Surface Adsorbate Fluctuations and Noise in
Nanoelectromechanical Systems
Y. T. Yang
†,‡
,
C. Callegari
†,§
,
X. L. Feng
†,
‖
, and
M. L. Roukes
*
Kavli Nanoscience Institute, Mail Code 114-36, California Institute of Technology, Pasadena,
California 91125, United States
Abstract
Physisorption on solid surfaces is important in both fundamental studies and technology.
Adsorbates can also be critical for the performance of miniature electromechanical resonators and
sensors. Advances in resonant nanoelectromechanical systems (NEMS), particularly mass
sensitivity attaining the single-molecule level, make it possible to probe surface physics in a new
regime, where a small number of adatoms cause a detectable frequency shift in a high quality
factor (
Q
) NEMS resonator, and adsorbate fluctuations result in resonance frequency noise. Here
we report measurements and analysis of the kinetics and fluctuations of physisorbed xenon (Xe)
atoms on a high-
Q
NEMS resonator vibrating at 190.5 MHz. The measured adsorption spectrum
and frequency noise, combined with analytic modeling of surface diffusion and adsorption–
desorption processes, suggest that diffusion dominates the observed excess noise. This study also
reveals new power laws of frequency noise induced by diffusion, which could be important in
other low-dimensional nanoscale systems.
Keywords
Nanoelectromechanical systems (NEMS); surface adsorbate; adsorption–desorption; diffusion;
resonator; noise
Physisorption of atomic or molecular species is ubiquitous in virtually all circumstances
where solid–gas interfaces exist (e.g., catalysis, adhesion). In physics, directly observing and
© 2011 American Chemical Society
*
Corresponding Author
, roukes@caltech.edu.
†
These authors contributed to this work equally.
‡
Present Addresses
Department of Electrical Engineering, National Tsing-Hua University, Hsinchu 30013, Taiwan, R.O.C.
§
Sincrotrone Trieste, 34149 Basovizza, Trieste, Italy.
Department of Electrical Engineering and Computer Science, Case Western Reserve University, Cleveland, OH 44106, United
States.
ASSOCIATED CONTENT
Supporting Information
. Experimental system and techniques employed; supporting theoretical and modeling work. This material is
available free of charge via the Internet at
http://pubs.acs.org
.
NIH Public Access
Author Manuscript
Nano Lett
. Author manuscript; available in PMC 2013 November 25.
Published in final edited form as:
Nano Lett
. 2011 April 13; 11(4): . doi:10.1021/nl2003158.
NIH-PA Author Manuscript
NIH-PA Author Manuscript
NIH-PA Author Manuscript
manipulating individual adsorbed atoms on ultraclean surfaces represents a significant
experimental feat.
1
Engineering of atomic and molecular structures on certain surfaces also
plays important roles in emerging applications including programmable self-assembly,
2,3
nanoscale lithography,
4
and nanotribology.
5
For solid-state transducers, especially high-
precision detectors and sensors, surface adsorbate can have important influences on device
performance.
6
Electromechanical resonators with high quality factor (
Q
), particularly quartz
crystal microbalances (QCMs), are widely used to measure the mass, and other physical and
electrochemical properties of the adsorbed layers, by monitoring changes of the QCM
resonance frequency,
Q
factor, or amplitude.
7,8
Resonant nanoelectromechanical systems
(NEMS) sensors have recently demonstrated significantly enhanced mass sensitivities, down
to the level of several, or even single, noble gas atoms,
9–11
representing an improvement up
to 10
12
-fold over a conventional QCM. Such sensitivity levels provide the new opportunity
of employing NEMS to directly probe the effects of surface adsorbate, on very small surface
areas (typically of the order of only ~100 nm × 1 μm) and in the submonolayer regime. This
approach can effectively complement and advance today’s prevailing experimental
techniques in surface science (e.g., spectroscopy and ellipsometry for surface
thermodynamics, atomic force microscopy, scanning tunneling microscopy, quasielastic
atom scattering for surface diffusion, and QCM for interfacial friction).
12
As an initial step
toward this new paradigm, in this work we employ a high-
Q
, very high frequency (VHF)
NEMS resonator to investigate the behavior of physisorbed xenon (Xe) atoms, by
performing precise measurements of the excess frequency noise that arises from statistical
fluctuations of the adsorbed atoms on the resonator surface.
