Final Year Project 2014/15

Atomic Force Microscopy and Nanoindentation The Design and Modelling of an Optimised

AFMNanoindentation Probe with variable

Stiffness

Rebecca Steven 1002786s Supervisor – Dr Phil Dobson

1002786

Dr Phil Dobson

The Design and Modelling of an Optimised Nanoindentation probe with Variable

Stiffness

Rebecca Steven

16 - 01 - 2015

Individual Project 5 ENG5041P

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Abstract This project will focus on the indentation side of AFM which has one primary issue; the stiffness of

the probe itself. The moment of contact between probe and material is detected by looking for the

first instance of deflection. Ideally a soft probe would be used as this would more easily deflect and

is better measured. However, a stiff probe is more desirable for testing the material and is more

accurate in determining its properties. Therefore a probe that has the ability to have an adjustable

or variable stiffness is desirable. A variable stiffness probe would have an initial stiffness low enough

to deflect easily but would then become stiff enough to carry out the test.

The design in this report utilises electrostatics to tune the stiffness of the probe. The final concepts

are centred on the placement of the electrodes. It was thought that the electrodes could be used to

hold a section of cantilever in place. This would shorten the length of cantilever free to deflect,

subsequently stiffening it. The final concept of having one electrode above the cantilever was

selected.

The problem has been approached by way of modelling and simulation. Initially the feasibility of using

an open source software called Elmer was evaluated. This software makes it possible to properly

simulate electrostatic forces but it was decided for simplicity that these forces would be

approximated in the more widely used programme Abaqus. The electrostatic forces were

approximated as springs between the cantilevers and the electrode. The electrostatic force was

calculated by hand and entered. A simulation was run to obtain the deflection (under a known force)

of the cantilever with and without the application of the electrostatic forces. The magnitude of these

deflections were then used to calculate the corresponding stiffness which was compared to the

design requirements.

Implementing this probe in industry would mean one probe needs to be produced instead of many.

Theoretically a probe has been designed that meets the requirements laid out in this report. The

probe has been designed to be compatible with standard equipment in industry. It has an initial

stiffness that qualifies as “soft” which can then be increased, by way of applying electrostatic forces,

to create a probe which can be defined as “stiff”. As well as having a variable stiffness, the probe

satisfies the requirements that it is manufacturable and is compatible with standard equipment used

in industry. It was then investigated whether altering the dimensions of the probe would have an

effect on the stiffness. Three more designs were thought up based on this idea. Further work is

required on this design to make it applicable in other areas of AFM testing. Alterations to the shape

or dimensions could bring the benefits of a variable stiffness probe to other AFM modes such as

contact or tapping.

ii

Acknowledgements

A brief note of thanks must be made to Dr Phil Dobson for his

support and advice when required.

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Contents

Abstract ....................................................................................................................................... i

Acknowledgements .................................................................................................................... ii

Contents ..................................................................................................................................... 1

Table of Figures .......................................................................................................................... 3

Table of Tables ........................................................................................................................... 4

List of notation ........................................................................................................................... 5

1. Introduction ........................................................................................................................... 6

1.1 AFM/Nanoindentation ..................................................................................................... 6

1.2 Background to the problem ............................................................................................. 8

1.2.1 Probe Materials and Set up ....................................................................................... 9

1.2.2 Other Works ............................................................................................................ 10

1.3 Design requirements ...................................................................................................... 12

2. Evaluation of Software ......................................................................................................... 13

2.1 Elmer .............................................................................................................................. 13

2.2 Abaqus ............................................................................................................................ 14

2.3 Microsoft Excel ............................................................................................................... 15

2.4 SolidWorks...................................................................................................................... 15

3. Design of an electrostatic cantilever system ....................................................................... 16

3.1 Definition of Cantilever .................................................................................................. 16

3.2 Probe Materials .............................................................................................................. 17

3.2.1 Coatings ................................................................................................................... 19

3.3 Electrodes ....................................................................................................................... 20

3.3.1 Horizontal Electrodes .............................................................................................. 20

3.3.2 Vertical Electrodes ................................................................................................... 23

3.4 Dimensions of Cantilever Design ................................................................................... 29

3.4.1 Compatibility with Holder Chip ............................................................................... 29

4. Design Assembly and Results ............................................................................................... 30

5. Cantilever Design with Widened End .................................................................................. 33

6. Cantilever Design with Vertical Fin ...................................................................................... 35

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7. Cantilever Design with Lateral Fins ...................................................................................... 37

8. Fabrication ........................................................................................................................... 39

9. Evaluation ............................................................................................................................ 41

10. Conclusion .......................................................................................................................... 44

References ............................................................................................................................... 45

Appendix A – Excel Calculations of Forces ............................................................................... 48

Appendix B – Integration Between Two Curves ...................................................................... 49

Appendix C – Budget Sensors’ Holder Chips ............................................................................ 51

Appendix D – Dimensions of Diving Board Cantilever Design ................................................. 52

Appendix E - Dimensions of Cantilever Design with Widened End ......................................... 53

Appendix F - Dimensions of Cantilever Design with Vertical Fin ............................................. 54

Appendix G - Dimensions of Cantilever Design with Lateral Fins ............................................ 55

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Table of Figures Figure 1. AFM Probe and Tip...................................................................................................... 6

Figure 2. Diagram of Laser Spot Detection. Not to Scale. ......................................................... 7

Figure 3. AFM Holder and Chip .................................................................................................. 9

Figure 4. One Dimensional Model (a) in initial position, (b) after applied voltage (c) on

contact ..................................................................................................................................... 10

Figure 5. Prototype Cantilever Calibatrion System .................................................................. 11

Figure 6. Electric Field Lines Between Cantilever and Electrodes ........................................... 11

Figure 7. Assembly with Equally Spaced Springs ..................................................................... 15

Figure 8. Three main cantilever types; (a) full moment connection, (b) diving board, (c)

elastic spring ............................................................................................................................ 16

Figure 9. Different cantilever shapes; (a) rectangular/diving board, (b) triangular, (c) dagger,

(d) V-shaped ............................................................................................................................. 16

Figure 10. Horizontal Electrodes and Cantilever on Holder Chip ............................................ 20

Figure 11. Horizontal Electrodes with Cantilever and Force Representation in Abaqus ........ 20

Figure 12. Area of Overlap, Highlighted in Red. Not to Scale. ................................................. 22

Figure 13. Graph of Pull in Voltage .......................................................................................... 22

Figure 14. Double Vertical Electrode and Cantilever on Holder Chip ..................................... 23

Figure 15. Double Vertical Electrodes and Cantilever ............................................................. 23

Figure 16. Cantilever Held Equally Between Two Electrodes .................................................. 24

Figure 17. Cantilever with Top Electrode on Holder Chip ....................................................... 25

Figure 18. Single Vertical Electrode and Cantilever ................................................................. 25

Figure 19. Cantilever Experiencing Repulsive Forces .............................................................. 25

Figure 20. Assembly of Cantilever, Electrode and Holder Chip ............................................... 30

Figure 21. Representation of Regular Cantilever ..................................................................... 30

Figure 22. Widened End Cantilever with Electrode on Holder Chip ........................................ 33

Figure 23. Representation of Cantilever with Widened End ................................................... 33

Figure 24. Cantilever with Fin and Electrode on Holder Chip.................................................. 35

Figure 25. Representation of Cantilever with Vertical Fin ....................................................... 35

Figure 26. Cantileverwith Lateral Fins with Electrodes on Holder Chip .................................. 37

Figure 27. Representation of Lateral Fins Cantilever .............................................................. 37

Figure 28. Representation of Etching Techniques ................................................................... 39

Figure 29. Silicon Wafer ........................................................................................................... 40

Figure 30. Cantilever with Vertical Fin ..................................................................................... 41

Figure 31. Cantilever with Lateral Fins..................................................................................... 41

Figure 32. Rectangular Cantilever ............................................................................................ 42

Figure 33. Cantilever with Widened End ................................................................................. 42

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Table of Tables Table 1. Increase in Stiffness with Double Vertical Electrodes................................................ 24

Table 2. Increase in stiffness with Single Vertical Electrode at 1V and 10μm ......................... 26

Table 3. Incremental Change in Stiffness at 1V and 10μm ...................................................... 27

Table 4. Incremental Change in Stiffness at 1.5V and 10μm ................................................... 27

Table 5. Incremental Change in Stiffness at 0.5V and 10μm ................................................... 28

Table 6. Incremental Change in Stiffness at 0.1μV and 3μm ................................................... 31

Table 7. Incremental Change in Stiffness at 0.2μV and 4μm ................................................... 32

Table 8. Change in Stiffness for Widened End Design ............................................................. 34

Table 9. Change in Stiffness after Increasing Voltage to 0.5μV ............................................... 34

Table 10. Change in Stiffness after Decreasing Length ........................................................... 34

Table 11. Change of Stiffness for Cantilever with Fin .............................................................. 35

