# Nanotechnology/AFM

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# Atomic Force Microscopy (AFM)

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The relation between torsional spring constant and lateral spring constant is in doubt. Please check ("Normal and torsional spring constants of atomic force microscope cantilevers" Green, Christopher P. and Lioe, Hadi and Cleveland, Jason P. and Proksch, Roger and Mulvaney, Paul and Sader, John E., Review of Scientific Instruments, 75, 1988-1996 (2004), DOI:http://dx.doi.org/10.1063/1.1753100) and ("Lateral force calibration in atomic force microscopy: A new lateral force calibration method and general guidelines for optimization" Cannara, Rachel J. and Eglin, Michael and Carpick, Robert W., Review of Scientific Instruments, 77, 053701 (2006), DOI:http://dx.doi.org/10.1063/1.2198768) for details.

Typical AFM setup. The deflection of a microfabricated cantilever with a sharp tip is measured be reflecting a laser beam off the backside of the cantilever while it is scanning over the surface of the sample.

# Methods in AFM

A wealth of techniques are used in AFM to measure the topography and investigate the surface forces on the nanoscale:

For imaging sample topography:

• Contact mode, where the tip is in contact with the substrate. Gives high resolution but can damage fragile surfaces.
• Tapping / intermittent contact mode (ICM), where the tip is oscillating and taps the surface.
• Non-contact mode (NCM), where the tip is oscillating and not touching the sample.

For measuring surface properties (and imaging them):

• Lateral force microscopy (LFM), when the tip is scanned sideways it will tilt and this can be measured by the photodetector. This method is used to measure friction forces on the nanoscale.
• Force Modulation Microscopy. Rapidly moving the tip up and down while pressing it into the sample makes it possible to measure the hardness of the surface and characterize it mechanically.
• Electrical force microscopy. If there are varying amount of charges present on the surface, the cantilever will deflect as it is attracted and repelled. kelvin probe microscopy will normally be more sensitive than measuring s static deflection.
• Kelvin probe microscopy. By applying an oscillating voltage to an oscillating cantilever in non-contact mode and measuring the charge induced oscillations, a map can be made of the surface charge distribution.
• Dual scan method - an other kelvin probe method described below.
• Magnetic Force Microscopy. If the cantilever has been magnetized it will deflect depending on the magnetization of the sample.
• Force-spectroscopy or force-distance curves. Moving the cantilever up and down to make contact and press into the sample, one can measure the force as function of distance.
• Nanoindentation. When pressing the cantilever hard into a sample it can leave an imprint and in the force distance curve while doing indentation can tell about the yield stress, elastic plastic deformation dynamics.
• Liquid sample AFM. By immersing the cantilever in a liquid one can also image wet samples. It can be difficult to achieve good laser alignment the first time.
• Electrochemical AFM.
• Scanning gate AFM
• Nanolithography
• Dip-pen lithography

## Reviews of Atomic Force Microscopy

SEM image of a typical AFM cantilever
• Force measurements with the atomic force microscope: Technique, interpretation and applications. Surface Science Reports 59 (2005) 1–152, by Hans-Jurgen Butt, Brunero Cappella,and Michael Kappl. 152 pages extensive review of forces and interactions in various environments and how to measure and control these with AFM.

# Cantilever Mechanics

Cantilever has width w, thickness t, length L and the tip height from the cantilever middle to to the tip is h.

The typical geometry of an AFM cantilever. Length l, thickness t, width w, and tip height h is measured form the middle of the beam

When the cantilever is bent by a point force in the z-direction ${\displaystyle F_{N}}$  at the tip will deflect distance z(x) from the unloaded position along the x-axis as [1]

${\displaystyle z(x)={\frac {1}{2}}{\frac {F_{N}L}{EI}}\left(x^{2}-{\frac {1}{3L}}x^{3}\right)}$

with cantilever length ${\displaystyle L}$ , Youngs modulus ${\displaystyle E}$ , and moment of inertia ${\displaystyle I}$ .