The core of our experimental apparatus (Figure 1a) consists of a NEMS resonator and a
micro gas nozzle, well aligned and in close proximity. The NEMS resonator is a doubly
clamped beam with dimensions 2.3 μm (
L
) × 150 nm (
w
) × 100 nm (
t
). It is surface-
nanomachined from a 100 nm silicon carbide (SiC) layer on silicon (Si) substrate, in a
process detailed elsewhere.
13
Metallization and passivation of the device consist of a 30 nm
film of thermally evaporated aluminum (Al) capped with 5 nm titanium (Ti). At least part of
the top Ti film upon exposure to air turns into native oxide (TiO
2
),
14
which is not only an
excellent passivation layer but also a popular surface for biochemical species and a
remarkable catalyst.
15,16
The device is electrically connected onto a dedicated chip carrier (with an embedded circuit
board) and mechanically and thermally anchored to a massive sample stage (brass, gold
plated). The sample stage is loaded into an ultrahigh-vacuum (UHV) cryostat, first pumped
down at room temperature overnight (to ~5 × 10
−7
Torr) and then cooled down in liquid
helium (LHe). The device temperature is precisely measured and regulated by a calibrated
on-board thermometer and a closed-loop temperature controller. We employ the
magnetomotive scheme
9,13
to excite the device into resonance and detect its response. The
orientation of the magnetic field is such that the out-of-plane flexural modes are excited; the
calculated modal mass of the resonator is
M
eff
≈
96 fg. Figure 1c displays the measured
response at a temperature
T
≈
58 K, showing the sharp (
Q
≈
5620) fundamental-mode
flexural resonance at a frequency
f
0
≈
190.5 MHz, corresponding to an effective stiffness
k
eff
≈
145 N/m. We have incorporated the NEMS resonance signal readout (Figure 1b) into
a low-noise, frequency-modulation phase-locked loop (FM PLL) circuitry.
9,17
This enables
frequency locking and tracking of the resonance, hence real-time monitoring of the
frequency variations that reflect the thermodynamic effects of adsorbates on the device
surface. Prior to investigating the real-time behavior of surface adsorbates, we perform
extensive tests and careful calibrations of the device response to a steadily increasing mass-
load using the FM PLL technique. Figure 1d shows the calibration of the excellent mass
responsivity of the device,
ℛ
|∂
f
0
/∂
M
| = 0.99 Hz/zg [i.e., a resonance frequency shift of
~1 Hz per zg (10
−21
g) of loaded mass].
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Adsorbates are delivered to the NEMS resonator via a micro gas nozzle (Figure 1a) fed by a
gas delivery system connected to a buffer chamber. The nozzle has a circular aperture of 300
μm in diameter. It operates at low gas density as an effusive cell
18
to produce an atomic
beam of Xe impinging onto the NEMS surface. The cell’s gas kinetics determines the Xe
atoms impinging rate
I
(atoms per unit area per unit time), which can be converted into an
effective pressure of the impinging gas phase above the device, via
p
=
I
(2
m
Xe
k
B
T
)
1/2
,
where
k
B
is the Boltzmann constant and
T
is the device temperature. In this study, the
calibrated constant flux delivered by the nozzle is
I
= 2.7 × 10
17
atoms/(m
2
·s). This
corresponds to a local effective pressure of
p
= 6.8 × 10
−8
Torr at a device temperature
T
=
60K. The cross section area of the device exposed to the well-aligned incoming atomic beam
is
A
D
=
Lw
= 3.45 × 10
−13
m
2
. A flat surface with this area can accommodate an ideal
close-packed monolayer of
N
a,est
≈
2.2 × 10
6
Xe atoms (with a Lennard-Jones diameter
d
Xe
≈
0.41 nm);
N
a,est
is thus the estimated number of available adsorption sites in the limit of
an ideal, flat, and clean surface.
We choose to introduce Xe gas to the NEMS surface due to its relatively large atomic mass
(
m
Xe
= 131.3 amu) and for its record as a classic prototype for studying physisorption in
surface science.
19
Existing Xe physisorption studies, however, are primarily limited to ideal,
smooth, often specially treated metal surfaces,
12,19
where a realistic description of the
surface processes can be tackled in quantitative analyses or computer simulations.