Table 12. Change in Stiffness after Altering Thickness ............................................................ 36

Table 13. Change in Stiffness for Lateral Fins Cantilever ......................................................... 37

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List of notation

Symbol Primary Unit

(unless otherwise

stated)

Acurve Area between deflection curve and axis µm2

𝐴𝑜𝑣𝑒𝑟𝑙𝑎𝑝 - Area of overlap between electrode and cantilever µm2

d - Distance between cantilever and electrode µm

E - Young’s Modulus GPa

F - Force µN

𝐹𝑒𝑎 - Electrostatically Attractive Force µN

𝐹𝑒𝑟 - Electrostatically Repulsive Force µN

I - Second moment of area µm4

k - Spring constant, stiffness, force constant N/m

𝑘𝑒 - Coulomb's law constant, 9.0x109 Nm2/C2

L - Length of cantilever µm

Q - Charge C

Qc - Charge of the cantilever C

Qe - Charge of the electrode C

t - Thickness of cantilever µm

V - Applied voltage µV

w - Width of cantilever µm

x - Distance along cantilever to point chosen µm

y - Deflection at distance x µm

δ - Deflection of cantilever µm

ε - Electrical Permittivity

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1. Introduction

1.1 AFM/Nanoindentation Nanoindentation is a variety of indentation hardness testing which is applied to samples that

are in the order of fractions of a nanometre [1]. It has the ability to characterise their nano

mechanical properties to a high precision. A hard tip of known properties is pressed into a

material at a known force, increasing the load on the probe as the tip penetrates further into

the sample until the user defined load is reached. The load may then be held constant for a

period of time or removed.

This type of testing is often performed in conjunction with Atomic Force Microscopy (AFM)

which was introduced in 1986 by Gerd Binnig [2], earning him and his colleagues a Nobel Prize.

AFM is a type of scanning microscopy with very high resolution, in the order of fractions of a

nanometre. The image can be detected by eye, on a photographic plate or captured digitally.

It is currently the foremost tool for imaging, measuring and manipulating at a nano scale.

AFM has proven to be a very effective tool since its invention, being used to investigate

nanonewton and sub nanonewton forces including inter- and intra- molecular forces. Another

application is investigating the physical and structural properties of cells, organelles and

polymers [3]. As an imaging or surface profiling tool AFM can be applied in three different

modes. These modes can be split into two categories; static or dynamic modes. During the

static mode, also known as contact mode, the probe, Figure 1 [4], is dragged across the

surface contours measuring them by deflection or by a feedback loop to keep the cantilever

in the desired position. Dynamic mode includes tapping mode and non-contact mode. In

tapping mode the cantilever oscillates up and down near its resonant frequency. Interaction

of the cantilever with forces on the sample cause the oscillation amplitude to decrease as the

tip gets closer to the sample. In non-contact mode the cantilever is also oscillated at the

resonant frequency or just above but his time long range forces such as van der Waals

decrease the resonant frequency.

Figure 1. AFM Probe and Tip

Although predominantly used for imaging, several types of interaction can be measured and

AFM is increasingly being used as an indentation tool. This allows simultaneous hardness

measuring and high resolution imaging using the same standard sharp AFM probe.

Indentation is a variant of static mode where the cantilever AFM probe acts as a spring and

bends in response to the force acting on the probe tip until static equilibrium is reached. AFM

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cantilevers are mainly manufactured using microelectrochemical technology and therefore

commonly formed from silicon, silicon nitride or polymers. The deflection of the cantilever is

proportional to the force acting on the tip, as derived from Hooke’s Law;

𝑘 = 𝐹

𝛿 (1)

The spring constant, in the case of this project, represents a cantilever. The spring constant

can also be referred to as the stiffness or force constant and is defined as the rigidity of an

object or the extent to which it deforms in response to an applied force. The more flexible an

object is the less stiff it is, a high Young’s modulus means a high spring constant and vice versa.

The deflection of the cantilever in the vertical direction is commonly measured by using a

laser spot focussed on the back of the cantilever, Figure 2 [5, 6]. The laser is reflected by the

cantilever onto an array of photodiodes, the differences between signals received from each

photodiode indicate the position of the laser spot. From the position of the laser spot the

deflection of the cantilever can be obtained. The movement of the laser spot greatly

magnifies the deflection and this method of measurement is known as an optical lever. The

feedback electronics keep the probe at a set distance, force or oscillation depending on the

mode being acted in.

Figure 2. Diagram of Laser Spot Detection. Not to Scale.

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1.2 Background to the problem

This project will focus on the indentation side of AFM which has one primary issue; the

stiffness of the probe itself. The moment of contact between probe and sample is detected

by using Laser Spot Detection and looking for the first instance of deflection [7]. Ideally a soft

probe would be used as this deflects at lower loads and contact can be detected earlier.

However, a hard probe is more desirable for testing the material and more accurate in

determining its properties. A probe that has the ability to have an adjustable or variable

stiffness is therefore desirable. Electrostatically actuated microelectromechanical systems

(MEMS) and cantilevers are commonplace in sensors and there is previous research to

introduce electrostatically actuated cantilevers into AFM. Electrostatically actuated

cantilevers can be used to calibrate the probe before testing begins. The objective of this

project is to design a cantilever AFM probe that has the ability to alter its stiffness. Designing

a variable stiffness cantilever would also have the effect of reducing the number of probes

required as currently the spring constant of the probe needs to be matched to the

approximated hardness of the sample.

Electrostatics is an area of physics dealing with the properties and occurrences of stationary

or slow moving electrical charges with no acceleration. These forces come from electric

charges exerting a force on each other. If there is an imbalance in this surface charge then the

objects will exhibit forces of either attraction or repulsion [8]. Building on this, it is possible to

electrically charge the cantilever and electrodes so that they exert a force on each other. If

one was positively charged and the other negatively charged then this force would be one of

attraction.

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1.2.1 Probe Materials and Set up

The actual probe is a consumable measuring device that has a sharp tip on the free end of the

cantilever, Figure 1 [7]. There are a range of different cantilever shapes to suit the different

applications of AFM. For indentation testing the most commonly used probe is known as a

rectangular or diving board AFM cantilever. The typical dimensions of these cantilevers are;

length of 100 - 500µm, width of 30 - 50µm and a thickness of 0.5 - 8µm. The typical force

constants of these cantilevers are 0.06 to 50N/m but these cantilevers can commonly have a

working force constant in the range of 0.01-100N/m [9].

The probe it usually made from polymers, micromachined silicon or micromachined silicon

nitride. AFM exploits silicon nitride’s material properties such as good fracture toughness,

and fatigue resistance, excellent wear resistance, low density and the ability to penetrate cast

iron, hard steel and nickel alloys [10].

The cantilever, or probe, protrudes from a holding plate also known as a holder chip, Figure

3 [7]. This chip allows operators to handle the probe with tweezers. The chip features

alignment grooves on the underside which slot into the machine being used.

Figure 3. AFM Holder and Chip

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1.2.2 Other Works

The concept of an electrostatically actuated device with moveable electrodes has been

described as a one dimensional model in a paper by Edward Keat Leem Chan [11]. This one

dimensional model consists of a movable top plate suspended above a tethered bottom plate

by flexible ties, Figure 4 [11]. The top plate represents an electrode while the bottom plate

represents a cantilever.

Figure 4. One Dimensional Model (a) in initial position, (b) after applied voltage (c) on contact

Application of a voltage to the electrodes generates an electrostatic force that works to pull

the plates together. This force pulls the suspended top plate downwards and by varying the

voltage, the position of this top plate can be controlled. As the top plate moves down, air is

pushed out of the gap creating a source of damping. When the gap between the plates

becomes equal to one third of the initial distance the plate becomes unstable due to the

spring constant of the plate becoming unbalanced by the electrostatic forces beyond this

point. This distance is known as the pull in distance and occurs at the pull in voltage. The static

imbalance means that the increasing electrostatic force caused by the decreasing gap

overcome the mechanical restoring force and the moveable plate comes into contact with

the fixed plate.

This concept will be built on in this project, designing and creating a three dimensional model

with two sets of electrodes. It is believed that the electrodes will make contact with a section

of the cantilever holding it in place. This will shorten the length free to deform and increase

the stiffness.

A prototype of an electrostatically actuated cantilever device, Figure 5, has been developed by Sunil Rana et al [3]. This device can calibrate the cantilever on a chip using electrodes and voltages up to 15V. The calibration process is fast, easy and non-destructive and can fabricate an AFM probe using the standard microfabrication processes. The calibration is carried out by

11

applying a known electrostatic force to the cantilever and measuring the corresponding deflection. The spring constant of the cantilever is then calculated from these results and compared to the theoretical spring constant value which the manufacturer has labelled the cantilever with. These values have an accuracy of 5%.