The tip deflection is

${\displaystyle \Delta _{z}=z\left(L\right)={\frac {1}{3}}{\frac {F_{N}}{EI}}L^{3}}$

giving a spring constant ${\displaystyle k_{N}}$  from

${\displaystyle F_{N}=k_{N}\Delta _{z}}$

so

${\displaystyle k_{N}=3{\frac {EI}{L^{3}}}}$

The angle of the cantilever in the x-z plane at the tip ${\displaystyle \theta _{x}}$ , which is what gives the laser beam deflection will be

${\displaystyle \theta _{N}=z^{\prime }\left(L\right)}$

${\displaystyle z^{\prime }\left(x\right)={\frac {1}{2}}{\frac {FL}{EI}}\left(2x-{\frac {1}{L}}x^{2}\right)}$

${\displaystyle z^{\prime }\left(L\right)={\frac {1}{2}}{\frac {F}{EI}}L^{2}={\frac {3}{2}}{\frac {z\left(L\right)}{L}}}$

The difference between a hinged and fixed beam's angle of deflection and the cantilever tip. The fixed beam will give a larger deflection signal

giving the relation

${\displaystyle \theta _{N}={\frac {3}{2}}{\frac {\Delta _{z}}{L}}}$

between the tip deflection distance and tip deflection angle. This is a factor 3/2 bigger than the result we would expect if the beam was stiff and hinged at the base, showing us that we get a bigger deflection of the laser beam when the beam is bending than when its straight.

AFM cantilever, with deflection angles and detector setup. The Z-deflection from the sample Z topography is giving a deflection in the xz-plane and measured by the top-bottom detector pair. Lateral forces on the cantilever give both torsion (yz-plane deflection and Left/ritgh detector signal) and a lateral deflection in the xy-plane that cannot be measured by the detector.

## The AFM detector signal

The cantilever can bend in several ways, which is detected by the quadrant photo detector that most AFMs are equipped with. Normal topography signal is given by 'normal' deflection of the cantilever tip in the x-z direction, ${\displaystyle \theta _{xz}=\theta _{N,}}$  and detected by the left-right (or A-B) detector coupling quadrants as ${\displaystyle V_{LR}=V_{1}+V_{3}-V_{2}-V_{4}}$ .

Lateral forces applied to the tip will bend the cantilever in the x-y and x-z plane too. Lateral deflection cannot be detected by the quadrant detector since it doesn't change laser beam deflection, and deflection is also rather small, as we shall see. Lateral forces also twist the cantilever tip producing torsional deflection in the y-z direction, ${\displaystyle \theta _{yz}=\theta _{tor},}$  which in turn produces the lateral force signal from the top-bottom detector measuring ${\displaystyle V_{LR}=V_{1}+V_{2}-V_{3}-V_{4}.}$

For deflection in the z-direction, 'normal' spring constant relating the force and deflection ${\displaystyle F_{N}=k_{N}\Delta _{z}}$  is

${\displaystyle k_{N}={\frac {1}{4}}Y{\frac {wt^{3}}{L^{3}}}}$

Expressed in the angle of deflection, ${\displaystyle \theta _{N}={\frac {3}{2}}{\frac {\Delta _{z}}{L}},}$  there is an angular spring constant

${\displaystyle F_{N}=c_{N}\theta _{N}}$

with ${\displaystyle c_{N}={\frac {2}{3}}k_{N}L}$ .

AFM cantilever and the forces acting between the tip and the sample.

# Contact, Tapping, and Non-contact Mode

If an oscillator experiences an attractive force pulling it away from it rest position, the resonance frequency will drop (at snap-in it will be zero). A repulsive force squeezing it will increase the resonance frequency.

The repulsive and attractive force regimes as the AFM tip approaches the sample.

If an AFM tip is moved to contact with a sample, the resonance frequency is first decreasing slightly due to attractive forces and then increasing due to the repulsive forces. Eventually the repulsive force become so high we cannot oscillate it and we have achieved contact.