19
In
ordinary and more practical situations involving nonideal, nonmetal, and disordered
surfaces, understanding of adsorbates dynamics is particularly poor; quantitative studies and
experimental approaches are greatly needed. On the surface of a resonating NEMS device,
as sketched in Figure 1e, the behavior of Xe atoms is intriguing and may include both
adsorption–desorption (out-of-plane) and surface diffusion (in-plane) processes. Because
conventional sensors lack the necessary sensitivity, no experiment has been carried out to
investigate the contribution of these processes to resonance frequency fluctuations. Note that
the frequency variations depend not only on the total number of adsorbed Xe atoms, but also
on their distribution and fluctuations on the device surface (e.g., along the length of the
device, in this study).
We first measure the adsorption spectrum
N
(
T
), i.e., number of adsorbed Xe atoms as a
function of the regulated device temperature
T
. As the device is cooled down,
N
(
T
)
increases. We call this process temperature-programmed adsorption (TPA, see Supporting
Information). Such variations are precisely measured as the resonance frequency change
f
m
due to the mass-loading effect of the adsorbates,
f
m
=
N
(
T
)
ℛ
m
Xe
. Figure 2 demonstrates
the measured adsorption spectrum and effective surface overage in the temperature range
T
≈
58–80 K. For a given incoming atom flux, the TPA process is highly repeatable and
reversible, by regulating the device temperature back and forth in the appropriate
temperature range. Unlike conventional Langmuir isotherm adsorption spectra (i.e., surface
coverage versus pressure of the gas phase, at constant surface temperature), the data here are
closer to isobars (number of adsorbates versus device temperature, at constant pressure of
the gas phase).
20
We note that, in our measurements with the above value of the atom flux,
when the device is cooled down to a temperature lower than
T
≈
56 K, the FM-PLL
consistently experiences such a sudden and steep frequency shift that it cannot sustain the
locking and tracking of the NEMS resonance; we call this phenomenon “PLL runaway”. We
presume that the inflection point before the onset of this behavior represents an equivalent
one-monolayer coverage, beyond which very fast accumulation of Xe atoms occur on the
surface, leading to the so-called “wetting”-like behavior (i.e., abrupt formation of multilayer
Xe films on device surface, which is dramatic and the associated frequency shift causes the
PLL runaway).
21
The measured frequency shift before PLL runaway is
f
mon
≈
590 kHz
and the corresponding number of atoms is
N
a,exp
≈
2.7 × 10
6
. This value strongly supports
our identification of PLL runaway with the completion of a monolayer. The slightly higher
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value (~20%) compared to our prediction
N
a,est
for an ideally flat surface is very reasonable
considering the small but nonzero roughness of the device surface, and the uncertainty in
determining the device geometry and actual surface area. After repeatedly experiencing the
PLL runaway and identifying the temperature at which it occurs, we focus our experiments
in the range
T
≈
58–80 K where the TPA is highly reversible and easily controllable.
The measured temperature-regulated surface coverage (Figure 2a),
(
T
) =
N
(
T
)/
N
a,exp
, helps
to reveal the thermodynamics between the surface adsorbates and the gas phase. The
adsorbing rate (atoms per unit area per second),
r
a
=
sI
, is a fraction of the atoms impinging
rate, with the ratio
s
being the sticking coefficient. We take
s
= 1 as customarily done for Xe
impinging on very cold surfaces.
22
We model the temperature dependence of the rate of
atoms leaving the surface as a thermally activated process following the Arrhenius law, that
is, by a surface-detaching rate
r
d
=
a
exp(−
E
a
/
k
B
T
), where
E
a
is the activation energy and
v
a
the attempt frequency. The steady-state balance between the gas phase and surface
adsorbates dictates
r
a
A
D
=
Nr
d
, hence
N
(
T
) = (
sIA
D
/
a
)exp(
E
a
/
k
B
T
). As shown in Figure 2b,
this simple model neatly fits the experimental data, yielding
E
a
≈
63.1 meV (±0.23 meV),
and
v
a
≈
1.23 × 10
4
Hz(±0.05 × 10
4
Hz). Note the
v
a
value from the fit is less reliable than
the
E
a
value (
v
a
is more sensitive to fitting uncertainties, also depending on accurate
determination of several experimental parameters). The extracted
E
a
is substantially smaller
than the desorption energy reported for Xe on TiO
2
surfaces,
23,24
in the range of
E
des
~ 200–
300 meV (note that reported desorption energy data for Xe on crystalline metals or graphite
are also in this range
22,25,26
). We recall that atoms can leave the device surface via either
desorption into vacuum or lateral diffusion (Figure 1d); both processes can be described by
an Arrhenius equation, where however the activation energy of the diffusion process is
generally much smaller. Indeed, we find the measured activation energy to be in good
agreement with typical reported values of the lateral diffusion energy barrier
E
diff
of Xe on
surfaces (e.g.,
E
diff
≈
52.1 meV,
25
54 meV,
27
moderately higher than the Xe–Xe binding
energy, ~20 meV). This suggests that in our experiment the dynamic balance of surface
adsorbates could be primarily between the adsorbing atoms from the nozzle and the atoms
diffusing off the NEMS resonator.
