theory to predict stiffness. Computations may be carried out analytically or via finite element analysis. Relevant parameters include length, thickness, width, film thickness, modulus of elasticity, and second moment of area. Depending on the method, compensation can be made for V -shaped beams (double arm cantilevers) [38] and the trapezoidal cross -section seen in some commerc ial cantilevers [39]. Accuracy is limited by the combined standard uncertainty from each measurement; thickness, length, and Youngs modulus are typically the largest error contributors. Intrinsic methods attempt to apply naturally occurring phenomena. An example is force measurement on DNA [40]. Since the bonding energy of DNA is specific and highly reproducible, the force to pull DNA apart is reproducible. Such artifacts are nearly identical in nature and can be mass -produced by the millions. A critical p arameter for such a calibration is temperature since, in general, increasing temperature reduces the additional energy required to rupture bonds. Dynamic methods use cantilever vibration to estimate stiffness from frequency shift or phase change. One of th e first techniques that applied this dynamic approach was developed by Cleveland et al. [41]. In this technique, a known mass was attached to the end of the cantilever and the corresponding reduction in natural frequency was measured. The location of the m ass on the beam is a critical parameter and removing the mass after calibration is not trivial. Mass has also been added using different materials such as thin gold films [42], water droplets dispensed from an inkjet [43], and even other cantilevers [44]. Other researchers altered the method so that mass addition

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31 was not required. A well known method proposed by Sader et al. [45] uses the resonant frequency together with the Q (damping) factor in a fluid, usually air. The use of fluid dynamics has become mo re common, especially in colloidal probe microscopy, because the cantilever has a sphere for a tip and spheres are convenient to model in fluid dynamics [46]. Another approach uses the equipartition theorem and thermal oscillations of the cantilever to det ermine cantilever stiffness [47]. With corrections made for the laser spot size [48] (a laser is used to determine cantilever deflection), this method has become popular due to its application ease. Static methods involve applying known forces directly to the cantilever and observing the resulting deflection. They are the most direct measurement of cantilever stiffness. Many devices have been used to apply a direct force to the cantilever. These include macro-sized (relatively large) cantilevers [49], piezoresistive levers [50, 51], and nanoindentation machines [52]. The measurement uncertainty in these experiments has been as high as 20%. The most sensitive parameter is claimed to be the load application point. Since the stiffness is related to length by an inverse cube law, small changes to load application point result in large errors in stiffness. To address this issue, Cumpson et al. [5355] at the National Physics Laboratory, Teddington Middlesex, UK, have developed a series of micromachined artifacts for cantilever calibration. These artifacts have fiducials for locating the load application point and are available in a variety of shapes to accommodate a wide stiffness range. Some even include a built -in mechanism for calibrating themselves [56]. Gate s et al. developed arrays of reference cantilevers with sufficient fabrication repeatability that a calibration performed on a single cantilever is representative of the entire array [57].

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32 There are also additional calibration challenges. The stiffness has been shown to change in response to wear at the tip [33] and over time even if unused [2]. Given the difficulties associated with stiffness calibration and force model development, this research introduces a new measurement paradigm. Instead of a single p oint measurement of the tip displacement, the full -field displacement of the 3D surface of the cantilever is obtained.

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33 CHAPTER 3 THREE -DIMENSIONAL IMAGING A New Platform Three dimensional imaging requires a platform different from traditional AFM. This chapter discusses the design of the prototype platform used in this study and presents its capabilities and restrictions. The platform consists of a scanning white light interferometer (SWLI), holders for the cantilevers, a translation stage, a rotation (tip-tilt) stage, and an adapter plate to connect these units to the SWLI table (Figure 3 1 ). Figure 31. Prototype platform assembly. It is composed of a scanning white light interferometer ( SWLI ), a translational stage, and a tip -tilt stage Scanning White Light Interferometer A key platform component is the SWLI, an optical 3D surface profiler that uses interference of a broad spectrum light source, or white light, to measure surface topography (Figure 32). Light reflected from the sample interferes with light reflected from a reference surface, but unlike coherent source interference, the white light interference only occurs over a small optical path difference. By translating the objective

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34 (which carries the reference surf ace) relative to the sample, a plot of the interference intensity for this path difference range can be captured on a pixel by pixel basis by the SWLI detector. The location of the modulated intensity region (due to the alternating constructive/destructive interference) indicates the relative height of the sample at that pixel. Each pixel on the detector corresponds to a lateral position on the sample; the field of view for the corresponding height map depends on the system magnification. A Zygo NewView 720 0 was used in this research. The selected system included a motorized translation/rotation (or X/Y/tip/tilt) table for sample alignment to the optical axis. As shown in Figure 3 1, the prototype platform is mounted to this table. Figure 32. SWLI schematic. A Mirau objective is represented, although other types are available. The left to -right offset between the interference intensity from positions 1 and 2 indicates their relative height difference (positive translation in the upward direction). There are constraints associated with SWLI measurements. First, a full height map takes several seconds to acquire, in general, since the objective must be translated. During this time the sample dynamics must be interrupted (this pseudo-static approach is applied in this study) or synchronized with the measurement (this strobing approach

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35 is not used here). Second, the lateral resolution is typically orders of magnitude coarser than the vertical resolution along the optical axis. For the experiments reported here, two objectives were used: a 5x Michelson with a lateral resolution of 2.2 m/pixel and a 20x Mirau with a 0.55 m/pixel lateral resolution (1x zoom and 640x480 pixel detector). In comparison, the NewView 7200 literature specifies a vertical resolution of 0.1 nm, although this value is dependent on the noise floor imposed by the measurement environment. Therefore, tests were performed to determine the repeatability for the experimental setup. A smooth silicon surface was placed on the stage at the cantilev er loading location and 130 scans were completed using a 40 m vertical scan range. The height repeatability for each pixel was then assessed. It was found that, on average, each pixel reported the same position within a standard deviation of 2.7 nm. There fore, a resolution of 2.7 nm was assumed. Third, the SWLI cannot detect large changes between adjacent points, so there is a maximum slope that can be detected. This slope varies depending on the selected objective/zoom; it was 4 deg for the 5x objective and 18 deg for the 20x objective (each at 1x zoom). Cantilever Holders Aluminum plates with dimensions of 60 mm x 60 mm x 3 mm were fixed to the stages (Figure 3-3). Cantilevers and test surfaces were adhered to the plates at the midpoint of three sides (a fourth side with a specialized geometry was also available to enable cantilever alignment in future work). This design allowed the holder to be unscrewed and rotated to select the next cantilever for experimentation while approximately maintaining the same position in the SWLI field of view. A similar configuration was used for tests with a cantilever contacting a rigid surface. Cantilevers

