BE.

309: Biological Instrumentation and Measurement Laboratory GEM4 Summer School

The teaching AFM: Part 1

Alignment, Calibration, and Noise

1 Objectives
1. Learn the function of our class AFM system’s components and the relationship between them.

2. Practice aligning the AFM optics.

3. Learn how to calibrate the AFM to relate its output signal to physical cantilever deflection.

4. Measure the mechanical vibration noise in the AFM system with the cantilever free and in
contact.

2 The Atomic Force Microscopy (AFM) System

This section describes the various components of the AFM you will use in the lab, and particularly
how they differ in operation from a commercial AFM. During the lab, we will talk about the
operational principles of a standard AFM. It may also be helpful to review some of the References
in Section 4 at the end of this module. A photo of our AFM setup is provided in Figure 4 at the
end of this section for you to refer to as you learn about the instrument.

2.1 Power-on
For our AFMs to run, you must turn on three
things: (1) the detection laser, (2) the photode-
tector, and (3) piezo-driver power supply. The
photodetector has a battery that provides reverse
bias, and the others have dedicated power sup-
plies (refer to Figure 4 at the end of this sec-
tion for where these switches are located). When
you finish using the AFM, don’t forget to turn
off the three switches you turned on at the begin-
ning: laser power supply, photodetector, and piezo
power supply. Figure 1: This is a schematic of the piezo disk used
to actuate the AFM’s sample stage. The circular
electrode is divided into quadrants as shown in (a)
2.2 Scanner system to enable 3-axis actuation. When the same voltage
is applied to all quadrants, the disk flexes as shown
To be useful for imaging, an AFM needs to scan its in (b), giving z-axis motion. Differential voltages
probe over the sample surface. Our microscopes applied to opposite quadrants, produce the deflec-
are designed with a fixed probe and a movable tion shown in (c), which moves the stage along the
sample, so whenever we talk about moving the tip x- and y-axes, with the help of the offset post, rep-
relative to the sample, we will always only move resented here by the vertical green line.
the sample. The sample is actuated for scanning
and force spectroscopy measurements by a simple
piezo disk, shown in Figure 1, which is divided

1
into quadrants and flexes to move the post on which the sample stage rests. The piezo disk is
controlled from the matlab scanning software, and you will learn more about this in the next
module.
For motions along the z-axis (vertically), there are three regimes of motion:

Manual (coarse): turning the knob on the red picomotor with your hand (clockwise moves the
stage up).

Picomotor (medium): using the joystick to drive the picomotor (pushing the joystick forward
moves the stage upward).

Piezo-disk (fine): actuating the piezo disk over a few hundred nanometers using the matlab
software.

For coarse positioning along the x- and y-axis, the micrometers on the positioning stage are used.
WARNING: The AFM probes can be broken by running them into the sample — avoid
“crashing” the tip into the surface, or worse bumping the stage into the die or fluid
cell. Use caution when moving the sample up and down.

2.3 Optical system

Our microscopes use a somewhat dif-
ferent optical readout from a stan-
dard AFM to sense cantilever deflection.
Rather than detecting the position of
a laser beam that reflects off the back
surface of the cantilever, we measure
the intensity of a diffracted beam. To
do this, a diode laser with wavelength
λ = 635nm is focused onto the interdig-
itated (ID) “finger” structure, and we
observe the brightness of one of the re-
flected spots (referred to as “modes”)
using a photodiode. This gives us infor-
mation about the relative displacement
of one set of fingers relative to the other
— this is useful if one set is attached
to the cantilever, and the other to some
reference surface.
As the cantilever deflects, and the
out-of-plane spacing between the ID fin-
gers changes, the reflected diffractive
modes change their brightness, as shown
in Figure 2. However, a complication of
Figure 2: A drawing of the interdigitated (ID) interferometric
using this system is the non-linear out- fingers, with the detection laser shown incident from the top of the
put characteristic of the mode intensi- figure. When the finger sets are aligned, as in the left box above,
ties. As the out-of-plane deflection of the even-numbered modes are brightest, and odd modes are darkest.
When they displace relative to each other by a distance of a quarter
the fingers increases, each mode grows
of the laser wavelength λ, the situation reverses, shown on the right.
alternately brighter and dimmer. The This repeats every λ/4 in either direction.

