1
Stress
-
Induced Variations in
the
Stiffness of
Micro
-
and
Nano
-
Cantilever Beams
R. B. Karabalin
1
, L. G. Villanueva
1
, M. H. Matheny
1
, J. E. Sader
2
, 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
ABSTRACT
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 perfo
rmed 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 longstanding and unphysical axial force
model, w
hich has been widely applied to interpret measurements using cantilever beams.
Our
findings
will
be of value in the development of
micro
-
and
nano
-
scale
resonant
mechanical
sensors.
2
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
und
erstanding
the mechanism by which
biomolecules affect the physics of the
se
mechanical structure
s
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 surface elasticity
[12, 13, 17, 19]
, increase in damping
[20]
, and
the imposition of surface stress
[6
-
11, 15]
. Numerous studies 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 interpretation
s
. It is
also
striking that, within this extensive body
of work
[1, 2]
, controlled
,
quant
itative 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 qualitative agreement with
measurements.
However, the axial force model as applied to cantilever beams has subsequently been shown to be
in violation of Newton’s 3
rd
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
.
T
his phenomenon is contingent on the beam length greatly
exce
eding its width and thickness, a fundamental assumption of Euler
-
Bernoulli beam theory; see
3
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 experimental studies have provided experimental
data, seemingly consistent with the axial force model in micro
-
and nanoscale cantilever beams
[6
-
9, 11, 15]
. However, it is important to note that surface stress change in all such expe
rimental
studies is achieved using processes that may cause unspecified changes in the mechanical
properties 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 stif
fness change can be systematically observed and theoretically predicted
[21, 25]
.
In this Letter we extend previous theoretical work, propose a new theoretical model and explain
small but non
-
negligible changes in the resonant frequencies of cantilever bea
ms 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 methodology, which show
s
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 vicinity of
its supporting clamp
[22], which
can affect its resonance frequency. Expressions for the relative
freq
uency shifts
of doubly
-
clamped and cantilever beams due to the application of surface
stress, and subsequent generation of net in
-
plane stress within the device, are shown in the second
f
/
f
R
4
column of Table 1; termed the
stress effect
. We emphasize that all r
esults in Table 1 are derived for
thin beams.
(stress effect)
(geometric effect)
Doubly
-
Clamped Beam
0
.
1475
L
h
2
1
s
T
E
h
Cantilever Beam
Table 1 |
Formulas determining the relative frequency shift of thin doubly
-
clamped beams (top row) and
cantilever beams (bottom row) when a load
s
T
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) fro
m 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,
ν
is the Poisson ratio, and
s
T
is an applied surface
stress.
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 elasticity. In the present
context, application of surface stress induces a change in th
e 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 1. 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 1 indicate that the resonant frequencies of doubly
-
clamped beams are
more sens
itive to surface stress changes than cantilever beams (considering typical devices
dimensions,
). For clamped
-
clamped beams, the stress effect listed in Table 1 dominates,
whereas for cantilever beams the situation is more complex: the stress effect is do
minant for thin
structures (
), with geometric effects prevailing for thicker devices. Importantly, if the
f
/
f
R
f
/
f
R
(1
)
(1 2 )(1
)
2(1
)
T
s
v
Eh
2
-
0.042
1
T
s
bb
L
h
Eh
(1
)
12
1
T
s
v
Eh
10
Lh
h
b
5
geometric effect dominates, the relative frequency shift
does not change with length.
Alternatively, if the stress effect prevails then
strongly
depends on length.
Figure 1 |
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
3
8.3 MHz
(length 6 μm, purple), 22.9 MHz (8 μm, green) and
14.8 MHz (10 μm, blue). Experimental details are provided in the Supporting Information.
b
, SEM
micrograph of the cantilever beams used for the experiments. Respective resonant responses are also
sh
own for each cantilever, yielding natural frequencies of 8.85 MHz (length 6 μm), 4.82 MHz (8 μm), 3.16
MHz (10 μm). 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 de
vices, 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.
