of 8
PHYSICAL REVIEW B
87
, 024304 (2013)
Nonlinearity in nanomechanical cantilevers
L. G. Villanueva,
1
,
2
R. B. Karabalin,
1
M. H. Matheny,
1
D. Chi,
1
J. E. Sader,
1
,
3
and M. L. Roukes
1
1
Kavli Nanoscience Institute and Departments of Physics, Applied Physics, and Bioengineering, California Institute of Technology,
Pasadena, California 91125, USA
2
Department of Micro- and Nanotechnology, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark
3
Department of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia
(Received 27 March 2012; revised manuscript received 8 June 2012; published 28 January 2013)
Euler-Bernoulli beam theory is widely used to successfully predict the linear dynamics of micro- and
nanocantilever beams. However, its capacity to characterize the nonlinear dynamics of these devices has not
yet been rigorously assessed, despite its use in nanoelectromechanical systems development. In this article, we
report the first highly controlled measurements of the nonlinear response of nanomechanical cantilevers using
an ultralinear detection system. This is performed for an extensive range of devices to probe the validity
of Euler-Bernoulli theory in the nonlinear regime. We find that its predictions deviate strongly from our
measurements for the nonlinearity of the fundamental flexural mode, which show a systematic dependence
on aspect ratio (length/width) together with random scatter. This contrasts with the second mode, which is always
found to be in good agreement with theory. These findings underscore the delicate balance between inertial and
geometric nonlinear effects in the fundamental mode, and strongly motivate further work to develop theories
beyond the Euler-Bernoulli approximation.
DOI:
10.1103/PhysRevB.87.024304
PACS number(s): 85
.
85
.
+
j, 05
.
45
.
a, 62
.
25
.
g
I. INTRODUCTION
Micro- and nanoelectromechanical systems (MEMS and
NEMS) are increasingly spawning a wide range of sensing
applications, including detection of mass,
1
,
2
force,
3
and spin.
4
In addition, they can also be used as time reference devices
5
and as basic tools to explore fundamental physical processes
6
and dynamical effects.
7
At small vibrational amplitudes
these systems behave as linear mechanical devices. However
as the amplitude increases, nonlinear effects are readily
manifested.
8
,
9
This becomes of central importance in all of the
aforementioned fields of application. For example, nonlinear
phenomena impose a fundamental limit for the minimum
detectable frequency shift
10
while simultaneously enabling
rich and complex dynamical behavior.
11
Arguably the most utilized mathematical description of the
deformation of MEMS and NEMS cantilever beams is Euler-
Bernoulli theory.
12
We observe that this theory accurately
predicts the resonant frequencies and other linear parameters
for the flexural vibration modes of thin cantilever beams
of aspect ratios (AR
=
length
/
width) greater than 2 (see
Appendix
A
). The generic Euler-Bernoulli theory implicitly
assumes the beam to be one dimensional and is formally valid
in the asymptotic limit of infinite AR. For beams of finite
AR and non-negligible thickness, it is sometimes necessary
to include the effects of transverse
13
or shear
12
deformation,
respectively, although these effects are second order and can
be often ignored in experimental design and application.
14
17
As introduced before, nonlinear behavior manifests for
finite amplitude of motion. This is true not only at the
micro- and nanoscale, but also for macroscopic structures
such as airplane wings.
18
,
19
Consequently, an effort to predict
the dynamics of the nonlinear response and the parameters
governing it has recently gained momentum.
20
22
Nonlinearity in the dynamic response of mechanical struc-
tures can have a multitude of origins,
8
,
23
including transduction
effects (actuation
/
detection),
21
material properties (nonlinear
constitutive relations),