Contact mode: Because the tip is in contact, the forces are considerably higher than in non-contact mode and fragile samples as well as the tip can easily be damaged. The close contact on the other hand makes the resolution good and scan speeds can be high.

The varying resonance frequency is the cantilever moves between the attractive end repulsive regions of the force distance curve can be used to measure the cantilever position and to keep it in the attractive or repulsive part of the force distance curve.

The tip oscillation frequency for tapping and non-contact mode AFM are to either side of the tip resonance frequency. The green signal is the oscillation amplitude while the yellow is the phase

Non-contact mode: If we oscillate the cantilever at a higher frequency than its free resonance and use the feedback loop to maintain a oscillation amplitude setpoint slightly lower than that of the free oscillation, it will move the tip down until the attractive forces lower the resonance frequency and makes the oscillation amplitude drop to the setpoint level.

Tapping mode: if we oscillate the cantilever at a lower frequency that its free oscillation, moving it towards the sample will first make it oscillate at a lower frequency which will make the stage move closer to try and raise the oscillation amplitude, and at eventually as it reaches repulsive forces will settle where resonance frequency cannot increase more without giving too high an amplitude.

 Use for k (N/m) f (kHz) Non contact (NC) 10-100 100-300 Intermittent contact (IC) 1-10 20-100 Contact 0.1-1 1-50

## Tapping Mode

Tapping-mode (also called intermittent contact mode) is the most widely used operating mode in which the cantilever tip can experience both attractive and repulsive forces intermittently. In this mode, the cantilever is oscillated at or near its free resonant frequency. Hence, the force sensitivity of the measurement is increased by the quality factor of the cantilever. In tapping-mode operation, the amplitude of the cantilever vibration is used in feedback circuitry, i.e., the oscillation amplitude is kept constant during imaging. Therefore it is also referred as amplitude modulation AFM (AM-AFM). The primary advantage of tapping mode is that the lateral forces between the tip and the sample can be eliminated, which greatly improves the image resolution. Tapping mode experiments are done generally in air or liquid. Amplitude modulation is not suitable for vacuum environment since the Q-factor of the cantilever is very high (up to 105) and this means a very slow feedback response.

# Lateral Force Microscopy

If the sample is scanned sideways in the y direction, the frictional forces will apply a torque on the cantilever, bending it sideways and this can be used to measure the frictional forces. The lateral force gives both a lateral and torsional deflection of the tip. Only the torsional can be detected in the photodetector.

For sideways/lateral bending, the lateral spring constant is corresponding to the normal spring constant but with width and thickness interchanged

${\displaystyle k_{lat}={\frac {1}{4}}Y{\frac {tw^{3}}{L^{3}}}=k_{N}{\frac {w^{2}}{t^{2}}}}$

and a similar eq's for angular deflection as above. With thickness typically much smaller than the width for AFM\ cantilevers, the lateral spring constant is 2-3 orders of magnitude higher than ${\displaystyle k_{N}.}$

For a sideways, lateral force ${\displaystyle F_{lat}}$  on the cantilever we will have a sideways deflection determined by

${\displaystyle F_{lat}=k_{lat}\Delta _{y-lat}}$

If the lateral force is applied to the AFM tip, ${\displaystyle F_{lat}}$ , it will give a lateral deflection but also apply a torque ${\displaystyle \tau }$  twisting the beam

${\displaystyle \tau =hF_{tor}}$

Twisting an angle ${\displaystyle \theta _{tor}}$  gives a torional tip deflection of ${\displaystyle \Delta _{y-tor}=\theta _{tor}h.}$

The relation for the torsional spring constant is (please check this equation)

${\displaystyle F_{tor}=k_{tor}\Delta _{y-tor}}$

with

${\displaystyle k_{tor}=G{\frac {wt^{3}}{3h^{2}L}}={\frac {Ywt^{3}}{8h^{2}L}}=k_{N}{\frac {1}{2}}\left({\frac {L}{h}}\right)^{2}}$

and then

${\displaystyle \tau =h^{2}k_{tor}\theta _{tor}={\frac {1}{2}}k_{N}L^{2}\theta _{tor}={\frac {1}{2}}k_{N}{\frac {L^{2}\Delta _{y-tor}}{h}}}$