To gain more insight beyond the above steady-state picture provided by the TPA and the
initial surface kinetics analysis, we further carefully probe the fluctuations of the surface
adsorbates by investigating the noise behavior of the VHF NEMS resonator while tuning the
surface coverage via the device temperature. We measure the frequency instability of the
NEMS resonator, quantified by the Allan deviation,
28
A
(
A
)=[1/(2(
N
− 1))
, with
f
i
being the average frequencies measured over an
averaging time
A
= 1 s by employing the FM-PLL technique. We first measure the baseline
instability without the Xe flux (Allan deviation
A,sys
); and then again with the Xe flux.
Figure 3 shows the measured Allan deviation as a function of device temperature. The axis
on the right shows the corresponding mass fluctuations (at zeptogram scale) set by the
frequency instability. The instability measured without Xe flux represents the noise floor of
the system and slightly decreases as temperature is lowered. This corresponds to the mild
but steady increase of the
Q
factor that we observe upon cooling the device in this
temperature range (see Supporting Information). With the introduction of surface
adsorbates, excess instability arises, clearly noticeable in Figure 3 for
T
< 70 K. Also visible
are a few small peaks distributed in the Allan deviation plot. The fact that these small peaks
appear in the data with surface adsorbates but not in the noise floor data suggests the
existence of some interesting surface processes. We hypothesize that they may be related to
the details of certain collective behavior of the adsorbed atoms, which could be dependent
on roughness and structural irregularity of the surface (e.g., terraces, defects).
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To understand the trend displayed by the measured data in Figure 3, we have developed
analytic models for the two processes, adsorption–desorption and surface diffusion, that we
consider relevant. We first examine the frequency noise due to adsorption–desorption. The
adsorption–desorption noise model (see Supporting Information) deals with nondiffusive
(i.e., laterally immobile) adsorbates at submonolayer coverage, in a fashion similar to an
earlier work modeling surface contaminants on a quartz crystal resonator.
29
Figure 2a shows
the measured effective surface coverage
(
T
) and the surface occupation deviation
occ
=
[
N
(
N
a
−
N
)]
1/2
/
N
a
= [
(1 −
)]
1/2
(note that the surface occupation variance is
, where
N
a
=
N
a,exp
is the number of
adsorption sites, and
r
ad
and
r
des
are pure adsorption and desorption rates per adsorption site
(in Hz), respectively. Clearly
occ
has a maximum at
= 1/2. Assuming pure adsorption–
desorption, a characteristic correlation time can be defined for this process as
r
= 1/
(
r
ad
+
r
des
) =
N
/(
sIA
D
), which is computed using the measured data and is plotted in the inset
of Figure 2b. This leads to the excess Allan deviation
A,ad–des
(
A
)
≈
0.29
occ
·(
N
a
)
1/2
(
m
/
M
)
(
A
/
r
)
1/2
in our experimental regime (see Supporting Information). Therefore, considering
only adsorption–desorption noise and system noise, the total expected instability is
. As displayed in Figure 3 (dark gray lines, note that
A,sys
A,ad–des
hence
A,ad,total
~
A,sys
, hardly distinguishable), the estimated adsorption–
desorption noise contribution is much lower even than the system noise floor, thus cannot
account for the measured frequency instability in the presence of a Xe flux.
This predicted low adsorption–desorption noise and the above kinetics data (
E
a
~
E
diff
E
des
) suggest that desorption cannot be the dominant mechanism of Xe atoms leaving the
device surface. Hence we have developed a model to analyze noise due to Xe atoms
diffusion on the device surface (see details in Supporting Information). We compute the
autocorrelation function
G
(
) of the fractional frequency fluctuations
y
(
t
)
f
(
t
)/
f
0
induced
by Xe atoms diffusion
(1)
with
D
=
L
2
/(2
a
2
D
) being the characteristic diffusion time,
a
= 4.42844 is a constant
determined by the shape of the fundamental mode of the device, and
M
is the device’s total
mass. The associated spectral density of fractional frequency fluctuation is
(2)
with
(
x
)
g
(√
x
)/√
x
, and
g
(√
x
) = cos(
x
) + sin(
x
) − 2
C
1
(√
x
)cos(
x
) − 2
S
1
(√
x
)sin(
x
), where
C
1
(
x
) and
S
1
(
x
) are the Fresnel integrals
30
(see details in Supporting Information), and
=
2
f
is the Fourier (offset) frequency.