2
intensity I of odd order modes vs. fin-
ger deflection ∆z has the form µ ¶
2π
I ∝ sin2 ∆z ,
λ
and for odd modes, the sine is replaced by a cosine. The plot in Figure 3(a) shows graphically the
intensity of two adjacent modes as a function of displacement.
This nonlinearity causes the sensor’s sensitivity to depend critically on the operating point along
this curve at which a measurement is done. To make useful measurements, the ID interferometer
therefore needs to be biased to a spot on the sin2 curve where the function’s slope is greatest -
midway between the maximum and minimum, as sketched in Figure 3(b). This can be done with
our devices by simply adjusting the position of the laser spot side to side on the finger grating –
the grating is not perfectly flat due to residual stress, and thus provides a simple biasing method.
At this point, it’s worth remembering the distinction between calibration, sensitivity and reso-
lution – terms which will be used frequently in the context of the AFM, but whose precise meaning
isn’t always made explicit. Be sure you’re clear on the differences between them.

(a) The non-linear intensities of the 0th and 1st order (b) The desired operating point for maximum de-
modes plotted as a function of cantilever displace- flection sensitivity is shown here on the sin2
ment. output characteristic of the ID fingers.

Figure 3: The characteristic output of the ID interferometric sensor.

3
Figure 4: The AFM setup, with major components indicated.

4
3 Lab Procedures
3.1 Laser alignment and diffractive modes
To get a cantilever position readout, the laser needs to
be well focused on the interdigitated fingers of the cantilever.
Use the white light source and stereo-microscope to look at
the cantilever in its holder. The laser spot should be visible
as a red dot (there may be other reflections or scattered laser
light, but the spot itself is a small bright dot). Adjust its
position using the knobs on the laser mount, until it hits the
interdigitated fingers (use the cantilever schematic in Figure 5
as a reference).
When the laser is focused in approximately the right posi-
tion, the paper “screen” around the slit on the photodetector
will allow you to see the diffraction pattern coming out of the Figure 5: Plan view of the imaging can-
beam splitter. Observe the reflections on this screen while ad- tilever geometry. The central (imaging)
justing the laser position until you see several evenly spaced beam dimensions are length L = 400µm,
modes. Make sure you aren’t misled by reflections from other and width b = 60µm. The finger grat-
parts of the apparatus — some may look similar to the diffrac- ings begin 117µm from the base and end
200µm from it.
tion pattern, but aren’t what you’re looking for.
Before engaging the AFM, start the piezo z-modulation scan in the matlab software using the
default frequency and amplitude of 2Hz and 8V. (Note: Be sure the mode switch on the rear of
the AFM head is flipped down to “force spec. mode,” and make sure to turn on the piezo power
supply using the color-coded switch on the table.)Carefully bring the tip near the surface, first
by hand, then very slowly with the joystick. When you make contact, you will see the modes on
the photodetector fluctuate in brightness. Because of the device geometry, only the central long
cantilever with the tip will make contact with the sample surface.
3.2 Calibration and biasing
If you observe the intensity signal on the oscilloscope in x-y mode, you should see something like the
plots shown in Figure 6: a flat line that breaks into a sin2 function at a certain x-value (whether it
starts upward or downward depends on the mode you choose). The flat line is the cantilever out of
contact, and the oscillating section is the cantilever bending, after making contact with the sample.
For the types of noise measurement that we will do, the signal needs to be at the maximum-
slope position along the output curve when it’s not in contact with the surface. If necessary, use
the offset on your voltage amplifier to position the sin2 so that it is centered around zero. Then,
set the out-of-contact bias point by moving the position of the laser focus on the fingers until the
flat section of your force spec. curve is approximately at zero volts, halfway between the maximum
and minimum, as in Figure 6(c).
To relate the mode intensity output to a physical deflection, we can take advantage of the
fact that a mode’s brightness goes from fully bright to fully dim as the fingers deflect through a
distance of λ/4. This way, by relating this displacement to the amplitude of the sin2 curve, you
can determine the cantilever sensitivity in nm/V.
You will also need to multiply the calibration by a correction factor to account for the location
of the diffraction fingers with respect to the tip of the cantilever (you can assume that the deflected
shape of the cantilever fits a second-order polynomial).

