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Micro and Nanosystems, 2009, 1, 22-29
Modeling, Design and Experimental Characterization of Bending
Resonant Circular Nano Cantilevers
N. Lobontiu*,1, B. Ilic2, T. Reissman2, E. Garcia2, Y. Nam3 and H.G. Craighead2
1
University of Alaska Anchorage, USA; 2 Cornell University, USA; 3Kangwon National University, Korea
Abstract: This work reports on a new type of nano cantilever, the symmetric circular one, and provides results of its outof-plane bending natural frequency by means of analytical modeling and experimental testing. The cantilever, with a 225
nm thickness and planar dimensions smaller than 10 m, can be implemented in dynamic nano electromechanical applications such as atomic force microscopy, surface topology characterization or resonance-shift mass detection. An analytic
model is proposed that considers the distributed stiffness and inertia properties through an accurate distribution function.
The model also captures the short-beam character of this structure and the related shear effects. Several silicon nitride
nano cantilevers have been surface-micromachined and tested by using in-vacuum laser excitation and interferometric
measurement of the bending resonant response. The model predictions and the experimental results indicated minimal errors. Based on this agreement, the analytical model was utilized to further analyze the influence of the basic geometric parameters on the cantilever’s natural frequency.
Keywords: MEMS, NEMS, cantilever, circular, resonator, bending.
1. INTRODUCTION
This report presents results on the modeling, design, experimental and simulation of the out-of-plane bending natural frequency for a novel nano cantilever design, the constant-thickness circular cantilever, which can be implemented in a variety of applications, such as resonant detection of minute amounts of attached substance or dynamic
atomic force microscopy (AFM) transduction. The main
advantage of this specific cantilever over existing designs
consists in its ability to produce a wide range of resonant
frequencies by changes in the circular notch radius and in the
notch center position. Another advantage is the analytical
model, which takes into consideration the short-beam character of this particular design by means of the exact deformation distribution function, and thus enables accurate simulation to be performed.
At micro and nano scale, under specified fabrication constraints, and for a given geometrical envelope, say of rectangular shape, the main modality of altering a cantilever’s
natural frequency is through shape modification. Shapes that
can be described by analytical curves present the advantage
that closed-form equations, which are often times sufficiently-accurate for conducting first-stage model simulation,
can be obtained for the amount of interest of a given structure, be it the stiffness in a static application, or a certain
resonant frequency in a dynamic problem. The procedure of
selecting specific geometric shapes is therefore extremely
important in producing designs that are capable of matching
pre-defined performance criteria (such as specified static
deflections or natural frequencies), as well as in refining the
geometrical set of parameters in view of optimizing the
structural response.
Cantilevers with dimensions in the micrometer and
nanometer range have been used lately for substance characterization or recognition through dynamic operation in viscous fluids [1-4], for imaging/sensing, data writing/storage
*Address correspondence to this author at the University of Alaska Anchorage, USA; E-mail: [email protected]
1876-4029/09 $55.00+.00
or material properties evaluation through scanning-probe
microscopy, atomic-force microscopy [5-10] or model identification [11]. Two physical principles are usually being
applied in micro/nano-scale cantilever-based transduction:
(a) the deflection change is statically monitored; or (b) the
shift in a relevant natural frequency (most often the one
corresponding to the out-of-plane, weak-axis vibrations) is
determined [12, 13]. The cantilever deflection alters as a
result of a variety of external factors such as microtemperature variation, molecular adsorption, microcorrosion, phase change or stress-induced surface property
changes [14-17]. Natural frequency shift methods have successfully been applied in detecting the quantity (to the level
of attogram) and position of matter that deposits on a cantilever [18-30] or in physical/chemical sensors [31, 32].
Fig. (1). Three-dimensional view of a circular nano cantilever.
Studied here is a new cantilever design, the symmetric
circular nano cantilever, which is pictured in the threedimensional rendering of Fig. (1). For a specified thickness,
this configuration provides dimensional and therefore operational variability through modifications of a few geometric
parameters, as it will be discussed shortly. Particular emphasis of this research falls on the out-of-plane bending resonant
frequency (about the y-axis, the weak axis), which is investigated through a mathematical model and experimental testing.
© 2009 Bentham Science Publishers Ltd.