From above we have

${\displaystyle {\frac {k_{lat}}{k_{tor}}}=2\left({\frac {wh}{tL}}\right)^{2}}$

The factor ${\displaystyle 2\left({\frac {wh}{tL}}\right)^{2}}$  is typically ${\displaystyle 2\left({\frac {20\ast 10}{2\ast 100}}\right)^{2}=2.0}$  - so about 1 but larger or smaller depending on whether its a contact or non-contact cantilever.

## Friction Loop Scan

Typical signal from a scan with the AFM in lateral force mode - a friction loop scan. At the turning points the tip sticks to the surface and the signal has a linear slope with the detector sensitivity. When the lateral tip-sample force exceeds the static friction force between the sample and substrate, the tip will start to slide with the dynamic friction force and s steady signal.

For optimal torsional sensitivity - but the following is not always correct since it depends highly on the contact forces you need etc: For high torsional sensitivity, ${\displaystyle wh>>tL}$ . Since we are in contact mode AFM, L must be large and t thin for a low ${\displaystyle k_{N}}$ . So better torsional sensitivity means wider cantilevers and definitely large tip heights.

## Coupled Lateral and Torsional deflection in the cantilever

But how much will a cantilever bend laterally and how much will it twist when applied a lateral force at the tip? An applied lateral force will move two degrees of freedom with a Hookes law behaviour - the torsion and lateral motion. The applied force ${\displaystyle F_{tor}}$  is also an applied ${\displaystyle F_{lat.}}$

The effective spring constant for pushing the tip in the y direction is then

${\displaystyle F_{tor}=k_{eff}\Delta _{y}}$

with ${\displaystyle \Delta _{y}=\Delta _{y,lat}+\Delta _{y,tor}}$  and

${\displaystyle k_{eff}={\frac {1}{{\frac {1}{k_{lat}}}+{\frac {1}{k_{tor}}}}}={\frac {k_{lat}}{1+{\frac {k_{lat}}{k_{tor}}}}}}$

and the deflection made in the torsional spring ${\displaystyle \Delta _{y,tor}}$  is

${\displaystyle \Delta _{y,tor}={\frac {\Delta _{y}}{1+{\frac {k_{tor}}{k_{lat}}}}}}$

and this approaches ${\displaystyle \Delta _{y}}$  when ${\displaystyle k_{tor} so the cantilever is more prone to tilting than lateral deflection. The lateral deflection can be found from

${\displaystyle \Delta _{y,lat}k_{lat}=\Delta _{y,tor}k_{tor}\Delta _{y,lat}={\frac {k_{tor}}{k_{lat}}}\Delta _{y,tor}}$

The torsional deflection angle is then

${\displaystyle \theta _{tor}={\frac {\Delta _{y,tor}}{h}}={\frac {k_{lat}}{k_{lat}+k_{tor}}}{\frac {F_{tor}}{hk_{eff}}}={\frac {k_{lat}}{k_{lat}+k_{tor}}}{\frac {1+{\frac {k_{lat}}{k_{tor}}}}{k_{lat}}}{\frac {F_{tor}}{h}}={\frac {1}{h}}{\frac {F_{tor}}{k_{tor}}}}$

as anticipated from the assumption that the torsional and lateral springs are coupled in series. So when a constant force is applied, the detector signal is a measurement of the force.

Question: during a friction loop scan, the tip is fixed by static friction on the surface and its a constant deflection, both torsional and lateral deflection must be included to find the actual deflected distance of the tip before its pulled free from to slide the surface and the lateral deflection could influence the beginning of the friction loop curve?