5
 (a) Bias too high. (b) Bias too low. (c) Bias set to optimal range.

Figure 6: Proper setting of the bias point for the measurements we’ll make in this lab.

4 Helpful References
1. Basic Operating Principles of AFM.

a. A website with a basic description:

http://www.weizmann.ac.il/Chemical_Research_Support/surflab/peter/afmworks
b. One with some more detail:
http://saf.chem.ox.ac.uk/Instruments/AFM/SPMoptprin.html
c. If these really stimulate your interest, this is a more comprehensive site on Scanning Probe
Microscopy (SPM), of which AFM is a subset:
http://www.mobot.org/jwcross/spm/

2. The paper that started it all.

G. Binnig, C. F. Quate, Ch. Gerber, “Atomic Force Microscope” Physical Review Letters
56(3):930-933, 1986.

3. Using interferometric ID fingers for position detection.

a. The original paper:

S. R. Manalis, S. C. Minne, A. Atalar, C. F. Quate, “Interdigitatal cantilevers for atomic
force microscopy,” Applied Physics Letters 69(25):3944-3946, 1996.
b. A more thorough treatment:
G. G. Yaralioglu, S. R. Manalis, A. Atalar, C. F. Quate, “Analysis and design of an in-
terdigital cantilever as a displacement sensor,” Journal of Applied Physics, 83(12):7405-
7415, 1998.

4. A nice review of using the AFM in boiology.

D. Fotiadis, S. Scheuring, S. A. Müller, A. Engel, D. J. Müller, “Imaging and manipulation
of biological structures with the AFM,” Micron, 33(4):385-397, 2002.

6
 BE.309: Biological Instrumentation and Measurement Laboratory GEM4 Summer Schoo

The teaching AFM: Part 2

Imaging with the AFM

1 Objectives
1. Learn to set up and prepare the AFM for imaging.

2. Image several different samples with the AFM.

3. Use the AFM to measure physical dimensions of imaged features.

4. Use the AFM to measure the elastic modulus and surface adhesion force of a sample.

2 The AFM Scan Control Software

The software that interfaces with the AFM is an application that runs in matlab. It is launched
by typing ‘scannergui’ in the matlab command window. Its main function is to systematically
scan the probe tip back-and-forth across the sample, recording the cantilever deflection information
at each point, line by line, and assembling that data into an image. Figure 1 shows a screen capture
of the scanner control window, and an overview of its operation is provided below.

2.1 Overview of Controls

Many of these are self-explanatory, such as the start imaging and stop buttons, as well as the
image area in the lower right, which displays the image currently being scanned. Some notes are
given below on features that are not immediately obvious.
To begin with, it’s easiest to simply use the default settings on all these controls, and to
experiment with changing them as you become more familiar with the tool.

Scan Parameters - The Scan size sets the length and width of the image in nanometers (always
a square shape), but its accuracy depends on having the correct value for Scan sensitivity
(which should already be set for you, but may require calibration). The Scan frequency (lines
per second) sets the speed of the tip across the surface, and together with the Number of lines
affects the amount of detail you will see in the image. Setting the Y-scan direction tells the
scanner whether to start at the top or the bottom of an image, and the trace/retrace selector
determines whether each line is recorded as the tip scans to the left or to the right.