Circular Nano Cantilevers
Fig. (2) shows the top-view schematic of a generic circular nano cantilever that is defined by the radius R, the minimum thickness wmin, the start angle min and the end angle
max.
Fig. (2). Schematic top view of a generic circular nano cantilever
with geometry and dimensions.
A large variety of circular geometric configurations can
be generated from a given rectangular plate of dimensions L
and W – see Fig. (3) – by modifying only three geometric
parameters pertaining to two identical circular regions that
are placed symmetrically with respect to the plate longitudinal direction, namely: the radius R, minimum transverse
distance wmin and common longitudinal position L0. Fig. (4)
shows three limit designs that can be realized by only chang-
Micro and Nanosystems, 2009 Vol. 1, No. 1
23
ing the value of the longitudinal parameter L0 of Fig. (3),
which results in alterations of the limit angles min and max.
The right corner-filleted design of Fig. (4a) is wellknown – it was first analyzed in terms of analytical compliances (the inverses of stiffnesses) in [33] in the 1960s and
more recently in [34], amongst others. Two other configurations, the half-length circular one and the symmetric design,
are shown in Figs. (4b and 4c).
The method of calculating the out-of-plane bending natural frequency is based on Rayleigh’s principle for conservative systems, which yields the resonant frequency as the
solution to an equation comprising the elastic potential energy and the kinetic energy. The calculation method that we
utilize herein is compact as it expresses both stiffness and
inertia properties of a vibrating continuous, beam-like structure by means of a unique function: the deflection distribution function. In the majority of similar research reports, the
simplified form of the distribution function corresponding to
a constant cross-section cantilever has been utilized. This
work employs the actual, exact-form distribution function
which depends on the particular geometry of the studied
structure by also taking into consideration the shearing effects (due to the short length of the circular cantilever) – and
these represent novel features of this report. The distribution
function can be expressed generically in terms of compliances for any cantilever profile which is expressed by an
analytical function.
Experimental testing has been conducted on several surface-micromachined circular-cantilever samples and comparison between experimental data and analytic model simulation revealed good correlation. Subsequent numerical
simulations, based on the analytical model, have also been
performed in order to derive conclusions on the relationship
between geometric parameters and the out-of-plane bending
natural frequency of the circular nano cantilever.
Fig. (3). Generation of a circular cantilever from a blank rectangular plate of dimensions L x W and two identical circles of radii R.
24 Micro and Nanosystems, 2009 Vol. 1, No. 1
Lobontiu et al.
Fig. (4). Limit configurations of circular cantilevers: (a) Right circular (corner-filleted) design (min = 0o, max = 180o); (b) Half-length
circular design (min = 0o, max = 90o); (c) Symmetric profile (min + max = 180o).
2. ANALYTIC MODEL OF OUT-OF-PLANE BENDING NATURAL FREQUENCY
A generic model is first derived of the out-of-plane bending natural frequency for a single-profile cantilever and then
specific geometry is used to calculate the bending natural
frequency of the circular cantilever design.
2.1. Generic Model for a Generic Single-Profile Cantilever
The out-of-plane bending natural frequency is formulated
in this section based on the cantilever model which is shown
in Fig. (5), and which represents the circular design reported
herein. The slopes and deflections are identified in Fig. (5)
that correspond to the free end – y and uz – and to a generic
point of abscissa x – y(x) and uz(x), respectively. Axial
deformations (ux(x) at the generic point of abscissa x and ux
at the free end) are produced by the force Fx, but these deformations are not shown in Fig. (5).
2
2u ( x ) E I y ( x) z 2 dx
x 2 = 0 l A( x)u z2 ( x)dx
l
(1)
0
where Iy(x) and A(x) are the variable cross-sectional moment
of inertia and area that are measured at the generic abscissa x
– Fig. (5), and are expressed as:
w( x)t 3
I y ( x) =
12
A( x) = w( x)t
(2)
with w(x) being the variable width and t the constant thickness.
The deflection at an arbitrary abscissa, uz(x), is expressed
in terms of the free-end deflection uz by means of a distribution function f(x) as:
u z ( x) = f ( x)u z
(3)
In this case, Rayleigh’s quotient – Eq. (1) becomes:
2
d 2 f ( x) E I y ( x) dx
dx 2 2
0
=
l
2
A( x) [ f ( x) ] dx
l
(4)
0
Fig. (5). Bent cantilever with representative slopes and deflections.
