Characterization of Attocube Nanopositioners for Vibration 

by Dimitri Alexson and Doran D. Smith 

ARL-TR-5315 September 2010 

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ARL-TR-5315 September 2010 

Characterization of Attocube Nanopositioners for Vibration 

Dimitri Alexson and Doran D. Smith 

Sensors and Electron Devices Directorate, ARL 

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14. ABSTRACT 

This report describes a series of vibration tests performed on nanopositioners manufactured by the company Attocube. Our 

goal was to determine if a cantilever mounted in a holder on top of an XYZ nanopositioner vibrated more than the Brownian 

motion of the cantilever at 4 K. We know that when the cantilever is mounted on a solid mechanical base that it does not 

vibrate more than the appropriate Brownian motion. The concern about the Attocube nanopositioner is that since it possesses 

mechanical residences in the 1-kHz area, it might vibrate too much and, hence, excite the cantilever. We find that the 

Brownian motion of the cantilever mounted on the XYZ nanopositioner is what it should be for the temperature; however, the 

frequency noise is greater than the Brownian motion would predict. At this time, this result is not understood. 

15. SUBJECT TERMS

Vibration, nanopositioner, attocube, cantilever, thermal 

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Doran D. Smith 

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iii 

Contents 

List of Figures iv

1. Introduction 1

2. Methods, Assumptions, and Procedures 3

3. Results and Discussion—Stiff Cantilever 8

4. Results and Discussion—Soft Cantilever 10

5. Conclusions 13

6. References 14

List of Symbols, Abbreviations, and Acronyms 15

Distribution List 16 
iv 

List of Figures 

Figure 1. Large amplitude IF calibration: four items are shown here: (1) raw data points are 

the red filled circles, (2) a straight line fit to the first three data points is the straight red 

line, (3) the sinusoidal fit to the data is green, and (4) the slope of the sinusoidal fit at 

zero is a green straight line. .......................................................................................................5

Figure 2. Running standard deviation of cantilever motion; the standard deviation squared of 

x(t) is equivalent to the measured value of <x(t)2

>. ..................................................................6

Figure 3. Running mean of cantilever motion; the two lines correspond to the running mean 

of the x and y components of the software lock-in output. ........................................................7

Figure 4. The puff ball; each chunk of data is plotted as a point at its corresponding x and y

values. ........................................................................................................................................7

Figure 5. Stiff cantilever <x(t)2> vs. temperature. The failure of the data to line on a 

straight line indicates the cantilever is being base driven. The straight line is a best fit to 

the data. ......................................................................................................................................8

Figure 6. FT of 10 s of stiff cantilever x(t). The Brownian motion peak occurs at 0 Hz and 

the baseline noise is due to IF shot noise. ..................................................................................9

Figure 7. Average FT of all stiff cantilever data collected. The area under the peak 

corresponds to <x(t)2> for the cantilever which sits on a noise baseline. .................................9

Figure 8. Soft cantilever <x(t)2> vs. temperature. Since the data all fall on a straight line 

down to the lowest temperature investigated, we know it is not significantly base driven 

and all of <x(t)2> can be attributed to Brownian motion. The straight line is a fit to the 

data and gives a spring constant of 0.000103 N/m. .................................................................10

Figure 9. Soft cantilever frequency noise vs. temperature. Data plotted as squares is for a 

cantilever driven close loop. The circles are the undriven cantilever measured during ring 

down. ........................................................................................................................................12
1 

1. Introduction 

The nano-magnetic resonance imaging (MRI) laboratory of the U.S. Army Research Laboratory 

(ARL) performs mechanically detected magnetic resonance at cryogenic temperatures, typically 

5 K. The system used to do this consists of a cryogenic probe immersed in liquid helium (He) 

and room-temperature electronic instrumentation. The major components of the probe consist of 

a vacuum jacket surrounding a probe head. The probe head has four components: (1) a radio 

frequency (RF) coil, (2) a cantilever with a tiny ferromagnet mounted on it, (3) an optical fiber, 

and (4) a sample mounted on a sample stage. The RF coil generates the RF magnetic field used 

to excite magnetic resonance in the sample. Magnetic resonance in the sample then excites the 

cantilever by interacting with the ferromagnet mounted on it. The motion of the cantilever is 

detected by the optical fiber. 