Scope View - As the tip scans back and forth, this plots the tip deflection data for each line.
Useful for quantitative feature height measurements.

Scanner Waveforms - Shows the voltage waveforms driving the piezo scanner, for each scan line
that is taken. Helpful for knowing where in the image the current scan line is located, and
the output level of the waveforms driving the scanner.

Z-mod Controls - These are only active during a z-mod scan, and have no effect when taking an
image. For more on this mode, see Section 2.4 below.

1
Figure 1: The Scanner GUI window. The AFM is scanning a 12 × 12µm area, at a rate of one line
per second, and is currently near the bottom of the image.

2
2.2 Cantilever probes for imaging
The probes we will use for imaging are shown in Figure 2 with
relevant dimensions. The central beam has a tip at its end,
which scans the surface. The shorter side beams to either side
have no tips and remain out of contact. The side beams provide
a reference against which the deflection of the central beam is
measured; the ID grating on either side may be used. When
calibrating the detector output to relate voltage to tip deflec-
tion, remember to include a correction factor to account for
the ID finger position far away from the tip.

2.3 Image Mode Operation

This is the primary operating regime of the AFM, and provides
Figure 2: Plan view of the imaging can-
a continuous display of the surface being scanned, as the probe tilever geometry. The central (imaging)
is gradually rastered up and down the image area. To use this beam dimensions are length L = 400µm,
mode, the switch on the back of the AFM must be flipped and width b = 60µm. The finger grat-
upward. It is important to remember that the maximum scan ings begin 117µm from the base and end
200µm from it.
area is only abut 15µm square, and adjusting the position of
the sample under the tip requires only the smallest movements
of the stage micro-positioners. Also keep in mind that there is
a delay after start imaging is pressed and before the scan begins, as the actuator drive signals
are buffered to the I/O hardware.

2.4 Z-mod (force spectroscopy) Operation

In this mode, the piezo moves the sample only along the z-axis – i.e. straight up and down (hence
z-mod, short for z-modulation). To use this mode, the switch on the back of the AFM must be
flipped downward. Besides being helpful during calibration and biasing of the readout laser, this
mode is used to perform force spectroscopy experiments, in which tip-sample forces can be measured
as the tip comes into and out of contact with the sample.
(Note that the red stop button is also used to stop a z-mod scan).

3
3 Lab Procedures
3.1 Sample Loading and Positioning
Correctly mounting a sample in the AFM is a key part of obtaining quality images. Our samples
are always mounted on disks, which are magnetically held to the piezo actuator offset post. The
AFM can image only a small area near the center of the opening in the metal cantilever holder, so
be sure that the area of interest for imaging ends up there.

(a) A photo of the underside of the can- (b) A close-up view of the opening into
tilever mount, with the sample disk which the sample rises, showing the can-
lowered for changing samples. tilever die and sample disk.

When changing or inserting a sample disk, the 3-axis stage must be lowered far enough for
the disk to clear the bottom opening of the cantilever mount, as shown in the figures above. This
requires a large travel distance, so exercise caution when bringing the sample back up to the
cantilever, and take care not to damage the probe.
In addition, as you change samples, it is critical to reposition the offset post as nearly centered
as possible on the actuator disk, to ensure true horizontal motion in the x-y plane (Centering the
sample disk at the top of the offset post is not critical; rather, it’s the position of the bottom end
of the post on the piezo scanner disk. For instance, in figure 3(b) above the sample disk is
visibly off-center).
Once the sample is in proximity to the cantilever, follow the procedures that you learned in the
previous lab module to align the laser, engage the tip with the surface, and view a force curve.