The circular nano cantilever is a single-profile configuration, as its variable width can be defined by one analytical
curve (the circular one). In the case of a variable crosssection beam, the out-of-plane bending natural frequency is
expressed by means of Rayleigh’s quotient as:
Eq. (4) is quite versatile as it enables calculating the outof-plane bending resonant frequency of both cantilevers
(fixed-free beams) and bridges (fixed-fixed beams) whose
cross-section can vary continuously – their cross-sectional
dimensions (width and thickness) are defined each by a single analytical curve, such as line segment, circular, elliptic or
other planar curve’s profile equation. The other prerequisite
in using Eq. (4) is knowledge of the bending-related distribution function f(x). In our case, the nano resonators are mod-
Circular Nano Cantilevers
Micro and Nanosystems, 2009 Vol. 1, No. 1
25
eled and fabricated as cantilevers that have one end built-in
(anchored to the substrate) and the other end free.
The compliances which are relevant in expressing the
bending distribution function of Eq. (6) are calculated as:
The distribution function can be determined by applying
a force at the cantilever’s free end and by calculating the
maximum deflection at the same location, uz, as well as the
deflection at a generic point on the cantilever, uz(x), which,
by elimination of the force, results in a relationship of the
type shown in Eq. (3). Another way of finding the distribution function is to apply a uniformly-distributed load along
the full-length cantilever in order to express the connection
of Eq. (3). The two approaches result in predictions that
yield little variations – see [35], and this is particularly valid
for constant cross-section members. The present work is
concerned with the concentrated-force approach of determining the bending distribution function.
l
x2
=
C
x
(
)
l
x EI y ( x) dx
l
1
Ca ( x) =
x EA( x) dx
l
x
C ( x) =
x EI y ( x) dx
c
l
x2
dx
Cl = EI y ( x)
0
For a constant cross-section cantilever, the bendingrelated distribution function is simply:
The compliance Cl is the linear compliance, and is calculated as the ratio between the free-end, out-of-plane deflection and the corresponding force applied at the same point
about the same direction; Cl(x) is another linear compliance
that is calculated as the ratio of the out-of-plane deflection at
a generic point of abscissa x to the force that is applied at the
cantilever free end; Cc(x), the cross compliance, is the ratio
of the slope (rotation angle) of the deformed cantilever being
measured at the point of abscissa x to the free-end, out-ofplane force. Ca(x), the axial compliance, represents the ratio
between the axial displacement at a point of abscissa x and
the axial force that would be applied at the cantilever free
end. The compliances Cr(x) and Cr, which enter Eqs. (7) and
(8), but which do not influence the distribution function of
Eq. (6), are rotary compliances; they relate the slopes at a
generic abscissa x and at x = 0 to the free-end moment My.
f * ( x) = 1 3 x 1 x3
+
2 l 2 l3
(5)
However, when the cantilever cross-section varies, Eq.
(5) is valid as an approximation only. The procedure of determining the bending distribution function in this case still
follows the approach described previously for the situation
where a point load is applied about the z-direction (out of the
plane) at the cantilever free end. It can be shown, see [34] for
more information, that the distribution function can generically be expressed in terms of compliances as:
f ( x) =
Cl ( x) + E
Ca ( x) xCc ( x)
G
Cl
(6)
2.2. Model of the Circular Cantilever
where E and G are the Young’s and shear modulii of the
cantilever material and x is the abscissa of a generic point
along the longitudinal axis of the cantilever that is measured
from the free end. Eq. (6) takes into consideration that the
symmetric circular cantilever is essentially a short beam (as
its length – an amount less than 2R – is not 3-5 times larger
than the largest cross-sectional dimension, which is the root
width, equal to R). The short-beam character of this structure
is modeled through the constant , which, for a rectangular
cross-section, is =5/6. The compliances (or spring constants) of Eq. (6) – see [34] for more details – relate the
cantilever deformations (slope and deflection) at the generic
point of abscissa x to the free-end force Fz and moment My
of Fig. (5) in the following manner:
0 Fz u z ( x) Cl ( x) Cc ( x)
0 M y y ( x) = Cc ( x) Cr ( x)
u ( x ) 0
0
Ca ( x) Fx x Cc
Cr
0
0 Fz 0 M y Ca Fx The symmetric-profile circular cantilever that is represented in Fig. (6) has been selected for modeling, microfabrication and experimental characterization.
