INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF MICROMECHANICS AND MICROENGINEERING
J. Micromech. Microeng. 15 (2005) 2249–2253 doi:10.1088/0960-1317/15/12/006
Effect of gold coating on the Q-factor of a
resonant cantilever
R Sandberg, K Mølhave, A Boisen and W Svendsen
Department of Micro and Nanotechnology, Technical University of Denmark,
Ørsteds Plads Bldg 345 East, 2800 Kgs. Lyngby, Denmark
E-mail: rks@mic.dtu.dk
Received 3 May 2005, in final form 13 September 2005
Published 28 October 2005
Online at stacks.iop.org/JMM/15/2249
Abstract
The resonance frequency and the Q-factor of the fundamental and higher
order flexural modes of silicon dioxide microcantilevers have been
characterized at different pressures and for different thicknesses of gold
coating. We present the experimental results and discuss the effect of the
gold film on the performance and sensitivity of the cantilevers when used as
mass sensors. An almost linear relationship between the Q-factor and the
resonance frequency of the uncoated cantilevers is observed, implying that a
higher sensitivity can be attained by actuation at higher order resonant
modes. We also find that even a thin gold coating may reduce Q-factors by
more than an order of magnitude.
1. Introduction where the viscous damping term is usually dominant at
atmospheric pressure.
The development and utilization of resonant micro and A gold coating of the cantilever is frequently utilized for
nanocantilevers has increased considerably within the past mass detection. The use of gold is due to the large selection of
few years because of their low cost and high sensitivity as materials that can be adsorbed via thiol chemistry as described
measurement and detection probes. in recent review articles by Lavrik et al [3] and Ziegler [4].
When used as a mass detector, the cantilever surface A specific area of application is DNA biochips, capable of
can be coated with a functionalization layer for adsorption detecting specific DNA sequences. Because of their great
of specific chemical compounds. Adsorbed molecules will potential, such devices have attracted much attention, both
add to the mass of the cantilever and thereby cause a change experimentally [5, 7] and theoretically [6].
in the resonance frequency which can be detected. In our characterizations of silicon dioxide cantilevers,
The sensitivity of the resonant cantilever when used as we have found that higher order resonant modes exhibit
a mass sensor depends on the spectral resolution, which is considerably higher Q-factors than the fundamental modes,
directly related to the Q-factor of the resonant mode, defined implying that the higher order modes potentially will yield
as [1] higher sensitivities in resonant cantilever sensor applications.
stored vibrational energy f0 As most cantilever sensor applications require operation
Q = 2π = , (1) at atmospheric pressure, great effort has been put into
energy lost per cycle of vibration f
describing the effect of pressure on the resonant properties
where f0 is the resonance frequency of the mode and f is
of cantilevers—both experimentally [8–11] and theoretically
the FWHM of the resonance peak in the frequency domain.
[12–15]. The different approaches have yielded good
The total energy loss per cycle may be expressed as a results for different aspects of the research area, but as
sum of different loss sources with corresponding Q-factors: the constellation of temperature, pressure and gold coating
internal material loss (Qi ), loss to the chip substrate through
effects is highly complex, many aspects are still to be
the cantilever support (Qs ) and viscous (and acoustic) loss [2]
discovered—especially that of internal cantilever losses, which
to the surrounding medium (Qa ). The total Q-factor is then
the theoretical models do not account for with sufficient detail
given by
to be directly applicable.
1 1 1 1 We have, therefore, studied these effects and present here
= + + , (2)
Q Qi Qs Qa the experimental results from the characterization of a series
0960-1317/05/122249+05$30.00 © 2005 IOP Publishing Ltd Printed in the UK 2249
R Sandberg et al
of monolithic silicon dioxide cantilevers with high intrinsic Table 1. Material data for SiO2 , Ti and Au.
Q-factors (on the order of 103 –104 ), onto which we have Material ρ (g cm−3 ) E (GPa)
applied gold coatings of varying thicknesses and measured
the resonant properties at different pressures. SiO2 2.15 70
Ti 4.5 116
Au 19.3 78
2. Experimental procedure Values are obtained from the material database
of CoventorWare and webelements.com.
