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Supplementary Information
Surface Adsorbate Fluctuations and Noise
in Nanoelectromechanical Systems
Y. T. Yang
†1
, C. Callegari
†2
, X. L. Feng
†3
, M. L. Roukes
*
Kavli Nanoscience Institute, Mail Code 114-36
California Institute of Technology, Pasadena, CA 91125, USA
This article provides supplementary information to
the main text of the title manuscript. We
present in Section I the experimental system
and techniques employed, along with supporting
experimental data. In Secti
on II we present the supporting theo
retical and modeling work.
These authors contributed to this work equally.
1
Present address: Department of Elect
rical Engineering, National Tsing-Hua Univ
ersity, Hsinchu, Taiwan 30013, R.O.C.
2
Present address: Sincrotrone Trieste, Basovizza Trieste, Italy.
3
Present address: Electrical Engineering, Case We
stern Reserve University, Cleveland, OH 44106, USA.
*
Corresponding author. Ema
il: roukes@caltech.edu
-2-
I. Experimental Techniques and Supporting Data
1. Experimental System and Apparatus
Figure S1. Experimental apparatus. (a) Illustration (design drawing) of the ultra high vacuum (UHV) cryostat with
various feedthroughs including very high and ultra high frequency (VHF/UHF) electronics and a gas delivery line.
(b) Close-up view of the sample stage area, with VHF/UHF
electronic connections, and mi
cronozzle structure. (c)
Illustration of the basic principle of the experiment, with the shutter-regulated micronozzle impinging Xe atoms
toward the surface of a vibrating VHF NEMS resonator. (d
) A picture of the gas nozzle-
shutter assembly employed
in this work.
gas
line
nozzle
source
Effusive
(Knudsen)
Nozzle
Shutter
VHF
NEMS
Resonator
Xe
Atoms
(a)
(b)
(c)
(d)
-3-
Figure S2. The gas nozzle-shutter assembly. (1) Gas line (stainless steel gas tube, outside diameter 1/16”, inside
diameter 1/32”) going to the microno
zzle. This supplies the Xe gas flux
from the buffer chamber (upstream and
outside of the UHV cryostat chamber)
to the gas nozzle. (2) Wires for heater
s. (3) Mounting holes for heaters. The
heaters are resistors, which are wired up and connected to
a temperature controller (LakeShore 331, outside of the
UHV chamber) for stabilizing the gas nozzle chamber temperature at 200K. (4) Micronozzle gas chamber (gold
plated brass). (5) Base/backbone plate (stainless steel).
(6) Insulating (Delrin) mounting block for the coil and
contacts, and for the counter balance. (7) Counter balanc
e (brass mass). (8) Pivot for the swing arm. (9) Insulating
(Delrin) block for contacts and wires. (10) Swing arm (G10 plastic). (11) Close sensing contact (copper-copper).
(12) Main gas chamber and house for heaters [gold plated br
ass, fit and sealed to (4)]. The connection between this
main gas chamber block to the base plate (5) has been made via 4 pads with small contact surface, to limit the
unnecessary heat transfer to the base
plate. (13) Shutter. (14)
Micronozzle aperture. (15)
Nozzle disk [fit and sealed
to (4), disks were made available with different aperture
sizes, and we tested disks with
aperture diameters of 30,
100, and 300
m]. (16) Coil for electromagnetic actuation of the shu
tter. The nozzle-shutter assembly is vertically
mounted in the UHV cryostat in a strong vertical magnetic field; a small amount of electrical current applied to this
coil induces a horizontal magnetic field, hence the elect
romagnetic actuation and deflection of the swing arm (10)
with respect to the pivot (8). (17) Thin wires for sensing contacts. (18) Open sensing contact (copper-copper). (19)
Pins for feedthrough wiring of heater and thermometer. (20) Insulating (Delrin) block for contacts and pins. (21)
Diode thermometer (LakeShore DT-470) and wires. The scale bar is 1cm.
(1)
Side
View
Front
View
(2)
(3)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(16)
(9)
(2)
(18)
(19)
(21)
(15)
(20)
(4)
(15)
(17)
-4-
2. NEMS Resonance Readout
Figure S3. Schematic of the open-loop VHF NEMS resonance signal transduction scheme. The VHF NEMS
resonance excitation and detection of
electromechanical response are base
d on magnetomotive transduction in a
strong magnetic field (
B
). The two-port measurement uses a HF/VHF
network analyzer (HP3577A). The readout
circuit is a reflectrometry scheme using a directiona
l coupler (CPL) and a low-no
ise amplifier (LNA).
3. Real-Time Resonance Frequency Locking
and Tracking, and the Noise Measurements