Historically, we have always mounted the sample on a three-dimensional (3-D) nanopositioner 

and the cantilever is mounted on a more rigid mechanical base. However, collaborators at 

Cornell University plan to build a probe that requires the cantilever to be mounted on the 

nanopositioner and the sample on the more rigid mechanical base. Since, at this time, they do not 

have a probe with sufficient flexibility to allow them to mount the cantilever on the 

nanopositioner and we do, ARL is making vibration measurements for Cornell. 

The 3-D nanopositioner used in our probe head is the XYZ 101 series nanopositioner by 

Attocube. Specifically, the nanopositioner is composed of a stack of three individual elements: 

one moving in the z direction, the ANPz101; one moving in the y direction, ANPx101; and one 

moving in the x direction, ANPx101. On the top of the 3-D nanopositioner is a sample stage. 

ARL designed a new sample stage to hold a cantilever instead of a sample. Cornell machined the 

new sample stage and we installed it in place of our sample stage. Without any further 

modification of our probe head, we were able to use our optical fiber to measure the motion of 

the cantilever mounted on top of the 3-D nanopositioner stack. 

Two approaches were used to measure the Brownian motion of the cantilever. For the first, the 

instantaneous position of the undriven cantilever versus time was recorded. From this, the root 

mean square (RMS) value of its position was calculated. The RMS value of the cantilever 

position should equal the cantilever’s Brownian motion for its base temperature. The second 

technique involved measuring the frequency noise of a driven cantilever. Hammel (1) and 

Marohn (2) have derived a theory that relates the driven cantilever’s frequency noise to the 

Brownian motion. Experimental results of the Marohn group has shown that in some situations 

driven cantilevers have more frequency noise than is predicted by their Brownian motion alone 

(3). 
2 

The Brownian motion of a cantilever can be very simply predicted using the equal partition 

theorem. The equal partition theorem states that 

kc x   kBT 2 (1) 

where kc is the cantilever spring constant, <x2

> is the RMS value of the instantaneous position, 

kB is Boltzmann’s constant, and T is the absolute temperature. The temperature can be measured 

with an independent temperature sensor, kB is a known constant, and <x2> is measured; 

therefore, the only unknown is kc. The cantilever spring constant can be determined by 

measuring the RMS value of the cantilever motion at several different temperatures. If <x2

> is 

plotted versus T, a straight line with a zero intercept should result. If the line bends toward the 

<x2> axis as T decreases, it means the cantilever is being base driven by vibrations. Base driving 

a cantilever means shaking the base of the cantilever, causing the tip to move, as opposed to 

moving the cantilever by applying forces directly to the tip of the cantilever. This approach 

serves as the basis for the first technique of measuring the Brownian motion of the cantilever. If 

the plot of <x2

> versus T is a straight line, then a valid value for the Brownian motion of the 

cantilever has been obtained, unpolluted by vibrations. 

The second technique for measuring Brownian motion involved measuring the frequency noise 

of a driven cantilever. A driven cantilever is a cantilever that is excited to a constant oscillation 

amplitude at its natural resonance frequency. We base drove the cantilever by exciting the X 

stage of the Attocube nanopositioner stack. 

To determine the frequency and frequency noise of the cantilever, the position, x, of the 

cantilever is recorded versus time x(t). Then x(t) is extracted from the recorded data with a 

software lock-in implemented in LabView*. The software lock-in is implemented by multiplying 

x(t) against a sine wave, sin(2*π*f0*t) and a cosine wave, cos(2* π *f0*t), where the local 

oscillator f0 is near the cantilever frequency fc. The result of each product results in sum and 

difference terms: 

 sin(2* π *fc*t)* sin(2* π *f0*t) = 1⁄2(cos(2* π *(f0-fc)*t)- cos(2* π *(f0+fc)*t)) (2)

 sin(2* π *fc*t)* cos(2* π *f0*t) = 1⁄2(sin(2* π *(f0-fc)*t)+sin(2* π *(f0+fc)*t)) (3)

The sum terms are each about 2*fc and the difference terms are slowly varying terms with a 

frequency of (f0-fc). A low pass digital filter is then used to filter the sum and difference terms. 

The 2*fc product is filtered out and only the difference terms are retained. The difference term 

resulting from the sine multiplication is called the X channel of the software lock-in; the cosine 

is the Y channel. For each x(t) point we then calculate the arctangent of the ratio of the sine and 

cosine terms, resulting in a phase angle. This phase angle is the phase of the difference terms. 