2.1. Cantilever devices
A chip containing six monolithic silicon dioxide cantilevers
with different lengths was used in the measurements. The and the Q-factor is given by the ratio between the resonance
silicon dioxide is 0.85 µm thick and was grown thermally on a frequency and the FWHM of the Lorentz peak.
Si (1 0 0) wafer. Using UV lithography and reactive ion etch, The resonance frequencies and the Q-factors of all
the oxide was patterned to form the cantilevers [16]. measurable modes of the six cantilevers were measured at
The cantilevers are all 10 µm wide with lengths of 89, a pressure of 50 Pa (henceforth denoted as vacuum) and at a
113, 137, 161, 185 and 209 µm. The lateral dimensions are pressure of 100 kPa in pure nitrogen (atmospheric pressure).
measured with an accuracy of about 1 µm using an optical In addition to this, and in order to get a more detailed
microscope. observation of the pressure dependence of the cantilever
In order to observe the effect of different coating dynamics, one uncoated and one 400 nm coated cantilevers
thicknesses, the SiO2 cantilevers were characterized, as were characterized at a series of 12 logarithmically spaced
described in section 2.2, without any gold coating, and with pressures between 50 Pa and 200 kPa.
three different thicknesses of gold, deposited on the top
surface. In the first deposition a 10 nm titanium layer was 3. Results
formed to ensure good adhesion to the SiO2 and a subsequent
layer of 100 nm Au were deposited. In the second deposition 3.1. Resonance frequencies
another 100 nm Au and in the third deposition, an additional
200 nm Au was formed. The cantilever characterizations were, The resonance frequencies of the cantilever modes can be
therefore, done for 0 nm, 100 nm, 200 nm and 400 nm Au deduced analytically as [18]
coating layers. 1/2
h E(z − z0 ) dz
2
ωn (κn )2
The depositions were done in an Alcatel E-beam fn = = N , (3)
evaporator with a chamber pressure of 10−4 Pa and a deposition 2π 2π2 i=1 (hi ρi )
rate of 10 Å s−1 for Ti and 5 Å s−1 for Au. where n is the mode number, and w are the length and width
of the cantilever, hi and ρi are the height and mass density
2.2. Characterization method of each of the N cantilever layers, E is Young’s modulus of
elasticity, which for a layered cantilever will be a piecewise
The resonance frequency and the Q-factor of the cantilevers constant function of z. The values used for E and ρ are listed
were measured by means of a Hewlett Packard model 4194A in table 1. z0 is the position of the neutral axis and κn is the
gain-phase analyser (GPA). The cantilever chips were placed modal parameter which has the numerical values
in a custom built vacuum chamber [17] with capabilities of
accurate control of pressure and temperature. A piezo-electric κn 1.875, 4.694, 7.855, 11.00, 14.14, 17.28, 20.42 (4)
transducer (PZT) driven by the output of the GPA was used for the first seven flexural modes.
to actuate the cantilever chips. Readout of the cantilever The measured values of fn,M for our cantilevers are in very
deflection was done by reflecting the focused beam of a HeNe good agreement with this expression; the relative difference
laser off the cantilever top surface and collecting it with a between the measured and the analytical values (fn,M −fn )/fn
focusing lens onto a position sensitive photo detector (PSD). is less than 5% and on an average 2.5% (RMS value over all
The converted and amplified photocurrent of the PSD was then values for coated and uncoated cantilevers).
used as an input on the GPA test channel.
