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JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 9, NO. 1, MARCH 2000 117
Quality Factors in Micron- and Submicron-Thick
Cantilevers
Kevin Y. Yasumura, Timothy D. Stowe, Eugene M. Chow, Timothy Pfafman, Thomas W. Kenny, Barry C. Stipe,
and Daniel Rugar, Member, IEEE
Abstract—Micromechanical cantilevers are commonly used for
detection of small forces in microelectromechanical sensors (e.g.,
accelerometers) and in scientific instruments (e.g., atomic force
microscopes). A fundamental limit to the detection of small forces
is imposed by thermomechanical noise, the mechanical analog of
Johnson noise, which is governed by dissipation of mechanical
energy. This paper reports on measurements of the mechanical
quality factor
for arrays of silicon–nitride, polysilicon, and
single-crystal silicon cantilevers. By studying the dependence
of
on cantilever material, geometry, and surface treatments,
significant insight into dissipation mechanisms has been obtained.
For submicron-thick cantilevers,
is found to decrease with
decreasing cantilever thickness, indicating surface loss mecha-
nisms. For single-crystal silicon cantilevers, significant increase in
room temperature
is obtained after 700 C heat treatment in
either N
or forming gas. At low temperatures, silicon cantilevers
exhibit a minimum in
at approximately 135K, possibly due to
a surface-related relaxation process. Thermoelastic dissipation is
not a factor for submicron-thick cantilevers, but is shown to be
significant for silicon–nitride cantilevers as thin as 2.3
m. [434]
Index Terms—Cantilever, force sensor, mechanical dissipation,
micromechanical resonator, quality factor, surface losses.
I. INTRODUCTION
T
HE majority of microfabricated sensors measure forces
applied to micromechanical flexures. Examples include
pressure sensors, which measure force on a diaphragm, and
accelerometers, which measure inertial force on a proof
mass. Many of these microfabricated sensors are capable of
measuring surprisingly small forces. For example, the Analog
Devices ADXL05 accelerometer features a proof mass of
approximately 10
kg and is capable of detecting an acceler-
ation as small as 5
10 times the acceleration of gravity in a
1-Hz bandwidth.
1
This acceleration represents a force of 0.5 pN
applied to the mass.
Manuscript received March 26, 1999; revised November 12, 1999. An earlier
versionofthis paperwas presentedat the 1998TranducersResearch Foundation
Hilton Head Workshop. Thisworkwas supportedby theNational Science Foun-
dation under CAREER Award ECS-9502046, by the National Science Founda-
tion under GOALI Award ECS-9422255, by the National Science Foundation
Instrumentationfor MaterialsResearchProgram underContractDMR9504099,
under aTerman Fellowship,and by the Officeof Naval Research under Contract
N00014-98-C-0070. Subject Editor, W. N. Sharpe, Jr.
K. Y. Yasumura, T. D. Stowe, E. M. Chow, and T. W. Kenny are with the
Departments of Applied Physics, Electrical Engineering, and Mechanical
Engineering, Stanford University, Stanford, CA 94305-4021 USA (e-mail
kevin@micromachine.stanford.edu).
T. Pfafman was with the Department of Mechanical Engineering, Stanford
University, Stanford, CA 94305-4021 USA. He is consulting in the Bay area.
B. C. Stipe and D. Rugar are with the IBM Research Division, Almaden Re-
search Center, San Jose, CA 95120-6099 USA.
Publisher Item Identifier S 1057-7157(00)02134-X.
Ordinarily, sensor performance is improved by reducing the
noise of the preamplifier used to convert the physical signals to
electrical signals, and by controlling other error sources such as
uncompensated thermal drift. There eventually comes a point,
however, where thermodynamics imposes a barrier to further
sensor improvement.For the caseof microcantilevers optimized
for use in force detection, thermomechanical noise sets a limit
to the ultimate force resolution [1].
Thermomechanical noise is a consequence of the cantilever
being in thermal equilibrium with its environment (i.e., a heat
bath with many microscopic degrees of freedom). Energy dissi-
pation in the cantilever causes the stored mechanical energy to
leak away and be converted into heat. The stronger the coupling
between the cantilever and heatbath, the faster the decay of can-
tilever motiontoward thermal equilibrium and the lowertheme-
chanical quality factor
of the oscillating mode. Conversely,
the coupling to the heat bath has the consequence that the can-
tilever will be subjected to constant random excitation by its in-
teraction with the many microscopic degrees of freedom in the
heat bath. This relationship between the energy dissipation and
random thermal excitation is embodied in the “fluctuation–dis-
sipation theorem” of statistical mechanics,which applies to me-
chanical systems just as it applies to the Johnson noise across
an electrical resistor [2, pp. 572–573]. The net result is that the
lowerthe mechanical
of the system, thelarger the force noise.