The measured noise spectra of fractional frequency fluctuations (Figure 4a) clearly
demonstrate that there is excess noise due to surface Xe adsorbates. In the case of no e, the
spectrum exhibits 1/
f
oise at very low offset frequency (~0.1–2 Hz) and flattens out above 2
Hz, indicating the instrumentation noise floor of the system. Figure 4b shows the adsorbate-
induced excess noise spectrum,
S
y,Xe
(
) =
S
y,total
(
) −
S
y,noXe
(
), extracted from such
measurements performed at four different temperatures in the range of interest. We compute
the diffusion time
D
by numerically fitting the measured spectral density data to eq 2, as
shown in Figure 4c. The measurement and analysis yield
D
~ 0.05–0.11s, and we obtain the
corresponding diffusion coefficient
D
=
L
2
/(2
a
2
D
) to be in the range of
D
~ 1.2–2.5 ×
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10
−8
cm
2
/s (we note this is at the same level as reported for Xe diffusion on some other
surfaces
25
). In Figure 4a and b, on the right axis we also show the corresponding adsorbed
Xe mass fluctuation noise
set by the measured fractional frequency
noise.
On the basis of the measured diffusion noise spectra, we estimate the excess Allan deviation
due to diffusion, by integrating the fitted noise spectral density (see Supporting
Information).
31
In our experimental regime (
A
D
), we have
A,diff
(
A
)
≈
0.83·√
N
(
m
/
M
)(
D
/
A
)
1/4
, and the total expected instability due to adsorbate diffusion is thus
. As Figure 3 demonstrates, this model yields excellent
agreement in both the temperature dependency and the magnitude of the experimental data.
Contrasted with the predictions of the pure adsorption–desorption model, this provides
strong support for a surface kinetics dominated by surface diffusion. Very interestingly and
importantly, the combined measurements and analyses have revealed a noise process that
has new power laws
S
y
(
) ~
with
= −1/2 and instability
A
(
A
) ~
A
with
= −1/4,
which to the best of our knowledge have never been documented nor considered in
conventional oscillators and clocks (i.e., all known oscillator noise spectra power laws
involve an integer exponent
=0, ±1, ±2...).
32
The revelation of these new weak power laws
is possible because for the first time we are now able to tune a very sensitive VHF NEMS
resonator to follow in real time the one-dimensional diffusion of adsorbate. It is important to
point out the general character of our diffusion noise analysis, which could straightforwardly
be extended by means of numerical simulations to account for the properties of different
adsorption sites and/or different coverage regimes.
In conclusion, we have investigated the kinetics and fluctuations of surface adsorbate on a
high-
Q
NEMS resonator vibrating at 190.5 MHz. By measuring and analyzing the
temperature-programmed adsorption spectrum and device frequency fluctuations, we have
found that adsorbate diffusion can dominate the observed excess noise. The sensitive
measurements have also led to the discovery of new power laws of frequency noise in low-
dimensional systems. These first measurements of surface adsorbates dynamics and
fluctuations using resonant NEMS may have important implications for at least the
following fundamental and technological applications: (i) The surface adsorbates-induced
noise translates into equivalent mass fluctuations on the device surface; this imposes limits
for the achievable sensitivity of frequency-shift based sensors. (ii) Following this study, it
has been attempted to control and engineer adsorbates distribution (e.g., concentration
gradient) and surface kinetics to enhance NEMS sensors’ functions and performance.
33
(iii)
The approach of using NEMS to probe submonolayer surface physics may prove to be a
useful supplement to the existing surface science tools, particularly when combined with in
situ techniques,
34
to pursue, for example, quantitative, real-time measurements of
mesoscopic processes such as interfacial damping and atomic tribology.
35
Further
engineering with innovative device geometries and resonant mode shapes can also help to
explore nanoscale Chladni figures
36
that may be important for molecular-scale biodetection
in fluids and other complicated surface phenomena. This study also opens new opportunities
for pushing the limits by investigating surfaces of NEMS made of other emerging materials
such as ultrathin nanowires,
37
nanotubes,
38
and atomically thin graphene sheets.
