of 16
SUPPLEMENTAL
MATERIAL
Dynamic Similarity of Oscillatory Flows Induced by
Nanomechanical Resonators
Elizabeth C. Bullard
1
, Jianchang Li
2
, Charles R. Lilley
3
, Paul Mulvaney
2
, Michael L. Roukes
2
and
John E. Sader
1,3*
1
Kavli Nanoscience Institute and Departments of
Physics, Applied Physics, and Bioengineerin
g
, California
Institute of Technology, Pasadena, California 91125
2
School of Chemistry and Bio21 Inst
itute, The University of Melbourne, Victoria 3010, Australia
3
Department of Mathematics and Statistics, The Univ
ersity of Melbourne, Victoria 3010, Australi
a
1.
Derivation of Equation
(4)
in
the
Main T
ext
The dynamic similarity theorem
in Eq. (7)
is
derived
from
Eq. (4). Equation (4)
is
obtained
by
extending the dimensional analysis of Refs. [
1, 2
] to include an extra physical quantity, the gas mean
free path. Here, we provide the details of
the
derivation
of Eq. (
4)
.
The principal assumptions are:
(1)
The
resonator
behaves as a linearly elastic solid;
(2)
Energy dissipation due to vibration occurs in the gas;
(3)
The oscillation amplitude of the
resonator
is small, so that all nonlinearities due to the
resonator
and
gas
are negligible.
The
maximum energy stored in the
resonator
directly follows:
E
stored
=
1
2
k
A
2
,
(
S
1
)
where
k
is the
resonator stiffness
, and
A
is the oscillation amplitude. The energy dissipated in the
gas
can be quantified by the (dimensionless) quality factor,
Q
2
π
E
stored
E
diss
ω
=
ω
R
,
(
S
2
)
where
E
diss
is the energy dissipated per oscillation cycle, at the
radial
resonant frequency
ω
R
.
Since the flow is linear, the energy dissipated
in the gas
per oscillation cycle depend
s
on the square
of the oscillation amplitude,
A
.
From Eq
s
. (
S
1) and (
S
2
), we
then
obtain
k
=
1
2
π
2
E
diss
A
2
ω
=
ω
R
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
Q
,
(
S
3
)
which is independent of
the oscillation amplitude.
SUPPLEMENTAL MATERIAL
Dynamic Similarity of Oscillatory Flows Induced by Nanomechanical R
esonators
E.C. Bullard,
J. Li, C.R. Lilley, P. Mulvaney, M.L. Roukes and
J.E. Sader
SI
-
2
In accord with the above
-
listed
assumptions, the energy dissipated per cycle
E
diss
must depend on
(i) the square of the
resonator
’s
oscillation
amplitude,
A
, (ii) the
gas
density
ρ
and shear viscosity
μ
,
(iii)
the gas mean free path
λ
, (iv)
the linear dimension (size) of the
resonator
, denoted
L
0
, (
v) the
relevant frequency o
f oscillation, which from
Eq. (
S
3
) is the resonant frequency in
gas
,
ω
R
, (v
i
) the
mode shape of the
resonator
, and (vi
i
)
its
geometry. Note that the last two quantities are dimensionless.
The functional form of
E
diss
, in
terms
of
these
parameters, can be rigorously determined using
dimensional analysis [
3
]. It follows that the product
ρ
m
μ
n
L
0
p
ω
R
q
λ
r
1
2
π
2
E
diss
A
2
ω
=
ω
R
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
,
(
S
4
)
must be dimensionless, where
m
,
n
,
p
,
q
,
r
are constants to be
evaluated
. Equating dimensions in Eq.
(
S
4
) leads to
three
independent dimensionless groups:
Π
1
ρ
L
0
3
ω
R
2
1
2
π
2
E
diss
A
2
ω
=
ω
R
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
,
Re
ρ
L
0
2
ω
R
μ
,
Kn
=
λ
L
0
.
(
S
5
)
The
second
and third
parameter
s are commonly referred to as the Reynolds and Knudsen number
s
,
respectively.
From
Buckingham’s
π
theorem
[
3
], it follows that there must exist a function
,
H
1
ρ
L
0
3
ω
R
2
1
2
π
2
E
diss
A
2
ω
=
ω
R
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
,
ρ
L
0
2
ω
R
μ
,
λ
L
0
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
=
0
,
(
S
6
)
which relates the energy dissipated per cycle
E
diss
to all
other parameters. Solving Eq. (
S
6
) for the first
argument and rearranging gives
1
2
π
2
E
diss
A
2
ω
=
ω
R
=
ρ
L
0
3
ω
R
2
Ω
Re,
Kn
(
)
,
(
S
7
)
where the dimensionless function
Ω
Re,
Kn
(
)
is to be determined.