Figure 5. Prototype Cantilever Calibatrion System

The electrode cantilever pair act like a capacitor and can be represented as two capacitors in

parallel. These two parallel capacitors represent the in-plane electric field and out-of-plane

electric field, Figure 6 [3]. The in-plane electrostatic force contributes the majority of the

electrostatic forces acting on the cantilever and are sufficient enough for analysis but to

obtain the total electrostatic force acting on the cantilever, the out-of-plane forces should

also be taken into account. It was considered that the assumtion made in this paper was

appropriate, therefore only the in plane electrostatic forces will be considered in the later

parts of this report.

Figure 6. Electric Field Lines Between Cantilever and Electrodes

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1.3 Design requirements

The probe must be designed to meet a set of requirements in order to be functional,

manufacturable and marketable.

To be functional, the cantilever must have a variable stiffness. The thicker and shorter the

cantilever the higher the force constant while the longer and thinner the cantilever the lower

the force constant. AFM cantilevers with a force constant greater than 40N/m are defined as

“stiff” while a cantilever with a force constant less than 1N/m is defined as “soft” [12].

Cantilevers with an intermediate force constant of 3-15N/m are defined as intermediately

stiff. These force constants are the normal force constant of the cantilevers. For the cantilever

design to be successful it must start with stiffness less than 1N/m and obtain stiffness greater

than 40N/m but less than 100N/m. The second moment of area of the probe needs to be low,

keeping the second moment of area low ensures that the design of the cantilever is flexible

and has a low stiffness.

To be manufacturable the probe must be able to be manufactured by the standard

micromachining process used in the semi-conductor industry. Electronic circuits are created

on a wafer by way of photolithographic and various chemical methods. Silicon is almost

always used in this process but compound semiconductors can also be produced. These

manufacturing processes have been adapted to produce AFM probes and because of this,

silicon or a silicon compound must be used as the main probe material.

As well as being fabricated from a suitable material, to be marketable the probe must meet

various dimension requirements. Commercially available holder chips are sized so it is

compatible with commercial scanning probe microscopes and most AFM machines widely

available in industry. The probe must therefore fit the holder chip in order to be suitable for

use in AFM machines. The rectangular or diving board cantilever is best suited to the AFM

procedure and has a size range of length 100-500µm, width 30-50µm and a thickness 0.5 to

8µm. The probe has to be within these ranges so as to be compatible with commercially

available holder chips.

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2. Evaluation of Software

2.1 Elmer Elmer was developed by the IT Centre for Science (CSC) in collaboration with Universities in

Finland along with Finnish institutes and industries. Its development started in 1995 but since

2005 the development and use of Elmer has gone worldwide [13].

Elmer is an open source software with the ability to run multiphysical simulations including

physical models of fluid dynamics, structural mechanics, acoustics, heat transfer and

electromagnetics with the latter being the one of most interest to the project. The models

are described by a set of partial differential equations set by the user which Elmer solves

through use of Finite Element Analysis (FEA).

To become familiarised with how the programme worked, tutorials for Elmer were found

online and the instructions for deflection in a pinned cantilever were followed [14]. After it

was determined that the tutorial worked, a basic cantilever was designed and processed using

the same steps but with different parameters. On the surface the software seemed to work

much the same as Abaqus however the process for inputting and assigning variables in Elmer

was more user friendly.

Although an interesting programme, Elmer presented many hurdles throughout its use. The

first hurdle encountered was that it is not possible to build or design in Elmer. Any designs

need to be modelled using a second software and imported. Elmer only reads selected files,

mainly mesh files, therefore the second software used to create the model needed to be able

to output a file type which is readable by Elmer. This made SolidWorks redundant as it did not

output any files that are readable by Elmer. Abaqus was the only programme available which

produced files that Elmer could read or open. Therefore a general cuboid was designed in

Abaqus CAE, a meshed .inp file was extracted and subsequently imported into Elmer.

Once the model was opened in Elmer the process to apply boundary conditions, material

properties and forces to the model was a lot simpler than Abaqus. The cuboid was constrained

at one end, converting it into a cantilever and the material properties (Young’s modulus,

Poisson’s ratio, density, electrical permittivity of material) were applied. However another

issue presented itself when trying to apply the force. AFM probes are subjected to a point

load at the indenter tip but in Elmer it was only possible to apply a load across one face. To

account for this, the value of the point load had to be calculated as a pressure load on the

end face.

A simulation was run using this force and a suitable deflection of the cantilever was obtained,

despite the load inaccuracies. The same cantilever was then deflected in Abaqus using the

same parameters to obtain a second deflection for comparison. The result from the Abaqus

simulation - using a point load - matched the result from Elmer - using a face load - and

therefore it was declared that Elmer was a reliable software and more complicated

simulations could be run.

14

Once it was established that Elmer would run models imported from Abaqus, tutorials for

Electrostatics were followed to apply the same approach. The process for applying

electrostatic boundary conditions was then applied to the same cantilever from before, this

time including an electrode. The electrode was modelled as a cuboid next to the cantilever

and the assembly was modelled and meshed in Abaqus before being imported into Elmer

On importing the assembly into Elmer, it was discovered that Elmer cannot read the .inp file

of an assembly. The cantilever and electrodes needed to be remodelled into one sketch

instead of an assembly before being imported to Elmer. Elmer could read the new .inp file but

a new problem was presented. When assigning properties and boundary conditions to the

two bodies, Elmer only saw one body. It was not possible for Elmer to distinguish between

these two bodies and therefore not possible to assign the cantilever different properties from

the electrodes.

Elmer had the potential to provide an electrostatic analysis using the proper parameters and

inputs instead of using an approximation but it was decided that using Elmer as the main

software for the simulations was not feasible. Too much time was being taken to process

simple issues that were easily resolvable in a different programme (such as Abaqus) as there

is a lot more information and support available.

2.2 Abaqus On opening Abaqus CAE there are three options; with Standard/Explicit Model, with CFD

Model and with Electromagnetic Model. The Electromagnetic Model was chosen and an

attempt was made to explore its feasibility to utilise for the project. A cuboid was created

with a view to modelling a cantilever and it was assigned material properties including values

for electrical conductivity and magnetic permeability. When it came to assigning the

boundary conditions magnetic vector potentials had to be calculated from the applied

electrode potential. The load was in the form of a body current density or surface current

density. Current density is calculated as “the amount of current flowing through a given cross

sectional area in a given time interval: usually measured in amperes per square centimetre”

[15].

Each time it was run, the simulation aborted citing an error in connection to the analysis.

There were no tutorials in the Abaqus database to help with steps for performing

electromagnetic analysis and there was not enough information available to understand how

to correct the error.

It was decided that an approximation of electrostatic forces would be created using the

Standard/Explicit Model. The two bodies would be connected using spring pairs at equally

spaced intervals representing a uniform electrostatic force, Figure 7. The electrostatic force

would be calculated and entered as the spring constant.

15

Figure 7. Assembly with Equally Spaced Springs

2.3 Microsoft Excel MATLAB was considered for the calculations but the linearity of the programme did not cope

with the order of the variables and equations needing solved. For this reason it was decided

that Microsoft Excel would be used as this does not require the variables to be inserted in

order. Excel is an invaluable tool for quickly repeating mathematical calculations and logging

results. It is a relatively easy programme to use as there is built in descriptions of the functions

and a search tool to help find the function best suited to the calculation. A spreadsheet was

created that calculated the charge of the bodies and electrostatic forces depending on the

dimensions input into the cells. This allowed calculating new electrostatic forces for different

designs or distances to be completed very quickly.

The nature of the cells in Excel made it a compact and neat looking programme for recording

and viewing large sets of data with ease. A new sheet was created for each design, keeping

the results separate. Within each sheet was a series of tables corresponding to a set of results

for each simulation relating to the design. This kept all the results in the one place but also

made them very easy to read and find a specific set of data.

2.4 SolidWorks SolidWorks was used to create initial, basic models of the cantilever designs in context as the

software is a lot faster and simpler to use and edit compared to Abaqus. To put the cantilever

designs into context, a holder chip was modelled using the dimensions from Figure 3 and each

cantilever design was modelled in place with its electrodes but no material properties were

assigned. The cantilevers are sized to the specifications of each design and they are also to

scale. Each model helps with understanding how the cantilever would fit onto the chip and

how it would look if it were to be manufactured. SolidWorks was also used to produce the

engineering drawings stating the dimensions of each probe.

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3. Design of an electrostatic cantilever system

3.1 Definition of Cantilever A cantilever is a fixed end beam, it is anchored at one end with a load which is carried to the

support where it is acted upon by a moment and shear [16]. There are three main types of

cantilever, Figure 8;

Figure 8. Three main cantilever types; (a) full moment connection, (b) diving board, (c) elastic spring

The full moment connection can be likened to a radio antenna or a flagpole. The diving board

can be treated as a beam with an overhang while the third type introduces a Robin Boundary

Condition to a beam element which adds a spring to the end of the board.

There are three types of load; concentrated, point or single force, distributed load also known

as uniformly distributed load or a linearly varying load and a couple. AFM cantilevers

experience all their load through the sharp tip therefore they experience a point load.