24
nonideal boundary conditions,
25
,
26
damping mechanisms,
27
,
28
adsorption/desorption processes,
29
and geometric/inertial effects.
30
,
31
Geometric nonlinearities
can appear in any mechanical structure when large deforma-
tions induce a nonlinear relation between strain and curvature,
thus modifying the effective stiffness of the structure. Inertial
nonlinearities are typically induced through the generation of
additional degrees of freedom in the motion, which serve to
enhance the effective mass of the structure.
The intrinsic (i.e., originating from the mechanical struc-
ture) nonlinear response of doubly clamped beams has been
shown to be dominated by a geometric nonlinearity due
to enhanced tension along the beam. Stiffening behavior is
observed,
5
,
32
which is accurately predicted by Euler-Bernoulli
theory.
8
In contrast, the nonlinear response of cantilever
beams has received comparatively little attention. Most articles
report theoretical investigations of the nonlinear response
of these structures.
21
,
31
,
33
These studies predict a stiffening
nonlinearity for the fundamental mode, while the higher order
modes are predicted to be softening in nature. Strikingly,
experimental assessment of the validity of such calculations for
the fundamental mode has been limited in geometrical range
and statistical analysis, and has not provided measurements
with linear transduction.
20
,
21
In this article, we address this issue and present detailed
experimental results for the intrinsic nonlinear resonant re-
sponses of nanomechanical cantilever beams. In particular,
we study the first and second flexural out-of-plane modes.
The fabrication of these devices and the transduction of their
motion are optimized in order to minimize the effect of other
sources of nonlinearity. We show that Euler-Bernoulli beam
theory yields predictions for the first mode that significantly
deviate from our experimentally observed data, especially
for cantilevers of low AR. In contrast, excellent agreement
between theory and measurement is observed for the sec-
ond mode. These results have significant implications for
024304-1
1098-0121/2013/87(2)/024304(8)
©2013 American Physical Society
L. G. VILLANUEVA
et al.
PHYSICAL REVIEW B
87
, 024304 (2013)
experimental design and interpretation, and are expected to
stimulate further improvement in theoretical modeling beyond
Euler-Bernoulli beam theory, as we discuss below.
II. THEORY
The type of structure that we use in our study is shown in the
schematic of Fig.
1
. Cantilever beams with U-shaped geometry
are chosen given their interest for various applications
34
,
35
and to facilitate linear detection, as will be detailed later. The
cantilever beams have a total length
L
and width
b
. The region
close to the clamp presents two legs of length
L
leg
and width
b
leg
. In our particular examples, the structures are designed to
have
L
=
3
L
leg
and
b
=
3
b
leg
. The linear dynamic analysis of
these cantilevers, according to Euler-Bernoulli theory, is easily
obtained using an analysis for beams with nonuniform cross
sections
12
,
36
(see Appendix
A
).
The geometric and inertial nonlinearities in our cantilever
structures according to Euler-Bernoulli theory are calculated
using Hamilton’s principle, the Galerkin method, and assume
that only one normal mode is active.
31
,
33
This yields Eq.
(1)
for the dynamics of the
n
th mode, where we have omitted the
index
n
for simplicity:
m
eff
̈
x
+
m
eff
ω
R
Q
̇
x
+
k
eff
x
+
β
geom
L
2
x
3
+
β
iner
L
2
(
x
̇
x
2
+
x
2
̈
x
)
=
G
cos(
ωt
)
,
(1)
where the dot denotes the time derivative,
Q
is the quality
factor,
G
cos(
ωt
) is the externally applied driving force, and
m
eff
,
k
eff
,
β
geom
, and
β
iner
are the effective mass, effective
elastic constant, geometrical nonlinear coefficient, and inertial
nonlinear coefficient, respectively, and they are given by
m
eff
=
1
0
μ
(
ξ
)
φ
(
ξ
)
2
dξ,
k
eff
=
1
0

