Stress-Induced Variations in the Stiffness of Micro- and Nanocantilever Beams
R. B. Karabalin,
1
L. G. Villanueva,
1
M. H. Matheny,
1
J. E. Sader,
2
and M. L. Roukes
1
1
Kavli Nanoscience Institute, California Institute of Technology, Pasadena, California 91125
2
Department of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia
(Received 1 November 2011; published 5 June 2012)
The effect of surface stress on the stiffness of cantilever beams remains an outstanding problem in the
physical sciences. While numerous experimental studies report significant stiffness change due to surface
stress, theoretical predictions are unable to rigorously and quantitatively reconcile these observations. In
this Letter, we present the first controlled measurements of stress-induced change in cantilever stiffness
with commensurate theoretical quantification. Simultaneous measurements are also performed on
equivalent clamped-clamped beams. All experimental results are quantitatively and accurately predicted
using elasticity theory. We also present conclusive experimental evidence for invalidity of the long-
standing and unphysical axial force model, which has been widely applied to interpret measurements
using cantilever beams. Our findings will be of value in the development of micro- and nanoscale resonant
mechanical sensors.
DOI:
10.1103/PhysRevLett.108.236101
PACS numbers: 68.37.Ps, 85.85.+j
Physical or chemical adsorption onto the surface of
micro- and nanoscale cantilevers is the basis for a rapidly
growing field of biological sensing in the mechanical
domain. Despite the remarkable achievements of the field
[
1
,
2
], it is surprising that understanding the mechanism by
which biomolecules affect the physics of these mechanical
structures remains elusive [
3
–
17
]. The adsorption process
can result in a multiplicity of concurrent effects including:
mass loading of the device [
4
,
5
,
18
], enhancement of sur-
face elasticity [
12
,
13
,
17
,
19
], increase in damping [
20
], and
the imposition of surface stress [
6
–
11
,
15
]. Numerous stud-
ies over the past 35 years have claimed that surface stress
can cause significant variations in device stiffness
[
3
–
14
,
17
]. In contrast, other works ignore such effects,
claiming that such stress loads do not affect the stiffness
of cantilever devices, and argue for alternative interpreta-
tions. It is also striking that, within this extensive body
of work [
1
,
2
], controlled, quantitative measurements of
the effects of surface stress on cantilever stiffness with
commensurate theoretical interpretation have yet to be
reported.
The original theoretical model of Lagowski
et al.
[
15
]
suggested that a
net axial force
is induced along the beam
axis upon application of stress—a load similar to the case
of doubly clamped beams derived within the framework of
Euler-Bernoulli beam theory [
21
]. Lagowski
et al.
found
that the
axial force model
was in quantitative and qualita-
tive agreement with measurements. However, the axial
force model as applied to cantilever beams has subse-
quently been shown to be in violation of Newton’s 3rd
law; i.e., it does not satisfy the fundamental physical
principle of force equilibrium [
16
,
22
,
23
], contrary to dou-
bly clamped beams. In short, application of surface stress
to a cantilever beam always induces stress of opposite sign
within the beam material, resulting in zero net axial force
along the beam. This phenomenon is contingent on the
beam length greatly exceeding its width and thickness, a
fundamental assumption of Euler-Bernoulli beam theory;
see Refs. [
16
,
22
,
23
]. Hence, Euler-Bernoulli beam theory
leads to the conclusion that cantilever beams should be
insensitive to surface stress.
Though it was proven unphysical, numerous experimen-
tal studies have provided experimental data, seemingly
consistent with the axial force model in micro- and nano-
scale cantilever beams [
6
–
9
,
11
,
15
]. However, it is impor-
tant to note that surface stress change in all such
experimental studies is achieved using processes that
may cause unspecified changes in the mechanical proper-
ties of the resonator [
15
,
24
]. It is thus not entirely evident
whether these previous measurements are due solely to
stress changes at the cantilever beam surface, or effects
of an unspecified nature. This situation contrasts strongly
to clamped-clamped beams whose stress-induced stiffness
change can be systematically observed and theoretically
predicted [
21
,
25
].
In this Letter we extend previous theoretical work, pro-
pose a new theoretical model and explain small but non-
negligible changes in the resonant frequencies of cantilever
beams due to application of stress. We also present the first
systematic measurements of both cantilever and doubly
clamped beams using a robust and highly repeatable meth-
odology, which shows remarkable agreement with theory.
This resolves the above-mentioned long-standing debate
and experimentally establishes the invalidity of the axial
force model for cantilever beams.