We then unwrap the phase, i.e., each time the phase jumps from +180° to –180°, we add 360° to 

it so that a continuous line is formed. The unwrapped phase angle terms are then divided into 

 *LabVIEW is a trademark of National Instruments Corporation. 
3 

small segments termed “chunks” with about two chunks per fc cycle. The slope of this line is 

determined for each chunk and the slope of the phase of a signal is its frequency. This results in a 

measure of the frequency of the difference terms versus time. Adding fc to these terms provides a 

measurement of the original fc versus time. From the fc time record, we can perform any desired 

statistical analysis. For example, forming the power spectral density (PSD) of the frequency of 

the difference terms gives a PSD of the frequency deviations (frequency noise) of the cantilever 

around fc. 

The Marohn group has reported observing a cantilever with 4 K Brownian motion to have 77 K 

frequency noise (4). Therefore, in this work, we measured both and used those measurements to 

characterize how well the Attocube nanopositioners performed as a cantilever support. 

2. Methods, Assumptions, and Procedures 

To characterize the nanopositioners, we used an existing magnetic resonance force microscopy 

(MRFM) system custom built by ARL. The MRFM system consists of room-temperature 

electronics and a cryogenic probe (P5, for the fifth MRFM probe ARL has built) directly 

immersed in liquid He. The cryogenic part of the probe consists of a vacuum jacket surrounding 

a vibration-isolated probe head. A superconducting magnet can apply up to a 9 T magnetic field 

to the probe head. Since the magnetic field will create eddy current damping in the metallic body 

of the probe head when the probe vibrates, all experiments were preformed in a zero magnetic 

field to maximize the vibrations the nanopositioners experience. 

The total mass of the probe head is approximately 0.5 kg. It is vibration isolated by suspension 

using three beryllium-copper (BeCu) springs. In the interior of each of the springs is inserted a 

Teflon† rod. When the probe head vibrates on the springs, the rubbing of the Teflon rods creates 

just enough friction to help damp out the vibrations. ARL has preformed experiments 

demonstrating that using the Teflon does decrease the amount of vibration the probe head 

experiences. 

Probe head P5 was designed with the cantilever mounted on the probe head’s mechanical ground 

and the sample held by a sample stage mounted on a XYZ stack of Attocube nanopositioners. 

The regular sample stage was replaced with a custom cantilever holder designed by ARL and 

machined by Eric Moore of Cornell University. The cantilevers were held in such a position that 

the probe’s original optical fiber-based interferometer (IF) could be used to measure its position. 

Aligning the cantilever to the fiber by driving the nanopositioners in 3-D was very convenient. 

On the sample stage was mounted a silicon (Si) diode temperature sensor and a Nichrome wire 

heater with about 20 ohms resistance. By passing 100 mA at 2.0 V through the heater, it was 

 †Teflon is a registered trademark of E.I. DuPont de Nemours & Co., Inc. 
4 

possible to heat the sample stage to 40 K. The cantilever holder was thermally connected to the 

probe head’s mechanical ground via a strip of Cu foil clamped between the bottom of the 

cantilever holder and the top of the nanopositioners. The Cu foil is used to thermally ground the 

cantilever holder since the thermal resistance and thermal time constant of the nanopositioner 

stack is unknown and measuring it was beyond the scope of this study. Later it may be 

determined to be unnecessary. 

The probe head was then thermally connected to the probe vacuum jacket via three strips of Cu 

foil. These Cu foils are in parallel with the springs and couple in unwanted vibrations. The 

competing requirements of vibration isolation and removal of the heat generated by the RF must 

be balanced. The Cu foils were each 99.9999% pure, 25 μm thick, 1 cm wide, and 10 cm long 

between their clamped ends (7). We typically only use one Cu foil in parallel with the springs, 

but in this test we elected to use three to couple more vibrations into the probe head. One foil 

would not be enough to conduct out the 0.2 to 1.0 W Cornell wants dissipated from their probe 

head without unduly heating the probe head. A single annealed foil of the above dimensions has 

a measured by us to yield a temperature rise of 4.0 K for 200 mW with the cold end near 5 K. 