The diving board is most commonly used in conjunction with a uniformly distributed load. The

diving board and elastic spring set ups are structurally equivalent depending on the

effectiveness of the spring and beam elements. The full moment cantilever will be the one

focussed on in this project as it best fits the criteria.

Cantilevers are widely found in construction, particularly bridges, sport stadium roofs,

chimneys and balconies also in fixed wing aircrafts. They are widely used in the field of MEMS

and are frequently made from silicon, silicon nitride or polymers.

The most important properties in an AFM cantilever are; the resonant frequency, the spring

constant and the dimensions. Micromachined cantilevers have a high resonant frequency

which is needed so that the cantilever does not interfere with its ability to respond. There are

many different cantilever shapes; rectangular, triangular, dagger and v-shaped [12].

Figure 9. Different cantilever shapes; (a) rectangular/diving board, (b) triangular, (c) dagger, (d) V-shaped

(a) (c) (b)

(d) (c) (b) (a)

17

Rectangular or diving board cantilevers are very versatile; they come in various dimensions

and are used for various applications in air or liquid. The simplicity of their geometry means

the equations and calculations relating to them are inherently less complicated than other

geometries.

Triangular cantilevers are primarily used for non-contact and tapping modes including high

speed scanning and are therefore are of no use in nanoindentation.

V-shaped probes can be used as a starting point when there is uncertainty about the

appropriate mode for the sample type. A long narrow v-shaped cantilever retains a low spring

constant ideal for contact modes while a shorter, wider v-shaped cantilever also retains a low

spring constant it is more suited to non-contact or tapping modes.

A plan view of a dagger shaped cantilever consists of a rectangular section at the holder chip

end and a triangular section at the tip [17]. Their use is limited due to the complex equations

and boundary equations relating to the geometry of the probe. The complexity is due to the

cantilever having one section with uniform cross section and two sections of varying linearity.

Their main application is in atomic force acoustic microscopy.

From the above information a rectangular or diving board cantilever was selected as they

have the simplest geometry for applying electrostatic forces. They range in length from 100-

500μm, widths of 30-50μm and thicknesses of 0.5-8μm. The rectangular cantilever and the

rectangular electrodes can be approximated as two infinite plates for deriving the equations

electrostatics. While the v-shaped geometry is more difficult, it should be investigated later

to utilise its low spring constant.

3.2 Probe Materials There are many materials that change their properties in response to the application or a

magnetic or electric field. The main materials that exhibit these properties are;

electrorheological materials, magnetorheological materials, electroactive polymers, stimuli

responsive gels and ferrofluids.

Electrorheological materials respond to the application of stress, most commonly applied in

the form of strong changes in the electric field, by changing their flow and deformation

behaviour. Such materials are typically fluids with a suspension of polymers in a low dielectric-

constant liquid and are more commonly known as smart suspensions. Over time the

suspension tends to settle out creating a loss of efficiency. The breakdown voltage of air is

~3kV/mm which is in the region of the electric field needed for the Electrorheological material

to operate. The apparent increase of the viscosity of the fluid at an applied electric field is too

small to have the desired effect on the stiffness of the cantilever.

Magnetorheological materials are another type of smart fluid which increase their viscosity

when subjected to a magnetic field. Although they exhibit the desired change in properties,

18

these fluids are unsuitable for this project as their main application is in shock absorbers in

the automotive industry and would be difficult to manipulate on the micro or nano level.

On application of an electric field electroactive polymers exhibit a change similar to

magnetorheological fluids but in shape and size instead of viscosity. However most require a

large actuation voltage in the order of hundreds to thousands of volts to produce a high

enough electric field to emit a response making this option very expensive and dangerous to

run.

Stimuli responsive gels are a variant of polymer which swell in aqueous solutions in a

reversible mechanical process. AFM testing is done in ambient air conditions where the

application of an aqueous solution would skew the test results. Applying an electric field to

the gel also changes its properties but the alterations are almost negligible compared to the

magnitude needed to alter the properties of an AFM probe.

Ferrofluids are ferromagnetic fluids similar to magnetorheological fluids but with

nanoparticles instead of micrometre scale particles giving the advantage of the particles not

settling out over time. They are another fluid that becomes strongly magnetized in the

presence of a magnetic field but being a fluid, they are again difficult to manipulate at the

nano or micro level and unsuitable for application in an AFM probe.

Although all of the above materials exhibit the desired change in properties in a reversible,

mechanical process each has its disadvantages for use on an AFM probe and will not be

investigated further.

AFM cantilevers are commonly formed from silicon nitride because despite being expensive

it is one of the only materials suitable, the other being silicon. Silicon tends to wear easily,

especially when being used on hard surfaces which is not ideal. The depleted region in silicon

presents a problem with flow of current and it is also reactive, tending to oxidise and produce

silicides. A better material for the cantilever would be silicon nitride as it is nearly chemically

inert and significantly harder than silicon with a Knoop’s hardness value of nearly half that of

diamond [10]. Silicon nitride satisfies three main requirements for the cantilever material; it

must be suitable for use on a nano or micro level, it must have suitable flexibility to allow for

the detection of the deflection under force and the material must also be able to be

manufactured cost effectively at such a small scale. Silicon nitride is chemically inert and also

very wear resistant. Silicon nitride combines high strength, creep resistance and oxidation

resistance to result in a better choice than most metals. Probes made from silicon nitride can

also be produced a lot thinner than silicon hence making them more flexible. The fabrication

of the cantilever is monolithic which yields a free standing, single crystal cantilever with an

integrated tip. Silicon nitride tips are inferior to those made from silicon, so coupling the

merits of both materials together gives a hybrid probe featuring the flexibility of a silicon

nitride cantilever and the sharpness from a silicon tip.

19

Silicon nitride is created by heating silicon between 1300⁰C and 1400⁰C in a nitrogen rich

atmosphere and can also be created by diimide however this method needs further treatment

to convert it into crystalline powder. Pure silicon nitride is difficult to produce because it does

not readily sinter and cannot be heated over 1850⁰C. Therefore the material properties are

dependent on the fabrication method with there being three different silicon nitrides

possible; reaction bonded silicon nitride (RBSN), hot pressed silicon nitride (HPSN) and

sintered silicon nitride (SSN). The density of silicon nitride is between 2370 kg/m3 and

3250kg/m3, HPSN and SSN have an approximate density of 3200kg/m3 which gives them

better physical properties but tend to be used in very demanding applications [10].

The relationship between Hooke’s law and elasticity, Equation (2), means a high Young’s

modulus when deflecting is undesirable. Silicon nitride has a Young’s modulus of between

166GPa and 297GPa. At rest a low stiffness is needed before it is artificially stiffened and

therefore 166GPa was chosen [10]. A Poisson’s ratio of 0.255 was chosen as this is an average

value [10]. A density of 2810kg/m3 was chosen because it is in the range of SSN and RBSN

densities. The nitridation in the process of making RBSN means that it does not need to be

machined after being fabricated and more complex shapes can be created. This will benefit

the project later on as more complicated cantilever designs will be investigated to see what

effect they have on the probe’s operation.

3.2.1 Coatings

Micromachined cantilevers are commonly made out of silicon or silicon nitride but can also

be made out of other materials such as glass or plastic. Cantilevers made out of different

materials interact with the sample in different ways and can reveal different properties. The

most common way to obtain cantilevers of different materials is to coat a standard silicon or

silicon nitride probe with the material of choice. This is commonly achieved by evaporation

or sputtering.

A partial gold, chromium or titanium coating on the detector facing side of the AFM

cantilevers enhances the laser reflectance by a factor of two while preventing light interfering

with the cantilever measurement. An overall gold coating is used when the probe acts as an

electrode and is approximately 70nm thick [18, 17]. The deposition process for gold coatings

is optimized for wear resistance and also compensates for stress. An aluminium coating again

prevents light from interfering within the cantilever while enhancing the laser reflectance by

a factor 2.5. If the coating is suited to the application, it is strongly recommended especially

when using thin cantilevers [19]. Hard magnetic coatings like cobalt permanently magnetise

the tip and which then has the ability to influence some soft magnetic samples.

20

3.3 Electrodes

At first it was thought that alterations to the geometry of the probe could cause an increase

of stiffness as the probe underwent deflection. It was deemed that this was not an effective

enough solution on its own as the variance in stiffness was too small. When coupled with the

application of electrostatic forces, altering the geometry could be a very effective way of

designing a variable stiffness cantilever.

The main concern at the start of the design process was deciding which plane to place the

electrodes in order to give the best results, ensuring the cantilever retains suitable freedom

of movement and appropriate change of stiffness. Initial designs were based around this and

can be split into two main ideas; horizontal electrodes and vertical electrodes. Initially the

two ideas were investigated in order to compare and select the optimal design.

3.3.1 Horizontal Electrodes

Initially it was thought that positioning two negatively charged electrodes either side of a

positively charged cantilever would attract the electrodes towards the cantilever and thus

immobilise it. Alternatively it can also be achieved by positively charging the electrode and

negatively charging the cantilever. This is represented in context with the holder chip in Figure

10 and can be approximated for the calculations to three cuboids, Figure 11.