The equipartition theorem gives a measure of how much
thermal energy is in each mode of a microcantilever [2, pp.
248–249]. The mean square vibration amplitude associated
with a mode of oscillation at temperature
is given by
(1)
where
is the cantilever spring constant and is the cantilever
displacement. We can calculate the equivalent force noise as-
sociated with mechanical dissipation by assuming that the can-
tilever behavesas a simple harmonic oscillator and imposingthe
requirement that random thermal excitations must produce the
mean square vibration amplitude given by (1).The mean square
vibration amplitude is the integral over all frequencies of the
force noise spectral density multiplied by the square of the me-
chanical transfer function
(2)
1
Analog devices technical data sheet for ADXL05 single-chip accelerometer
with signal conditioning
1057–7157/00$10.00 © 2000 IEEE
118 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 9, NO. 1, MARCH 2000
where the transfer function is
(3)
With the assumption that the force noise spectrum is white (i.e.,
frequency independent), (1)–(3) lead to a spectral density of
, where is the cantilever res-
onance frequency. This spectral density results in a force noise
in a bandwidth
of
(4)
For a simple rectangular cantilever, this minimum detectable
forcecan also beexpressedintermsofthe cantileverdimensions
(5)
where
is the Young’s is the mass density of the cantilever
material,
is the cantilever width, is the cantilever length, and
is the cantilever thickness.
From (5), for the minimum detectable force, a strategy can be
found to design ultrasensitive cantilevers: make them narrow,
thin, and long. This strategy is effective only if high mechan-
ical
is maintained. Unfortunately, relatively little is under-
stood about the mechanisms responsible for energy dissipation
in micron- and submicron-thick microstructures. To better un-
derstandthe dissipation mechanisms,we havestudiedcantilever
factor as a function of controllable cantilever properties such
as material, geometry, and surface treatment. In this paper, we
present the results from this ongoing study.
II. L
OSS MECHANISMS
For a cantilever operating in vacuum, vibrational energy can
be dissipated via coupling to the support structure (clamping
loss) and by internal friction. Internal friction results from a va-
riety of physical mechanisms, including motion of lattice de-
fects, thermoelastic dissipation (TED), phonon–phonon scat-
tering, etc. [3]. Traditionally, internal friction is considered as a
bulk (volume) effect, but surface effects can dominate for sub-
micron-thick cantilevers or for resonators with very high
[4],
[5].
Cantilever
is defined as , where is
the stored vibrational energy and
is the total energy lost
per cycle of vibration. Since
can be written as
, where represents the energy lost due to the var-
ious dissipation mechanisms, we can write the inverse
as the
sum
(6)
where we have explicitly included terms to characterize
clamping loss, TED, bulk internal friction (other than TED),
and surface effects.
Clamping loss was studied theoretically by Jimbo and Itao
[6],[7] using a two-dimensional theory thatmodeledthesupport
structure as an infinitely large elastic body. Their calculations
result in the estimate
. For all of the
cantilevers studied here,
, giving
. Since this is significantly larger than any of our measured
values, we conclude that clamping loss does not limit for
our structures.
The effect of TED will be considered in detail in Section VII.
In general, for submicron-thick cantilevers operating at kilo-
hertz frequencies, TED is negligible. However, as we shall dis-
cuss later, TED can become the dominant source of energy dis-
sipation in thicker silicon–nitride cantilevers.
In order to better understand the effect of internal friction and
the relative importance of volume and surface contributions, we
nowconsider a simple model of a vibrating cantilever where we
treat the stress and strain in the cantilever as scalars (ignoring
tensor properties) and assume that the cantilever vibration am-
plitude is small compared to the length of the cantilever. The
energy dissipation is modeled by considering a complex-valued
Young’s modulus
, where is the conventional
(real-valued) Young’s modulus and
is the dissipative part,
which we will assume to be small compared to
. For dissi-
pative processes that occur on the atomic scale (such as mo-
tion of lattice defects),
can be considered to be a property
of the material and its defects, though it may have substantial
frequency and temperature dependence depending on the types
of processes involved. For more macroscopic processes, such as
TED,
can also have explicit dependence on cantilevergeom-
etry.
For a cantilever vibrating sinusoidally such that the strain is
given by
, where is time, the stored
energy can be written in terms of the peak elastic energy [8]
(7)
where σ is the longitudinal stress and
is the peak strain.
For a simple rectangular cantilever, the volume integral is over
the cantilever thickness (
to ), width (
to ), and length ( to ).