EI

(
ξ
)
φ

(
ξ
)
2
=
m
eff
ω
2
R
,
(2)
β
geom
=
2
1
0

EI

(
ξ
)[
φ

(
ξ
)
φ

(
ξ
)]
2
dξ,
β
iner
=
1
0
μ
(
ξ
)
(
ξ
0
φ

(
ζ
)
2
)
2
dξ,
FIG. 1. (Color online) Schematic showing the type of device
investigated in this study: a U-shaped cantilever beam of total length
L
and width
b
. The region close to the anchor presents two legs of
length
L
leg
and width
b
leg
.
where
μ
(
ξ
) is the mass per unit length as a function of
normalized distance
ξ
along the beam axis,

EI

(
ξ
)isthe
bending rigidity,
φ
(
ξ
) is the normalized mode shape, and the
primes denote spatial derivatives. Note that this theoretical
formulation is generally applicable to cantilevers with spatially
varying cross sections like the devices used in this work.
Using secular perturbation theory,
8
we can solve Eq.
(1)
and
extract the amplitude response in the vicinity of the resonant
frequency
ω
R
:
x
2
(
ω
)
(
G
2
k
eff
)
2
(
ω
ω
R
ω
R
3
8
α
L
2
x
2
(
ω
)
)
2
+
(
1
2
Q
)
2
,
(3)
where
α
is the dimensionless nonlinear coefficient, which
depends on both inertial and geometric nonlinearity,
α
=
β
geom
k
eff
2
3
β
iner
m
eff
.
(4)
Note that
β
geom
iner
>
0
,
and therefore the final nonlinearity
of the structure is determined by two competing effects:
geometric and inertial nonlinearities. The former stiffens
the structure at large amplitudes, while the latter leads to
a softening effect. For the cantilevers used in this work,
using the mode shapes that are derived in Appendix
A
,we
obtain
α
1
=
0
.
044
±
0
.
001 and
α
2
=−
18
.
6
±
0
.
15 for the
first and second flexural modes, respectively. Variations in the
parameters
α
1
and
α
2
are due to fabrication nonuniformities,
as will be discussed below.
To experimentally assess the validity of these calculations,
we utilize a system that employs a highly linear transduction
technique to actuate and detect the motion. The resonators
are made from well-characterized materials, allowing us to
stay within their linear range of mechanical response. Also,
the fabrication process (based on bulk micromachining) yields
cantilever beams with well-defined clamping regions.
III. FABRICATION
The fabrication of the devices starts with 725
μ
m double
sided polished, 100-mm-diameter, silicon wafers. We deposit
a 500-nm-thick layer of low stress LPCVD (low pressure
chemical vapor deposition) silicon nitride (SiN) on both sides
of the wafer [Fig.
2(a)
]. We then pattern the SiN on one
side of the wafer (back side) using photolithography and dry
etching, prior to performing an anisotropic silicon etching in
KOH (potassium hydroxide) [Fig.
2(b)
]. This step defines SiN
membranes on one side of the wafer (front side).
Once the membranes are defined, we perform electron beam
lithography using a double layer of poly(methyl methacrylate)
(PMMA) in order to lift off the metal layer. We evaporate a
bilayer of Cr (5 nm, adhesion layer) and Au (50 nm) which is
subsequently patterned using the liftoff of the PMMA double
layer processed before [Fig.
2(c)
]. A second lithography and
liftoff process is then performed to define the metal contacts
with a thicker metal layer (Cr
/
Au, 5
/
150 nm).
Finally, usingthegoldas ahardmask, weperformamilddry
etching of the silicon nitride layer, which defines the released
structures with no undercut at the clamping region [Figs.
2(d)
and
1
). The resulting structures are a trilayer stack of SiN
(510
±
5 nm thick), chromium (adhesion layer, 5 nm), and gold
024304-2
NONLINEARITY IN NANOMECHANICAL CANTILEVERS
PHYSICAL REVIEW B
87
, 024304 (2013)
(a)
(b)
(c)
(d)
Silicon
Silicon Nitride
Gold
FIG. 2. (Color online) Schematic for the fabrication process flow.
Side view is shown in the left column and the corresponding top view
is depicted in the right column. (a) SiN is deposited on both sides
of a Si wafer. Back side SiN is patterned to define windows for the
subsequent anisotropic silicon etching in KOH, yielding membranes
on the front side (b). We then deposit (c) two bilayers Cr/Au by
means of two subsequent liftoff processes: one to be used in the
detection of motion (5 nm
/
50 nm Cr
/
Au) and another one to define
the contacts (5 nm
/
150 nm Cr
/
Au). (d) Using the gold as a hard
mask, we perform a mild dry etching of the silicon nitride layer
which defines the released structures with a proper clamping region,
i.e., with no undercut (see also Fig.
1
).
(20
±
10 nm). Note that the final gold layer has a decreased
thickness as a consequence of the dry etching that is performed.
Some examples of the released structures are shown in
Figs.
3(a)
and 3(b), where we can see the two legs near the