Figure S4. Diagram of the experimental setup and the measurement system. The gas delivery sub-system include
Xe gas source (lecture bottle cylinder), a buffer chamber and a specialized gas feedthrough line leading to the micro-
nozzle which is positioned and aligned in the vicinity of the NEMS device; it also includes various accessories such
-5-
as pressure gauges, regulators, and various coarse and fine
tuning control valves. The
electromechanical shutter is
controlled by a relay circuit. For real-time resonance frequency locking and tracking, the NEMS resonance readout
circuit (described in Figure
S2) is embedded into a closed loop with feedback, which consists of a frequency-
modulation phase-locked loop (FM-PLL).
1
VCO: voltage controlled oscillator, BPF: band-pass filter. As shown,
time-domain frequency instability is measured by using a
high-precision frequency counter. Noise spectra are
measured by using a dynamical signal an
d noise analyzer. Also, as shown in Figure S2, nodes (I) and (II) here in
Figure S3 indicate the alternative configuration for switching
back to the open-loop res
onance detection circuit.
4. Temperature-Programmed Adsorption (TPA)
Figure S5. A typical set of measured data for the demonstration of temperature-programmed adsorption (TPA). (a)
A fresh measurement of resonance frequency as a function of device temperature when there is no impingement of
Xe gas at all. This measurement gives a baseline
f
~
T
dependence as shown by the fit to the data. (b) Measured
frequency trace raw data when there is impingement of Xe gas flux and hence temperature-programmed adsorption.
The raw data of measured resonance frequency tracked by the FM-PLL is plotted together with the previously
calibrated
f
~
T
dependence. Here the measured
f
as a function of
T
is due to both the pure temperature dependence
and the loading of surface adsorb
ates (Xe atoms). Note the onset of the ‘PLL runaway’ near
T
56K. Beyond this
point, the data trace in the lower temperature range is ou
t of lock and not meaningful. In fact, each time this
happens, we heat up the device to restore to the fresh surf
ace condition. The Xe gas flux is stopped. We switch to
the open-loop resonance detection circuit scheme (Figure S3) with network analyzer to find the NEMS resonance
signal again, and we manually bring the FM-PLL back into lock.
20 30 40 50 60 70 80 90
190.4
190.6
190.8
191.0
191.2
191.4
191.6
Measured Data (No Xe)
Fit of
f
~
T
Dependence
Frequency (MHz)
T
(K)
50 55 60 65 70 75 80 85
190.4
190.6
190.8
191.0
191.2
f
~
T
Dependence (No Xe)
Measured Data (With Xe)
Frequency (MHz)
T
(K)
Onset of 'PLL Runaway'
50 55 60 65 70 75 80 85
0.0
0.1
0.2
0.3
0.4
0.5
0.6
f
mon
=590kHz
Frequency Shift (MHz)
T
(K)
Data
out
of
Locking
Data
out
of
Locking
(a)
(b)
(c)
-6-
5.
Q
Factor Temperature Dependence
Figure S6. Measured device quality factor (
Q
) as a function of device temperature. No Xe gas impingement in this
measurement. There is a mild but steady
Q
increase as
T
is decreased in the range of 20~80K. A similar tendency
has also been observed in other VHF/UHF NEMS resonators.
2
Here this measurement verifies the weak
temperature dependence of
Q
. This is also in good qualitative consistence with the observed mild reduction of
frequency instability (noise floor without Xe gas flux) as th
e temperature is decreased (see Figure 3 in main text),
because theoretically the frequ
ency instability should be pr
oportional to the inverse of
Q
.
3
20
30
40
50
60
70
80
5000
5500
6000
6500
Measured
Q
T
(K)
-7-
II. Theoretical Analyses and Modeling
1.
Noise due to Surface Diffusion
Here we describe our treatment of the noise proc
ess due to adsorbates diffusion on the surface of
NEMS device.
Figure S7. (a) Schematic illustration of mode-shape and location-dependent mass loading response for a doubly
clamped beam. (b) The square of normalized mode
shape and its optimized Gaussian approximation.
For the doubly clamped beam NEMS resonator ope
rating in its fundamental flexural mode,
we first consider its mode shape and the pos
ition-dependent frequency shift induced by mass
loading effect of an adsorbed atom. As illustrated in Figure S7, the
normalized
mode shape is
kx
kx
x
u
cosh
1173
.
0
cos
8827
.
0
)
(
, (S1)
for
x
[-
L
/2,
L
/2], with
u
(0)=1,
k
=4.73/
L
and
L
being the beam length.
4
For our interest in
developing an analytic
approach, we have found that approxi
mating the square of the normalized
mode shape [
u
(
x
)]
2
with a Gaussian functi
on and extending the integr
ation interval from [-
L
/2,
L
/2] to (