In contrast to the case of doubly clamped beams, the
application of surface stress to a long and thin cantilever
beam does not generate a net axial force because the
longitudinal displacement is not constrained. However, a
cantilever generates a net in-plane stress in the immediate
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vicinity of its supporting clamp [
22
], which can affect its
resonance frequency. Expressions for the relative fre-
quency shifts
f=f
R
of doubly clamped and cantilever
beams due to the application of surface stress, and subse-
quent generation of net in-plane stress within the device,
are shown in the second column of Table
I
; termed the
stress effect
. We emphasize that all results in Table
I
are
derived for thin beams.
Application of a load to any elastic body changes its
geometry due to elastic deformation. This
geometric effect
is typically ignored in the classical theory of linear elas-
ticity. In the present context, application of surface stress
induces a change in the beam length, width, thickness, and
density, which alter the resonant frequency of both
clamped-clamped and cantilever beams. Expressions for
the relative frequency shift due to this geometric effect are
shown in the third column of Table
I
. To account for the
true geometry of the devices, finite element analysis is
performed below. The complete effect is given by the
sum of the stress and geometric effects.
The formulas in Table
I
indicate that the resonant fre-
quencies of doubly clamped beams are more sensitive to
surface stress changes than cantilever beams (considering
typical devices dimensions,
L>
10
h
). For clamped-
clamped beams, the stress effect listed in Table
I
domi-
nates, whereas for cantilever beams the situation is more
complex: the stress effect is dominant for thin structures
(
h
b
), with geometric effects prevailing for thicker
devices. Importantly, if the geometric effect dominates,
the relative frequency shift
f=f
R
does not change with
length. Alternatively, if the stress effect prevails then
f=f
R
strongly depends on length.
To examine the predictions of the above theoretical
model, we perform systematic measurements on nanoscale
cantilever and doubly clamped beams of identical geome-
tries. Our measurements are achieved by fabricating multi-
layered mechanical bars out of piezoelectric material,
utilizing recent advances in nanoelectromechanical sys-
tems (NEMS) fabrication techniques [
26
]. Thus, stress
changes in our devices are internally produced piezoelec-
trically, which allows for reproducible and controllable
variation. It should be noted that in our experiment the
stress is applied to a layer of finite thickness, rather than to
the surface of the devices. To account for any differences
that this may induce, we conduct rigorous numerical finite
element simulations of both devices that includes their full
structure and load (see [
27
]). Doubly clamped beams with
identical dimensions provide a benchmark for the mea-
surements on cantilevers.
Cantilever and doubly clamped nanoscale beams are
fabricated on the same chip from a 320 nm 4-layer stack
(comprised of: 20 nm aluminum nitride (AlN), 100 nm
molybdenum (Mo), 100 nm AlN, and 100 nm Mo), using a
process described elsewhere [
26
]. A standard wafer curva-
ture measurement yields built-in stresses in all four layers
close to zero. SEM micrographs of the devices are
shown in Fig.
1
. All beams possess an identical width
b
¼
900 nm
and total thickness
h
tot
¼
320 nm
, whereas
their lengths
L
are 6, 8, and
10
m
. The absence of
curvature in cantilever beams indicates the absence of a
stress gradient in the structural layers. Note that the only
difference between cantilevers and doubly clamped beams
is that the boundary condition at one end is changed—
material and geometric properties are otherwise identical.
An electric field is produced in the active piezoelectric
layer by applying a dc voltage between the top and bottom
molybdenum layers. This induces an axial stress along the
beam axis via its inherent piezoelectric properties.
Ensuring the active piezoelectric layer is offset from the
neutral axis of the beam enables the fundamental flexural
mode of the device to be actuated through application of an
additional ac voltage. Measurements of the resulting nor-
mal deflection are performed in vacuum using optical
interferometry (see [
27
]). Resonance measurements are
shown in Figs.
1(a)
and
1(b)
for doubly clamped beams
and cantilevers, respectively. We monitor the resonant
frequency while changing dc voltage by means of a
computer-controlled phase locked loop. Controllable var-
iations in the stiffness of both types of devices due to stress
changes are observed.
We initially study the effect of an applied stress on the
fundamental resonant frequencies of doubly clamped
beams. Formulas connecting resonance frequency shifts to
surface stress are presented in Table
I
(see Supplemental
Material [
27
] for corresponding expressions for piezoelec-
tric loads). Measurement results of doubly clamped beams
are presented in Fig.
2
, which clearly exhibit a linear
variation in resonant frequency with voltage (stress) in
the piezoelectric layer, as predicted theoretically. No
variation of quality factor with bias voltage is observed.