Substituting Eq
.
(
S
7) into Eq. (
S
3
)
gives
k
=
ρ
L
0
3
Ω
Re,
Kn
(
)
ω
R
2
Q
,
(
S
8
)
which is the required expression
,
Eq. (4)
,
of the main text.
SUPPLEMENTAL MATERIAL
Dynamic Similarity of Oscillatory Flows Induced by Nanomechanical R
esonators
E.C. Bullard,
J. Li, C.R. Lilley, P. Mulvaney, M.L. Roukes and
J.E. Sader
SI
-
3
2
.
Device Fabrication
We used 4" silicon wafers coated with 100nm, 300nm, or 500nm of low stress silicon nitride (SiN)
on both sides to fabricate the devices.
P
hotolithography
was used
to defi
ne etch windows on the
backside of each wafer, followed by a dry etch to remove the Si
N.
A
KOH etch remove
d
the silicon
from the selected areas to form SiN membranes on the front side of the wafer.
E
lectron beam
lithography
was used to
defi
ne cantilevers on the edges of the membranes. We deposited either a layer
of Cr or a Cr/Au layer as an
etch mask. We dry etched the
silicon nitride membranes to defi
ne the
cantilevers. We then removed the metal layer(s) with a wet etch. All of the devices were then coated
with a Cr/Au layer for reflectivity; a thermal evaporator was used for the 100nm and 500nm devices,
while an electron beam evaporator was used for t
he 300nm devices. Prior to deposition of the metal
layer on the devices, we performed a test deposition on a SiN coated SOI wafer, patterned with
rectangles with photolithography. After liftoff, we measured the step height of the metal layer with an
atomic
force microscope. The step height was used to calibrate the subsequent deposition on the
devices. The 500nm devices have a 3nm layer of Cr with a 100
-
110nm thick layer of Au on top. The
300nm devices have a 3nm layer of Cr with a 60
-
66nm layer of Au on to
p. The 100nm devices have a
3nm layer of Cr with a 20
-
22nm layer of Au on top. The reflective layer was not deposited prior to the
etch of the SiN membrane because the etch would have also etched the Au, leaving a layer of unknown
thickness. The thickness
of the metal layers must be with 10% of the desired thickness to ensure
geometric similarity between the devices, as well ensuring that the density of the devices remains
constant. Images of the 100nm set of devices are shown in Figure
S
1.
SUPPLEMENTAL MATERIAL
Dynamic Similarity of Oscillatory Flows Induced by Nanomechanical R
esonators
E.C. Bullard,
J. Li, C.R. Lilley, P. Mulvaney, M.L. Roukes and
J.E. Sader
SI
-
4
Figure S1:
SEM micrographs (color
ized for clarity
) of 100nm devices, of length
L
=
10
μ
m. Yellow is gold; purple is SiN. All images are taken at 7500x magnification. Devices
labeled in accordance with Figure 1.
A: Triangular cantilever;
B: (a) Square paddle head
c
antilever;
C: Cantilever with legs;
D:
length/width=10
rectangular cantilever;
E:
length/width=5
rectangular cantilever;
F:
length/width=3
rectangular cantilever.
SUPPLEMENTAL MATERIAL
Dynamic Similarity of Oscillatory Flows Induced by Nanomechanical R
esonators
E.C. Bullard,
J. Li, C.R. Lilley, P. Mulvaney, M.L. Roukes and
J.E. Sader
SI
-
5
3
.
Experimental Setup
We used optical interferometry to measure the quality factors of
the devices with both a network
analyzer and a spectrum analyzer. We placed the devices in chamber and changed the pressure from
3mT to 760T.
Figure S
2
:
Schematic of apparatus used for gas pressure measurements of cantilever
devices.
Optical
Interferometry
A picture of the optical setup is shown in Figure
S
2. The simple
interferometer is very similar
to the
desig
n used by Rugar
et al.
[
4
]. The fi
rst component of the detection system is the laser.
We used an
amplitude stabilized HeNe laser to
m
inimize fluctuations
in the beam intensity.
Fluctuations in the
intensity of the laser are indistinguishable from
fl
uctuations caused by
a change in the path length;
consequently, the amplitude stabilization reduces the noise.