clamp that permit sensitive detection of the cantilever motion.
The structures are designed to have a width of
b
=
4
.
5
μ
m,
and the width of each leg to be
b
leg
=
1
.
5
μ
m. Deviations
from these values between devices are of order
±
50 nm. A
range of cantilevers of different lengths is fabricated, with AR
ranging from 2 to 13, and the legs designed to be one-third
of the total length. Alignment tolerance causes dispersion
of approximately
±
1
μ
m in the total cantilever length and
the length of the legs (an example of this can be seen in
Appendix
C
). This variation slightly affects the theoretical
estimation of the nonlinear coefficients. Therefore, every
device is individually inspected using a scanning electron
microscope (SEM) to accurately determine the dimensions
and hence the theoretical nonlinear coefficients.
IV. EXPERIMENTAL RESULTS
Actuation is performed by means of a piezoshaker stage op-
erating in linear regime, taking precautions against the effects
of electric leakage. Measurements are performed using two
techniques: (i) a highly sensitive optical detection scheme for
the detection of very small amplitudes (e.g., thermomechanical
noise); and (ii) the highly linear metal-based piezoresistive
readout technique
32
,
37
for the detection of larger amplitudes.
Thermomechanical noise data is used to calibrate the optical
detection responsivity (
V
optical
/
m) in its linear range, for
small amplitudes, using the equipartition theorem. We then
obtain the metal-based piezoresistive detection responsivity
(
V
PZM
/
m) by comparing the resonant responses for both
readout methods, keeping the drive levels low to maintain
linearity of the optical detection. Finally, large amplitude
motion and nonlinear behavior are captured using metal-
based piezoresistive detection,
37
which is linear and now
calibrated. More details on the experimental procedure are
shown in Appendix
B
. We would like to emphasize here
that calibration of the motion is not performed using the
nonlinear coefficient, as has been proposed in the past,
38
but
using an independent phenomenon: the Brownian motion of
the cantilevers. Examples of the observed dynamic responses
for different drives are shown in Figs.
3(c)
3(f)
,forthefirst
and second flexural modes and for two cantilever devices of
different AR. While the second mode presents a softening
nonlinearity in both cases, Figs.
3(e)
and
3(f)
, the behavior of
the first mode can vary from being a stiffening nonlinearity
[Fig.
3(d)
] to one that is softening [Fig.
3(c)
], as the AR is
varied.
To facilitate quantitative comparison between different
devices, we extract the dimensionless parameter
α
from the
measurements using a double fitting procedure: (i) we first
fit the full-resonant response to Eq.
(3)
, and then (ii) we fit
the frequency positions of the maxima,
ω
max
, for each drive to
(
ω
max
ω
R
)
R
3
αx
2
max
/
(8
L
2
). These two procedures yield
the same parameter values, and thus provide a consistency
check on fit procedure robustness. The values for
α
for the
first mode are given in Fig.
4(a)
, whereas those for the second
mode are in Fig.
4(b)
. Both figures display a solid gray line that
denotes the predicted theoretical value from Euler-Bernoulli
beam theory, taking into account the nonconstant cross section
(see Sec.
II
). For the two smallest ARs (i.e., AR
=
2 and 3)
the thermomechanical motion of the second mode cannot be
detected given the high stiffness of those modes, and thus
their nonlinearity cannot be characterized. The same was true
for modes higher than the second. Nonetheless, such higher
order modes have been measured previously on macroscale
devices,
23
,
39
yielding good agreement with Euler-Bernoulli
theory.
Figure
4
displays the differences in the nonlinear behavior
exhibited by the first two cantilever modes. We summarize the
observations as follows.
First mode
. Figure
4(a)
clearly shows a systematic decrease
in the nonlinear parameter
α
for the fundamental mode, as
AR is reduced, with the behavior changing from stiffening
to softening: experimental values approach the theoretical
(stiffening) prediction for large AR. The solid line gives the
theoretical prediction of Euler-Bernoulli theory; the dotted line
delineates softening and stiffening behavior; and the dashed
line is presented only as a visual aid.
Second mode
. The experimental data in Fig.
4(b)
contrast
strongly with those for the first mode [Fig.
4(a)
]. The dashed
line represents the average of the experimental data and the
boundaries of the colored zone define one measured standard
024304-3