Two cantilevers were used to characterize the nanopositioners. The stiff cantilever was made of 

silicon nitride (SiN) (5) with a spring constant of 0.01 N/m. It was mass loaded with a 75-μm 

glass sphere resulting in a cantilever resonant frequency of 1045 Hz. The soft cantilever was 

made of crystalline Si (6) with a spring constant of 0.0001 N/m, was not mass loaded, and had a 

resonant frequency 2398 Hz. The stiff cantilever is more sensitive to vibrations for two reasons: 

(1) the stiffer a cantilever is, the easier it is to base drive; and (2) typically there are more 

environmental vibrations coupled into the probe head at lower frequencies, with a significant 

falloff above 3 kHz. 

To determine if the cantilever motion observed at 4 K was all Brownian motion or part base 

driving from probe head vibrations, we measured the Brownian motion as a function of 

temperature. The majority of thermal expansion of most material has occurred by the time it 

reaches the 4–20 K temperature range; however, not completely. As we vary the temperature 

from 5 to 20 K, it is necessary to retune the IF. The IF is tuned by varying the temperature of the 

IF’s laser diode (8), which changes its wavelength. This procedure is repeated at each 

temperature since the determination of the amplitude of the Brownian motion is dependent upon 

accurate calibration of the IF sensitivity. 

ARL determines the IF sensitivity by a calibration process called “Large Amplitude IF 

Calibration.” The calibration process involves base driving the cantilever from amplitudes small 

with respect to a fringe length, one half of the IF wavelength, to an amplitude larger than one 

fringe length in even steps. If the IF behaves ideally, the interference fringe should have a 

sinusoidal shape as the IF wavelength is varied or as the cantilever is driven to larger amplitudes. 

This means for small amplitudes (the amplitudes relevant for measuring Brownian motion) the 

response is indistinguishable from linear, but as the amplitude becomes a sizable fraction of a 
5 

fringe length the IF response begins to roll over, as seen in figure 1. We fitted a few points in the 

linear region to a straight line to obtain the IF sensitivity. We also fitted the entire amplitude 

response to a sine wave. We compared the slope of the fitted sine wave to the linear fit near zero. 

They should agree. This serves two purposes. First, it provides a check to determine if the 

mathematics used to derive IF sensitivity near zero was preformed correctly. Secondly, it 

determines how much distortion is in the IF sensitivity. We find the fringe shape is frequently 

close to sinusoidal, but not always. We believe that sometimes there are multiple optical paths 

interfering, possibly due to features on the cantilever’s surface. 

Figure 1. Large amplitude IF calibration: four items are shown here: (1) raw data points are the red 

filled circles, (2) a straight line fit to the first three data points is the straight red line, 

(3) the sinusoidal fit to the data is green, and (4) the slope of the sinusoidal fit at zero is a 

green straight line. 
6 

To determine an accurate value of the Brownian motion of the cantilever, we must determine the 

average value of <x(t)2

> accurately. Data collection must be long enough so that the measured 

value of <x(t)2

> has converged to its actual value. We measure the Brownian motion of our 

cantilevers without feedback, so it is a long, slow process. Each independent measurement of 

<x(t)2> requires about three times the damping time of the cantilever. Therefore, for a high Q 

cantilever, 20 min of data collection time can be required. Theoretically, to get <x(t)2

> to 

converge, it is the error in the mean that we must get to a sufficiently small value. However, in 

practice, we watch the time-dependent values of the standard deviation (figure 2) and mean of 

x(t) (figure 3). When the standard deviation becomes flat and the means of both the in-phase and 

out-of-phase channels converge to zero, we assume that we have collected data for long enough. 

For well-behaved cantilevers, i.e., not base driven, this assumption has been shown to result in 

plots of <x(t)2> versus T that were straight lines through zero. 

Figure 2. Running standard deviation of cantilever motion; the 

standard deviation squared of x(t) is equivalent to the 

measured value of <x(t)2

>. 
7 

Figure 3. Running mean of cantilever motion; the two lines correspond 

to the running mean of the x and y components of the software 

lock-in output. 

An additional check on the quality of the data is plotted in what is termed the “puff ball,” as 

shown in figure 4. Here, we plot every chunk of data as a point using its x and y values. For wellbehaved data, the puff ball should be a well-formed round circle without too many outlying 

points. If that is not the case, something undesired has occurred during the data collection. 