The exact form of
depends on the cantilever geometry
and vibrational mode shape. For a cantilever with rectangular
cross section, the strain is uniform across the width of the can-
tilever and increases linearly in
away from the neutral plane
(
). Accordingly, we write
(8)
where
is the strain that occurs along the top surface of
the cantilever (where the strain is maximum). This expression
holds even for higher order vibrational modes of a rectangular
cantilever. Combining (7) and (8) gives
(9)
YASUMURA et al.: FACTORS IN MICRON- AND SUBMICRON-THICK CANTILEVERS 119
Now consider energy dissipation. For a bulk (volume) loss
mechanism, the energy lost per cycle can be written as [8]
(10)
Here,
signifies integration over a complete cycle of vibration.
Note that if
and are constants (as they would be for a
cantilever made of a single material), the volume integrals on
the right-hand sides of (7) and (10) differ only by a constant
factor. Thus, the
factor is independent of cantilever geometry
and vibrational mode shape and is given by
(11)
Next consider the effect of a thin surface layer that exhibits
enhanced dissipation. The dissipation enhancement may be due
to the disruption of the atomic lattice at the surface or due to a
thin layer of surface contamination. In most cases, the surface
layer will not substantially change the stored energy in the can-
tilever, but it can significantly enhance the dissipated energy.
Characterizing the surface layer by thickness
and complex
modulus
, the energy lost per cycle due to
the surface layer is
(12)
where the volume of integration is confined to the surface layer,
which we assume to be on the top, bottom, and sides of the
cantilever. If
is small compared to the dimensions of the can-
tilever, (12) can be converted to a sum of surface integrals and
then evaluated using (8) to yield
(13)
Using (9) and (13) to evaluate the
factor, we obtain
(14)
If we define the
factor of the material that comprises the sur-
face layer as
, then (14) may be rewritten as
(15)
Fora thin wide cantilever where
, is proportional
to cantilever thickness and given by
(16)
Based on the results in (15) and (16), we would, therefore, ex-
pect to see a strong thickness dependence to the cantilever
factor should the dominant loss mechanism be surface related.
III. C
ANTILEVER FABRICATION
Three materials have been used to fabricate our can-
tilevers—silicon nitride, polysilicon, and single-crystal silicon.
The first, silicon nitride, was chosen because of its durability,
Fig. 1. Array of 2000-Å-thick silicon–nitride cantilevers. Shown is part of
an array of
m cantilevers of length varying from 150 to 300
m.
Cantilever arrays made from silicon nitride and polysilicon use this array
pattern.
ease of fabrication, and general use as a processing material.
The second material, polysilicon, was chosen because of its
wide use as a micromechanical sensor material. A number of
fabrication processes rely on a top polysilicon layer from which
a sensor or device is fabricated, making a study of dissipation
in polysilicon resonators of wide interest. The last material,
single-crystal silicon, was chosen because of its expected low
internal friction as exhibited by larger bulk oscillators [3], [9]
and low internal stress, allowing for the fabrication of ultrathin
cantilevers with little or no curling [10].
Silicon–nitride cantilevers, as shown in Fig. 1, were fabri-
cated from low-stress silicon nitride grown using low-pressure
chemical vapor deposition on
100 silicon wafers. After film
deposition, the cantilevers were patterned by photolithography
and then defined using an SF
dry etch. Tetramethylammonium
hydroxide (TMAH) was then used to etch away the exposed sil-
icon and undercut the cantilevers, thereby releasing them. After
rinsing in water and methanol, the cantilevers were dried using
aCO
critical point drier [11].
Polysilicon cantilevers were fabricated from polysilicon-on-
insulator wafers. Starting with a
100 silicon wafer, a 4000-Å-
thick layer of thermal oxide was grown. Next, a polysilicon
layer of the desired thickness was deposited and cantilevers
were etched using an SF
plasma etch. Protective layers of low-
temperatureoxideandsiliconnitride were then deposited to pro-
tect the cantilevers during subsequent backside patterning and a
TMAH etch. The topside silicon nitride layer was then removed
and a buffered oxide etch (BOE) was used to free the oxide-en-
cased cantilever structures. Finally, a critical point drying step
was performed.
Single-crystal silicon cantilevers start with
100 sil-
icon-on-insulator wafers. A thermal oxidation was performed
to thin down the top silicon layer to the desired cantilever
thickness. BOE was then used to remove the top oxide layer
exposing the top silicon layer for cantilever patterning. As in
the polysilicon process, low-temperature oxide and silicon–ni-
tride were deposited for frontside protection and as a backside
masking layer during the backside TMAH etch. The cantilevers
were then released with BOE followed by critical point drying.
Fig. 2 shows an array of 1700-Å-thick single-crystal silicon
cantilevers. Further details of the fabrication process can be
found in the work of Stowe et al. [10].
120 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 9, NO. 1, MARCH 2000
Fig. 2. Array of 1700-Å-thick single-crystal silicon cantilevers. The
cantilevers have necks of width 5
m and lengths from 80 to 260 m.