To examine the predictions of the above theoretical model, we perform systematic
measurements on nanoscale cantilever and doubly
-
clamped beams of identical geometries. Our
measurements are achieved by fabricating multilayered mechanical bars out of piezoel
ectric
material, utilizing recent advances in nanoelectromechanical systems (NEMS) fabrication
techniques [26]. Thus, stress changes in our devices are internally produced piezoelectrically,
which allows for reproducible and controllable variation. It shou
ld 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
f
/
f
R
f
/
f
R
0
2
0
4
0
6
0
8
.
8
3
8
.
8
8
4
.
8
0
4
.
8
3
3
.
1
5
3
.
1
7
F
r
e
q
u
e
n
c
y
(
M
H
z
)
A
m
p
l
i
t
u
d
e
(
a
.
u
.
)
1
4
.
7
1
4
.
9
3
8
.
2
3
8
.
4
0
1
0
2
0
2
2
.
7
5
2
3
.
0
0
F
r
e
q
u
e
n
c
y
(
M
H
z
)
A
m
p
l
i
t
u
d
e
(
a
.
u
.
)
6
m
m
8
m
m
1
0
m
m
6
m
m
8
m
m
1
0
m
m
(
a
)
(
b
)
y
x
z
6
simulations of both device
s that includes their full structure and load (see Supplementary
Information). Doubly
-
clamped beams with identical dimensions provide a benchmark for the
measurements 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 curvature
measurement yields built
-
in stresses in all four layers c
lose to zero.
SEM micrographs of the
devices are shown in Fig. 1. All beams possess an identical width
b
= 900 nm and total thickness
h
tot
= 320 nm, whereas their lengths
L
are 6, 8 and 10 μm.
The absence of curvature in cantilever
beams indicates the absence of stress gradient in structural layer.
Note that the only difference
between cantilevers and doubly
-
clamped beams is that the boundary condition at one end is
changed
–
material and geomet
ric 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
p
roperties. 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 normal deflecti
on are performed in vacuum
using optical interferometry (see Supporting Information). Resonance measurements are shown in
Fig. 1a and Fig. 1b 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 variations in the stiffness of both types of devices due to stress changes are
observed.
7
Figure 2 | Frequency shift results for doubly
-
clamped beams. a
, Relative frequency shift
f
/
f
R
for the three doubly
-
clamped beams in Fig. 1a, 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 the Supporting Information.
T
he 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
-
2.5 pm/V.
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 1 (see Supporting Information for corresponding expressions for piezoelectric
loads). M
easurement results of doubly
-
clamped beams are presented in Fig. 2, which clearly
-
5
0
0
5
0
∆
f
(
k
H
z
)
d
o
u
b
l
y
-
c
l
a
m
p
e
d
b
e
a
m
s
-
3
-
2
-
1
0
1
2
3
D
C
B
i
a
s
(
V
)
6
4
2
0
-
2
-
4
-
6
C
o
r
r
e
s
p
o
n
d
i
n
g
s
u
r
f
a
c
e
s
t
r
e
s
s
(
N
/
m
)
6
m
e
x
p
6
m
F
E
M
8
m
e
x
p
8
m
F
E
M
1
0
m
e
x
p
1
0
m
F
E
M
6
m
e
x
p
6
m
F
E
M
8
m
e
x
p
8
m
F
E
M
1
0
m
e
x
p
1
0
m
F
E
M
(
a
)
(
b
)
-
4
0
-
2
0
0
2
0
4
0
-
4
∆
f
/
f
(
1
0
)
R
d
o
u
b
l
y
-
c
l
a
m
p
e
d
b
e
a
m
s
8
exhibit a linear variation in resonant frequency with voltage (stress) in the piezoelectric layer, as
predicted theoretically.
No variation of quality factor with bias voltag
e is observed.
This provides
strong validation for the robustness of the experimental methodology and the ability to tune the
stiffness of doubly
-
clamped beams. Use of the resonant frequency shift expressions enables the
axial force (or the corresponding s
urface stress, as commonly reported) to be calibrated against the
applied voltage (see Supporting Information), which we determine to be 2.3 N/m per Volt.