3.2 Imaging
The general approach to imaging is to (1) set the overall output signal range and offset while in
the z-mod regime, (2) stop the z-mod scan and bring the probe into contact with the surface, and
(3) carefully adjust the cantilever deflection to give the desired bias point, and (4) start the image
scan.
Correct biasing is key to obtaining good images. If you are not yet comfortable with choosing
and setting an appropriate bias point for the cantilever’s optical readout, it is worth reviewing that
material from the previous module.
For imaging, the bias does not necessarily need to be set at maximum sensitivity when out of
contact. Rather, the signal should be at maximum slope (in the middle of its travel range) when
the probe is engaged, and exerting a small amount of force on the surface. Remember that the
more force the probe applies to the surface, the more wear and damage to the probe and surface
can result.

4
 The samples available for imaging are calibration gratings and E. coli bacteria on a silicon
dioxide surface. Unlike commercial instruments, our AFMs do not have an integrated optical
viewing system, so when positioning the sample it is difficult to determine exactly what spot on
the sample the probe tip will scan. Use the stereo-microscope (at moderate magnification) and a
fiber-light to observe the tip and position it as well as you can. Temporarily turning off the sensing
laser will make it easier to see. You can also move the sample “on the fly”, as you’re imaging, but
it takes some practice to make small enough stage movements. You will also likely have to readjust
the bias point (also fine to do on the fly).

Measuring Image Dimensions

When it’s necessary to determine the size of imaged features, the Scan size control is your most
handy reference. This gives the size of the image square, which can be quickly related to features
in the image. If needed, the scan size can be changed to make imaged features fit in the image
window as desired.
NOTE: the Scan sensitivity is pre-set to an approximate value, but if you would like more precise
lateral feature measurements, it’s worth verifying the calibration of the Scan sensitivity. This can
be done by scanning a reference with precisely known feature sizes (such as a calibration grating),
and adjusting the Scan sensitivity until the actual image size matches the Scan size setting.
Finally, for feature height (z-axis) measurements, they can be made by observing the Scope
View of the AFM software. The waveform displayed as the AFM scans over a feature shows
the feature height dimensions as voltages — the calibration that you’ve done to these voltages to
nanometers of tip displacement will give you feature sizes.

3.3 Elastic Modulus Measurements

[For these measurements, we’ll use the shortest and stiffest cantilevers available to us, which will
give the best signal. These have very similar geometry to the long cantilevers in Figure 2, but have
a length of 250µm, and a width of 50µm. The fingers begin 43µm from the base and end 125µm
from it. Ask your lab instructor to provide you with a short cantilever when you are ready.]
As you know, some of the most useful applications of AFMs in biology take advantage of their
ability to measure very small forces. We’ll use this capability to measure the elastic moduli of
some soft samples, to simulate mapping cell wall elastic properties, similarly to the 2003 paper by
Touhami et al. in Langmuir.
As seen in this paper, samples with different elastic moduli change the slope of the in-contact
portion of the force curve, when using the optical lever sensor. For our non-linear ID sensor, the
equivalent of the changing slope is a changing period for the sin2 function. Just as softer samples
cause lower slope with the optical lever, softer samples give the output function of the ID sensor a
longer period, with greater spacing between the peaks (see figure 4(a) below).
The approach for measuring modulus is to first take a force curve on a hard reference sample,
considered to have infinite hardness. We will use a bare silicon nitride surface. This allows us to
determine how the x-axis signal corresponds to stage movements.
You’ll want to bias this measurement similarly to measuring noise — the output should be at
the middle of the output range when out of contact (See Fig. 4(b)).
After your measurement of the hard sample, switch to the more compliant PDMS elastomer
samples, and run force curves on them.
Make sure to capture the plots of the force curves for later analysis, described in Section ??.
To get good force curves:

5
 (a) Force curves for samples of varying hard- (b) Preferred biasing and calibration on a hard
ness. The green (solid) curve is for a hard surface for measuring elastic modulus, and
sample, the red (dashes) curve is softer, and corresponding physical stage movement
blue (dots) curve is softest.

1. Don’t change the biasing or laser position between samples – if you do, the force curves you
get can’t be compared one to another.

2. Careful initial biasing at the middle of output range is worth it — this will make a big
difference in ease of data analysis.