Following the laser is an
optical isolator,
which prevents any light re
fl
ected from components
further along the beam path from entering the
laser; the re
fl
ected light could destabilize
the laser. The next component is a half wave plate in a
T-Cube Power Supply
Sample
Vacuum
Chamber
Achromatic Lens
f=150mm, 2w
0
=12.4um
Motorized
XYZ Stage
Quarter Wave
Plate
Manual
XYZ
Stage
Photodetector
Lens
f=100mm
HeNe Intensity
Stabilized Laser
Optical
Isolator
Beam
Expander
x20
1" Polarizing
Beamsplitter
Cube
d=0.7mm
d=14mm
Neutral
Density
Filter
Half Wave
Plate
Scroll Pump
Gas Inlet
SUPPLEMENTAL MATERIAL
Dynamic Similarity of Oscillatory Flows Induced by Nanomechanical R
esonators
E.C. Bullard,
J. Li, C.R. Lilley, P. Mulvaney, M.L. Roukes and
J.E. Sader
SI
-
6
rotation mount; the polarization is
adju
sted for the maximum amount of light to enter the interferometer.
Following the
half
w
ave plate is a neutral density fi
lter, used to attenuate the beam power. We used a
ND=1.5
fi
lter while making measurement for the 500nm
and 300nm devices and a ND=1.0 fi
l
ter while
making measurements for the 100nm devices. We chose these levels of attenuation to limit
heating of
the device under measurement, in order to prevent drift in the resonant frequency. A 20
×
beam
expander follows. When the beam exits the laser, it
has a
1
/
e
2
diameter of 0.7mm. The 20
×
beam
expander increases the
1
/
e
2
diameter to 14mm; the beam expander is required to minimize the spot size
at the sample. The minimum spot size at the sample,
2
w
0
,
is set by the diffraction limit. The minimum
spot size is given in Equation 1
[
5
]:
2
w
0
=
1.83
f
λ
D
,
(S9)
where
f
is the focal length of the lens,
λ
is the wavelength of light, and
D
is the diameter of
the input
beam. Due to constraints on the
diameter of the vacuum chamber, the minimum
focal length of the lens
is 150mm, which cor
responds to a spot size of 12.4
μ
m at the sample.
The next component in the optical setup is the polarizing beam splitter cube. The PBS
splits the
beam into its two ort
hogonal polarization components. The PBS also prevent ghost
re
fl
ections, which
would occur with a plate beam splitter. Following the PBS is a quarter
wave plate in a rotation mount;
the quarter wave plate is required to rotate the polarization of
the beam
by 90
°
so that the beam
re
fl
ected from the sample is directed to the photodetector.
The light is then focused through an
achromatic lens mounted in a kinematic mirror mount
on a motorized XYZ stage. We have chosen an
achromatic lens to reduce aberrations
in the
beam. The lens is also chosen to ensure the highest
numerical aperture possible. The XYZ
stage is used to move the beam during alignment.
The beam then enters the vacuum chamber through a quartz window with an anti
-
refl
ective coating.
A portion of t
he beam hits the device and is re
fl
ected back along the
optical path until it reaches the
PBS, where it is directed to
the photodetector. The remain
der of the beam is re
fl
ected by the
piezoshaker underneath the chip containing the cantilevers.
Prior to ent
ering the photodetector, the
beam traverses a len
s with a focal length of 100mm,
to reduce the beam size, since the diameter of the
active area of the photodetector is 0.8mm.
The photodetector is a New Focus 1801 photodetector,
which has a bandwidth of DC
-
125MHz
and a noise
floor of 3.3 pW/
Hz. The photodetector is mounted
on an XYZ translation stage
for alignment with the beam.
SUPPLEMENTAL MATERIAL
Dynamic Similarity of Oscillatory Flows Induced by Nanomechanical R
esonators
E.C. Bullard,
J. Li, C.R. Lilley, P. Mulvaney, M.L. Roukes and
J.E. Sader
SI
-
7
Measurements
We measured the quality factors of the devices over a range of pressures. We measured
two
quantities: the
intrinsic
quality factor of the devices
,
Q
int
,
and the quality factor due to gas
damping
,
denoted
Q
.
We used the optical setup to detect the motion of the devices and a piezoshaker to
actuate
them. We mounted a piezoshaker with
silver paste onto a PCB and then mounted
the chip containing
the cantilevers to the piezoshaker with superglue. We then wire bonded
the piezoshaker to the PCB.