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36 and rigid surfaces were bonded to the holders with adhesive to enable the required topdown view for the SWLI measurem ents. Figure 33. Example cantilever placement on the aluminum holders (top views). The individual cantilevers are too small to be seen at this scale, but the monolithic base chips are labeled A1-C2 (a) A -C (b), and a rigid sample is labeled S (b). If the holders were unscrewed and rotated in the indicated direction, the next experiment would be C1 against C2 (a) and C against S (b). Stages The positioning stage (Thorlabs MAX 301) was a threeaxis, parallel kinematics, flexure based design with 4 mm of coarse motion (thumbscrew actuation) and 20 m of fine motion driven by piezoelectric actuators with strain gauge feedback. The motion was controlled by a Thorlabs BPC103 control ler. The second stage was a manual tip-tilt platform (Thorlabs ATP002) positioned on a base assembly (Thorlabs AMA501) which enabled equal height, side by -side use with the positioning stage. The tip-tilt platform provided 4 deg of roll and pitch and acted as the fixed stage in the experiments. Any cantilever on this stage could be tilted into alignment with the positioning stage and then held fixed for the duration of the experiment. The positioning stages vertical linearity was tested from 7 m to 20 m within the 0 -20 m range of the piezoactuators. A silicon surface was adhered to the holder on the positioning stage. A second silicon surface was adhered to the holder on the tip/tilt

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37 stage. The SWLI was used to measure the height difference between the two surfaces as the positioning stage was translated along the optical axis using 1 m increments. The result (Figure 3-4) reveals a scale error of 79 nm/m. Though this quantity was available, the experimental results performed in this study did not requi re motion of the positioning stage, i.e., only features within the SWLI scan range were used, so this scale factor was not required in the data analysis. Figure 34. Positioning stage scale error This was determined by comparing stage vertical position to displacement determined from SWLI measurement. Platform Assembly The platform assembly rested on the SWLI motorized table. During use, the platform positioning stage was first aligned to the SWLI optical axis using the table. Then, the tiptilt stage was aligned to the positioning stage. Since the SWLI angular detection limit can be low (depending on the objective and effective magnification), proper alignment is important to maximize the available measurement range. Additionally, aligning to the optic al axis ensures that the vertical motion from the positioning stage produces no lateral motion in the field of view.

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38 Visualization of Quasistatic Bending The prototype platform enables the direct visualization of cantilever deformations without the need f or interpretation via a single point measurement of the cantilever tip motion used in conjunction with beam models. In this section, cantilever defle ction measurements are provided. Various loading conditions are represented including no loading, rigid sur face loading, and cantilever on -cantilever loading. The results are discussed and the primary imaging limitations noted [58]. Residual Stress, Batwings, and Differencing Figure 35. The Olympus OMCL -AC240TS cantilever The deformed shape has approxim ately 0.5 m of initial tip deflection. The cantilevers presented here had a nonplanar shape even in the absence of an external load. Cantilevers are typically designed to be flat, but residual stresses from the surface coating often lead to unwanted deformation. This deformation is typically small, but is observable in the SWLI measurements. Figure 35 shows the reflective backside of an Olympus OMCL -AC240TS (1.8 N/m stiffness) cantilever (i.e., the side the laser would normally reflect from in an AFM). I t is seen that the cantilever deformation is approximately 0.5 m at the tip with no external load applied. An imaging artifact

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39 sometimes referred to as batwings is also observed. These spikes along the periphery of the cantilever are false height readin gs and tend to occur at steep height transitions. The speck near the tip (also visible in the section view) is surface contamination and also causes the bat wing effect. Additional cantilever measurements are shown in Figure 36. The data were collected us ing a 20x objective at 1x zoom in the 320x240 pixel mode. Figure 36. Olympus BL -RC150VB AB cantilevers Stiffness is 0.03 N/m (a) and 0.006 N/m (b) The deformation levels are ~ 0.25 m (a) and ~1 m (b) in the absence of external loads. When studying deformation of beams it is necessary to separate the contribution due to the external force(s) from those caused by other sources such as residual stress. For linear elastic beam models, deformation follows the rule of superposition. Therefore, in each of the analysis results presented here, the cantilever was measured before applying external loads. This reference image was then subtracted from all subsequent images to isolate the relative deformation caused by the applied forces. This differencing does not remove the bat wings, however, as these locations tend to report random erroneous heights from image to image.

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40 Snap -in When an AFM cantilever is placed in close proximity to a surface, an attractive force develops which pulls the tip closer to t he surface. At a critical gap size, the attractive force overcomes the cantilevers restorative elastic force and the tip snaps into contact with the surface. This attractive force represents one of the many forces an AFM cantilever experiences. Snap-in be havior was measured for an Olympus OMCLAC240TS (1.8 N/m) cantilever positioned near a smooth silicon surface (Figure 3-7). For these measurements, the silicon surface was mounted on the threeaxis positioning stage and moved toward the cantilever in 50 nm steps. The sequence of three plots included in Figure 37 displays the cantilever deformation profile for three stage locations. In position 1, the silicon surface is sufficiently far from the cantilever so that no appreciable attractive force is present. In position 2, snapin has occurred. The cantilever deflection is approximately 0.1 m at the tip. In position 3, the stage has continued to move vertically past the cantilevers undeformed position so that it is deflected upwards. In Figure 3 7b, the cen terline cantilever profiles are provided for the same three stage locations. The tip deflection versus stage motion is provided in Figure 3 -7c; measurement points are identified by the small circles in 50 nm increments and the three locations from Figures 3 -7a and 37b are presented by the large circles. Note that the information in Figure 37c is all that is available from standard AFM single point deflection metrology. All images were captured using a 20x objected at 1x zoom in 320x240 pixel mode. The SWLI data was differenced to isolate the deformation caused by snap -in and it was filtered to remove the bat wing artifacts.