# Measuring the cantilever dimensions

The vibration frequency of the fundamental mode of the cantilever is an easily measurable quantity in the AFM and can be used to evaluate a cantilever is within specifications. Its given by

${\displaystyle f[Hz]={\frac {t\beta _{i}^{2}}{4\pi L^{2}}}{\sqrt {\frac {Y}{3\rho }}}={\frac {\left(1\ast 10^{-6}\right)\left(1.875\right)^{2}}{4\pi \left(100\ast 10^{-6}\right)^{2}}}{\sqrt {\frac {\left(160\ast 10^{9}\right)}{3\ast 2330}}}=1.\,338\,5\times 10^{5}}$

Easily measurable quantities in AFM: length L, resonance freq f, tip length ${\displaystyle l_{tip},}$ width ${\displaystyle w}$ ,

Not so easy: thickness t, cross section (often there are inclined sidewalls), force konst ${\displaystyle k_{norm},}$  tip length from middle of the cantilever since we dont know the thickness.

# Noise Sources in AFM

## Thermal noise

${\displaystyle E={\frac {1}{2}}kx^{2}={\frac {1}{2}}k_{B}T}$

${\displaystyle \left\langle x\right\rangle _{rms}={\sqrt {\frac {k_{B}T}{k_{spring}}}}}$

for a 1 N/m cantilever this amounts to ${\displaystyle \left\langle x\right\rangle _{rms}={\sqrt {k_{B}300}}=0.6.}$ Å. So a 1 Å noise level requires ${\displaystyle {\frac {k_{B}300}{10^{-20}}}=0.4}$  N/m which is not a very low spring constant for a contact mode cantilever.

So thermal noise can become a problem in some AFM cantilevers at room temperature!!

# Electrical Force Microscopy

The Kelvin Probe Microscopy Method and Dual Scan Method can be used to map out the electrical fields on surfaces with an AFM.

## Kelvin Probe Microscopy Method

The principle of Kelvin probe microscopy (KPM). The lock-in amplifier generates a signal on the tip and the electrostatic tip-surface interaction is readout by the laser and the lock-in amplifier adjusts accordingly.

In the Kelvin probe microscopy (KPM) method a voltage is applied between the AFM tip and the surface. Both a DC and AC voltage is applied to the tip so the total potential difference ${\displaystyle V_{tot}}$  between the tip and surface is:

${\displaystyle V_{tot}=-V_{S}+V_{DCt}+V_{ACt}\cdot \sin(\omega \cdot t),}$

where ${\displaystyle V_{S}=V_{S}(x,y)}$  is the local surface potential, ${\displaystyle x,y}$  is the position of the tip, ${\displaystyle V_{DCt}}$  is the DC signal on the tip, ${\displaystyle V_{ACt}}$  is the amplitude of the AC signal, and ${\displaystyle \omega }$  is the frequency of the AC signal.

The frequency of the AC signal is much lower than the resonance frequency of the cantilever (a factor of 10) so the two signals can be separated by a lock-in amplifier. Via the electrostatic forces the setup measures the surface potential. If one assumes that the electrostatic force (${\displaystyle F}$ ) between the tip and surface is given by [2]

${\displaystyle F={\frac {{\frac {\partial C}{\partial z}}\cdot V_{tot}^{2}}{2}},}$

where ${\displaystyle C}$  is the capacitance and ${\displaystyle z}$  is the distance between the tip and the surface.

If a parallel plate capacitor is assumed

${\displaystyle C={\frac {A_{c}\epsilon _{0}}{z}}}$ ,

where ${\displaystyle A_{c}}$  is the area of the tip. The derivative of capacitance is

${\displaystyle {\frac {\partial C}{\partial z}}=-{\frac {A_{c}\epsilon _{0}}{z^{2}}}}$ .