When the position of the laser beam on the cantilever
3.2. Q-factor dependence on the resonance frequency
and the oscillation level of the PZT are optimized, to yield
the highest signal-to-noise ratio without distorting the width In the characterization of each cantilever, a clear tendency
or position of the small-vibration resonance peak, resonance towards higher Q-factors for the higher order modes was
frequencies up to approximately 2 MHz of flexural cantilever observed. The Q-factors of the shorter cantilevers were also
modes can be detected. This frequency is mainly limited by higher than those of the longer cantilevers. This is illustrated
the rise time of the PSD. For the 209 µm long cantilever in figure 1, which shows the Q-factors in vacuum of all the
this corresponds to the 7th, flexural mode, while only three measured modes of the uncoated cantilevers. The observed
flexural modes are detectable for the 89 µm long cantilever. Q-factor variation can be described by a frequency
The measurements are carried out by using the GPA to sweep dependence, where higher resonance frequency of a mode
the actuation–detection frequency over a range, and fitting the implies a higher Q-factor of that mode.
measured amplitude with a Lorentz function. The resonance The Q-factors for resonance frequencies below 1 MHz
frequency is taken as the peak position of the Lorentz curve appear to be linearly dependent on resonance frequency, but
2250
Effect of gold coating on the Q-factor of a resonant cantilever
Mode 1
4 Mode 2
4
10 10
Mode 3
Mode 4
Mode 5
Mode 6
Q –factor
Q – factor
L= 89 µm
L=113 µm
3
L=137 µm 10
3
10 L=161 µm
L=185 µm
L=209 µm
4 5 6
10 10 10 0 50 100 150 200 250 300 350 400
Frequency/Hz Au coating thickness/nm
Figure 1. Q-factors in vacuum of all measured flexural modes for Figure 2. Q-factors in vacuum of the first six flexural modes of the
each of the six uncoated cantilevers, graphed as a function of the cantilever with length = 209 µm as a function of gold coating
resonance frequencies of the modes. The dotted line (Q ≈ thickness.
0.123f0 0.855 ) is a power law fit to the points below 1 MHz. The
3
dashed-dotted line is the theoretical Q-factor limit from molecular 10
air damping.
are best fitted by a power law as Q ≈ 0.123f0 0.855 (dotted
line). The apparent falloff at Q ≈ 15000 for frequencies
above 1 MHz is assumed to be the maximum attainable
Q –factor
Q-factor, limited by internal friction in the SiO2 material. 2
10
The almost linear dependence between the Q-factors and
Mode 1
the resonance frequencies is supported by the theory presented
Mode 2
by Blom et al [12]. Blom derives an expression for the Mode 3
Q-factor caused by air damping in the molecular Knudsen Mode 4
region, which is applicable for our cantilevers at pressures of Mode 5
approximately 25 Pa and below. This is plotted in figure 1 and Mode 6
is seen to correspond very well to the measured values. The 1
10
deviation may be accounted for by inaccuracies in determining 0 50 100 150 200 250 300 350 400
Au coating thickness/nm
the absolute pressure and the Q-factors.
Figure 3. Q-factors in 100 kPa N2 of the first six flexural modes of
the cantilever with length = 209 µm as a function of gold coating
3.3. Effect of gold coating in vacuum thickness.
By characterizing the cantilevers in a vacuum environment, the
loss due to viscous damping (Qa ) can be eliminated and the the value on average by the 100 nm coating, while the Q-
relative effect of a gold coating on the intrinsic properties of factor of modes 4 and above decrease by at least an order of
the cantilever can be explored. Figure 2 shows the measured magnitude.
Q-factors of the first six flexural modes of the cantilever with
length = 209 µm as a function of gold coating thickness. 3.4. Effect of gold coating at atmospheric pressure
The internal friction in gold is much higher than in SiO2
The Q-factors at atmospheric pressure (100 kPa nitrogen)
[19] and the gold coating, therefore, gives rise to a higher
are significantly lower than in vacuum. In contrast to the
internal loss and thus a reduced Q-factor. The most significant observations in vacuum, the Q-factors at atmospheric pressure
reduction in the Q-factor appears already for the 100 nm increase slightly with increasing gold coating thickness, as
coating and further increase of the coating thickness leads only seen in figure 3.1 As the energy loss per cycle of vibration is
to a small additional reduction. This suggests that the Q-factor assumed to be increased by the coating, as seen in figure 2,
of resonant cantilever sensors is impaired significantly even by we see from equation (1) that the increasing Q-factor can
very thin film gold coatings and that a significant amount of be explained as an increase in total vibrational energy. This
thermoelastic damping occurs at the surface boundary between makes good sense considering the increased mass density of
SiO2 and Au. the gold-coated cantilever. The mass of the 850 nm thick SiO2
The same tendency in Q-factor reduction that is illustrated cantilever is almost doubled by the addition of 100 nm Au,
in figure 2 is observed for all of the characterized cantilevers.