Figure 10. Horizontal Electrodes and Cantilever on Holder Chip

Figure 11. Horizontal Electrodes with Cantilever and Force Representation in Abaqus

21

Immobilising or restraining a section of the cantilever shortens the length of cantilever that is

free to deform, therefore increasing the stiffness according to the following relation [20];

𝑘 = 3𝐸𝐼

𝐿3 (2)

A rectangular cantilever and two rectangular electrodes were created with width 40μm,

thickness 4μm and length 225μm [12]. The electrode and cantilever faces were treated as two

infinite plates to obtain the equation for attractive electrostatic forces between the two [21];

𝐹𝑒𝑎 = 𝜀 𝐴𝑜𝑣𝑒𝑟𝑙𝑎𝑝

2𝑑2. ∆𝑉2 (3)

The electrodes were placed at a starting distance of 5μm from the cantilever, the potential

difference between the electrodes and cantilever was 1V and the permittivity value of the air

was taken as 1. The area of overlap at rest was 0.9nm2 giving an electrostatic attractive force

of 18N, Excel calculations can be found in Appendix A.

Ignoring the electrodes and applying a force of 1μN to the cantilever in the direction of the

arrow, Figure 11, gives a vertical deflection. It can be assumed that the lateral stiffness of the

cantilever is large enough that no lateral motion occurs and the vertical force is the only force

affecting the deflection [22].

Both the end of the cantilever and end of the electrodes opposite the force were kept fixed

in place. The top faces of the electrodes were also constrained in to ensure the electrode did

not deflect in the vertical direction. The deflection of the cantilever will decrease the area of

overlap between the electrode and the cantilever which, according to Equation (3), will

decrease the electrostatic force acting between the two. To ascertain the effect, the

electrostatic force after deflection needs to be found by calculating the change in the area of

overlap. To calculate the change in the area of overlap the function for deflection of the

cantilever was obtained [23];

𝑦(𝑥) = 𝐹𝑥2

6𝐸𝐼(3𝐿 − 𝑥) (4)

in order to view the deflection of the cantilever as a curve on a graph. If deflection of the

cantilever can be treated purely as a curve then the static electrode can be treated as the axis

and integration between two curves can be implemented to find the area between the two,

highlighted in red in Figure 12.

22

Figure 12. Area of Overlap, Highlighted in Red. Not to Scale.

The value obtained from integration was then subtracted from the area of the electrode to

give a new area of overlap 0.894nm2, full integration can be found in Appendix B.

Using the area of overlap after deflection, the electrostatic force was calculated to be 17.9N.

From this is can be determined that the change in electrostatic force during the deflection is

negligible and will therefore be discounted.

The pull in voltage of the system also needs to be taken into account. Edward Keat Leem Chan

equates the electrostatic forces to the mechanical forces in his paper to obtain a new

equation which can be solved. This equation can be solved for displacement, showing that

the distance at which the electrode will pull in to meet the cantilever is beyond two thirds of

the original distance, Figure 13 [11].

Figure 13. Graph of Pull in Voltage

An electrode 5μm from the cantilever needs to travel 3.33μm to reach the point it would be

expected that the electrode would reach instability and travel to make contact with the

cantilever.

When simulating this design the electrodes did not attract to the cantilever like expected but

remained stationary as the cantilever deflected. Increasing the voltages acting on the two

bodies and as a direct result increasing the electrostatic forces between them showed no

change in the result. The electrostatic forces considered in this simulation acted

perpendicular to the direction of deflection. They had no effect and hence the electrodes are

unable to be attracted to the cantilever.

23

3.3.2 Vertical Electrodes

Building on the previous model, the electrodes were moved out of plane with the deflection

but into plane with the electric field. Electrodes were positioned equally spaced above and

below the cantilever, Figure 14, both with a width equal to that of the cantilever. Again, for

ease of calculations the electrodes and cantilever have been taken as three cuboids, Figure

15.

Figure 14. Double Vertical Electrode and Cantilever on Holder Chip

Figure 15. Double Vertical Electrodes and Cantilever

The electrodes were constrained at the same end as the cantilever but were free to be

attracted towards it. The springs were again placed at an initial distance of 25μm apart giving

9 pairs along the cantilever and electrodes that represent a constant electrostatic force at an

applied voltage of 1V. The stiffness of the spring pairs acting between the bodies was

calculated by using the equation for attractive electrostatic forces, Equation (3). The force

applied to deflect the cantilever was 100μN. The simulation was run with the new, vertical

24

electrode placement and this time it could be seen that the cantilever was equally attracted

to both electrodes, Figure 16, being held stationary between the two.

Figure 16. Cantilever Held Equally Between Two Electrodes

When using two electrodes, the results showed a significant increase in stiffness, Table 1.

However, having an electrode on the bottom hampers the probes ability to come into contact

with the sample, hindering the experiment.

Table 1. Increase in Stiffness with Double Vertical Electrodes

Number of

springs

Deflection

(μm)

Electrode length

(μm)

Force Applied

(μN)

Stiffness

(N/m)

0 7.5620 0 100 13.224

9 0.1963 225 100 509.424

The simulation was repeated at the same electrode distance of 10μm, voltage of 1V and force

of 100μN but this time only using only a top electrode, Figure 17. As the deflection increased,

the distance between the two bodies would decrease and according to Equation (3) the

electrostatic force would also increase. It was thought that this increase in electrostatic force

would attract the cantilever towards the electrode and this would shorten the free length of

cantilever available to deflect. This reasoning was due to the assumption that the springs

started in tension and when the simulation was started would tend towards a state of

equilibrium. As they tended towards equilibrium, the cantilever was expected to be pulled

towards the electrode due to the fact that the spring constant of the cantilever was

significantly less than that of the spring.

25

Figure 17. Cantilever with Top Electrode on Holder Chip

Figure 18. Single Vertical Electrode and Cantilever

The results for one electrode showed that the springs did not pull the cantilever to the

electrode like expected but held the cantilever at a fixed distance away from the electrode,

Figure 19. This was due to the incorrect assumption that the springs started off in tension

when in fact they started the simulation in an equilibrium state.

Figure 19. Cantilever Experiencing Repulsive Forces

In this simulation it was noted that the cantilever was not attracted to the electrodes but held

at a fixed distance therefore it was determined that the force acting on the cantilever was not

26

attractive. Due to the springs being a resistive force the cantilever was experiencing repulsive

forces instead. The section of the cantilever with the electrostatic force applied became held

at a fixed distance from the electrode. This still shortened the length of the cantilever that

was free to deflect but not in the way expected. Using the same parameters, the spring

constant was then calculated using the equation for repulsive electrostatic forces [24];

𝐹𝑒𝑟 = 𝑘𝑒𝑄𝑒𝑄𝑐

𝑑2 (5)

Where the Coulomb’s Constant, 𝑘𝑒 is a proportionality constant of value 9𝑥109𝑁𝑚2/𝐶2 and

the charge of the cantilever and electrode can be calculated using the equation below [24];

𝑄 = 𝐴𝑉

𝑑 (6)

The simulation with one electrode was run again with the new repulsive force in order to

obtain the new values for change in stiffness. The overall increase in stiffness of cantilever

with only the top electrode, Table 2, was on average half the increase seen than with two

electrodes but was sufficient enough to achieve a spring constant greater than 40N/m.

Table 2. Increase in stiffness with Single Vertical Electrode at 1V and 10μm

Number of

springs

Deflection

(μm)

Electrode length

(μm)

Force Applied

(μN)

Stiffness

(N/m)

0 7.562 0 100 13.224

9 0.335 225 100 298.329

The simulation with one electrode was repeated with the springs at the same intervals of

25μm along the length of the cantilever but this time being “turned on” incrementally. Spring

set 1 is 25μm from the fixed end of the cantilever giving a potential electrode length of 25μm

while spring set 2 is 25μm from set one, giving an electrode length of 50μm.

Introducing the spring sets incrementally allows for one set to be turned on before running

the simulation again to investigate the length of cantilever that, when fixed, generates the

desired change in stiffness. The electrode must not span the entire length of the cantilever;

therefore a voltage needs to be found which immobilises a length of cantilever shorter than

the full length of the cantilever which gives the desired stiffness change. It is shown in Table

3 that the desired stiffness is achieved halfway between spring sets 5 and 6 meaning an

electrode with minimum length 138μm is required.

27

Table 3. Incremental Change in Stiffness at 1V and 10μm

It was then investigated what effect increasing or decreasing the voltage had on the length of

the electrode and subsequently the length of cantilever visible for measurement. This was

done by first increasing the voltage to 1.5V, Table 4, then decreasing the voltage to 0.5V, Table

5.

Table 4. Incremental Change in Stiffness at 1.5V and 10μm

It can be seen that increasing the voltage to 1.5V increases the electrostatic force, decreasing

the length of electrode needed to achieve the desired stiffness change to a length of 125μm.