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41 Figure 37. Cantilever snapin demonstration. a) Height map of the cantilever at various stage locations, 13; b) corresponding two-dimensional profiles, 1 -3; and c) traditional plot of stage motion versus tip deflection with 1-3 labeled. The arrow indicates the sequ ence of commanded stage motions. Other Experiments Figure 38 Two Veeco 1930 (1.3 N/m, 35 m wide) cantilevers in contact. The left cantilever lever tip is positioned above the right cantilever lever tip. One of the benefits of full -field imaging is t he ability to capture height maps of the cantilever and sample simultaneously. For example, in cantilever oncantilever experiments using an AFM, measurements can only be made for the instrumented

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42 cantilever leaving the deflection of the other cantilever t o be inferred. Using the prototype platform, both cantilevers can be viewed simultaneously. Figure 3-8 shows two Veeco 193000 (1.3 N/m, 35 m wide) cantilevers with one cantilever pushed against the tip of the other. There is approximately 9 m of vertical offset between their bases and both cantilevers exhibit approximately 4.5 m of end deflection. The objective used was 20x, 0.8x zoom, in 640x480 pixel mode. Figure 39 Data dropout. Due to SWLI measurement limitations this is common ly observed on t he cantilever periphery ( area 1 left image). This results in errors when determining the plan view dimensions. Including the sample in the measurement reveals lost data ( area 2 right image). The sample can also be used as an aid in SWLI measurements of a c antilevers plan view dimensions. Due to SWLI measurement limitations, data dropout is common at the cantilever periphery. However, by placing a smooth sample underneath the cantilever as a backdrop, the lost pixels can be revealed (Figure 3-9). Additional ly, the sample can aid in tip wear analysis. Though not demonstrated here, it is feasible to measure the distance from the sample to the surface of the cantilever to estimate progressive tip wear as depicted in Figure 3-10. Finally, an example of the platf orms capability to identify anomalous bending behavior is presented in Figure 311. Here, an Olympus OMCLAC160TS (42 N/m) cantilever was pressed against a smooth silicon surface. The deflection profile was

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43 linear, rather than the expected cubic behavior. It is assumed that a crack at the cantilever base caused this hinge-like bending behavior. The cantilever broke off the base chip before this could be confirmed, but the fracture location supports the crack assumption. The imaging parameters included a 20 x objective, 1x zoom, and 320x240 pixel mode. Figure 310 Tip wear estimat ion. U sing SWLI measurement s that include the sample surface in the view a n initial measurement, h1, can be compared to a current measurement, h2, to find the tip wear h1-h2. Figure 311. Anomalous bending behavior. This Olympus OMCL -AC160TS (42 N/m) cantilever shows a linear bending profile rather than the expected cubic behavior.

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44 CHAPTER 4 THREE -DIMENSIONAL FORCE MO DELING The 3D deflection data provided by SWLI significantly increases the amount of information available to determine force. To illustrate the mathematical method, the normal force component is determined first. A horizontal rectangular beam is loaded through conta ct at the stylus (Figure 4 -1). Forces both perpendicular, F, and parallel, P, to the beam axis develop where the relationship between the two forces is unknown (for example, a proportional friction model is not assumed). The deflection equation is determined from the Euler Bernoulli beam model. Assuming y=y(x), u=0, and EI is constant, four integrations of the beam equation ( EI d4y/dx4=u=0) yields, ( ) = + + + (4 -1) where C1 is determined by force balance with the shear force and C2 4 are normally determined by applying boundary conditions. Rather than applying boundary conditions and condensing the deflection equation to y(x=L) as in the single point measurement paradigm, the SWLI data provides y(x) for 0 x L so the full deflection equation is utilized. Equation 4-1 is re written as a general third order polynomial, ( ) = + + + (4 -2) where B0 3 are the polynomial coefficients. These coefficients can be estimated by a least -squares fit to the 3D data. Therefore, the normal force component F is directly related to the third order polynomial coefficient B3, = 6 (4 -3) The transduction coefficient ( 6 EI ) has many advantages over stiffness. First, it is independent of the beam length, while stiffness has a sensitive relationship to beam

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45 length (1:3 length:stiffness error for a rectangular cantilever). Second, the equation holds regardless of load application point (this avoids another 1:3 error rate). Similarly, the measurement location does not affect the estimate of B3 so there is no error for detector misalignment. Third, the normal force is completely decoupled from other forces (immune to stylus contact effects). Figure 41. A cantilever loaded at the free end by two independent forces P and F Figure 42. Force measurement does not depend on the global geometry of the cantilever. Both cantilevers shown utilize the same force equation except that the bottom cantilever should be measured on both arms (ideally each arm carries half the load). double arm triangular rectangular cantilever top view similar measurement locations cantilever

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46 The result is a direct application of the shear force equation for beams ( -EI d3y/dx3=V) The shear force is constant along the length of the beam and is directly related to the third derivative of the beam deflection. This explains why the technique is independent of location along the beam and is independent of boundary conditions. Also, the entire beam does not have to be measured (though it improves the signal to noise ratio). Instead a small section of the beam can be measured a nd, if the section is small enough (ideally, d3y/dx3 is infinitesimal), the shape change across the length of the beam is unimportant. This is in contrast to stiffness which is dependent on the global shape of the beam to be solved analytically (shape is often restricted to prismatic beams). Therefore, many beam geometries are easily adapted to the 3D force technique such as double arm triangular beams (Figure 42). Figure 43. The 3D view of the forces acting on the cantilever. A similar development yields expressions for the parallel force component, P, and torsional force component, T to identify the full 3D force vector appli ed to the cantilever Front view