Figure 4. The puff ball; each chunk of data is plotted as a point at 

its corresponding x and y values. 

For this work, data were collected using a 16 bit analog-to-digital converter (ADC) at 100 kilosamples per second for 12 s at a time. All 12 s of data underwent analysis as previously 
8 

described and then the first two seconds were discarded since it took a little less than 2 s for all 

the digital filters in the software lock-in to achieve steady-state. We determined that 60–80 data 

collection cycles were required to obtain a reasonable convergence of the data. 

3. Results and Discussion—Stiff Cantilever 

Figure 5 shows <x(t)2> versus temperature for the stiff cantilever. Clearly, it is not a straight line 

through zero. A data point taken at 77 K (not shown) had a x(t) of 82 K, basically Brownian 

motion, while all the data from 5 to 22 K show the effect of vibrations. We conclude that the 

cantilever is base driven by vibrations, resulting in an effective temperature of about 30 K. 

Figure 5. Stiff cantilever <x(t)2> vs. temperature. The failure of the 

data to line on a straight line indicates the cantilever is being 

base driven. The straight line is a best fit to the data. 

This was the first time we used a stiff cantilever with the software lock-in. The amplitude of the 

Brownian motion of the stiff cantilever was 10 times less than that of the soft cantilevers we 

typically use. This resulted in x(t) being close to the noise floor. For what we believe to be 

equivalent bandwidths, we found the Stanford Research Instruments SRS830 lock-in did a better 

job of filtering out the noise than our software lock-in did. Previously, the SRS830 was the only 

lock-in we had used on stiff cantilevers. Without more study, we cannot say what caused the 

difference. However, by adjusting the bandwidths, our software lock-in achieved the same 

signal-to-noise ratio (SNR) as the SRS830. We suspect the difference was due to different filter 

shapes, and hence, different effective bandwidths. 
9 

Because of this SNR problem, we preformed one additional check on the data. We Fourier 

transformed (FT) the output of the software lock-in. The FT for one 10 s data set in the vicinity 

of the cantilever resonance is shown in figure 6. This figure contains two features: a noisy 

baseline and a peak due to x(t). The average FT for 60 separate data collection cycles is shown in 

figure 7. The noisy baseline has converged and the x(t) peak is well defined. By fitting a straight 

line to the noise baseline, we subtracted the noise off the FT peak. The area under the FT peak is 

known to be equal to <x(t)2> if the source of x(t) is all Brownian motion (9). This should give us 

a better measure of the true value of <x(t)2> versus temperature. Performing the FT analysis on 

the data did not get it to lie on a straight line. Again, we conclude the cantilever is being base 

driven by environmental vibrations. 

Figure 6. FT of 10 s of stiff cantilever x(t). The Brownian motion peak occurs at 

0 Hz and the baseline noise is due to IF shot noise. 

Figure 7. Average FT of 60 10-s data collections runs for the stiff cantilever. The 

area under the peak corresponds to <x(t)2

> for the cantilever which sits on a 

noise baseline. 
10 

4. Results and Discussion—Soft Cantilever 

The Cornell soft cantilevers are known to have a spring constant of ~0.0001 N/m. At a 

temperature of 4.2 K, this cantilever should have a Brownian motion value of 0.76 nm RMS. Our 

first measurement of the soft cantilevers at 5.2 and 22.5 K resulted in RMS values of x(t) of 1.81 

and 1.91 nm RMS, respectively. Thus, this cantilever also appears to be base driven. 

However, before concluding that the nanopositioners vibrate too much, we decided to turn down 

the IF laser optical power. We had typically been operating the IF with ~400 nW onto the 

cantilever, so we turned it down to ~200 nW. This corrected the problem. Figure 8 shows 

<x(t)2> versus temperature for the soft cantilever. Since all the data fall on a straight line down 

to the lowest temperature investigated, we know the cantilever is not significantly base driven or 

driven by the IF optical power. When contrasting this to the data plotted in figure 5, we 

determined that all of <x(t)2

> can be attributed to Brownian motion. The straight line is a fit to 

the data, and using the Equipartition theorem, gives a spring constant of 0.000103 N/m. We 

conclude that we have been using too much IF power for the soft Cornell cantilevers. 