We placed the device in a vacuum chamber pumped with a scroll
pump with a base pressure of
3m
T.
We varied the pressure in the chamber by adding N
2
gas through a needle valve attached to the
chamber. We measured the pressure with a MKS
317 Pirani gauge, accurate to 1% of the indicated
decade, with two digits of precision.
We measured the intrinsic quality factor,
Q
int
, of each device at a few mT
with an Agilent 4395A
Net
work/Spectrum/Impedance Analyzer in network analyzer mode. We dr
ove the devices with
the
piezoshaker and measured the photodetector output. We used a network analyzer to
measure
Q
int
because measurements of the thermomechanical noise spectrum with
the 4395A in spectrum analyzer
mode had a variance of about 10%
-
15%
of the measured
quality factor
.
Such
inconsistent
measurements would require several (about 10) measurements to acquire an accurate value
for
the
quality factor
. An accurate value for the
intrinsic
quality factor
,
Q
int
,
is required
to
convert the
measured
(
total
)
quality factor
Q
total
at larger gas pressures
to the
quality factor
due to gas damping
,
Q
;
Figure S
3
:
FFT of
piezoshaker excited device showing resonance peak and laser noise
peaks.
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
10
8
10
6
10
4
10
2
Frequency (Hz)
Magnitude (au)
10
5
Device resonance peak
SUPPLEMENTAL MATERIAL
Dynamic Similarity of Oscillatory Flows Induced by Nanomechanical R
esonators
E.C. Bullard,
J. Li, C.R. Lilley, P. Mulvaney, M.L. Roukes and
J.E. Sader
SI
-
8
see Section 4
.
The measurements made with the network analyzer had a variance of
less than 1%,
leading to signifi
cantly faster measurements. We fitted the resonance peak to the response of a
damped
harmonic oscillator using a non
-
linear
least squares fi
t in MATLAB.
Howe
ver, we were unable to use the NA to make the
quality factor
vs pressure
measurements
because
both the response of the piezoshaker at higher pressures and laser noise altered the shape
of the
response, rendering it no longer a Lorentzian. The thermomechanical noise
spectrum measured with the
SA was
always Lorentzian, occasionally with peaks from the laser noise.
The laser noise is present
even when the beam splitter is replaced with a mi
rror to directly
send the beam to the photodetector.
The laser has noise peaks at the following frequencies
in kHz: 36.7, 73.3, 110, 147, 183, 220, 257, 293,
330, 367, 403, 440, 477. A fast Fourier
transform of a ring down measurement collected with an
Agi
lent 54845A oscilloscope for a
cantilever with a resonant frequency of 204.6kHz is shown in Figure
S
3; the laser noise peaks
are clearly visible. We could easily remove the peaks f
rom the laser noise
during the fi
tting
of the thermomechanical noise spectru
m. The thermomechanical displacement noise
spectral
density for a cantilever is given by Equation 2
[
6
]
:
S
x
th
ω
(
)
=
ω
R
ω
R
2
ω
2
(
)
2
+
ω
R
ω
/
Q
total
(
)
2
4
k
B
T
M
eff
Q
total
,
(S10
)
where
ω
R
is the
radial
resonant frequency of the cantilever,
Q
t
o
t
a
l
is the
measured
quality factor
(
due to
all dissipation processes present
)
,
k
B
is Boltzmann's
constant,
T
is the temperature, and
M
eff
is the
Figure S
4
:
Thermomechanical noise spectrum for
Device
E, thickness
500nm,
Run 1,
showing fit to Eq. (S10).
237.5
237.6
237.7
237.8
237.9
238
238.1
238.2
238.3
0
1
2
3
4
5
6
7
8
9
x 10
12
Frequency (kHz)
Amplitude (V
2
/Hz)
Data
Fit
(
f
R
=237.8975 kHz,
Q
total
=1589
±40)
SUPPLEMENTAL MATERIAL
Dynamic Similarity of Oscillatory Flows Induced by Nanomechanical R
esonators
E.C. Bullard,
J. Li, C.R. Lilley, P. Mulvaney, M.L. Roukes and
J.E. Sader
SI
-
9
eff
ective mass of the cantilever. For a
cantilever with an aspect ratio
L
/
b
>
3
,
M
eff
0.24
M
[
7
].
Shown in Figure
S
4 is the thermomechanical noise peak for
Device E
with
thickness
500nm
,
Run 1,
at
P
= 3.7mT.
We fi
tted the thermomechanical noise peak to
Eq.