TABLE I. Formulas determining the relative frequency shift of thin doubly clamped beams (top row) and cantilever beams (bottom
row) when a normalized load,
ð
1
Þ
T
s
=
ð
Eh
Þ
, is applied. Both stress (left column) and geometric effects are presented. While
the latter effect is of equivalent magnitude for both boundary conditions, the former originates from: (i) a net axial force in the case of
doubly clamped beams, and (ii) from in-plane stress generation near the clamp for cantilevers.
L
,
b
, and
h
are the length, width, and
thickness of the resonator, respectively,
E
is the Young’s modulus,
is the Poisson ratio, and
T
s
is an applied surface stress.
f=f
R
(stress effect)
f=f
R
(geometric effect)
Doubly Clamped Beam
0
:
1475
ð
L=h
Þ
2
fð
1
þ
2
Þð
1
þ
Þ
=
ð
2
½
1
Þg
Cantilever Beam
0
:
042
ð
b=L
Þð
b=h
Þ
2
fð
1
þ
2
Þ
=
ð
1
Þg
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This provides strong validation for the robustness of the
experimental methodology and the ability to tune the stiff-
ness of doubly clamped beams. Use of the resonant fre-
quency shift expressions enables the axial force (or the
corresponding surface stress, as commonly reported) to be
calibrated against the applied voltage (see [
27
]), which we
determine to be
2
:
3N
=
m
per
V
. In agreement with theory
(see Table
I
), the relative frequency shift shown in Fig.
2(a)
depends strongly on the beam length, scaling as
L
2
(
f
R
is the
original resonant frequency), whereas the absolute shift in
resonant frequency
f
is found to be independent of the
beam length [see Fig.
2(b)
].
In Fig.
3
we present complementary results for the
resonant frequency shift in cantilever beams. These nano-
mechanical devices have identical dimensions and material
properties to the doubly clamped beams in Fig.
2
. Linear
tuning of the resonant frequency with the applied piezo-
electric layer voltage is also observed. However, these
frequency shifts are 2 orders of magnitude smaller than
those exhibited by doubly clamped beams, even though the
piezoelectric stress loads are identical—this constitutes
direct experimental evidence of the invalidity of the un-
physical axial force model for cantilever beams; see
Eq. (S9). Not only the magnitude but also the scaling
behavior of the resonant frequency shift
f
with the
beam length is inconsistent with the axial force model
(see [
27
]).
Notably, measurements of
f=f
R
for cantilever beams
display independence on the cantilever length and are
identical in sign to those obtained for doubly clamped
beams; see Figs.
2(a)
and
3(a)
. This observation is com-
pletely consistent with the geometric effect listed in
Table
I
, which is expected to dominate since these devices
are relatively thick (
b=h
3
). Importantly, the formulas
given in Table
I
are derived under the assumption of a thin
beam (
h
b
), where the stress load is applied to an
infinitesimal layer at the surface. To account for the true
device geometry and properties, as noted above, we calcu-
late the combined contribution from geometric and stress-
induced effects using full 3D finite element simulations
(see [
27
]). The agreement between predictions from these
simulations (shown as scatter plots in Figs.
2
and
3
) and the
experimentally measured data is within 15%. The apparent
slight asymmetry between positive and negative voltages is
within the experimental error (see [
27
]). These results
constitute the first repeatable measurements and theoretical
quantification of stress-induced changes in the frequency
shifts (stiffness) of cantilever beams. They also provide
compelling experimental evidence for the invalidity of the
axial force model, which has been widely applied to the
interpretation of measurements made using cantilever
beams.
It remains to understand why several previously reported
cantilever measurements display results that are inconsis-
tent with the present findings [
6
–
9
,
11
,
14
,
15
]. These studies
report stress-induced stiffness changes that appear to be in
agreement with the unphysical axial force model that
erroneously predicts effects much larger in magnitude
than the results reported here. These previous, now anoma-
lous, measurements were typically performed using
0
20
40
60
8.83
8.88
4.80
4.83
3.15
3.17
Frequency (MHz)
Amplitude (a.u.)
14.7
14.9
38.2
38.4
0
10
20
22.75
23.00
Frequency (MHz)
Am
p
litude(a.u.)
6
m
8
m
10
m
6
m
8
m
10
m
(b)
(a)
y
x
z
FIG. 1 (color online). Resonant response of piezoelectric beams. a, SEM micrograph of the doubly clamped beams used for the
experiments. On top of the micrograph, we show resonant responses of each of the beams, yielding resonant frequencies of 38.3 MHz
(length
6
m
, purple), 22.9 MHz (
8
m
, green) and 14.8 MHz (
10
m
, blue). Experimental details are provided in [
27
]. b, SEM
micrograph of the cantilever beams used for the experiments. Respective resonant responses are also shown for each cantilever,
yielding natural frequencies of 8.85 MHz (length
6
m
), 4.82 MHz (
8
m
), 3.16 MHz (
10
m
). Both types of beams have the same
composition (320 nm of total thickness) and width (900 nm). Lengths are 6, 8, or
10
m
for both types of devices, causing the
boundary conditions to be the only difference, thus allowing proper comparison of the experimental results for the two configurations.