Fig. 3. Diagram of the room-temperature measurement system. Shown are
the piezoelectric disk, cantilever die, and laser doppler vibrometer. Not shown
is a viewportthrough which theincident laser passes throughbefore striking the
target cantilevers. The objects in this diagram are not to scale.
IV. MEASUREMENTS
Unless otherwise specified, all measurements were per-
formed at room temperature. A diagram of the experimental
system used for room-temperature measurements is shown
in Fig. 3. The cantilevers were placed in a vacuum chamber
on a stack of piezoelectric disks. A viewport on the side
of the chamber allows the use of an external laser doppler
vibrometer to measure the cantilever motion. The laser doppler
vibrometer is a commercially available system which measures
the velocity-dependent doppler shift of the reflected laser
radiation.
2
Low-temperature measurements were performed
in a separate apparatus that uses a fiber-optic interferometer
detection system.
A “free ring-down” technique was used tomeasure cantilever
.Thecantileverswerefirstdrivenon-resonancetoasteady-state
amplitude. The drive excitation was produced by applying an
oscillating voltage to the piezoelectric disks on which the can-
tilever die was mounted. The drive was stopped abruptly and the
cantilever motion was measured as the amplitude decayed. The
ring-down was then fit to an exponential function. From the fit,
the
decay time constant of the ring-down was obtained.
The decay time constant allows calculation of the cantilever
accordingto .The quality oftheexponentialfitto the
decaydataallowsustobesureoftheaccuracyofeach
measure-
ment.Thedatapresentedinthispaper,unlessotherwiseindicated,
2
302 sensor head with OVF3001 controller, Polytec P.I., Costa Mesa, CA.
Fig. 4. Mechanical factorversus pressurefor a 5100-Å-thicksilicon–nitride
cantilever. Above
1 mtorr, the is pressure limited. Below this pressure, the
is limited by intrinsic dissipation mechanisms.
Fig. 5. Mechanical factor versus cantilever length for silicon–nitride
cantilevers of thickness 2000, 5100, 7000 Å, 1.2 and 2.3
m. Cantilever width
is 10
m for all five thicknesses.
are averages of multiple ring-down measurements for each can-
tilever. The error for each individual cantilever ring-down mea-
surement is on the order of a few percent.
One important source of energy dissipation in micromechan-
ical oscillators is air damping. Fig. 4 shows a plot of
versus
pressure for a 5100-Å-thick silicon–nitride cantilever. For this
cantilever, we can see that the
is pressure limited above a
pressure of approximately 1 mtorr. Below 1 mtorr, the dissipa-
tion associated with air damping becomes negligible compared
to intrinsic loss mechanisms. The work described in this paper
was performed under vacuum at a pressure of 10
torr. This
is below the pressure limited transition point for the cantilevers
studied.
In order to test the repeatability and long-term stability of a
cantilever’s
, the was measured for a polysilicon cantilever
(
m, m, and m) over a period of
2 h. Over this 2-h time span the
had a standard deviation of
1%, demonstrating excellent measurement reproducibility. On
the other hand, after exposure to laboratory air for several days,
cantilever
’s can change by 10% or more, depending on the
cantilever thickness. Presumably, these changes are due to oxi-
dation and contamination of the cantilever surface.
Measurements of
for arrays of silicon–nitride cantilevers
were carried out for thicknesses of 2000, 5100, 7000 Å, 1.2 and
2.3
m, lengths from 90 to 300 m, and widths of 5, 10, 25, and
50
m. Fig. 5 shows the data for the 10- m-wide cantilevers.
YASUMURA et al.: FACTORS IN MICRON- AND SUBMICRON-THICK CANTILEVERS 121
Fig. 6. Mechanical factor versuscantileverthickness for fivesilicon–nitride
cantilever thicknesses (2000, 5100, 7000 Å, 1.2 and 2.3
m). Each of the five
clusters of points represents hundreds of individual
measurements.
Fig. 7. Mechanical factor versus cantilever length for silicon–nitride
cantilevers of thickness 1.2
m. Cantilevers of widths 5, 10, 25, and 50
m
are shown.
Several trends are clear from this data. First, the is roughly
independent of the cantilever length for all of the thicknesses,
except the 2.3-
m-thick cantilevers. Both bulk (volume) and
surface-dependentdissipative processes are expectedto produce
length independent
’s. The lower for the shortest of the
2.3-
m-thick cantilevers is believed to be caused by TED and
will be discussed in greater detail later in this paper.
Fig. 6 shows a plot of cantilever
versus cantilever thick-
ness. In Fig. 6, there is an increase in the mechanical
as the
thickness of the cantilever increases. A linear fit to the four
thinnest cantilever thicknesses is shown. For the four thinnest
cantilever thicknesses, the strong thickness dependence is in-
dicative of surface loss mechanisms, as discussed earlier.