3. After you’ve brought each sample into contact and are satisfied with the Z-modulation range,
run the scan at slow speed (e.g. 0.5Hz) for the cleanest force curves.

Data Analysis
According to Touhami et al., the depth δ of an indentation made by a conical tip (approximately
true for ours) is related to the applied force F by
µ ¶
2 E
F = tan α δ2 ,
π 1 − ν2

where α is the half-angle of the conical tip, and E and ν are the elastic (Young’s) modulus and the
Poisson’s ratio of the substrate material, respectively.
Substituting in appropriate values for α (35.3◦ ) and ν (0.25), we are left with

F = 0.60Eδ 2 ,

in which we need only the force and indentation δ values to calculate modulus.
The force F is calculated by treating the cantilever as a Hookian spring, which obeys the law
F = kz, where k is the spring constant and z the tip deflection. For the 250µm long cantilever,
assume a spring constant of 0.118 N/m.
Finally, all that remains is to calculate indentation depths for the soft materials from the
difference in the period of the sin2 output between their force curves and the one for the hard
sample. Corresponding forces are derived from the cantilever deflection. Don’t forget at all points
to include the factor that relates cantilever tip deflection to finger deflection.

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 BE.309: Biological Instrumentation and Measurement Laboratory GEM4 Summer School

The teaching AFM: Part 3

Thermomechanical noise and Boltzmann’s constant

1 Objectives
1. Use your knowledge of the AFM system and associated instrumentation to record the vibra-
tional noise spectrum of a cantilever probe.

2. Estimate the value of Boltzmann’s Constant kB from the cantilever vibrational spectrum.

2 Background
2.1 Theory: Thermomechanical Noise in Microcantilevers
For simplicity of analysis, we model the cantilever as a harmonic oscillator with one degree of
freedom, similar to a mass on a spring, as discussed in lecture. According to the Equipartition
Theorem, the thermal energy present in the system is simply related to the cantilever fluctuations
as follows:
1 1 D E
kB T = k ∆z 2 ,
2 2
where h∆z 2 i is the mean-square deflection of the cantilever, T is the absolute temperature, k is the
cantilever spring constant, and kB is Boltzmann’s Constant (yes, this notation can be confusing —
take care to keep these ks straight).
The characteristic transfer function of the second-order resonant system has the form
s v
u
4kB T u 1
|G(ω)| = u³ ´ ,
Qkω0 t 1 − ω 2 2
+ 1 ω2
ω02 Q2 ω02

in whichω0 and Q are the (angular) resonant

frequency and quality factor, respectively. At
low frequencies, (ω ¿ ω0 ) this expression yields
what’s called the “thermomechanical noise limit”
(see Figure 1 for an illustration):
s
4kB T
δ= .
Qkω0 Figure 1: A data plot of a cantilever’s noise spec-
trum, with an ideal transfer function G(ω) fit on top
These relations suggest several possible ap- (dark line). Note that G(ω) is flat at low frequen-
proaches that can be taken for determining kB , cies, at the thermomechanical limit, as indicated. In
for which you will need the values of several pa- contrast, real data has more 1/f -type noise present
rameters. These include (1) the quality factor Q at lower frequencies (see Section 3.2).
and resonant frequency ω0 of the resonator (2) the
cantilever’s mean-square deflection h∆z 2 i, and (3)
its spring constant k.

1
 1. The quality factor and resonant frequency can be obtained from taking the noise PSD
of the vibrating cantilever. If your intuitive sense for them is good, you can estimate these
quantities directly from the plot, or determine them more precisely by fitting an ideal transfer
function to the noise data, and extracting the fitting parameters (more on this in Section ??).

2. The mean-square deflection is readily available from either time-domain or PSD data of
the cantilever thermal noise. Recall that these are related through Parseval’s theorem as
follows: D E Z ∞
2
∆z = S(ω)dω ,
0
where S(ω) is the PSD function of ∆z. By now, you know enough MATLAB spectral analysis
techniques to make these measurements.