Figure 8. Soft cantilever <x(t)2> vs. temperature. Since the data all fall on a 

straight line down to the lowest temperature investigated, we know it 

is not significantly base driven and all of <x(t)2> can be attributed to 

Brownian motion. The straight line is a fit to the data and gives a 

spring constant of 0.000103 N/m. 
11 

Figure 8 was generated without performing an FT analysis of the software lock-in output. We 

were able to skip this step since the soft spring constant is 100 times softer than the stiff spring 

constant. Resulting in the soft cantilever’s Brownian motion being 10 times greater than the stiff 

cantilever. With the greater motion, the IF noise floor becomes insignificant. 

Eric Moore and John Marohn (3, 4) of Cornell University have observed that cantilevers 

displaying 4 K Brownian motion can have significantly higher frequency noise than would be 

predicted by Brownian motion alone. Therefore, since the soft cantilever achieves Brownian 

motion for <x(t)2> at the lower IF power, we preformed frequency noise measurements on it to 

test this observation. Using the Marohn work, we extracted an effective temperature from the 

cantilever. From Marohn’s white paper (2), we used the following equation: 

0

2 2

2

( )

 



k

k T P f

xrms

therm B

x  . (2) 

The Q of the cantilever was measured at each temperature and found to decrease as the 

temperature rises. 

Some of our initial measurements indicated temperatures as high as 120 K. The software 

package Extract used to analyze the data has been extensively tested and was believed to be 

working correctly (10). However, we decided to perform an additional independent test of its 

performance. Following Marohn (11), we recorded the Brownian motion history x(t) of a 5.2 K 

undriven cantilever to a disk. Using Extract, we read x(t) back into the program, added to it a 

pure sine wave to simulate the driven cantilever, and then had Extract analyze the data as if it 

had just acquired actual driven cantilever data. Extract’s analysis returned an effective cantilever 

temperature of 4.8 K. We believe 4.8 K is close enough to 5.2 K to conclude that Extract is 

working correctly. The attractiveness of the approach of recording the Brownian motion to disk 

instead of attempting to simulate it is that it incorporates the actual damping time of the 

cantilever into the Brownian motion and the correct Brownian motion amplitude for the 

cantilever’s temperature. This avoids having to determine whether or not simulated data 

accurately reflects the very small bandwidth that a high Q cantilever represents. 

After once again verifying that Extract is working correctly, we obtained frequency noise data 

with effective temperatures significantly above that the Brownian motion of the cantilever. We 

collected data for the driven cantilever three ways: (1) via a closed loop using the SC-Solutions 

frequency controller, the scFDC; (2) via an open loop, where the cantilever was driven with the 

Stanford Research Instruments signal generator SRS345; and (3) by turning off the amplitude 

drive, grounding the Attocube nanopositioner and recording the data as the cantilever rang down. 

Approach three allows us to measure the frequency noise of the cantilever with no drive 

electronics attached to it. However, attached to the top of the probe is still ~6 m of unshielded 

instrumentation cable that is grounded back in the instrumentation rack. This grounded-at-one- 
12 

end cable acts as a receiving antenna for environmental noise, of course. Modification of the 

instrumentation and further testing will be required to determine the effect of grounding the 

wires to the piezoelectric drive elements at the top of the probe. 

Figure 9 summarizes our results. It shows the apparent temperature of the driven cantilever 

extracted using the Marohn approach versus the actual base temperature of the cantilever for the 

driven and undriven cantilever. Two points are evident from this figure: (1) our closed loop drive 

electronics is contributing significant frequency noise (we do not yet know if the open loop 

driven cantilever is significantly different from the closed loop driven cantilever) and (2) the 

undriven, freely ringing down cantilever appears to respond to the thermal excitation while 

saturating at lower base temperatures; however, the slope of the thermal response is wrong. It 

should be 1, while here it is somewhat over 2. 

Figure 9. Soft cantilever frequency noise vs. temperature. Data plotted 

as squares is for a cantilever driven closed loop. The circles 

are the undriven cantilever measured during ring down. 

It should be noted that we are aware that it is possible that we are analyzing the data for the 

undriven cantilever incorrectly. The apparent temperature of the cantilever derived from the 

frequency noise is a sensitive function of x(t) because x(t) appears squared in the equation for the 
13 

temperature. We use the RMS value of x(t) over the entire 10 s of data collection period for xrms. 