Fig. 7 shows data for silicon–nitride cantilevers of thickness
1.2
m. The cantilever widths are 5, 10, 25, and 50 m. The
’s of these cantilevers are independent of the cantilever width.
Just as the cantilever
is expected to be length independent for
both volume and surface dissipative processes, the cantilever
should also be independent of cantilever width when
[see (16)].
Single-crystal silicon cantilevers show geometrical depen-
dence similar to the silicon–nitride devices. Fig. 8 shows data
for single-crystal silicon cantilevers of thickness 600, 1700, and
2400 Å. Consistent with the data in Figs. 5 and 6, we see that
the
is roughly independent of cantilever length and increases
with increasing cantilever thickness.
Fig. 8. Mechanical factor versus cantilever length for single-crystal silicon
cantilevers of thickness 600, 1700, and 2400 Å.
Fig. 9. Mechanical factor versus cantilever lengthfor a die of 1700-Å-thick
single-crystal silicon cantilevers showing the effect of heat treatment.
V. H EAT TREATMENT
As seen in Figs. 6 and 8, the mechanical
’s of both
silicon–nitride and single-crystal silicon cantilevers depend
on cantilever thickness, suggesting surface-dominated dissipa-
tion mechanisms. Possible sources of surface dissipation are
adsorbates on the cantilever surface or surface defects created
during the cantilever fabrication process. Both of these sources
might be expected to respond to heat treatment. To test this
hypothesis, 1700-Å-thick single-crystal silicon cantilevers
were heated for 1 h at 700
C in a nitrogen atmosphere. As
shown in Fig. 9, the
increased by about a factor of three. In
addition, 1-h heat treatment in 700
C forming gas (Ar with
4.25% H
) produced consistent factor of two increases in for
700-Å-thick silicon cantilevers.
VI. L
OW-TEMPERATURE BEHAVIOR
Low-temperature behavior of microcantilevers is of interest
in experiments designed to achieve the lowest possible force
noise. Fig. 10 is a plot of
versus temperature for a heat-
treated 700-Å- thick single-crystal silicon cantilever with a res-
onance frequency of 3.8 kHz and spring constant of 4
10
N/m. The room temperature of this cantilever was approxi-
mately 1
10 . Upon cooling, the cantilever first decreases,
reaching a broad minimum of 4
10 centered at approxi-
mately 135 K. By 77 K, the
recovers its room temperature
value and then increases rapidly as the temperature is lowered
further. At 4 K, the
reaches 2.5 10 , roughly 2.5 times
122 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 9, NO. 1, MARCH 2000
Fig. 10. Mechanical factor versus temperature for a Å,
m, and m single-crystal silicon cantilever. The resonance
frequency of this cantilever was 3.8 kHz. The solid curve is meant as a guide to
the eye.
Fig. 11. Mechanical factor versus oscillator thickness for single-crystal
silicon oscillators at
K. Line fit is to the five thinnest oscillators.
the room-temperature value. Similar data was obtained for non-
heat-treated cantilevers. Even though the room temperature
’s
of heat-treated cantilevers were a factor of two larger than for
nonheat-treated cantilevers, both heat- and nonheat-treated can-
tilevers had comparable
’s at 4 K.
Similar
minima have been observed in all of our silicon
microcantilevers, both heat treated and nonheat treated, and in
micron-thick commercial silicon cantilevers. Previous studies
on large single crystals of silicon have reported internal friction
peaks in the range of 115–124 K [12], [13]. Since our observed
dissipation peak is centered at higher temperature and is some-
what broader than that found in the bulk studies, it is likely to
have a different origin. Some type of relaxation process related
to surface imperfections, oxidation, or adsorbed contaminants
is a possible cause.
Although the low temperature
of the 700-Å-thick can-
tilever in Fig. 10 reaches 25000 at 4 K, it is still significantly
lower than has been observed in larger silicon oscillators.
Fig. 11 shows a compilation of data from our work and from
the literature at 4 K. As in the case of silicon–nitride cantilevers,
the
of single-crystal silicon oscillators is highly dependent
on the oscillator thickness, suggesting surface-dominated loss
mechanisms. Since this plot has data for different kinds of
oscillators, it should not be over interpreted, but the general
trend is evident—an increase in the mechanical
as the
oscillator dimensions increase. Included in Fig. 11 is data
for ultrathin silicon cantilevers [10], [14], doubly supported
high-frequency silicon resonators [15], commercially available
silicon cantilevers [16], double torsional oscillators [17], and
suspended bulk silicon crystals [12]. Shown in Fig. 11 is a
line fit to the five thinnest oscillators (
, , ,
Å, and 1.5 m). These micron- and submicron-scale
oscillators have a roughly linear
versus dependence. The
larger oscillators deviate from this linear dependence possibly
because loss mechanisms other than surface mechanisms begin
to dominate and limit the
as the oscillator size increases.
