SI for: High sensitivity deflection detection of nano-cantilevers
Babak Sanii and Paul D. Ashby
Molecular Foundry, LBNL, Berkeley, CA 94618, USA
(Dated: 04 September 2009)
PACS numbers:
I.
PLANE WAVE EXPANSION
We reformulate the imaginary portion of the exponent of equation 2 to the plane wave expansion as follows:
e−i(z
√
ko 2 −α2 +αy)
= e−iko ρcos(θ−γ)
∞
X
=
i−n ein(θ−γ) Jn (ko ρ)
(1)
(2)
n=−∞
II.
APPROXIMATING EQUATION 4 OF THE MAIN TEXT
The integral in equation
p 4 of the main text is intractable; earlier analysis solved the integral by approximating
both sin−1 (α/ko ) and ko2 − α2 with the first two terms of their respective Taylor series[1, 2]. However, these
approximations are only valid when the wavelength is much smaller than the half-spotsize. Here we approximate the
integral’s value using Gauss/Lobatto quadrature[3] and limit the integration such that sin−1 (α/ko ) is real, to avoid
branch cut discontinuities.
III.
SENSITIVITY IN WATER
FIG. 1: A map of the sensitivity to noise ratio of the detection scheme in water as a function of 1/e spot diameter (39
samples, 200-2000nm) and nanowire radius (40 samples, 25-500nm), normalized and interpolated nearest-neighbor between
samples. Sensitivity is defined as the steepest slope of the calculated translation profile (e.g., Fig 1B), and the relative noise is
approximated by the square root of the total signal incident on the split photodiode.
2
IV.
DETERMINING MATERIAL PARAMETERS FROM THERMAL VIBRATION SPECTRA
The thermal fluctuation profile in Fig. 3B was fit with a Levenberg-Marquardt nonlinear regression[4] to a viscous
fluid model of a cylindrical cantilever in air[5]. The physical dimensions of the nanowire cantilever were determined
from SEM images (Fig. 2B) and nominal values for the density (1.1778km/m3 ) and viscosity (1.8527 × 10−5 N s/m2 )
of air were used as fixed parameters of the fit. Our best-fit determines the density of the nanowire to be 7139.9km/m3 .
We find the best-fit Young’s modulus of 47.7GPa, which is within the previously measured range[6].
V.
FORWARD SCATTERING VS. BACKWARD SCATTERING PLOTS
FIG. 2: Top-left: A map of the sensitivity to noise ratio of the detection scheme as a function of 1/e spot diameter (40 samples,
200-2000nm) and nanowire radius (40 samples, 25-500nm), normalized and interpolated nearest-neighbor between samples.
Top-right: A similar plot looking at the backscatter signal. Bottom-left: The ratio of the forward to backward scattering. D:
a map of where the ratio is greater than one (red region).
VI.
DETERMINING WHEN NANOWIRE THERMAL FLUCTUATIONS IN WATER ARE
DETECTABLE BY OUR SYSTEM
With a fixed amount of input laser power, there is a limit to the smallest nanowire movement we can detect. This
corresponds to a range of nanowire sizes whose thermal fluctuations we can discern. We calculate this by normalizing
the theoretical sensitivity in water with a measured sensitivity point (shown on the graph with a red X, radius
= 113nm,length = 19.1 µm), scaling it to 5mW of detected power (accounting for backscattering losses from the
nanowire), and considering only the spotsize of that experiment (≈ 1650nm). The
√ shot noise is considered in addition
to the input noise of the Asylum Research MFP-3D AFM Controller (300 nV / Hz). In this way an expected noise
floor as a function of nanowire radius was determined.
Next, we calculated the thermal fluctuations in water of a range of nanowire radii and lengths per Sader’s viscous
model[5] and compared the off-resonance thermal fluctuation amplitude with the expected noise floor. The hashed
3
region in figure 4 is where the noise floor is greater than the expected thermal fluctuations. We note that increased
laser power expands the range of accessible nanowires. Preliminary results indicate that the ultimate limit may be
the onset of heating damage of the nanowire or boiling of the surrounding water.
VII.
INFLUENCE OF NANOWIRE RADIUS VARIATIONS ON RESULTS
Matching the measured force noise to the theoretical viscous fluid model is complicated by the non-uniform thickness
of the nanowire. The theoretical model uses the nanowire’s radius, length and material properties to determine
its thermal fluctuations in water. To calculate force noise, we measure the thermal position-noise of a nanowire
(below resonance where the mechanical gain is unity) and multiply it by the stiffness we determine from its physical
dimensions, as measured by SEM images. However, because the nanowire’s radius is not uniform, we roughly estimate
the error in this result by considering nanowires of radii of one standard deviation on either side of the mean (see
Table I for two examples). The stiffness of the nanowire cantilever varies as its radius to the fourth power.
There are other considerations not included in the analysis. For example, nanowire imperfections such as crystal
boundaries and surface roughness will likely contribute to discrepancies between theoretical predictions and experimental results (e.g., Fig 4, inset, of the main text).
[1]
[2]
[3]
[4]
[5]
[6]
S. Kozaki, IEEE Transactions on Antennas and Propagation 30, 881 (1982).
S. Kozaki, Journal of Applied Physics 53, 7195 (1982).
W. Gander and W. Gautschi, BIT Numerical Mathematics 40, 84 (2000).
Y. Bard, Nonlinear parameter estimation (Academic Press New York, 1974).
J. Sader, Journal of applied physics 84, 64 (1998).
M. Yazdanpanah, Ph.D. thesis, University of Louisville (2006).
4
Nanowire 1: 13.3µm long (after red arrow)
81nm Mean Radius
(std.dev.of ± 9nm)
Signal/Noise Ratio
20√
Measured Thermal Noise
2.8pm/ √Hz
Theoretical Thermal Noise
3 ± 1pm/ Hz
Spring Constant
2 ± 1mN/m
√
Theoretical Force Noise
6.0 ± 0.2f N/
√ Hz
Calculated Force Noise
6 ± 3f N/ Hz
Nanowire 2: 19.4µm long
Signal/Noise Ratio
Measured Thermal Noise
Theoretical Thermal Noise
Spring Constant
Theoretical Force Noise
Calculated Force Noise
82nm Mean Radius
(std.dev.of ± 5nm)
12.4√
9.6pm/ √
Hz
10 ± 2pm/ Hz
0.7 ± 0.4mN/m
√
6.9 ± 0.3f N/
√ Hz
7 ± 4f N/ Hz
TABLE I: Measured and theoretical force noise calculations. We determine the signal-to-noise ratio by converting the thermal
position fluctuation into volts and comparing it to the input noise. The theoretical thermal noise is determined with a viscous
fluid model[5], and the calculated force noise is the measured thermal noise multiplied by the spring constant. Uncertainty is
estimated by repeating the analysis for nanowire radii of one standard deviation on either side of the mean. We justify modeling
Nanowire 1 as only the extension past the red arrow by noting that the radius of the 6.1µm long base is 185 ± 25nm, and so
it is ≈ 250 times stiffer than the extension.