3. The spring constant (stiffness) can be analytically calculated from geometrical parameters
in two ways (see Section 2.2 for cantilever dimensions). From basic mechanical beam-bending
analysis of a rectangular cantilever the stiffness k can be expressed as

Ebh3
k= ,
4L3
in which E is the elastic (Young’s) modulus of the beam material, and L, b and h are the
length, width, and thickness of the beam, respectively (b is used for the width to avoid
confusion with angular frequency ω).
This method, however, does not always yield accurate results — can you suggest why? An-
other analytical model for the spring constant was devised by Sader and coworkers1 , and it
relies on measuring the cantilever’s resonant frequency:
2
k = 0.2427ρc hbLωvac ,

where ρc is the mass density of the material, L, b, and h are the same geometrical parameters
as above, and ωvac is the cantilever’s resonant frequency in vacuum. For the purposes of these
calculations, you can assume that the resonant frequency that you will measure in air ωair
is 2% lower than ωvac . (remember the factor of 2π when interconverting between ω and f in
your equations).
Suitable material parameters to use for the low-stress silicon nitride (Six N), out of which
these cantilevers are made are ρc = 3400kg/m3 , and E = 250GPa. As mentioned before, for
complete cantilever dimensions, see Figure 2 in Section 2.2.

1
J. E. Sader, et al, “Calibration of rectangular atomic force microscope cantilevers,” Review of Scientific Instru-
ments, 70(10):3967-3969, 1999.

2
2.2 Cantilevers for Thermal Noise Measurements
The probes you will use for this lab are different than what
you’ve used for imaging and force measurements. Their plan
view is shown in Figure 2.
For noise measurement purposes, we’d like a clean vibra-
tional noise spectrum, which is best achieved using a matched
pair of identical cantilevers. The configuration with a central
long beam and reference side-beams has extra resonance peaks
in the spectrum that make it harder to interpret. With the
geometry in Figure 2 the beams have identical spectra which
overlap and reinforce each other. Using a pair of identical
beams also helps to minimize any common drift effects from
air movements or thermal gradients.
There are two sizes of cantilever pairs available. Choose Figure 2: Plan view showing the
either size to make your measurement. For the long de- geometry of a differential cantilever
vices, L = 350µm and the finger grating starts 140µm and pair. We assume that because the
ends 250µm from the cantilever base. For the short devices, beams are fabricated so close to-
L = 275µm, and the fingers are between 93µm and 175µm from gether, their material properties and
dimensions are identical.
the base. The width and thickness of all of the cantilevers is
b = 50µm and h = 0.8µm, respectively.

3 Lab Procedures
3.1 Alignment, Calibration, Biasing
By now you’re familiar with aligning, calibrating, and biasing. The major difference in this case is
performing the z-modulation scan for calibration.
Because this device is a pair of identical cantilevers, simply bringing it down to a surface will
deflect both beams equally. A z-mod scan will show approximately zero deflection of one beam
relative to the other. Instead, we want to bend only one of the beams, while keeping the other
unbent. To do this, you’ll have a sample with a sharp step edge. The goal is to position the
cantilever pair above this edge such that one of the beams will be on the surface, and the other
will hang in free space. A z-mod scan should then deflect only one of the beams, giving us the
calibration curve we want.
A few additional remarks to guide you:

– Remember to flip the “imaging/z-mod” mode switch on the back of the AFM to the proper
position.

– Reflections of the laser from the edge of the substrate can interfere with the diffractive modes.
If this is the case, try repositioning the sample edge, perhaps using only the corner to bend
one of the cantilevers, until the sin2 shape improves.

– As you’ve done several times, bias the force curve for maximum sensitivity when out of
contact — the flat portion of the curve should be placed midway between the maximum and
minimum.