Scale bars:
2
m
.
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surface adsorption or modification processes. While these
processes are known to induce surface stress, additional
unspecified and uncontrolled effects are also possible.
There is certainly the distinct possibility of over-layer
formation, since material is adsorbed to [
12
,
13
,
17
,
19
]or
removed from the beam surface [
15
]. Some studies have
shown excellent agreement between classical composite
beam theory and such measurements [
12
,
13
,
17
,
19
], while
other measurements remain unexplained [
6
–
9
,
14
,
15
]. Our
findings strongly suggest that previous reports of stress-
induced changes in cantilever beams originate from other
uncontrolled surface phenomena.
Miniaturization of NEMS technologies is key to en-
hanced sensitivity and ultrafast measurements. Our study
establishes that as cantilever thickness is reduced, a trans-
duction mechanism arises that is different to the one
observed in the present devices. Specifically, we predict
that cantilever devices made of ultrathin materials (such as
graphene) allow for gigantic tunability in their resonance
properties. Such materials would exhibit an additional
inverse-squared thickness dependence, enabling very
strong modification of their stiffness. This could ultimately
permit the development of UHF bandpass cantilever
filters, with broad control of their frequency range for
applications in biosensing, telecommunications and medi-
cal technologies.
We would like to thank X. L. Feng, J. Xiang, and M. J.
Lachut for useful suggestions and discussions. We would
also like to thank E. Defay
̈
, G. Le Rhun, and C. Marcoux
from CEA-LETI for providing us with Aluminum Nitride
material. L. G. V. acknowledges financial support from the
European Commission (PIOF-GA-2008-220682) and
Professor A. Boisen. J. E. S. acknowledges support from
the Australian Research Council grants scheme.
6
μ
mexp
6
μ
m FEM
8
μ
mexp
8
μ
m FEM
10
μ
mexp
10
μ
mFEM
-3
-2
-1
0
1
2
3
DC B ias (V)
6
4
2
0
-2
-4
-6
6
μ
mexp
6
μ
mFEM
8
μ
mexp
8
μ
mFEM
10
μ
mexp
10
μ
mFEM
-1.0
-0.5
0.0
0.5
1.0
f/f(10 )
-4
cantilever beams
R
Corresponding surface stress (N/m)
(a)
(b)
-1.0
-0.5
0.0
0.5
1.0
f(kHz)
cantilever beams
FIG. 3 (color online). Frequency shift results for cantilever
beams. a, Relative frequency shift
f=f
R
for the three cantilever
beams in Fig.
1(b)
, showing independence of length, which is
consistent with the theoretical model. Both experimental (lighter
colors) and FEM results (scattered plots, darker colors) are
presented. FEM data are a result of combining both stress and
geometric effects; these show excellent agreement with mea-
surements. b, Absolute frequency shift
f
(in kHz) for the same
three beams. Comparison of these results with those in Fig.
2
demonstrates that the effect of stress on the resonant frequency is
much smaller for cantilevers than doubly clamped beams.
-50
0
50
f(kHz)
doubly-clamped beams
-3
-2
-1
0
1
2
3
DC B ias (V)
6
4
2
0
-2
-4
-6
Corresponding surface stress (N/m)
6
μ
mexp
6
μ
mFEM
8
μ
mexp
8
μ
mFEM
10
μ
mexp
10
μ
mFEM
6
μ
mexp
6
μ
mFEM
8
μ
mexp
8
μ
mFEM
10
μ
mexp
10
μ
mFEM
(a)
(b)
-40
-20
0
20
40
-4
f/f(10 )
R
doubly-clamped beams
FIG. 2 (color online). Frequency shift results for doubly
clamped beams. a, Relative frequency shift
f=f
R
for the three
doubly clamped beams in Fig.
1(a)
, showing the dependence of
f=f
R
on length (as predicted by theory). b, Absolute frequency
shift
f
(in kHz) for the same three beams. Experimental
(lighter colors) and FEM results (scattered plots, darker colors)
display excellent agreement. The measurements are shown as a
function of the applied dc bias (in volts) and as a function of the
corresponding surface stress, as calculated in [
27
]. The stress
calculation requires estimation of the piezoelectric coefficient
d
31
; this was obtained by a linear fit of these experimental
results, yielding a value of
2
:
5pm
=
V
.
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supplemental/10.1103/PhysRevLett.108.236101
for
details.
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