This means there is a larger length of cantilever visible for laser spot deflection.

Spring Set Deflection

(μm)

Electrode length

(μm)

Force Applied

(μN)

Stiffness

(N/m)

0 7.562 0 100 13.224

1 7.542 25 100 13.259

2 7.233 50 100 13.826

3 6.171 75 100 16.205

4 4.578 100 100 21.844

5 3.084 125 100 32.425

5.5 2.256 138 100 44.326

6 1.949 150 100 51.308

7 1.155 175 100 86.580

8 0.636 200 100 157.307

9 0.335 225 100 298.329

Spring Set Deflection

(μm)

Electrode length

(μm)

Force Applied

(μN)

Stiffness

(N/m)

0 7.562 0 100 13.224

1 7.518 25 100 13.301

2 6.912 50 100 14.468

3 5.368 75 100 18.629

4 3.679 100 100 27.181

5 2.365 125 100 42.283

6 1.426 150 100 70.126

7 0.785 175 100 127.389

8 0.385 200 100 259.538

9 0.176 225 100 567.859

28

Table 5. Incremental Change in Stiffness at 0.5V and 10μm

Spring Set Deflection

(μm)

Electrode length

(μm)

Force Applied

(μN)

Stiffness

(N/m)

0 7.562 0 100 13.224

1 7.557 25 100 13.233

2 7.471 50 100 13.385

3 7.093 75 100 14.098

4 6.225 100 100 16.064

5 4.950 125 100 20.202

6 3.601 150 100 27.770

7 2.450 175 100 40.816

8 1.590 200 100 62.893

9 1.011 225 100 98.912

It can be seen that decreasing the voltage to 0.5V decreases the electrostatic force, increasing

the length of electrode needed to achieve the desired stiffness change to a length of 175μm.

This means there is a smaller length of cantilever visible for laser spot deflection.

The fact that the electrode does not need to span the length of the cantilever means the free

end will be visible for laser spot measurement. For the use of laser spot detection the free

end of the cantilever needs to be visible at all times to the laser and hence cannot be

obstructed by the top electrode. It is beneficial to have a larger length available for laser spot

deflection as a larger area of cantilever means more of the laser will be reflected, giving a

better reading. To have a larger area visible the above tables show a larger voltage results in

a shorter electrode.

29

3.4 Dimensions of Cantilever Design

Looking at the equation for calculating stiffness in a cantilever, Equation (2), it can be seen

that the main element influencing the stiffness, k, is the length of the cantilever. At a set

Young’s Modulus a longer cantilever will have a lower k and therefore be softer just as a

shorter cantilever will be stiffer as it has a higher k. The relationship between length and

stiffness depends on the relationship of stiffness with the second moment of area which itself

is dependent on the width and thickness of the cantilever [20];

𝐼 = 𝑤𝑡3

12 (7)

The dominant factor in the second moment of area is the thickness of the cantilever, as it is

cubed, in the equation above. Therefore the best way to minimise stiffness is to minimise the

thickness of the cantilever as the stiffness is critically dependant on the thickness of the

cantilever.

To have a soft cantilever the cantilever needs to be long and thin, the width of the cantilever

having the least effect on the stiffness out of the three dimensions. This information

influenced the dimensions chosen for the cantilever. A width of 50μm was chosen as it is an

above average width for cantilevers of this application [12]. A thickness of 2.5μm is a smaller

than average size for this application but chosen in order to design a softer cantilever than

most. These dimensions give a second moment of area 651x10−21m4. The three values were

inputted into Equation (2) and the length was increased until k was less than 1N/m. Probes

with a force constant less than 1N/m are defined as soft, which was the initial target for the

conditions of the cantilever. A cantilever length of 450μm gives a stiffness of 0.356Nm which

is within the length range of 100μm to 500μm for a rectangular cantilever.

3.4.1 Compatibility with Holder Chip

Holder chips for AFM probes are manufactured from micromachined monolithic (single

crystal) silicon and have excellent uniformity which helps provide high quality, clear imaging

[12, 25]. Providing holder chips are sourced from a reputable company (for example from

Budget Sensors, Appendix C) it will accommodate most industry size and commercially

produced cantilevers of which the above cantilever design is in the range. Although chips can

vary in width, thickness and length, the standard chip dimensions for a rectangular probe are

3400μm by 1600μm by 300μm. . Holder chips can also come with or without alignment

grooves on the bottom to better help with fitting into the machine. These holder chips are

compatible with use in most commercial; Scanning probe microscopes, DI/Veeco AFMs, TM

Microscopes, JOEL, Molecular imaging and other commercially available AFM machines. It is

possible to manufacture a probe and attach it to a holder chip by way of soldering.

30

4. Design Assembly and Results When assembled, Figure 20, the distance between the cantilever and the electrode must be

greater than the maximum assumed deflection in order to avoid obstructing the movement

of the cantilever. An approximation of the probe can be taken to simplify simulations, Figure

21. The distance of 3μm between the electrode and the cantilever was chosen by deflecting

the cantilever under the set force of 1μN, which gave a maximum deflection of 2.77μm.

Details of the dimensions can be found in Appendix D.

Figure 20. Assembly of Cantilever, Electrode and Holder Chip

Figure 21. Representation of Regular Cantilever

The charge used in this type of application is usually in the order of 10-9 and sometimes 10-6

coulombs, at 1μV the electrodes and cantilever have a charge of 7.5nC giving at electrostatic

force between them of 56 250N. This force was deemed far too large and therefore the

voltage was reduced further to 0.1μV giving and electrostatic force of 562.5N

Using these values a simulation was run with spring sets at intervals of 50μm down the length

of the bodies to determine the length of electrode needed. It must be noted that the

electrode needs to be shorter than the cantilever to allow for ease of measuring the deflection.

The results can be seen below, Table 6;

31

Table 6. Incremental Change in Stiffness at 0.1μV and 3μm

Spring Set Deflection

(μm)

Electrode length

(μm)

Force Applied

(μN)

Stiffness

(N/m)

0 2.772 0 1 0.361

1 2.325 50 1 0.430

2 1.626 100 1 0.615

3 1.064 150 1 0.940

4 0.645 200 1 1.550

5 0.353 250 1 2.830

6 0.166 300 1 6.013

7 6.06E-02 350 1 16.502

8 1.33E-02 400 1 75.358

9 1.47E-03 450 1 681.199

Cantilevers with a spring constant less than 40N/m are defined as stiff [12], this cantilever

reaches this stiffness between spring set 7 and set 8. Inserting more spring sets half way

between 7 and 8 would find the exact length of cantilever that is needed to achieve 40N/m

but this is unnecessary in terms of the stiffness. Commercial cantilevers are produced with a

spring constant up to 100N/m so having an electrode 400μm in length will produce a

cantilever within the stiff range.

Shortening the electrode from 450μm to 400μm decreases the area of overlap and decreases

the charge of the two bodies but this decrease is small enough to have a negligible effect on

the electrostatic force acting between the two bodies. An electrode of 400μm leaves 50μm

of cantilever visible for laser spot deflection.

Increasing the electrostatic force should decrease the length of electrode needed, in turn

increasing the section of cantilever visible for measurement. This is achieved by increasing

the voltage to 0.2μV and increasing the distance to 4μm to give a new force of 711.91N, Table

7.

32

Table 7. Incremental Change in Stiffness at 0.2μV and 4μm

Spring Set Deflection

(μm)

Electrode length

(μm)

Force Applied

(μN)

Stiffness

(N/m)

0 2.772 0 1 0.361

1 2.306 50 1 0.434

2 1.613 100 1 0.620

3 1.052 150 1 0.951

4 0.636 200 1 1.572

5 0.347 250 1 2.879

6 1.626 300 1 0.615

7 5.87E-02 350 1 17.036

7.5 3.00E-02 375 1 33.333

7.625 2.39E-02 381.25 1 41.771

7.75 1.98E-02 388 1 50.505

8 1.24E-02 400 1 80.580

9 1.31E-03 450 1 766.284

33

5. Cantilever Design with Widened End The reflectivity of AFM cantilevers is typically enhanced by the deposition of a thin coating,

often a gold film on the detection side. A wider cantilever will also help with the measurement

of deflection as it will more completely reflect the laser spot, eliminating laser falling which is

the main loss of brightness and light focus. This increase in width will also increase the initial

stiffness of the cantilever according to Equation (2). This design with the wider end can be

seen in context with the holder chip and electrode, Figure 22.

Figure 22. Widened End Cantilever with Electrode on Holder Chip

This can be approximated into two cuboids, Figure 23. The maximum width of a rectangular

cantilever is 50μm [12] so the wider part of the cantilever was set 50μm wide for a length of

70μm. At 70μm long, this section has a stiffness of 94.52N/m and is therefore considered to

be stiff. The rest of cantilever was narrowed from a width of 50μm to 30μm, decreasing the

stiffness of this section significantly and creating a cantilever that resembled a spatula or

paddle. The lower end of the range of widths for a rectangular cantilever is 30μm and to

achieve a stiffness less than 1N/m the length of the thin section was reduced from 450µm to

275μm. Details of the dimensions can be found in Appendix E.