,
) considerably simplifies the calculations we present here. Minimizing the
‘distance’ norm between the two functions by mean
s of a least square technique, we have found
-0.4
-0.2
0.0
0.2
0.4
0.0
0.2
0.4
0.6
0.8
1.0
Exact Solution
Gaussian Apprx.
|
u
(
x
)
|
2
x
/
L
(a)
(b)
-8-
an excellent approximation

2
2
exp
)
(
L
ax
x
u
(see Figure S10) with an optimized
parameter
a
=4.42844.
When an atom (or molecule) of mass
m
lands at position
x
on the surface of device (with total
mass
M
), and assuming the added mass only changes
the kinetic energy but
not the potential
(flexural) energy of the device, we have the frac
tional frequency variati
on (with respect to the
initial reference frequency
f
0
or
0
),




2
2
2
2
2
0
0
2
1
1
2
1
x
u
M
m
dx
x
u
L
x
u
M
m
f
f
eff
L
L

. (S2)
Here we have


M
a
dx
x
u
L
M
M
L
L
eff
2
2
2
1
, which yields the effective modal mass
M
M
eff
40
.
0
. (The effective mass is 0.397
M
if we use the original solution, confirming that
the Gaussian approximation is
excellent with discrepancy
<1%). For a quasi-continuous
distribution of adsorbates mass over the length
of the NEMS device, we can simply replace

2
x
u
m
with
 
2
x
u
x
mC
, with
C
(
x
) being the concentration of
the adsorbed atoms (number
density per unit length).
We consider pure diffusion of one species in one
dimension. The concentration fluctuation

t
x
C
,
obeys the following diffusion equation,


2
2
,
,
x
t
x
C
D
t
t
x
C
. (S3)
The Fourier transform


t
k
C
t
x
C
k
,
~
,
F
obeys


t
k
C
Dk
t
t
k
C
,
~
,
~
2
, (S4)
and hence


0
,
~
,
~
2
k
C
e
t
k
C
t
Dk
, (S5)
-9-
where
k
is the wave number.
The device frequency fluctuation due to the mass fluctuation is given by








dx
x
u
t
x
C
M
m
a
dx
x
u
L
dx
t
x
C
x
u
M
m
f
t
f
t
y
L
L
L
L
L
L
2
2
2
2
2
2
2
2
2
0
,
2
1
1
,
2
1
)
(
. (S6)
We can now calculate the autocorrelation function
of the fractional frequency fluctuations due to
surface diffusion,

  
     
  
'
,
'
,
'
2
'
,
'
,
'
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
0
dx
x
x
x
u
x
u
dx
a
M
m
dx
t
x
C
t
x
C
x
u
x
u
dx
a
M
m
f
t
f
t
f
G
L
L
L
L
L
L
L
L
, (S7)
where

t
x
C
t
x
C
x
x
,
'
,
,
'
,
.
Employing Fourier transform techni
ques (recall the classical appl
ication of Fourier transform
and its properties to solving diffu
sion-type differential equations),
5
we calculate and carry out
transforms explicitly to yield

D
x
x
D
L
N
x
x
4
'
exp
2
1
,
'
,
2
, (S8)
in which
N
is the average total number of adsorbed at
oms on the device surface. Further, we use
the above result for

,
'
,
x
x
to calculate the double integral
in Eq. (S7), again with the
integration limits extended to (

,
), and we obtain
6
-10-

D
M
m
aN
D
a
L
a
M
m
NL
G
1
1
2
2
4
2
1
2
2
2
2
2
(S9)
where we define the char
acteristic diffusion time
D
a
L
D
2
2
2
. The advantage of being able
to carry out analytically
the calculation of the autocorrelation function well justifies the Gaussian
approximation of the mode shape. It
is interesting and worth noting that

G
is of the form

2
1
d
D
with
d
=1, the dimensionality of the problem in
the present study. In fact, we note
that our analysis here is in good analogy to
the well-known theory of fluorescence correlation
spectroscopy (FCS),
7
,
8
in which the correlati
on function scales as
1
1
D
, consistent with the
dimension (
d
=2) of the FCS process.
Now we employ the Wiener-Khinchin theorem
9
to obtain the corresponding double-sided
(DS) spectral density of fractional frequency noise,


  




D
D
D
D
D
D
i
DS
y
M
m
aN
g
M
m
aN
d
M
m
aN
d
G
d
e
G
S






2
2
0
2
,
4
2
2
2
/
1
cos
2
2
cos
. (S10a)
The more useful single-sided (
SS) spectral density (folding the
negative Fourier frequency part
over onto the positive side, correspo
nding to practical measurements)
10
,
11
is thus




D
D
DS
y
y
M
m
aN
S
S

2
,
2
2
. (S10b)