VII. TED
Another possible source of energy loss in cantilever
microstructures is TED [18], [19]. Thermoelastic energy dissi-
pation is caused by irreversible heat flow across the thickness
of the cantilever as it oscillates. The rate of energy dissipation
due to this heat flow is dependent upon the cantilever geometry,
material, and temperature. Roszhart [20] has shown that TED
can be a dominant source of energy loss in single-crystal silicon
cantilevers as thin as 10
m. Although TED generally becomes
less important as thickness is reduced, we show below that
it can be significant in silicon–nitride cantilevers as thin as
2.3
m.
If a load is applied to a cantilever that causes it to deflect
in a flexural mode (a mode that entails localized volume
changes), the regions under compression will warm while
the regions under expansion will cool. This process creates a
temperature gradient within the cantilever. Energy will then
flow from the warmer regions to the cooler regions, resulting
in an irreversible energy loss. The rate of energy loss depends
upon the material properties: thermal conductivity, coefficient
of thermal expansion, and heat capacity. It will also depend
upon the resonance frequency of the oscillation mode and the
thermal time constant for heat transfer from the warmer regions
of the cantilever to the cooler regions. Following Roszhart [20],
the thermoelastically limited
factor can be expressed
as
(17)
The term
(18)
contains the material dependencies of the TED process: the co-
efficient of thermal expansion α, the cantilever temperature
,
modulus of elasticity
, mass density , and specific heat .
The other term
(19)
depends on the ratioof the cantileverfrequency
to a character-
istic frequency
, which quantifies the rate of heat flow across
YASUMURA et al.: FACTORS IN MICRON- AND SUBMICRON-THICK CANTILEVERS 123
TABLE I
M
ATERIAL CONSTANTS FOR SILICON
NITRIDE AND SINGLE-CRYSTAL SILICON* [21]–[26]
*The modulus of elasticity for silicon nitride was de-
termined from a curve fit of
versus for an array of
2.3-
m-thick ( m) cantilevers. For simple beam
cantilevers oscillating in a flexural mode,
.
Curve fitting gave a value of 126
7 10 N m for the
modulus of elasticity.
Fig. 12. Plot showing values for 600-, 1700-, and 2400-Å-thick silicon
cantilevers. Also shown are the resonators studied by Roszhart. The cantilevers
studied by Roszhart are being limited by TED. The thermoelastically forbidden
region iswhere the cantilever
’s wouldbe limited byTED and,therefore, have
’s that fall upon the TED curve.
the thickness of the cantilever. TED is maximized at .In
terms of material properties,
is expressed as
(20)
where
is the cantilever material thermal conductivity and
is the cantilever thickness. Using the parameters in Table I,
for 600-Å-thick single-crystal silicon cantilevers is 40 GHz,
while 2.3-
m-thick silicon–nitride cantilevers have a character-
istic frequency of 390 kHz. We would, therefore, expect to see
the effect of TED in 2.3-
m-thick cantilevers that have reso-
nance frequencies in the hundreds of kilohertz range.
Fig. 12 shows a plot of
versus relative frequency ratio
for single-crystal silicon at 300K. values greater than
are forbidden, as they would be limited by TED. Four
sets of data are shown in Fig. 12. The first is data from the
work of Roszhart, showing the excellent agreement between
theory and the measured
valuesof his 10–17.5- m-thick can-
tilevers. Also shown are the
values for 600-Å-thick (
Fig. 13. Plot of measured versus relative frequency ratio for
silicon–nitride cantilevers. Data for five different silicon–nitride thicknesses
are shown. The thickest, i.e.,
m, cantilevers are approaching the
thermoelastically limited
values.
GHz), 1700-Å-thick ( GHz),and2400-Å-thick (
GHz) single-crystal silicon cantilevers. Having a relative
frequency ratio of 10
to 10 , these ultrathin silicon can-
tilevers are not being limited by TED.
for these can-
tilevers is in therangeof10
–10 .Themeasured ’sof10 are
orders of magnitude below the thermoelastic
limit. In order
for 1700-Å-thick silicon cantilevers at 300K to become ther-
moelastically limited, their relative frequency ratio would need
to increase by six orders of magnitude (i.e., their lengths must
decrease by three orders of magnitude.) This would result in
cantilevers with lengths of only
3 m. It can, therefore, be
safely concluded that, except under extreme design conditions,
600-, 1700-, and 2400-Å-thick single-crystal silicon cantilevers
will not have thermoelastically limited
’s.