In
agreement with theory (see Table 1), the relative frequency shift shown in Fig. 2a depends strongl
y
on the beam length, scaling as
(
is the original resonant frequency),
whereas the absolute shift
in resonant frequency
is found to be independent of the beam length (see Fig. 2b).
In Fig. 3 we present complementary results for the resonant freq
uency shift in cantilever beams.
These nanomechanical devices have identical dimensions and material properties to the doubly
-
clamped beams in Fig. 2. Linear tuning of the resonant frequency with the applied piezoelectric
layer voltage is also observed. Ho
wever, these frequency shifts are two 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 unphysical axi
al
force model for cantilever beams; see Eq. (S9). Not only the magnitude but also the scaling
behavior of the resonant frequency shift
with the beam length is inconsistent with the axial
force model (see Supporting Information).
f
f
9
Figure 3 | Frequency shift results for cantilever beams. a
, Relative frequency shift
f
/
f
R
for the three cantilever beams in Fig. 1b, 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 combinin
g
both stress and geometric effects; these show
excellent
agreement with measurements.
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 st
ress on the resonant
frequency is much smaller for cantilevers than doubly
-
clamped beams.
Notably, measurements of
/
R
ff
for cantilever beams display independence on the cantilever
length and are identical in sign to those obtained f
or doubly
-
clamped beams; see Figs. 2a and 3a.
This observation is completely consistent with the geometric effect listed in Table 1, which is
expected to dominate since these devices are relatively thick (
b
/
h
~
3
). Importantly, the formulas
6
m
e
x
p
6
m
F
E
M
8
m
e
x
p
8
m
F
E
M
1
0
m
e
x
p
1
0
m
F
E
M
-
3
-
2
-
1
0
1
2
3
D
C
B
i
a
s
(
V
)
6
4
2
0
-
2
-
4
-
6
6
m
e
x
p
6
m
F
E
M
8
m
e
x
p
8
m
F
E
M
1
0
m
e
x
p
1
0
m
F
E
M
-
1
.
0
-
0
.
5
0
.
0
0
.
5
1
.
0
∆
f
/
f
(
1
0
)
-
4
c
a
n
t
i
l
e
v
e
r
b
e
a
m
s
R
C
o
r
r
e
s
p
o
n
d
i
n
g
s
u
r
f
a
c
e
s
t
r
e
s
s
(
N
/
m
)
(
a
)
(
b
)
-
1
.
0
-
0
.
5
0
.
0
0
.
5
1
.
0
∆
f
(
k
H
z
)
c
a
n
t
i
l
e
v
e
r
b
e
a
m
s
10
given in Table 1 are derived under the assumption of a thin beam (
), 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 calculate the combined contribution from geometric and stress
induced effects using full 3D finite elem
ent simulations (see Supplementary Information).
The
agreement between p
redictions from these simulations (shown as scatter plots in Figs. 2 and 3) and
the experimentally measured data
is
with
in
15%
.
The apparent
slight asymmetry between positive
and negat
ive voltages is within the experimental error
(see Supplementary Information).
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 previous
ly
reported
cantilever
measurements display
results that are inconsistent 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
mu
ch larger in magnitude than the results reported here.
These
previous, now anomalous,
measurements were typically performed using 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 clas
sical 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 uncontroll
ed surface phenomena.
h
b
11
Miniaturization of NEMS technologies is key to enhanced sensitivity and ultrafast
measurements. Our study establishes that as cantilever thickness is reduced, a transduction
mechanism arises that is different to the one observed in th
e 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 medical technologies.
We would like
to thank
X. L. Feng, J. Xiang, M. J. Lachut for useful suggestions and
discussions. We would also like to thank E. Defaÿ, G. Le Rhun, C. Marcoux from CEA
-
LETI for
providing us with Aluminum Nitride material. L.G.V. acknowledges financial su
pport from the
European Commission (
PIOF
-
GA
-
2008
-
220682) and Prof. A. Boisen. J.E.S. acknowledges
support from the Australian Research Council grants scheme.
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