3
3.2 Recording Thermomechanical Noise Spectra
Once you’re happy with your calibration and biasing, withdraw the lever’s tip from the surface,
making sure that the bias point stays where you set it. Use the LabVIEW ”Spectrum Analyzer”
to record the thermal noise signal coming from the freely vibrating cantilever. Once you are happy
with how the spectrum looks, save it to a .txt file of your choise.
You only need to measure the noise spectrum down to about 50-100Hz. Below this frequency,
1/f -type or “pink” noise dominates. You are welcome to measure this if you are interested, but
it is of limited use for determining kB . For very low-frequency measurements, anti-aliasing and
proper input coupling becomes very important. If you are interested in this, your lab instructor
can provide guidance.
Some guidelines for getting a good noise spectrum:
• Choose a sampling frequency at least 2× higher than the highest frequency of interest, or
about 10× higher than the first resonance peak of the cantilever.
• Use AC coupling on your voltage amplifier, and use a gain of 100-1000.
• If necessary, add a low-pass anti-aliasing filter (recall Module 1) at an appropriate frequency
to eliminate high-frequency components being “folded” over into the frequency region of
interest.
• Recall from the previous lab that, if you prefer, you can also measure the time-domain signal
directly, and later calculate its PSD in MATLAB. You can decide which technique you prefer.

3.3 Data Analysis with MATLAB

Once you bring your saved PSD data into MATLAB ([Fvec,PSDvec]=load(‘filename’) is the syntax
you want), you can manipulate it as you wish. To fit the second-order transfer function G(ω) to
the noise data, we’ll use the lsqcurvefit routine from MATLAB’s optimization toolbox, which
does a least-squares curve fit, as you may have guessed. We’re aiming to do something similar to
what you see in Figure 1, where an ideal function is overlaid on real noise data.
To make the fit converge easily, we’ll separate the nonlinear f0 and Q parameters from the
linear scaling factor. When doing the fitting, it is helpful not to use the whole frequency range
of your data. Instead, crop your PSD data to a suitable range around the resonant peak — the
vectors xdata and ydata used below are cropped PSD frequency and magnitude data, respectively,
extracted from Fvec and PSDvec.
First, you’ll need a MATLAB function transfunc to generate the unscaled transfer function
(i.e. the thermomechanical noise scaling factor is 1 here – refer to the equations on page 1):
1
|G(ω)| = r³ ´2 .
ω2 1 ω2
1− ω02
+ Q2 ω02

The function takes the f0 (note that this is real frequency in Hz, and not angular frequency in
rad/sec) and Q parameters as input, with a vector of frequencies, and outputs corresponding PSD
magnitude data:
function [output]= transfunc(params,xdata)
% params [f_0 Q]
x=xdata/params(1); % x-matrix to contain freq. points normalized to f/f_0
output=sqrt(1./((1-x.^2).^2 + (x/params(2)).^2));

4
 Then, create a function scaling to do the linear scaling, and calculate the thermomechanical
noise level (the left-divide operation actually does a least-squares fit):

function [y]=scaling(params,xdata,ydata)
unscaled=transfunc(params,xdata);
A=unscaled\ydata; % note the left-divide here!
y=unscaled*A;

Finally, use the lsqcurvefit routine, supplying an appropriate initial guess for f0 and Q:

options=optimset(‘TolFun’,1e-50,‘tolX’,1e-30);
p=lsqcurvefit(‘scaling’,[f_guess Q_guess], xdata, ydata, [ ], [ ], options, ydata);

This will return the best f0 (again in Hz, not in rad/sec) and Q parameters after the fit as a two-
element vector p. Now you just need the scaling pre-factor, which you can get by left-dividing the
full-range fit function by the PSD magnitude data (the left-divide again gives you a least-squares
fit “for free”):

A=(transfunc(p,full_xdata))\full_ydata;

Here full xdata and full ydata are the full-range frequency and magnitude PSD vectors,
rather than just the cropped sections used for the fit algorithm. You can now see how well the fit
worked, by plotting it on top of the original PSD data:

Gfit = A*transfunc(p,full_xdata); loglog (full_xdata, Gfit);