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47 V(F,P,T) (Figure 4-3). First, the parallel force component, P, is derived. Using the same generic loading case in Figure 4 -1, the moment equation is, = = ( ) + (4 -4) where r is the tip height of the stylus. The deflection equation is then, ( ) = + + + (4 -5) where C3 4 are zero for a fixed base boundary condition (motion and slope are zero) and C2 is determined from the moment equation. Unlike shear, the moment depends on measurement location (not constant over the length of the beam). This is important when the location of the base and free end are not well known, which occurs when completing a SWLI measurement with a field of view smaller than the cantilever length. Here, the coordinate system of the measurement will not correspond to the coordinate system of the deflection equation which is centered at the base of the cantilever (Figure 4 -4). Therefore, a polynomial fit to the deflection data cannot be used directly. Instead, the deflection equation is re written for coordinate system independence so that a mapping can be performed between the polynomial fit of the data and the deflection equation. For example, both the deflection equation and polynomial fit c an be mapped to, = ( )+ ( ) (4 -6) where A1 2 are coordinate independent coefficients, and y0 and x0 translate the polynomial from the origin to an arbitrary coordinate system. Expanding the transform polynomial to relate to the defle ction equation coefficients and polynomial fit coefficients yields,

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48 = + ( 3 ) + ( + 3 ) + ( ) (4 -7) = = (4 -8) = = 2 (4 -9) Note that the A1 coefficient does not vary with coordinate system, providing a second check on the invariance of measurement location for F Solving for P in terms of the polynomial fit coefficients yields the measurement -independent result, = 3 3 (4 -10) This result fully decouples P from the normal force F However, determining P does require additional knowledge about the cantilever geometry, including tip height, r, and beam length, L Note that beam length is not cubed and is therefore less sensitive to error than the single point approach. Figure 44. SWLI measurement coordinates. The SWLI has its own coordinate system (xSWLI, ySWLI) which is offset from the global coordinate system defined in the Euler -Bernoulli deflection equation (xEB, yEB) by an unknown amount ( x0, y0) The final force component, T applies torsion to the beam causing twist. The twist equation can be derived using St. Venants torsion theory which assumes the rate of side view SWLI

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49 change of angular twist over the length of the beam d is constant. The resulting equation for twist is, = (4 -11) where is the angular twist, R is the applied torque, x is the position along the length of the beam, G is the materials shear modulus, and J is the torsion constant. Converti ng to deflection yields, ( ) = (4 -12) where z is the location across the width of the beam. This equation assumes a particular coordinate system (origin at the base) so it can be generalized as, = ( ) ( ) + (4 -13) = + + (4 -14) A surface fit to the 3D data yields a coordinate-independent estimate of the coefficient D which is directly related to the applied torque, = (4 -15) = (4 -16) where T i s the final force component of the 3D force vector applied to the tip.

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50 CHAPTER 5 VALIDATION OF THE FO RCE MODEL Normal Loading Equation 4 3 was evaluated for the case of a meso -scale horizontal beam (no tip) with a normal mass -based force applied (i.e., F was ideally the only force component). The beam was fabricated from a 50.8 mm diameter, double -sided polished <111> silicon wafer by diamond cutting at the flat. The design dimensions for the beam were 1 mm wide, 5 mm long, and 0.3 m m thick. After fabrication, the dimensions were measured using the SWLI. The average width was 0.941 mm and the length was 4.820 mm (the SWLI lateral resolution was 4.4 m for a 5x objective at 0.5x zoom with a 640x480 detector). A thickness measurement wa s completed by placing a glass surface beneath the cantilever and using the SWLI to measure the distance from the top of the glass surface to the top of the cantilever. The thickness value, t was 0.305 mm but varied depending on location by 9 m A stitc hed SWLI image of the entire cantilever is shown in Figure 5 1 Figure 51. SWLI image of the fabricated cantilever. The enlarged field of view was enabled by stitching multiple images together. It is seen that the width of the beam varies slightly along its length. The wafer edge roll off is observed at the free end (left end) of the beam.

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51 The normal forces for the beam were produced using a series of 10 masses (Figure 52a); the individual masses were each suspended from a tether which was looped over the beam (Figure 52b). The mass values, mmeas, were measured using a Mettler Toledo AB265S/FACT precision balance (0.1 mg resolution). The beam was fixed to a holder using CrystalbondTM heat activated adhesive and the holder was bolted to the tip -tilt s tage. a) b) Figure 52. Fabricated masses. a ) Masses used for vertical loading experiments b ) cantilever attached to the tip-tilt stage and loaded near the free end. Measurements are valid at any section between the base and the load point. The measurement procedure was initiated by align ing the cantilever base to the SWLI optical axis (the optical axis was treated as parallel to the gravity vector, although the degree of alignment was not determined). To perform this alignment, a 5x objective wi th 0.5x zoom was applied to give a 2.82 mm by 2.12 mm field of view and 4.4 m/pixel lateral resolution. The cantilever was then laterally positioned under the objective using the motorized stage. The cantilever was longer than the field of view so only a section of the cantilever was measured; stitching was not applied for the force measurements. The following steps were completed for each mass Attach the mass to the cantilever Complete a first SWLI measurement Remove the mass Complete a second SWLI measurement