Figure 23. Representation of Cantilever with Widened End

34

The initial deflection would take place in the thinner, softer section of cantilever. The

electrode would be the same length and width as this section and on application of the

electrostatic forces would immobilise the thinner section. This would leave the wider, stiffer

section free to deflect.

At a distance of 4μm and a voltage of 0.1μV the cantilever was subjected to a deflecting force

of 1µN. This gave a resultant electrostatic force of 75.63N and an increase in stiffness can be

observed, Table 8. The increase in stiffness was not as large as expected. To account for this

the voltage applied was increased from 0.1μV to 0.5μV which increased the electrostatic force

to 1890.63N. This time the stiffness increased from 15.528N/m to 33.42N/m, Table 9. This is

still not greater than 40N/m so cannot be defined at reaching the target of stiff.

Table 8. Change in Stiffness for Widened End Design

Table 9. Change in Stiffness after Increasing Voltage to 0.5μV

Increasing the voltage further gave an electrostatic force that was deemed too large therefore

the geometry of the probe was altered. Decreasing the length of the wide end of the

cantilever increased its stiffness which, when deflected, gave the desired result, Table 10.

Table 10. Change in Stiffness after Decreasing Length

Springs Deflection (μm) Force Applied (μN) Stiffness (N/m)

Off 1.7100 1 0.585

On 0.0155 1 64.350

To increase the electrostatic force and subsequently the stiffness of this design the voltage

was increased. It can be seen from Equations (2) and (7) that stiffness is more greatly affected

by the thickness of the component. In hindsight, a simpler way of increasing the stiffness for

this design and following designs would have been to increase the thickness or length of the

cantilever instead of the voltage.

Springs Deflection (μm) Force Applied (μN) Stiffness (N/m)

Off 2.0430 1 0.489

On 0.0644 1 15.528

Springs Deflection (μm) Force Applied (μN) Stiffness (N/m)

Off 2.0430 1 0.489

On 0.0299 1 33.422

35

6. Cantilever Design with Vertical Fin It was noticed that on application of the electrostatic force the cantilever would undergo a

resultant deflection in the opposite direction from the applied force. The fin was added on

the back of the cantilever which was to allow the cantilever to deflect without interference

from the electrode, Figure 24. The fin should act as a damper, taking the majority of the force

from the electrode allowing the remainder of the cantilever to deform while still being

stiffened.

Figure 24. Cantilever with Fin and Electrode on Holder Chip

An approximation of this design without the holder chip can be taken, Figure 25. The

cantilever was the same as the original design at 50µm wide, 2.5µm thick and 450µm. The fin

was also 50μm wide, 2.5μm thick but shorter at 350μm long, leaving 100μm of cantilever free

to deflect and measure. The cantilever is fixed in place at one end and the electrode is fixed

in place while the fin is only fixed to the cantilever. Details of the cantilever dimensions can

be found in Appendix F.

Figure 25. Representation of Cantilever with Vertical Fin

At a distance of 3µm and a voltage of 0.1µV the electrostatic force is 340.28N. The change of

stiffness on application of the electrostatic force is very small, Table 11, like the previous

design at the same voltage.

Table 11. Change of Stiffness for Cantilever with Fin

Springs Deflection (μm) Force Applied (μN) Stiffness (N/m)

Off 2.7740 1 0.360

On 0.0454 1 22.022

36

As discussed above, the most effective way of increasing the stiffness of a cantilever is by

increasing its thickness. The thickness of this cantilever and fin was increased from 2.5µm to

3µm. Keeping all other parameters the same the simulation was run again. The initial stiffness

of the cantilever almost doubled when the thickness was increased but it is still in the

acceptable range of less than 1N/m and greater than 40N/m, Table 12.

Table 12. Change in Stiffness after Altering Thickness

Springs Deflection (μm) Force Applied (μN) Stiffness (N/m)

Off 1.6050 1 0.623

On 0.0232 1 43.029

37

7. Cantilever Design with Lateral Fins Another way of removing the electrode’s direct effect on the cantilever is to move it laterally.

Moving electrodes to wings attached to the cantilever should have a similar effect to the

vertical fin, Figure 26.

Figure 26. Cantileverwith Lateral Fins with Electrodes on Holder Chip

The wings were fixed at one end while the cantilever was free to deflect as it does not actually

come into contact with the holder plate. An approximation of this was modelled in Abaqus,

Figure 27.

Figure 27. Representation of Lateral Fins Cantilever

Starting with the cantilever 50µm by 2.5µm by 450µm the wings were added 50µm from the

tip end. These wings were initially 50µm wide and extended 10µm past the cantilever end but

having them this width gave an initial stiffness greater than 1N/m. Decreasing the width of

the wings to 40µm and having them a distance of 40µm away from the cantilever gave a

stiffness less than 1N/m. The electrodes were the same length and width as the wings and at

a distance of 2.5µm from the cantilever and a voltage of 0.1µm they exerted an electrostatic

force of 589.82N. This force resulted in the desired stiffness change to greater than 40N/m,

Table 13. Details of the dimensions can be found in Appendix G.

Table 13. Change in Stiffness for Lateral Fins Cantilever

Springs Deflection (μm) Force Applied (μN) Stiffness (N/m)

Off 1.7560 1 0.569

On 0.0223 1 44.883

38

This design has a starting stiffness less than 1N/m and obtains a stiffness greater than 40N/m

like desired. Moving the electrodes away from the cantilever and having them act on the fins

instead eliminates any interference between the electrostatic forces and the deflection of the

cantilever. The electrodes will not affect the cantilever but will allow it to continue to deflect

without being interfered with.

39

8. Fabrication There are several different micromachining processes that can be used to make the

cantilevers depending on their material and tip shape. The requirement for a high resonant

frequency and low spring constant of the cantilevers leads to silicon micromachining

techniques being used [26]. AFM cantilevers made from silicon or silicon nitride are

manufactured by way of silicon micromachining, Figure 28 [19]. A silicon wafer is processed

in a series of stages; deposition, photolithography and etching, to produce a material with

dimensional tolerances in the range of 1μm. The typical silicon wafer contains very pure

silicon which is grown into monolithic or monocrystalline ingots. These ingots are then sliced

into wafers and polished to become flat and uniform.

Figure 28. Representation of Etching Techniques

Silicon micromachining benefits the AFM industry as it has the ability to manufacture sharper

tips compared with the electrochemical etching process for the tips used in scanning

tunnelling microscopy (STM). In silicon micromachining it is possible to simultaneously batch

fabricate thousands of cantilevers while retaining their mechanical properties, making it an

inexpensive process.

The most commonly used tip shapes are pyramidal and conical. There are two main

manufacturing processes used for machining silicon nitride cantilevers with these tip shapes.

The strictest requirements are for the sharpness of the tip and the cantilever thickness as it

was proved in an earlier section that the thickness has the greatest effect on the force

constant of the cantilever. In micromachining, the thickness can usually be controlled to

within 25% of a nominated value and the tip radius is dependent on the manufacturing

process but can be controlled to within 10nm to 50nm.

To make a silicon nitride cantilever with a pyramidal tip, a square pyramid a few microns

across is etched on a silicon wafer, Figure 29 [4]. The etching can be done with a preferential

silicon etchant such as potassium hydroxide which etches in a certain crystallographic silicon

plane. The pit now acts as a mould for the silicon deposition stage. The deposition of the

40

silicon nitride determines the thickness of the cantilever. In this process the chemical balance

of the silicon nitride is adjusted to create a stress free film. Any residual stresses will mean

the finished cantilever will have a permanent deflection making it difficult to align and

measure.

Figure 29. Silicon Wafer

Cantilevers with conical tips are typically dry etched. A silicon dioxide dot of 1μm thick is

placed on a silicon wafer by wet oxidisation. These wafers have been cut to the required size

using a diamond cutter and prepared by being cleaned in a solution to remove organic and

metallic impurities. The gases in in the dry etching machine do not attack the silicon dioxide

while the silicon is vertically and laterally etched under the silicon dioxide mask. The

composition of these gases determines the ratio of vertical to lateral etching.

Photolithography is then carried out to define different patterns [27]. The etching process is

stopped before the silicon covers the dot, the silicon is then oxidised. This process produces

a sharp tip every time as some of the silicon is used up during oxidisation, revealing a sharp

tip on removal of the oxide. The section that is the cantilever is boron doped which provides

a conductive, stress free cantilever.

41

9. Evaluation The cantilever with the vertical fin, Figure 30, is the most difficult design to get a low enough

starting stiffness. The inclusion of the second level (or “fin”) where the electrode acts

increases the initial stiffness from 0.356 to 0.623 but it is still well below 1N/m. On application

of the electrostatic force, the change in cantilever stiffness is not as marked at with the other

designs but still meets the criteria of greater than 40N/m with a stiffness value of 43.029N/m.