1 The Q-factor measurements at atmospheric pressure are subject to a low
From the figure, we see that the higher order modes are much
signal-to-noise ratio; especially for the low order modes. The apparent
more severely affected by the gold coating than the low order decrease in the Q-factors for modes 1 and 2 from 100 nm to 200 nm coating
modes. The Q-factor of mode 1 drops to approximately half is, therefore, assumed to be the result of measurement inaccuracy.
2251
R Sandberg et al
4 4
M5, L=209µm
10 Mode 2 10 M4, L=161µm
Mode 3
M3, L=113µm
Mode 4
Mode 5 No coating
No coating 100 nm Au
400 nm Au 400 nm Au
Q – factor
3 3
10 10
2 2
10 10
(a) (b)
2 3 4 5 2 3 4 5
10 10 10 10 10 10 10 10
Pressure (Pa) Pressure (Pa)
Figure 4. (a) The Q-factors of modes 2–5 of an uncoated cantilever and a cantilever with a 400 nm Au coating, illustrating the general
pressure dependence. (b) The Q-factors of three modes of different cantilevers having almost identical resonance frequencies. The drawn
lines should not be interpreted as an interpolation between the measured values but rather as a visual aid in distinguishing the different data
points.
implying that the vibrational energy for a constant oscillation and the Q-factor. A high resonance frequency implies a
amplitude is almost doubled as well. high Q-factor. For sensor applications, an obvious conclusion
would therefore be that using a higher order mode for detection
3.5. Pressure dependence of Q-factor will yield a higher sensitivity.
We furthermore observed that the relative reduction in the
The detailed pressure dependence of the Q-factors of an Q-factor for increasing pressure did not depend on resonance
uncoated and a 400 nm gold-coated cantilever is illustrated mode number or cantilever length but only on the thickness of
in figure 4(a). The uncoated cantilever has a length of = the gold coating. The Q-factors of uncoated cantilevers were
195 µm and the coated cantilever has a length of = 185 µm. much more reduced by viscous damping than the Q-factors of
Their resonance frequencies deviate by approximately 10%. coated cantilevers.
The Q–P plot shows a general pressure dependence that In vacuum, the Q-factor is severely reduced by the
resembles a straight line (i.e., Q ≈ βP α ). Though this deposition of even a thin gold film, especially for higher
description is not entirely accurate, it is useful for a qualitative order modes. This sets a limit on the high sensitivity of
treatment of the modes that have only been characterized at gold-coated nanocantilever sensors, and calls for alternative
vacuum and atmospheric pressure, as seen in figure 4(b). functionalization methods, such as partial gold coating of
The almost parallel curves in figure 4(a) suggest that the the cantilever tip or even functionalization of separate objects
relative Q-factor decrease is identical for all modes and only attached to uncoated cantilevers [20].
varies with the coating thickness. To verify this, we have Finally, it should be mentioned that the cantilevers
identified three different order modes on different cantilevers characterized here are all much longer than their width, and
with almost identical resonance frequencies: much wider than their thickness. The observed Q-factor
mode 5 for = 209 µm with f5 = 1027 kHz, dependences on pressure and coating thickness may be specific
to this w h geometry, and different conclusions may
mode 4 for = 161 µm with f4 = 1038 kHz,
be drawn for significantly different geometries, which will
mode 3 for = 113 µm with f3 = 1065 kHz. therefore be investigated further.
The Q-factors of these three modes are plotted in
figure 4(b) for vacuum and atmospheric pressure and for References
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