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52 The two SWLI measurements were differenced to isolate the deflection caused by loading. As described before, differencing isolates the deformation caused by the applied force from other sources However, differencing requires lateral alignment between images. Therefore, the mass was applied first in the measurement sequence since it required the longest setup time. This reduced the time between steps 2 and 4 to a few seconds, which mitigated the effect of lateral drift due to stage settling and thermal effects. In addition to drift -related lateral motion, vertical deflection of the cantilever from an applied force also leads to a small shift (foreshortening) of the cantilever along the beam axis. Because the beam deflections were small in this study this shear effect was neglected. A deflection profile was obtained for each applied force by taking a section view through the beam center in the difference map (similar to the approach used for the AFM cantilevers shown in Figure 3 7) The section view data was then fit in a least squares sense using a cubic polynomial to obtain the coefficient B3. Equation 4 3 was adjusted for F = mest g and divided by the measured mass mmeas to assist in the data analysis, = = 1 (5 -1) where mmeas is the mass measured by the precision balance in kg mest is the mass estimated by the cantilever force equation in kg E<110>=168.9 GPa [ 59 ], I=wt3/12 for the rectangular cross -section, t= 0.305 mm, and g = 9.8 1 m/s2. There was a noticeable taper in the width so the average value within the measurement field of view was used (w =0.926 mm) Equation 5-1 should equal 1 within the measurement uncertainty The uncertainty was estimated from the uncertainties of each variable in E quation 51 and is

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53 discussed later in the chapter. The results are displayed in Figure 5 3 with error bars based on the uncertainty analysis and multiplied by a coverage factor of 2. As seen in the figure, the estimated mass under predicts the measured mass by a mean of 9.6% but is within the error bars Figure 53. Results of E quation 5 -1 for the 10 masses. The error bars are based on the full uncertainty analysis multiplied by a coverage factor of 2. The variability of B3 with profile location in a single measurement was also investigated. Eleven deflection profiles near the beam centerline were selected from a single image and used to individually calculate B3. The standard deviation was found to be 0.5% of the mean value. This deviation is an indicator of the model limits since the Euler -Bernoulli equations describe a two-dimensional beam. Noise in the SWLI height map from the single measurement result would also contribute. Next, the cantilever was loaded and 11 separate measurements were sequentially completed before removing the load. Only the centerline profile was extracted from each measurement and was used to calculate B3. The standard deviation was found to

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54 be 1.5% of the mean. This provides an indication of the li mit imposed by the environmental noise over the required measurement time of approximately 4 minutes. Figure 54 B eam loaded under identical conditions, but measurement field of view was varied. The boxes represent the partially overlapping measurement regions along the beams axis. For identical loading, the shear should be identical regardless of location. Ideally the force calculated using E quation 5 -1 is independent of the measurement location for constant loading conditions To test this behavior a measurement (denoted 1) was completed for a field of view near the free end and compared to a measurement (denoted 2) completed near the base; see Figure 5 -4 The values of B3 differed, however, due to the tapering beam width (see Figure 51). To enabl e a direct comparison of the two results, it was necessary to consider the nonconstant beam width and corresponding variation in the moment of inertia along the beams axis. This was accomplished using the average widths, wavg, of the beam for the two selected fields of view The result was evaluated by considering the ratio of the two measurement results 1 and 2 (same mass, but different location). After simplification, the ratio is, ( )( ) = = = 1 (5 -2)

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55 which is nominally equal to 1. E xperiment ally the ratio was determined to be 1 03 based on a mass load of m =8.617 g and the following values: ( B3)1= 37.02 m2, (wavg)1=0.88 mm, (B3)2= -33.89 m2, and ( wavg)2=0.93 mm. This deviation from unity is reasonable given the 1.5% standard deviation obtained from the repeated B3 tests for a single load with the profile extracted along the beams centerline. In a second study of the polynomial fit -based force determination appro ach, a horizontal Olympus OMCL -AC240TS probe (rectangular cross -section, 30 m x 2.8 m, 240 m length, k =1.8 N/m, approximately 14 m tip height, less than 10 nm tip radius) was deflected vertically against a rigid, smooth silicon surface. The experimental B3 value was 2.11 104 m2 from the centerline deflection profile. If the manufacturer specified EI value of 8.29 1012 N m2 is applied, the resulting vertical force is F= 6 B3EI = 6( -2x11x104)(8.31x1012) =1.0 5x106=1.05 N. Based on the measured deflection of 0.59 m at the probes free end (from the SWLI height map), the force obtained from the manufacturers spring constant is F =1.8(0.59x106)=1.06x106=1.06 N. This gives a 1% agreement and provides a preliminary validation of the method for a typical AFM probe. Data was collected using a 20x objective, 1x zoom, 320x240 pixel detector. In a final study, the full surface data of a cantilever was used to calculate the polynomial coefficients rather than taking a section view down the center of the cantilever axis This w as done in preparation for the torsion experiments which required the three-dimensional deformation data (see Equation 4-14). To accommodate a full surface fit least -squares curve fitting was used again. The equation for the surface is a linear combination of each deformation

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56 ( ) = + + + + + (5 -3) Since the generic equations for bending and torsion contain both x and constant terms, these components are added together in Equation 5 3 and are represented by new coefficients C1 and C0. For normal loading, t he only component of interest was B3. Given a number of ( x z y ) points, Equation 5 3 can be written in matrix form, 1 1 ( ) = ( ) (5 -4) where m is the number of measurements across the width of the cantilever, n is the number of measurements down the length of the cantilever, and m x n is the total number of measurements. This can be compactly written as [XZ]{B}={Y} and solved for in the least squares sense using the pseudo inverse { } = [ ]{ } (5 -5) A B Figure 55. The aluminum tee bonded to a 1.5 mm wide cantilever. A) a microscope image of the tee, B) a SWLI height map of the tee. The images are stitched to enlarge the field of view. The groove near the cantilever is for alignment purposes. 2.5 mm 1 mm 5.5 mm 3 mm 3 mm 3 mm 4 mm 15 mm

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57 A 1. 4 6 mm wide by 0.311 mm thick by ~ 5 mm long cantilever was mass loaded to test this approach. Additionally, a removable tee was bonded to the cantilever to be used for the torsion experiments described later. The tee, shown in Figure 55, h ad three sockets to accept a stainless steel sphere whi ch, in turn, held the masses This configuration enabled repeatable positioning of the masses into the socket s For normal loading the center hole was used. Eight masses (3.260 g, 5.426 g, 7.552 g, 9.734 g, 11.877 g, 14.059 g, 16.122 g, and 18.285 g) were applied to the cantilever with five trials per mass. T he coefficients wer e determined using Equation 5-5 and th e mass es w ere c omputed us ing Equation 5 -1. The result s are shown in Figure 5-6 where the error bars are based on the uncertainty analysis and m ultiplied by a coverage factor of 2. T he estimated mass under predicts the measured mass by a mean of 6.2% but is within the error bars. Figure 56. Results of E quation 5 -1 for the 8 masses using the surface fit approach. The error bars are based on th e full uncertainty analysis multiplied by a coverage factor of 2.