(7)
Similarly, the free-end point deformations and loads are
connected by the relationship:
u z Cl
y = Cc
u 0
x (9)
(8)
Fig. (6). Top view of circular symmetric cantilever with geometry.
Equations (4), (6) and (9) are valid for any variable crosssection cantilever which is formed of a single profile (defined by a unique geometric curve). For the symmetric circular cantilever of Fig. (6), the limit configuration occurs for
26 Micro and Nanosystems, 2009 Vol. 1, No. 1
Lobontiu et al.
min = 0o and max = 180o, as indicated in Fig. (4a). It can be
seen that two basic parameters fully define the planar configuration of a generic symmetric circular cantilever, as the
one of Fig. (6), namely: the radius R and the minimum width
wmin. Alternatively, the start and end angles min and max, as
well as the length l can be used to uniquely define a circular
cantilever; the latter parameters can be found from the basic
ones as:
2
3R 2 2 Rwmin wmin
min = cos 1
2R
max = min
2
2
l = 3R 2 Rwmin wmin
cromachining were utilized to produce the circular resonators. The excitation was generated by a diode laser system
and the motion detection was performed by using an interferometric system - [36], the testing being performed in
vacuum in order to avoid air damping losses and the corresponding inaccuracies. Figs. (8 and 10) show the experimentally-determined bending natural frequencies (resonant
peaks) that have been detected for two of the fabricated
specimens which have been tested.
(10)
The moment of inertia and cross-sectional area are variable for the cantilever configuration analyzed here, and they
are expressed as:
w( )t 3
I y ( ) =
12
A( ) = w( )t
(11)
Fig. (7). Tilted photograph of a fabricated symmetric circular nano
cantilever with R = 7 m, wmin = 0.2 m and t = 225 nm.
where:
w( ) = wmin + 2 R (1 sin )
(12)
The abscissa x can be expressed as:
x=
1
2
3R 2 2 Rwmin wmin
R cos 2
(13)
and therefore the differential involved in all compliance
integrals is:
(14)
dx = R sin d
In working with the second derivative of the distribution
function – as required by Rayleigh’s quotient of Eq. (4) – the
chain rule of differentiation (Leibniz’s rule) has to be used:
df d df
df
1
1
df
dx = d dx = d dx = R sin d
d
d 2x 2
d2 f
d2 f
1
1
1
d f df d 2 df = 2 2 2 =
2
2
2
dx
tan d d
dx d
R sin d
dx
d d Fig. (8). Experimental resonant peak for the nano cantilever of Fig.
(7).
(15)
The exact (analytical) distribution function is not provided here as its expression is quite lengthy. Subsequent
formal complications arise when expressing the out-of-plane
bending resonant frequency of Eq. (4), which however results in a closed-form equation, not being furnished herein
because of its exceeding length, but which is used for further
analytical calculations that are reported further on.
3. FABRICATION AND EXPERIMENTAL TESTING
OF CIRCULAR NANO CANTILEVERS
Several circular nano cantilevers of double symmetry
have been microfabricated, such as the ones photographed in
Figs. (7 and 9). Low-stress silicon nitride and surface mi-
Fig. (9). Tilted photograph of another fabricated symmetric circular
nano cantilever with R = 7 m, w min = 1 m and t = 225 nm.
Micro and Nanosystems, 2009 Vol. 1, No. 1
Circular Nano Cantilevers
27
loss mechanisms that lower the experimental natural frequency. It can also be seen that the predictions by the simplified model – Eq. (5) are always larger than the ones of the
full model – Eq. (6), which is also a pertinent conclusion, as
the full model will generate lower stiffnesses (through consideration of the shear effects) compared to the simplified
model. Errors as large as 11.6% are registered between the
two models’ results, the average relative error being 7.5%.
The relative errors between the experimental data and the
simplified model results can be as large as 13.5% (with a
10.5% error average), whereas the maximum error between
experimental and full model data is 6.4%, with an average of
3.2%.