Combining the force (${\displaystyle F}$ ) and ${\displaystyle V_{tot}}$  yields:

${\displaystyle F={\frac {\frac {\partial C}{\partial z}}{2}}\cdot {\Big (}-V_{S}+V_{DCt}+V_{ACt}\cdot \sin(\omega \cdot t){\Big )}^{2}=}$

${\displaystyle {\frac {\frac {\partial C}{\partial z}}{2}}\cdot {\Big (}(V_{DCt}-V_{S})^{2}+V_{ACt}^{2}\cdot \sin(\omega \cdot t)^{2}+2\cdot (V_{DCt}-V_{S})\cdot V_{ACt}\cdot \sin(\omega \cdot t){\Big )}.}$

Using Pythagorean identities

${\displaystyle {\big [}\cos(x)^{2}+\sin(x)^{2}=1{\big ]}}$

and de Moivre's formula

${\displaystyle {\big [}(\cos(x)+i\cdot \sin(x))^{n}=\cos(n\cdot x)+i\cdot \sin(n\cdot x){\big ]},}$

We find

${\displaystyle V_{ACt}^{2}\cdot \sin(\omega \cdot t)^{2}=V_{ACt}^{2}\cdot \ {\big (}1-\cos(\omega \cdot t)^{2}{\big )}=}$

${\displaystyle {\frac {1}{2}}\cdot V_{ACt}^{2}\cdot {\big (}2-2\cdot \cos(\omega \cdot t)^{2}{\big )}=}$

${\displaystyle {\frac {1}{2}}\cdot V_{ACt}^{2}-{\frac {1}{2}}\cdot V_{ACt}^{2}\cdot \cos(2\cdot \omega \cdot t).}$

This inserted in the equation for the force (${\displaystyle F}$ ) gives [3]:

${\displaystyle F={\Big (}{\frac {\frac {\partial C}{\partial z}}{2}}\cdot {\big (}(V_{DCt}-V_{S})^{2}+{\frac {1}{2}}\cdot V_{ACt}^{2}{\big )}+{\big (}2\cdot (V_{DCt}-V_{S})\cdot V_{ACt}{\big )}\cdot \sin(\omega \cdot t)-}$

${\displaystyle ({\frac {1}{2}}\cdot V_{ACt}^{2})\cdot \cos(2\cdot \omega \cdot t){\Big )}}$

${\displaystyle F=k_{1}+k_{2}\cdot \sin(\omega \cdot t)+k_{3}\cdot \cos(2\cdot \omega \cdot t),}$

where

${\displaystyle k_{1}={\frac {\frac {\partial C}{\partial z}}{2}}\cdot {\big (}(V_{DCt}-V_{S})^{2}+{\frac {1}{2}}\cdot V_{ACt}^{2}{\big )}}$ ,

${\displaystyle k_{2}=(2\cdot (V_{DCt}-V_{S})\cdot V_{ACt})}$ , and ${\displaystyle k_{3}=-{\frac {1}{2}}\cdot V_{ACt}^{2}}$ .

The frequency ${\displaystyle \omega }$  is set by an external oscillator and can therefore be locked by the lock-in amplifier. The signal detected by the lock-in amplifier (the ${\displaystyle k_{2}}$  part) is minimized by constant varying ${\displaystyle V_{DCt}}$ . When this signal approaches zero, this corresponds to ${\displaystyle V_{DCt}=V_{S}}$  i.e. mapping ${\displaystyle V_{DCt}}$  vs. the sample surface ${\displaystyle (x,y)}$  gives ${\displaystyle V_{S}(x,y)}$ .

## Dual Scan Method

The principle of the Dual Scan (DS) method where first a topography line scan is made, then the tip is lifted a distance d and another line scan is made with the source drain voltage turned on.

In the Dual Scan (DS, or sometimes called lift-mode method) one first makes a line scan with no potential on either the AFM tip or the sample in either tapping or non-contact mode. Next, the AFM tip is raised several tens of nanometers (30-70 nm) above the surface. A new line scan is made at this height, but this time with a potential on the sample also in non-contact mode. This is repeated over the desired scan area until the whole area has been scanned. For imaging the surface potential, the phase of the cantilever vibration is mapped out. The principle is shown in the figure where d is the distance between the tip and the surface in the second scan. The phase shift is dependent on the force (${\displaystyle F}$ ) acting on the tip [4]

${\displaystyle \phi =\tan ^{-1}({\frac {k}{Q\cdot {\frac {\partial F}{\partial z}}}})}$

where ${\displaystyle Q}$  is the quality factor of the cantilever, ${\displaystyle k}$  the spring constant, and ${\displaystyle z}$  the distance between the tip and the surface.