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58 T orsion Loading A B Figure 57 Tee attached to cantilever. A) Attaching a tee to the cantilever enabled offset loading, B) an array of cantilevers with one cantilever being loaded using the tee. Equation 416 was evaluated on multiple cantilevers using offset loading as shown in Figure 5 -7a An aluminum tee was fabricated and bonded to the cantilevers (Figure 5 -5 ). Three holes were drilled through the tee as load zones for a stainless steel sphere. The resulting ball and socket arrangement enabled precise placement of the sphere on the tee. A hook inserted through a hole in the sphere enabled it to support the masses. The tee was bonded to the free end of the cantilevers using CrystalbondTM heat activated adhesive. Th e assembly shown in Figure 57 b enabled offset loading with a constant torque arm. New cantilevers were fabricated with dimensions and torque arms shown in Table 5 -1 Positive arms generated counter -clockwise twist and negative arms generated clockwise tw ist. The t orque arm was define d as the perpendicular distance from the center of the hole to the cantilever beam axis. Since the tee was bonded manually, placement of the tee onto the cantilever was not tightly controlled resulting in cantilever = =

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59 to -cantilever variability in the torque arm alignment Additionally, the field of view of the microscope was too small to simultaneously capture the holes and the cantilever in the same measurement and stitching was time intensive. Therefore, the torque arm was esti mated based on the orientation of the alignment grooves to the cantilever axis and the known dimensions of the tee. Table 5 1. Cantilever geometries. The thickness, t was approximately 311 m. Cantilever number Length (mm) Width nominal (mm) Width measur ed, w (mm) Torsion multiplier, [60] Positive torque arm (mm) Negative torque arm (mm) 1 5 1 0.959 0.263 5.339 NA 2 5 1.5 1.454 0.286 5.371 5.605 3 5 2.5 2.462 0.306 5.528 5.448 4 5 3 2.962 0.310 5.355 5.622 5 5 0.7 0.656 0.233 5.221 5.756 6 13 1.5 1.455 0.288 5.378 5.598 7 13 3 2.963 0.310 5.347 5.630 8 13 0.7 0.657 0.233 5.590 5.387 The cantilevers were each loaded with at least three different masses five times per mass in the same manner as for the normal loading experiments. The first mass was the smallest in the set (2.166 g). The third was the largest mass the cantilever could ac cept while still reflecting light back to the SWLI (note the SWLI has a maximum surface angle that it can measure which depends on the magnification). The other mass es w ere selected to be between the first two. The coefficients were estimated using Equation 5 5 with the coefficient of interest being D Equation 4 -16 was adjusted to assist in data analysis similar to Equation 5 -1, = = 1 (5 -6) where mmeas is the mass measured by the precision balance in kg mest is the mass estimated from the cantilever force equation in kg 54.0 G 64. 7 GPa depending on

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60 width and thickness [61], J = t3 ( and w are given in Table 5-1 ) t and r is the torque arm given in Table 5-1 The results for positive torque and negative torque are shown in Figures 5 -8 and 59, respectively The estimated mass under predicts the measured mass for positive torque loading by 7.4 % and under predicts the measured mass for negative torque loading by 11 2%. The mean of the two results is 9.3%. Figure 58. Torsion results when mass is loaded at the positive torque arm. The error bars are based on the full uncertainty analysis multiplied by a coverage factor of 2. A second study was performed to determine if the normal and torsional components could be determined simultaneously with the same measurement. Figure 5 -7a shows how an offset load produce s both twist and bending deformation. Because t he t ee increased the e ffective length of the cantilevers which subsequently increased the measurement sensitivity to external noise t wo alterations to the setup were required to lower the noise levels First, foam insulation was placed around the work area to

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61 block air currents. S econd, the tethers for the masses were replaced by a rigid, stainless steel link Motion at the test mass was rigidly linked to the sphere which caus ed the sphere to rock in its socket and quickly damped vibrations Figure 59. Torsion results when mass is loaded at the negative torque arm. The error bars are based on the full uncertainty analysis multiplied by a coverage factor of 2. A 1. 46 mm wide by 0.311 mm thick by ~ 5 mm long cantilever was used along with 5 masses (3.260 g, 5.426 g, 7.552 g, 16.122 g, and 18.285 g) applied at both positive and negative torque arms (5.464 mm and 5.515 mm respectively) with five trials per mass The coefficients were determined using Equation 5-5, the mass estimated, and the results analyzed by Equation 5-6 Using the normal force coefficient yields the results shown in F igure 5 -10 and using the torsion force coefficient yields the results shown in Figure 511 with the error bars based on the uncertainty analysis and multiplied by a coverage factor of 2. The mean estimated mass using the normal force

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62 coefficient under predicts the measured mass by 8.0% and the mean estimated mass using the torsional force coefficient under predicts the measured mass by 11.9% (mean positive torque arm of 12.5% and mean neg ative torque arm of 11.2%) Figure 510. Torsion loading results for mass loaded at the positive and negative torque arms. The error bars are based on the full uncertainty analysis multiplied by a coverage factor of 2. Figure 511. Normal loading r esults for mass loaded at the positive and negative torque arms. The error bars are based on the full uncertainty analysis multiplied by a coverage factor of 2.