This cantilever satisfies the variable stiffness requirement of the design however it is the most

difficult to manufacture. The standard process of manufacturing the probes out of silicon or

silicon nitride is by patterning and etching. Patterning includes photolithography and is a flat

process where a pattern is etched or a new material is deposited onto the material

underneath. Etching is another flat technique which removes part of the deposited material

from the wafer. Neither of these processes supports the fabrication of the fin making the

cantilever difficult to manufacture.

Figure 30. Cantilever with Vertical Fin

The cantilever design with the lateral fins, Figure 31, achieves the desired change in stiffness

from 0.569N/m to 44.89N/m. As it achieves the desired stiffness change from below 1N/m to

greater than 40N/m it satisfies the design requirement for variable stiffness. Unlike the design

with the vertical fin, this design is completely flat making it more suitable for manufacturing.

The lateral fins with the cantilever can be manufactured in silicon nitride using the standard

etching, coating and under etching techniques. Despite satisfying these two requirements,

this design does not adhere to the conditions of the commercially available holder chip. To fit

the selected holder chip the cantilever must be of a rectangular or diving board design

between 30μm and 50μm wide. Although the cantilever is a variation of the diving board

design it is far too wide. Each fin has a width of 40μm. Taken on their own a single fin would

fit the holder chip as it is within the tolerance of 30μm - 50μm but putting both fins together

the cantilever design is too wide.

Figure 31. Cantilever with Lateral Fins

42

It would seem that the better design is either the standard rectangle, Figure 32, or the

cantilever with the widened end, Figure 33. Both designs are diving board cantilevers

manufacturable by the standard micromachining processes.

Figure 32. Rectangular Cantilever

The cantilever with the widened end has a minimum width of 30μm. This width is on the

lowest limit but can still be accommodated by commercially available holder chips. While the

regular cantilever is on the upper end of the limit at 50μm, it is also able to fit the

commercially available holder chips.

Figure 33. Cantilever with Widened End

Both cantilevers also satisfy the requirements for a variable stiffness probe. The cantilever

with the widened end has a starting stiffness which is deemed “soft” and achieves a stiffness

defined as “stiff”. The narrower section of cantilever allows for the cantilever to deflect better

as it has a lower stiffness. The electrode dimensions are dictated by the size and shape of this

section. The rectangular cantilever also satisfies the design requirement for a variable

stiffness cantilever starting and “soft” and ending as “stiff”. It has a starting stiffness which is

defined as stiff and on application of 0.2V becomes “stiff” at 766.28N/m. This is a much

greater stiffness than any of the other designs have been able to achieve. The rectangular

cantilever has a uniform cross section which makes it possible to vary the length of the

electrode across the whole length of the cantilever unlike the design with the widened end.

An advantage over the paddle design is that by varying the length of the electrode a greater

range of probe stiffness can be achieved.

Although the cantilever with the widened end and the regular cantilever both satisfy the

design requirements and are very similar, the regular cantilever is a more standard probe

shape for which the manufacturing process has been tailored over many years. The regular

43

cantilever is also a simpler design lending itself to easier alteration of the width, thickness or

length to subsequently change the stiffness of the design. Changing the cantilever’s

dimensions will change the size of electrode needed and electrostatic force and more work is

needed to investigate if these alterations will improve efficiency. If the dimensions were to

be kept the same, the dimensions of the electrode can be altered more easily for this design

than for the cantilever with the widened end. Taking this into account, altering the applied

voltage is still the most obvious way for altering the electrostatic forces on the current design.

For these reasons, the rectangular cantilever is deemed to be the most suitable design for the

application of a variable stiffness AFM probe.

44

10. Conclusion Theoretically a probe has been designed that meets the requirements for a variable stiffness

AFM nanoindentation probe. Implementing this probe in industry would result in one probe

needing to be produced with the same stiffness and dimensions instead of producing many

varying probes. Having a soft probe makes measuring the deflection of the probe easier, while

tuning the probe to a higher stiffness better carries out the material test. It is a benefit that

the probe was designed to be compatible with the equipment already in use. This means that

the probe can be used by equipment already considered standard in industry and owned by

many companies without creating the unnecessary expenses of having to design or purchase

new machinery. Further work is required on this design to make it applicable in other areas

of AFM testing. Alterations to the shape or dimensions could bring the benefits of a variable

stiffness probe to the other AFM modes such as imaging, contact or tapping. A variable

stiffness V-shaped cantilever should be investigated as they are designed to have a low spring

constant. This is good because it reduces the work needed to make a variable stiffness probe.

The probe already has a stiffness less that 1N/m so work is only needed to increase the

stiffness.

As the report shows, it is possible to produce an AFM nanoindentation probe with a variable

stiffness. The standard rectangular probe is the most efficient and effective way of achieving

this. It is believed that the assembly of electrode and cantilever that has been designed will

work. All the main design requirements laid out at the beginning have been met by this probe

design and the probe should function fully. The probe has been designed using the standard

material silicon nitride which can be manufactured by the usual micromachining process. The

dimensions of the probe fit commercially available equipment including holder chips and

microscopes. Most importantly the probe has an initial stiffness less than 1N/m and an

achievable stiffness greater than 40N/m. Therefore the probe satisfies the values which

constitute a “soft” and a “stiff” probe making it a variable stiffness AFM probe.

45

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48

Appendix A – Excel Calculations of Forces

Inputs Variables Charge Forces

F 1.00E-06 N L 3.50E-04 m Q1 7.5E-10 C

E 1.66E+11 GPa h 2.50E-06 m Q2 7.5E-10 C F(attractive) 0.00 N

I 6.51E-23 m4 w 5.00E-05 m k 9.00E+09 Nm2/C2

εo 1 ΔV 1.00E-07 V Q 7.50E-010 C F(repulsive) 562.50 N

d 3.00E-06 m

x-area 2.25E-08 m2

49

Appendix B – Integration Between Two Curves

𝐴𝑐𝑢𝑟𝑣𝑒 = ∫ (𝑓(𝑥) − 𝑔(𝑥)) 𝑑𝑥𝑥2

𝑥1

𝐴𝑐𝑢𝑟𝑣𝑒 = ∫ (0 − 𝐹𝑥2

6𝐸𝐼(3𝐿 − 𝑥) ) 𝑑𝑥

𝑥2

𝑥1

𝐴𝑐𝑢𝑟𝑣𝑒 = ∫ (− 3𝐿𝐹𝑥2

6𝐸𝐼+

𝐹𝑥3

6𝐸𝐼 ) 𝑑𝑥

𝑥2

𝑥1

𝐴𝑐𝑢𝑟𝑣𝑒 = [− 𝐿𝐹𝑥3

2𝐸𝐼.1

3+

𝐹𝑥4

6𝐸𝐼 .

1

4]

𝑥1

𝑥2

𝐴𝑐𝑢𝑟𝑣𝑒 = [− 𝐿𝐹𝑥3

6𝐸𝐼+

𝐹𝑥4

24𝐸𝐼 ]

𝑥1

𝑥2

𝐴𝑐𝑢𝑟𝑣𝑒 = [− 𝐿𝐹𝑥3

6𝐸𝐼+

𝐹𝑥4

24𝐸𝐼 ]

0

225𝑥10−6

𝐴𝑐𝑢𝑟𝑣𝑒 = [− 225𝑥10−6 . 1𝑥10−6 . 225𝑥10−63

6 . 232𝑥109 . 2.133𝑥10−22+

1𝑥10−6 . 225𝑥10−64

24 . 232𝑥109 . 2.133𝑥10−22 ] − [0 ]

𝐴𝑐𝑢𝑟𝑣𝑒 = 6.47𝑥10−12m2

𝐴𝑜𝑣𝑒𝑟𝑙𝑎𝑝 = 𝐴𝑟𝑒𝑎 − 𝐴𝑐𝑢𝑟𝑣𝑒

𝐴𝑜𝑣𝑒𝑟𝑙𝑎𝑝 = 225𝑥10−6 𝑥 4𝑥10−6 − 6.47𝑥10−12

𝐴𝑜𝑣𝑒𝑟𝑙𝑎𝑝 = 9𝑥1010 − 6.47𝑥10−12

𝐴𝑜𝑣𝑒𝑟𝑙𝑎𝑝 = 8.935𝑥10−10m2 or 0.894nm2

50

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0 50 100 150 200 250

Def

lect

ion

(μ

m)

Length (μm)

Representation of Deflection of Cantilever as a Curve

51

Appendix C – Budget Sensors’ Holder Chips

52

Appendix D – Dimensions of Diving Board Cantilever Design

53

Appendix E - Dimensions of Cantilever Design with Widened End

54

Appendix F - Dimensions of Cantilever Design with Vertical Fin

55

Appendix G - Dimensions of Cantilever Design with Lateral Fins