Since at 5.2 and 26 K, the ring down time is 16.6 and 13.0 s, respectively, there are significant 

changes in xrms during data collection. 

Perhaps this accounts for the differences between the driven and undriven cases (unlikely), or 

why the slope is somewhat over two instead of one (possibly). Further analysis is needed.

5. Conclusions 

We learned that for the software implemented lock-in, it is important to prefilter the data with a 

bandpass filter with a bandwidth of 10% of fc before multiplying it by the local oscillator. 

Otherwise, too much noise is introduced into the x and y values. High pass filtering at 500 Hz is 

not sufficient. 

We concluded that we have been using too much IF power for the soft Cornell cantilevers. 

Attempts to turn the IF optical power down below 200 nW resulted in operating the laser too 

near threshold and introduced significant frequency noise. At a future time, we will introduce 

more attenuation after the laser diode so the laser does not have to operate near threshold and 

perform tests with even lower optical power. 

We also determined that just because a cantilever achieves a 5 K Brownian motion does not 

mean its frequency noise is 5 K. We have been operating our IF system at an optical power of 

400 nW, and this has been introducing over an order of magnitude more frequency noise that 

would be expected from Brownian motion, assuming the current cantilever is typical. This issue 

explains why our cantilever-enabled readout of magnetization inversion transients (CERMIT) 

data did not look as good as the amplitude data from years ago using SiN cantilevers with spring 

constants of only 0.01 N/m. 

However, the failure of <x(t)2

> to equal Brownian motion at 400 nW differs from two previous 

measurements of <x(t)2

> versus T for a cantilever mounted on the probe head mechanical 

ground. For both of those measurements, <x(t)2> equaled Brownian motion. All cantilevers were 

Cornell soft cantilevers with spring constants of ~0.0001 N/m, but they were not the same 

cantilever. The question remains, does the current cantilever absorb more light for some reason? 

Further testing will have to be done to explore this. 
14 

6. References 

1. Obukhov, Y.; Fong, K. C.; Daughton, D.; Hammel, P. C. Real Time Cantilever Signal 

Frequency Determination Using Digital Signal Processing. J. Appl. Phys. 2007, 101, 

034315-1–034315-5. 

2. Marohn, J. Frequency Noise, white paper, Cornell University, 24 February 2008. 

3. Moore, E.; Marohn, J. Private communication. Cornell University, 2010. 

4. Moore, E. Private communication. Cornell University, 2010. 

5. Park scientific instruments, model #: MLCT-NOHW, part #: 00-103-0941, 

wafer #: 170-40-8. 

6. Hickman, S. cantilever fabricated by, Cornell University, 2008. 

7. Gmelin, E.; Asen-Palmer, M.; Reuther, M.; Villar, R. Thermal Boundary Resistance of 

Mechanical Contacts Between Solids at Sub-ambient Temperatures. J. Phys D: Appl. Phys.

1999, 32, R19–R43. 

8. Bruland, K. J.; Garbini, J. L.; Dougherty, W. M.; Chao, S. H.; Jensen, S. E. Thermal Tuning 

of a Fiber-optic Interferometer for Maximum Sensitivity. Rev. Sci. Instr. 1999, 70 (9), 3542–

3544. 

9. Marohn, J. Harmonic Oscillator Brownian Motion, white paper, Cornell University, 27 

March 2001. 

10. Smith, D. D. Description of Software Package Extract for the Characterization of the 

Amplitude and Frequency Noise Properties of Cantilevers Used for Nano-MRI; ARL-TR4995; U.S. Army Research Laboratory: Adelphi, MD, September 2009. 

11. Marohn, J. Spin Readout and Imaging Protocol Development for Magnetic Resonance Force 

Microscopy of Organic Thin-Film Samples; SBIR final report; Contract DAAD19-02-D-001, 

2007. 
15 

List of Symbols, Abbreviations, and Acronyms 

3-D three-dimensional 

ARL U.S. Army Research Laboratory 

ADC analog-to-digital converter 

BeCu beryllium-copper 

CERMIT cantilever-enabled readout of magnetization inversion transients 

FT Fourier transform 

He helium 

IF interferometer 

MRFM magnetic resonance force microscopy 

MRI magnetic resonance imaging 

PSD power spectral density 

RF radio frequency 

RMS root mean square 

Si silicon 

SiN silicon nitride 

SNR signal-to-noise ratio 
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