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63 Uncertainty Analysis The measurement uncertainty reported in the figures was determined by the propagation of errors of Equations 5-1 and 56, = + + + + (5 -7) = + + + + + (5 -8) where terms with asterisks can be expanded further. The terms I and J expand to, = = + 3 (5 -9) = + (5 10) where w is the beam width, t is the beam thickness resolution i s associated with the SWLI height resolution and standard dev is based on measurements of thickness at multiple locations on the beam. The terms B3 and D expand to, = + (5 11) = + (5 12) where the least -squares term is associated with the covariance matrix obtained duri ng fitting and the standard dev term is based on the repeated trials of each mass load. The term r, the torque arm, expands to, = + (5 13)

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64 where resolution is associated with the SW LI lateral resolution and alignment is associated with the extrapolation of the hole location using the alignment grooves instead of a stitched image of the entire assembly. Representative values are provided in Table 52 where the largest uncertainty contributor, thickness (standard dev) is also the term that gets multiplied by three in Equation 5 9. Table 5 2. Uncertainty contributions. Name Variable name V alue Youngs modulus 0.005 Shear modulus 0.0 1 Cantilever width 0.003 -0.013 Cantilever thickness (resolution) ( ) 1.7x105 Cantilever thickness (standard dev) ( ) 0.029 Normal force coefficient (least -squares) 0. 005 Normal force coefficient (standard dev) 0.02 Torsional force coefficient (least -squares) ( ) 0.000 2 Torsional force coefficient (standard dev) ( ) 0.0 2 Gravity 0.004 Torque arm (resolution) ( ) 0. 002 Torque arm (alignment) ( ) 0.0 2 Mass measurement 1.1x105 6.1x105

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65 Youngs modulus for single crystal silicon depends on crystal orientation. For (111) silicon with the cantilever beam axis on the (111) plane, Youngs modulus is independent of orientation with a value of 168.9 GPa. However, manufacturing limits prevent perfect alignment to the (111) plane. The manufacturer specified alignment tolerance to the (111) plane was 0.5 deg. Using the stiffness tensor for silicon and applying tensor rotations, the maximum error in modulus for wafer misalignment was determined to be 0.5% of 168.9 G P a. A similar process was used to determine the uncertainty for shear modulus due to misalignment to the (111) plane. However, s hear modulus does depend on the orientation of the beam axis on the (111) plane The beam axis was nominally oriented along the < -1 -12> direction by cutting perpendicular to the wafer flat (the wafer flat was along the <110> direction) The flat had a tol erance of 1 deg to the <1 10> direction and t he cutting process had a tolerance of 2 deg. Finally, the shear modulus depend ed on the width to thickness ratio of the cantilever. The combined uncertainty for shear modulus was less than 1% of the nominal shear modulus (54 G The cantilever width uncertainty was specified as two times the SWLI lateral resolution (2 x 4.4 m). Dividing by the cantilever width gives a range from 0.3% to 1.3% Similarly, cantilever thickness uncertainty was specified a s two times the SWLI vertical resolution (2 x 2.7 nm) which was negligible for thicknesses of 0.305 and 0.311 mm. However, thickness had a much greater variability when measuring at multiple locations on the same cantilever. Thickness measurements were mad e by placing a glass slide under the cantilever and measuring from the top of the glass to the top of the cantilever surface. This distance was affected by contaminants between the glass and

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66 cantilever and by variability in wafer thickness. Also, measuring the thickness risked damaging the cantilevers. Therefore, a maximum variability of 9 m was assumed based on multiple measurements of a few cantilevers at different locations on each cantilever. For a 0.311 mm thick cantilever th e uncertainty is 2.9%. The coefficient matrix used in the force equations was obtained from a least squares fit to the three-dimensional deformation data, i.e. a series of (x,y,z) points. The uncertainty in the deformation data is related to the uncertainty in the coefficient matrix by the covariant matrix. Rewriting Equation 53 more generally, ( ) = ( ) + ( ) + ( ) + ( ) + ( ) + ( ) (5 14) a new symmetric matr ix, can be defined with elements (5 15) where l and k are the row and column numbers of respectively and vary from 1 to 6, m is the number of measurements across the width of the cantilever, n is the number of measurements down the length of the cantilever, and m x n is the total number of measurements Since measurements are obtained by the SWLI camera, each ( x z ) location is a different pixel where i and j are the pixel indices. The term ij is the y coordinate measurement uncertainty at a given pixel ( i j ) and was assumed to be the same for all pixels. This enabled to be estimated by, + + + + + (5 16) where N = n x m and 6 corresponds to the number of free parameters (the coefficients). The covariant matrix is the inverse of The elements of the covariance matrix are the variances and covariances of the fitted coefficients,

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67 = . . . (5 17) A typical result of the covariance matrix using torsion loading of 7.552 g on a 1.46 mm wide by 0.311 mm thick by ~5 mm long cantilever on the positive torque arm is, = 816 1 3 456 0 004 0 000 0 001 0 000 0 015 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 0 000 . 0 001 0 000 . 0 000 10 (5 18) with values for | B3| and | D | of 18.2556 and 0.4874 respectively. Taking (1)11/ B3 and (1)55/ D gives the uncertainty in these terms of 0.5% and 0.02% respectively. The coefficients also had measurement -to -measurement deviation when loaded with the same mass multiple times Therefore, the standard deviation was computed for every measurement sequence and found to be around 2% of the mean for both B3 and D Gravity was assumed to be 9.81 m/s2 but alignment to gravity was not checked so a conservative 5 deg deviation was assumed leading to an uncertainty of 0.4% of 9.81 m/s2. The torque arms had a minimum uncertainty due to the SWLI lateral resolution (2 x was used) Additionall y, the alignment grooves were used to extrapolate the torque arm distance. Using the extrapolation procedure and comparing to a direct measurement of the torque arm by stitching multiple images together, it was found the torque arm could be off by as much a s 10 Finally, all masses were weighed by a precision balance with a resolution of 0.1 mg so t he uncertainty was set to two times this value (2 x 0.1 mg) .