For small phase shifts the phase can be written as

${\displaystyle \phi \approx {\frac {Q\cdot {\frac {\partial F}{\partial z}}}{k}}.}$

The derivative of the force can be written as:

${\displaystyle {\frac {\partial F}{\partial z}}={\frac {1}{2}}{\frac {\partial ^{2}C}{\partial z^{2}}}\cdot V_{s}^{2},}$

where ${\displaystyle V_{s}}$  is the surface potential and ${\displaystyle C}$  is the capacitance between the tip and the surface [5] . The second derivative of the capacitance is

${\displaystyle {\frac {\partial ^{2}C}{\partial z^{2}}}={\frac {2A_{c}\epsilon _{0}}{z^{3}}}.}$

Combining equations for the phase and the derivative of the force yields the phase dependence of the phase shift

${\displaystyle \phi \approx {\frac {Q\cdot {\frac {\partial ^{2}C}{\partial z^{2}}}\cdot V_{s}^{2}}{k\cdot 2}}.}$

To find the surface potential, one must estimate the other parts of equation for the phase. The spring constant (${\displaystyle k}$ ) can be determined if the dimensions (a regular cantilever) and material of the cantilever are known:

${\displaystyle k={\frac {E\cdot w\cdot h^{3}}{4\cdot L^{3}}},}$

where ${\displaystyle E}$  is the Young modulus, ${\displaystyle w}$  is the width of the cantilever, ${\displaystyle h}$  the height, and ${\displaystyle L}$  is the length. The quality factor (${\displaystyle Q}$ ) of the cantilever can be found by measuring the shape of resonance peak. The second derivative of the capacitance can be estimated by assuming that the tip of the AFM is a plate with radius ${\displaystyle r}$  so the derivative of the capacitance is given by:

${\displaystyle {\frac {\partial ^{2}C}{\partial z^{2}}}={\frac {2\cdot \pi \cdot r^{2}\cdot \epsilon _{0}}{z^{3}}},}$

where ${\displaystyle \epsilon _{0}}$  is the vacuum permittivity. This way of estimating the other parts of the equation for the phase is quite accurate according to [6]. One can also estimate the values by measuring them at a surface with a known potential and at different known heights and then one simply calculate backwards for that particular AFM tip.

## Discussion

Both the DS and KPM methods have their strengths and weaknesses. The DS method is easier to operate, since it has fewer interlinked parameters, needing to be adjusted. The KPM method is faster, as it does not require two scans (an image with the DS method with a resolution of 512 ${\displaystyle \times }$  512 pixels and a scan rate of ${\displaystyle 0.8}$  Hz takes about half an hour). The DS method will normally obtain much better lateral resolution in the potential image compared to the KPM method. This is due to the fact that the signal depends on the second derivative of the capacitance, which in turn depends on the distance in ${\displaystyle {\frac {1}{r^{3}}}}$  compared to the KPM method where the dependence is only ${\displaystyle {\frac {1}{r^{2}}}}$ . This rapidly reduces the problem of the tip sidewall interaction. On the other hand, the KPM method has better sensitivity because it operates much closer to the surface.

## References

• A. Bachtold, M. S. Fuhrer, S. Plyasunov, M. Forero, E. H. Anderson, A. Zettl, and P. L. McEuen; Physical Review Letters 84(26), 6082-6085 (2000).
• G. Koley and M. G. Spencer; Applied Physics Letters 79(4), 545-547 (2001).
• T. S. Jespersen and J. Nygård; Nano Letters 5(9), 1838-1841 (2005).
• V. Vincenzo, and M Palma, and P. Samorí; Advanced Materials 18, 145-164 (2006).
• Veeco, "Electrostatic Force Microscopy for AutoProbe CP Research Operating Instructions", 2001 (Manual).
• D. Ziegler and A. Stemmer; Nanotechnology 22, 075501 (2011).