(S10
)
using a non
-
linear least squares
fi
t in
MATLAB
; see Figure S4
.
We measured
quality factor
Q
total
vs
p
ressure
for each device with the
spectrum analyzer, with
three
measurements below 10mT. The rest of the measurements were
logarithmically spaced
for four measurements each decade, with the exception of 100T
-
1000T; the
measurement was
terminated at 760T. For the 500nm devices, a measurement was made at 150T
instead of
170T to enable comparison with the quality factor of the 100nm devices at 760T. For the
300nm devices, an additional measurement was made at 250T for the same reason.
Figure
S5a
shows
the thermomechanical noise peak
for Device D
measured at
P
= 1T.
Figure
S
5
b
gives
the
measured
quality factor
Q
t
o
t
a
l
vs pressure
for a measurement run
on this device
, as well as
frequency
f
R
vs
pressure
in an inset to verify the assumption that
ω
R
ω
vac
for these devices.
Results for all devices are
given in
S
ection
5
.
Measurements were performed on two devices for each device type (A
-
F) and thickness (100nm,
300nm, 500nm), for a total of 36 devices. Each device was measured twice, for a total of 72
measurement runs. Results for one of each type of device are given in
Figure S6, including both
measurement runs. The measurements were consistent between devices and only one device of each
type and thickness is included for clarity.
Figure S5:
Measurement on 500nm 10:1 aspect ratio device,
Device D
, Run 1
. (a) Fit of
thermomechanical
noise peak
,
Eq.
(S10),
measured at
P
=
1T. (b
)
Measured q
uality factor
Q
t
o
t
a
l
v
s pressure, with resonant frequency
f
R
vs pressure
shown in inset.
220.3
220.4
220.5
220.6
220.7
220.8
220.9
221
0
0.5
1
1.5
2
2.5
3
3.5
4
x 10
12
Frequency (kHz)
Amplitude (V
2
/Hz)
Data
Fit
(
f
R
=220.7 kHz,
Q
total
=1000
±30)
Pressure (T)
Quality factor,
Q
total
10
2
10
1
10
0
10
1
10
2
10
3
220
220.2
220.4
220.6
220.8
Pressure (T)
f
R
(kHz)
Frequency vs Pressure
1000
100
200
300
500
700
0.01
0.1
1
10
100
1000
(a)
(b)
SUPPLEMENTAL MATERIAL
Dynamic Similarity of Oscillatory Flows Induced by Nanomechanical R
esonators
E.C. Bullard,
J. Li, C.R. Lilley, P. Mulvaney, M.L. Roukes and
J.E. Sader
SI
-
10
4
.
Quality Factor Measurements
Measurements of the quality factors
for
one of each type of device (Device A
-
F) and thickness
(100nm, 300nm, 500nm),
as a function of
N
2
gas pressure, are
given in Figure S6.
The quality factors
due to the gas only
,
Q
,
are obtain using the relation:
1
Q
total
=
1
Q
+
1
Q
int
,
(S11
)
Figure S6:
Measured quality factors
,
Q
total
,
as a function of gas pressure for all devices
.
SUPPLEMENTAL MATERIAL
Dynamic Similarity of Oscillatory Flows Induced by Nanomechanical R
esonators
E.C. Bullard,
J. Li, C.R. Lilley, P. Mulvaney, M.L. Roukes and
J.E. Sader
SI
-
11
where
Q
total
is the quality
factor measured in Figure S6 and
Q
int
is the measured
intrinsic
quality factor
(at very low gas pressure
), see
above
.
Use of Eq. (10) in Table I requires gas pressures that are intermediate to those measured
for the 500
nm device.
To
determine
Q
at
these
gas pressures,
data
reported in Table I
are
obtained by fitting
and
interpolating
measured data
of
Q
vs
gas
pressure on a double logarithmic
-
scale
[2]
.
This is performed
for both the 100nm and 500nm data; see Figure S7. This fit procedure is not used elsewhere.
The scaled quality factor
s
,
H
Kn
(
)
,
determined from the raw
data in Figure S6
and Eq.
(S11
)
,
are
given in Figure S
8
.
The c
hosen l
ength scale
s
for the Knudsen number
are
: Devices A, D, E, F: leg
width at clamp; Devices B, C: Width at free end.
Figure S
7:
Measured quality factors,
Q
, vs gas pressure
(dots)
for Device E with fits
shown
[
lines
]
. (a) 100nm device; (b) 500nm device.