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68 CHAPTER 6 CONCLUSIONS All results unde r predicted the m easured mass ( 6 % to 12% ). The thickness was the largest uncertainty contributor at 3% (Table 5-2) providing a 9% uncertainty in the estimated mass compared to the measured mass. This uncertainty alone nearly closes the gap in the under predictions Thickne ss measurement is a significant challenge that is not unique to this research. Not only is it difficult to measu re but its uncertainty is amplified by being cubed in the force equation. For the first torsion study (parameters in Table 51), the estimated mass was systematically higher for the positive torque arm than for the negative torque arm (Figure s 5 8 and 5 9) This is c ontrast ed with the secon d torsion study where the opposite trend was observed (Figure 5-11) The reason for the difference is t he method for computing the torque arms. In the first torsion study, because many canti levers were tested, the torque arms were extrapolated based on the alignment grooves. In the second torsion study, the torque arms were directly measured by stitching mult iple images together to obtain both the tee and cantilever in a single view. Note that in the first study, the gap between the mean positive torque arm and the mean negative torque arm was 3.8% while in the second study, the gap was 1.3% and in the opposit e direction ( total difference of 5.1%). The torque arm concept used in this research is analogous to AFM tip height. Determining these values is a significant challenge in cantilever force research. In the study where the normal force coefficient simultane ously with the torsion force coefficient (Figure 5 10 and Figure 5-11) were computed from a single test the estimated mass using the normal force could be directly compared to the estimated

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69 mass from the torsional force. The two results both under estimat e d the measured mass but by different amounts. The normal force under estimated by 8% and the torsion force by 11.9% (difference of 3.9%). Common parameters in the force equations contribute equally and in the same direction to both results and, therefore the difference must be because of uncommon parameters. For example, shear modulus uncertainty does not contribute to the normal force result but does contribute to the torsion force result. Summary Measuring the deflection of an AFM cantilever at a singl e spatial point requires the use of a scalar stiffness as the force transduction constant. Stiffness, however, is a poorly defined quantity for a cantilever sensing an unknown 3D force vector. If, instead deflection of the cantilever is measured at multip le points simultaneously the force vector can be determined unambiguously. Measurements made this way are immune to many of the problems associated with traditional single point measuring schemes. There is no dependency on whe re the load is applied or what portion of the beam is measured and there is no cubic dependency of measurement location on length. Each force component can be independently determined whereas in single point techniques the total force vector must be decomposed using prior knowledge about the relationships between the force components. A prototype platform was designed to measure the full -field deflection profile of cantilevers under load. The primary platform component was a scanning white light interferometer ( SWLI ), which measu res surface contours through the low coherence interference of white light. This platform was used to test a new 3D force model. Forces were applied to single crystal silicon cantilevers to cause bending and torsion. From the 3D deformation, the force s wer e determined unambigu ously

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70 Future Work This study identifies many follow on research topics A natural extension is to validate the model for the third force component, P. T he n the complete force model c ould be verified using AFM -scale cantilevers. The o bvious challenge for micrometer scale cantilevers is identifying sources of suitably small reference forces and determining the material properties and the thicknes s of multi layered cantilevers. Finally the prototype platform could be improved. For examp le, the SWLI measurement rate is on the order of seconds. Adapting the SWLI for shorter measurement times or selecting an alternative measurement approach could reduce dynamic disturbances such as vibrations. W ith a new high frequency measurement transducer a new dynamic 3D force model could be investigated.

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71 LIST OF REFERENCES [ 1 ] Langlois E, Shaw G, Kramar J, Pratt J and Hurley D 2007 Spring constant calibration of atomic force microscopy cantilevers with a piezosensor transfer standard Rev of Sci Instrum 78 093705 [ 2 ] Emerson R and Camesano T 2006 On the importance of precise calibration techniques for an atomic force microscope Ultramicroscopy 106 413 [ 3 ] Kim M and Pratt J 2010 SI traceability: Current status and future tr ends for forces below 10 microNewtons Measurement 43 169 [ 4 ] Matei G Thoreson E, Pratt J, Newell D and Burnham N 2006 Precision and accuracy of thermal calibration of atomic force microscopy cantilevers Rev of Sci Instrum 77 083703 [ 5 ] Heim L, Kappl M and But t H 2004 Tilt of atomic force microscope cantilevers: Effect on spring constant and adhesion measurements Langmuir 20 2760 [ 6 ] Wang F 2009 Comment on Influence of atomic force microscope cantilever tilt and induced torque on force measurements Applied Physics 106 096103 [ 7 ] Hutter J 2005 Comment on Tilt of atomic force microscopy cantilevers: effect on spring constant and adhesion measurements Langmuir 21 2630 [ 8 ] Edwards S, Ducker W and Sader J 2008 Influence of atomic force microscope cantilever tilt and induced tor que on force measurements Applied Physics 103 064513 [ 9 ] Butt H, Cappella B and Kapple M 2005 Force measurements with the atomic force microscope: Technique, interpretation and applications Surface Science Reports 59 1 [ 10 ] Chung K, Pratt J and Reitsma M 2010 Later al force calibration: Accurate procedures for colloidal probe friction measurements in atomic force mi croscopy Langmuir 26 2 [ 11 ] Hutter J and Bechhoefer J 1993 Calibration of atomic -force microscope tips Review of Scientific Instruments 64 7 [ 12 ] Binnig G, Quate C and Gerber C 1986 Atomic force microscope Physical Review Letters 56 9 [ 13 ] Binnig G, Rohrer H, Gerb er C and Weibel E 1982 Surface studies by scanning tunneling microscopy Physical Review Letters 49 1 [ 14 ] Gdde nhenrich T, Lemke H, Hartmann U and Heiden C 1990 F orce microscope with capacitive displacement detection Vacuum Science and Technology 8 1

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76 BIOGRAPHICAL SKETCH The author was born in Miami, F L, in 1982. In 2001, he started his bachelors degree at the University of Florida (UF). Before graduation, as part of a n undergraduate research fellowship, he worked at the National Institute of Standards and Technology (NIST), the United States standards agency. There, he became interested in small force metrology and afterward worked in th is field for his masters degree project. In 2007, he earned his masters degree from UF and then interned at NIST. Later, he started his doctoral degree which extended his masters work to the full threedimensional force model presented in this study.