39
Supplementary Material
Refer to Web version on PubMed Central for supplementary material.
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Acknowledgments
We thank S. Stryker for help in engineering the experimental apparatus. We thank C.A. Zorman and M. Mehregany
for custom-made high-quality thin SiC layers. X.L.F. is grateful to M. C. Cross and L. G. Villanueva for helpful
discussions, and to Y. Wu for help with the illustrations. We acknowledge the support from DARPA/MTO and
SPAWAR under the Grant N66001-02-1-8914.
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Figure 1.
Adsorbates fluctuations on the surface of a VHF nanoelectromechanical resonator. (a)
Illustration of an effusive atomic beam delivering a minute amount of Xe atoms onto the
surface of a cold NEMS resonator. (b) Experimental system diagram and schematic of
measurement circuits (with acronyms as detailed in Supporting Information). Shown is the
closed-loop case; in case of open loop NEMS resonance detection, a network analyzer is
connected between nodes (I) and (II). (c) Resonance response near
f
0
= 190.5 MHz
measured via reflectometric network analysis using magnetomotive transduction. The fit
(red lines) gives a quality factor
Q
≈
5620. (d) Calibrated mass responsivity of the device,
ℛ
|∂
f
0
/∂
M
| = 0.99 Hz/zg, measured in continuous mass-loading mode. Inset: scanning
electron microscope (SEM) image of the VHF NEMS resonator; scale bar is 1 μm. (e)
Illustration of Xe atoms-NEMS interactions with atoms kinetics including adsorption–
desorption (out-of-plane) and surface diffusion (in-plane).
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Figure 2.
Surface kinetics of adsorbed Xe atoms on the NEMS device via resonant measurements. (a)
Measured effective surface coverage (
, left axis), along with the calculated surface
occupation deviation (
occ
, right axis) based on measurement data. (b) Measured number of
adsorbed atoms (data: blue circles) and the fit to the Arrhenius equation (dashed line). The
fit yields
E
a
≈
63.1 ± 0.23 meV, and
v
a
≈
(1.23 ± 0.05) × 10
4
Hz. Inset: the characteristic
correlation time assuming a pure adsorption–desorption process,
r
= 1/(
r
ad
+
r
des
) =
N
/
(
sIA
D
).
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Figure 3.
Frequency instability induced by surface adsorbate fluctuations. Red circles are the
measured Allan deviation in the case of constant flux of impinging Xe atoms and blue
squares in the case of no Xe flux (system noise floor). In the range of
T
≈
58–80 K, the
excess frequency fluctuation due to adsorbed Xe atoms is clearly visible. The dashed lines
are excess Allan deviation due to diffusion (violet with small triangles, see entries in Table
1, last column) and pure adsorption–desorption (dark gray), respectively. The solid lines are
total instability levels calculated by adding in quadrature the system instability to that due to
diffusion (violet with large triangles) and adsorption–desorption (dark gray), respectively.
Allan deviation is measured with an averaging time
A
= 1 s to ensure adequately dense data
with respect to the regulated temperature change. Data symbols are average values and error
bars are standard deviations, each computed from ensembles of 500 raw data points.
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Figure 4.
Noise spectra of VHF NEMS resonator due to adsorbed Xe atoms. (a) Measured noise
spectral density of fractional frequency fluctuation at
T
= 59.2 K with and without Xe atoms.
(b) Excess noise spectral density of fractional frequency fluctuations due to Xe adsorbates,
measured at four different temperatures. (c) Least-squares fit of the measured excess noise
spectral density data to the predicted function based on the diffusion noise model (see
Supporting Information) to extract diffusion time constants and coefficients at various
temperatures (results collected in Table 1).
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Table 1
Diffusion Time Constants and Coefficients
a
T
(K)
N
(
T
)
D
(s)
D
(cm
2
/s)
A,diff
(
A
= 1s)
58.0
2.19 × 10
6
0.114
1.18 × 10
−8
1.16 × 10
−6
59.2
1.71 × 10
6
0.064
2.12 × 10
−8
8.86 × 10
−7
60.7
1.27 × 10
6
0.055
2.44 × 10
−8
7.36 × 10
−7
63.4
7.69 × 10
5
0.053
2.53 × 10
−8
5.69 × 10
−7
a
Data derived from the combined analysis in Figure 4 (diffusion noise model) and Figure 2 (loading curve).
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