The same analysis can be performed for thin silicon–nitride
cantilevers. Fig. 13 shows a plot of
versus for the sil-
icon–nitride cantilevers used in this study. The last set of can-
tilevers, i.e.,
m, are approaching the thermoelastically
forbidden region. Two additional curves are shown in Fig. 13.
They represent a conservative 10% variation in the material
properties used in the calculation of room-temperature TED for
silicon–nitride cantilevers. Notice that the 2.3-
m-thick can-
tilevers have
’sthat are decreasingwith increasingrelative fre-
quency—suggesting that the transition from surface-dominated
lossto thermoelasticallydominatedloss occurs at
for silicon–nitride cantilevers at room temperature. This is the
point at which the surface-limited
equals the thermoelasti-
cally limited
.
VIII. L
OSS PARAMETER
In order to make quantitative comparisons of different can-
tilever geometries and materials, a parameter that we shall call
the “loss parameter” γ must be introduced. Recall from (5) that
the strategy to make sensitive force sensing microcantilevers
was to make them narrow, thin, and long (small
) while
maintaining high mechanical
. This general strategy tells us
that
is not the best parameter to use when comparing dif-
ferent oscillators. Even though millimeter-size silicon oscilla-
tors can obtain
’s of 6 10 at room temperature [9], the
large mass and high spring constant of these oscillators make
124 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 9, NO. 1, MARCH 2000
Fig. 14. Loss parameter versus cantilever length comparison
plot. Arrays of silicon–nitride, polysilicon, and single-crystal silicon cantilevers
are shown with their thicknesses and widths. The equivalent minimum force
detectable is shown on the right-hand side for the cantilevers at
K.
them poor force detectors. In order to determine whether a can-
tilever and, more generally, any oscillator, would be a sensitive
force measurement device, all of the cantilever dependent terms
in (4) must be taken into account. By using
,we
can compare cantilevers of different geometries and materials.
In terms of the cantilever geometries and material properties,
can be expressed as
(21)
for the first flexural mode of a simple beam cantilever. For
more complicated structures, the correct spring constant and
resonance frequencies for each mode of oscillation must be
used. This analysis can, therefore, be extended to include tor-
sional and double torsional oscillators as well as more complex
systems where an effective spring constant and resonance
frequency can be either calculated or measured directly.
Fig. 14 shows a summary plot of the loss parameter
for sil-
icon–nitride, polysilicon, and single-crystal silicon cantilevers.
The equivalent minimum force detectable at room temperature
is also shown in Fig. 14 for comparison. A 1700-Å-thick single-
crystal silicon cantilever of length 275
m, width 5 m, and
of 1.8 10 has a loss parameter kg/s.
This corresponds to a room temperature force noise of 5.0
10 N/ Hz or 50 aN/ Hz. Let us now compare this can-
tilever to a thicker silicon–nitride cantilever of approximately
the same length. A 5100-Å-thick silicon–nitride cantilever of
length 270
m, width 10 m, and of 1.5 10 has a loss
parameter
kg/s and a force sensitivity of
233 aN/
Hz. Thus, the silicon cantilever has almost five times
lowerequivalentforce noise thanthe thicker silicon–nitride can-
tilever at room temperatureeventhough the
’sarecomparable.
In Fig. 14, we can also see that the relationship between can-
tilever stiffness and length causes the longer cantilevers to have
better force resolution than the shorter cantilevers. This is ex-
pectedbecause longer cantilevershave lower
whilehaving
the same approximate
.
IX. C
ONCLUSIONS
The goal of this paper was to survey the dissipation charac-
teristicsof micron- andsubmicron-thickmicrocantilevers andto
elucidate various dissipation mechanisms. For both silicon–ni-
tride and single-crystal silicon cantilevers, a monotonic reduc-
tionin
with decreasing thicknesswasobserved.This behavior
is consistent with surface- dominatedenergydissipation. Exper-
iments with 700
CN and forming gas heat treatments of sil-
icon cantilevers indicated that surface contaminants or defects
can be removed via such treatments, allowing significant in-
crease in room temperature
. Surface effects are also likely to
beresponsible for thedipin
observednear135Kin our silicon
cantilevers. TED wasfound to be significantin2.3-
m-thicksil-
icon nitride cantilevers with frequencies above 50 kHz, but was
not significant for any of the thinner cantilevers that we studied.
It is clear that further experiments should be carried out. For
example, performing experiments in ultrahigh vacuum would
allow better control over surface characteristics and allow the
effects due to surface oxidation and contamination to be distin-
guished. Also,
measurements over a broader set of cantilever
thicknesses need to be performed to more firmly establish the
thickness dependence. Finally, the role of other surface charac-
teristics, such as deposited filmsand surface roughening,should
be explored.
A
CKNOWLEDGMENT
The authors that H.Jerman forhelpful discussions. This work
made use of the National Nanofabrication Users Network facili-
ties supported by the National Science Foundation under Award
ECS-9731294.