5. ANALYTIC MODEL SIMULATION
The analytical model has further been utilized to study
the relationships between the two defining planar geometric
parameters, the radius R and the minimum width wmin, and
the natural frequency of interest. Fig. (11) is the twodimensional plot of the out-of-plane bending natural frequency as a function of R when considering that wmin
= 0.5 m, in addition to t = 225 nm, E = 160 GPa, = 2100
kg/m3 and = 0.25. As shown in Fig. (11), the natural frequency decreases non-linearly with the increase of R for
constant minimum width and thickness.
Fig. (10). Experimental resonant peak for the nano cantilever of
Fig. (9).
4. COMPARISON OF ANALYTICAL MODEL AND
EXPERIMENTATION RESULTS
The analytical model predictions and the experimental
results that have been obtained for several circular nano
cantilevers are now being compared.
Fig. (12) shows the variation of the out-of-plane bending
natural frequency with wmin for a constant radius of R = 8m
and all the other numerical values mentioned above. It can be
seen that for the 0.1 m – 1 m range of wmin, the natural
frequency increases in an almost linear fashion, whereas for
wmin > 1 m, the increase of fn is nonlinear. As a combined
conclusion, in order to increase the out-of-plane bending
natural frequency of symmetric circular cantilevers it is necessary to decrease the circular radius R and to increase the
minimum width wmin.
Table 1 comprises experimental data and analytic model
results of the out-of-plane bending natural frequency of several symmetric circular nano cantilever designs. The thickness of all microfabricated samples was t = 225 nm and the
material properties for low-stress silicon nitride that were
used for the analytic simulation are: Young’s modulus,
E = 160 GPa, mass density, = 2100 kg/m3 and Poisson’s
ratio = 0.25. Table 1 presents analytical results obtained
both by using the simplified distribution function of Eq. (5)
and the precise (full-model) one of Eq. (6) – which takes into
consideration the cross-section variation and the shear effects
produced by the cantilever’s short length. As Table 1 shows,
the predictions by the analytical models are higher than the
experimental results, and this is consistent with the various
Table 1.
5. CONCLUSIONS
The work introduces the new symmetric circular nano
cantilever, which is defined by a constant thickness in the
Experimental and Analytical Natural Frequencies (AF- Full Analytic Model Data, AS – Simplified Analytic Model Data,
E – Experimental Data)
Design
R
wmin
#
[μm]
[μm]
Resonant Frequency [MHz]
Experimental
Relative Errors [%]
Analytic
Simplified
Full
AF-AS
E-AS
E-AF
1
6
0.2
3.99
4.52
4.02
11.6
11.7
0.7
2
6
0.5
4.53
4.81
4.65
3.3
5.8
2.6
3
6
1
4.72
5.27
4.90
7.0
10.4
3.7
4
7
0.2
2.92
3.31
3.00
9.4
11.8
2.7
5
7
0.5
3.37
3.72
3.48
6.5
9.4
3.2
6
7
1
3.60
3.95
3.73
5.6
8.9
3.5
7
8
0.2
2.62
3.03
2.80
7.6
13.5
6.4
8
8
0.5
2.91
3.25
2.99
8.0
10.5
2.7
9
8
1
2.83
3.22
2.94
8.7
12.1
3.7
28 Micro and Nanosystems, 2009 Vol. 1, No. 1
Fig. (11). Natural frequency as a function of the notch radius.
order of 200 nm and with planar dimensions that do not
exceed 10 m. In its out-of-plane (weak-axis) bending mode
vibration, the cantilever can be utilized as a dynamic transducer in atomic force microscopy, surface topology characterization or resonance-shift mass detection applications. An
analytic model that gives the out-of-plane bending natural
frequency is derived by taking into consideration several
factors which contribute to generating an accurate model
such as: the variable geometry of the planar profile, the distributed stiffness and inertia properties (by means of a precise deflection distribution function), and the inherent shortbeam nature of this cantilever design together with the associated shear deformations. Silicon nitride nano cantilevers
have been surface-micromachined and subsequently tested in
a vacuum environment by applying laser excitation and interferometric detection of the out-of-plane bending resonant
response. The errors between model predictions and experimental data were very small, which enabled using the analytical model to further analyze the relationship between the
basic geometric parameters of the cantilever’s planar profile
and its relevant natural frequency.
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Received: December 02, 2008
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