R
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Kevin Y. Yasumura received the B.A. degree
in physics from the University of California at
Berkeley, in 1994, and the M.S. degree in applied
physics from Stanford University, Stanford, CA, in
1998, and is currently working toward the Ph.D.
degree at Stanford University. His doctoral work
involves studying mechanical energy dissipation in
microcantilever oscillators and the application of
these oscillators to the detection of small forces.
Timothy D. Stowe received the B.S. degree in
applied and engineering physics from Cornell
University, Ithaca, NY, in 1993 the M.S. degree in
applied physics from Stanford University, Stanford,
CA, in 1996, and is currently working toward the
Ph.D. degree at Stanford University. His doctoral
work involves dissipation force microscopy in
regards to external tip-sample dissipation.
Mr. Stowe is a John and Fannie Hertz Fellow.
Eugene M. Chow received the B.S. degree in engi-
neering physics from the University of California at
Berkeley, in 1995, the M.S. degree in electrical en-
gineering from Stanford University, Stanford, CA, in
1997, and is currently working toward the Ph.D. de-
gree in electrical engineering at Stanford University.
His research interests are in the fabrication, appli-
cation, and design of microfabricated sensors, with
particular interest in scanning probes and through-
wafer interconnects.
Timothy Pfafman received the B.S. degree in math-
ematics and physics from the Massachusetts Institute
of Technology, Cambridge, in 1986, and the Ph.D.
degree in physics from the University of California
at Berkeley, in 1991.
In 1991, he was with the Los Alamos National
Laboratory, where he was involved with calorimetric
X-ray detectors for astrophysics applications. In
1996, he joined the Design Division, Department
of Mechanical Engineering, Stanford University,
Stanford, CA, as a Research Associate, where he was
involved with general micromachining, sensor development for small satellites,
software tools for micromachining, and ultimate frisbee. He is currently a
software and hardware consultant in Silicon Valley.
Thomas W. Kenny received the B.S. degree in
physics from the University of Minnesota at Min-
neapolis–St. Paul, in 1983, and the M.S. and Ph.D.
degrees in physics from the University of California
at Berkeley, in 1987 and 1989, respectively.
He was with the Jet Propulsion Laboratory, where
his research focused on the development of electron-
tunneling-based high-resolution microsensors. Since
1994, he has been an Assistant Professor with the
Mechanical Engineering Department, Stanford Uni-
versity, Stanford, CA.He currently oversees graduate
students in the Stanford Microstructures and Sensors Laboratory, whose re-
search activities cover a variety of areas such as advanced tunneling sensors,
piezoresistive sensors, cantilever arrays, fracture in silicon, and the mechanical
properties of biomolecules and cells. This group is also collaborating with re-
searchers from the IBM Almaden and Zürich Research Centers onnuclear mag-
netic resonance microscopy as well as AFM thermomechanical data storage.
Dr. Kenny is a Terman Fellow.
Barry C. Stipe received the B.S. degree in physics
from the California Institute of Technology,
Pasadena, in 1991, and the Ph.D. degree in physics
from Cornell University, Ithaca, NY, in 1998.
For his doctoral thesis, he combined low-tem-
perature scanning tunneling microscopy (STM)
with vibrational spectroscopy, which included the
imaging, excitation, manipulation, and spectroscopic
characterization of individual chemical bonds within
single adsorbed molecules using inelastic tunneling
electrons.
His research interests include the study of matter at the atomic and molecular
level with the use of proximal probe techniques combined with high-resolution
spectroscopies. Since August 1998, he has been with the IBM Research Divi-
sion, Almaden Research Center, San Jose, CA, with the aim of imaging and
studying individual spins with atomic resolution by magnetic resonance force
microscopy.
DanielRugar (M’87)receivedthe B.A.degreein ap-
plied physics (magna cum laude) from Pomona Col-
lege, Pomona, CA, in 1975, and the Ph.D. degree in
applied physics from Stanford University, Stanford,
CA, in 1982.
From 1982 to 1984, he was the Hunt Fellow of the
Acoustical Society of America and a Research Asso-
ciate at Stanford University, where he was involved
with acoustic microscopy and phonon dispersion in
superfluid helium. In 1984, he joined the IBM Re-
searchDivision, AlmadenResearch Center, San Jose,
CA, where he is currently Manager of nanoscale studies. He has published over
80 papers and holds 13 patents. He has worked on many aspects of high-density
data storage and scanning probe microscopy, and his current research interests
include new techniques for ultrahigh density data storage, magnetic resonance
force microscopy, and ultrasensitive force detection.
Dr. Rugar is a member of the American Physical Society and currently serves
as a distinguished lecturer for the IEEE Magnetics Society.
