of 16
SUPPORTING INFORMATION
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
SI-1
I.
Effect of Surface Stress on Doubly
-Clamped Beams
A. Axial force model (Stress effect in Table I)
In this section, we derive the required formulas for the effect of surface stress on the
resonant
frequency of doubly
-clamped beams due to the axial tension built up along the beam (
stress effect
in
Table I). In the main manuscript we discussed that a net axial force is induced in doubly
-clamped
beams, along their major axis. This net axial fo
rce is given by the integral of the resulting axial stress
over the beam cross
-section. For the resonators used here, this coincides
with the integral of the axial
stress in the active piezoelectric layer only; other parts of the beam do not contribute to
the net axial
force because the beam
-ends are restrained from moving. The net axial force determines the resulting
effect on resonance frequency and/or stiffness of the device.
Figure S1 |
Graphical representation of a doubly-
clamped (a) and a
cantilever
(b) beam with the dimensions used in the paper, as well as the axis
definition. In (a) a compressive stress (positive stress by convention) is
shown, whereas in (b) a tensile (negative by convention) stress is shown.
L
b
h
σ
+
z
y
x
a.
b.
L
b
h
σ
-
z
y
x
Supporting Information:
“S
tress-Induced Variations in the Stiffness of
Micro
- and Nano
-C antilever Beams
R. B. Kara
balin, L. G. Villanueva, M. H. Matheny, J. E. Sader, M. L. Roukes
SI-2
The effect of surface stress on the re
sonance frequency of a doubly
-clamped beam is calculated
to leading order for small surface stress loads, and thus gives a linear relationship between the resonant
frequency shift and the applied surface stress change. It is assumed that the beam
structure
possesses a
rectangular cross
-section whose width
b
greatly exceeds its thickness
h
, i.e., the beam is formally a thin
plate
S1
. The length of the structure is
L
(see Fig. S1).
We use the general theoretical formalism presented elsewhere
S2
to calculate the effect of
surface stress, and thus decompose the problem into two subproblems:
Subproblem (1)
: Deformation of an unrestrained plate under application of a net surface stress load.
Subproblem (2)
: Beam structure with no surface stress load
and a specified in
-plane displacement to
satisfy the required clamped boundary conditions at the ends.
Superposition of these two subproblems gives the required in-
plane deformation of the original
problem, with exact satisfaction of free edge and clamped boundary conditions; see ref. S
2 for details.
Application of a net surface stress change to the surface
of the structure, with the clamped
boundary conditions removed, results in an isotropic strain whose displacement field is:
,
(S1)
where
u
and
v
are displacements in the
x
and
y
directions, respectively;
are the Cartesian
coordinates in the plane (see Fig. S1);
ν
and
E
are the Poisson ratio and Young’s modulus of beams
material respectively;
h
is the device thickness; and
σ
s
T
is the total applied surface stress
S2
. This is the
required solution to Subproble
m (1).
To account for the clamped displacement conditions at the end of the structure, in accord with
Subproblem (2), a displacement load in the
x
-direction must be applied to its ends; displacements in the
y
-direction are not important since the beam length greatly exceeds its width, in accord with Saint
-
Venant's principle
S3
. From Eq. (S1), this axial displacement is:
,
(S2)
where
L
is the beam length. Such an axial displacement induces an axial tensil
e load:
,
(S3)
( )
(
)
( )
1
,,
T
s
uv
x y
Eh
νσ
= −
(, )
xy
(
)
1
T
s
L
u
Eh
νσ
=
(
)
1
T
axial
s
Fb
νσ
= −
Supporting Information:
“S
tress-Induced Variations in the Stiffness of
Micro
- and Nano
-C antilever Beams
R. B. Kara
balin, L. G. Villanueva, M. H. Matheny, J. E. Sader, M. L. Roukes
SI-3
where
b
is the beam width. The axial load in Eq. (S3) will lead to a change in stiffness and hence
resonant frequency. Since the beam length greatly exceeds width and thickness, this effect is calculated
from Euler-
Bernoulli beam theory. The governing equation for the deflection function
w
, for a beam of
linear mass density
μ
and areal moment of inertia
I
, is:
,
(S4)
which is solved with the usual clamped boundary conditions at the beam ends,
.
We assume an explicit time dependence of
, where
ω
is the angular frequency and
t
is
time, i.e.,
. Multiplying both sides of Eq. (S4) by the deflection function
w
,
scaling
x
by the beam length
L
, and integrating over the beam length yields the following
exact result
for square of the radial resonant frequency:
,
(S5)
where
is the scaled axial distance.
To calculate the leading order effect of surface stress change on the frequency shift, we use the
deflection function for a doubly
-clamped beam in the absence of surface stress. Solving Eq. (S4) under
this condition then gives:
,
(S6)
where
is the
n
-
th positive root of:
,
(S7)
with
corresponding to the fundamental mode.
Substituting Eq. (S6) with
into Eq. (S5), and using Eq. (S3), then
yields the required
result:
,
(S8)
4
22
4
22
0
axial
w
ww
EI
F
x
xt
∂ ∂∂
μ
∂ ∂∂
− +=
'0
ww
= =
exp(
)
i t
ω
( , )
( ) exp(
)
wxt W x
t
i
ω
=
ω
2
=
EI
L
4
μ
′ ′
W
(
x
)
(
)
2
d
x
0
1
W
(
x
)
( )
2
d
x
0
1
+
F
axial
L
2
EI
W
(
x
)
(
)
2
d
x
0
1
W
(
x
)
( )
2
d
x
0
1
x
x
L
=
( )
(
)
cosh
cos
cosh
cos
sin
sinh
sinh
sin
nn
nn
n n
nn
DD
W x
Dx
Dx
Dx
Dx
DD
= −+
n
D
cosh
cos
1
nn
DD
=
1
n
=
1
n
=
ω
ω
R
= 0.1475
1 -
ν
( )
σ
s
T
E h
L
h
2
Supporting Information:
“S
tress-Induced Variations in the Stiffness of
Micro
- and Nano
-C antilever Beams
R. B. Kara
balin, L. G. Villanueva, M. H. Matheny, J. E. Sader, M. L. Roukes
SI-4
where the original radial resonance frequency
, the beam mass density
is
ρ
, and the radial frequency shift is defined
. We emphasize that Eq. (S8) is valid in the
asymptotic limit of small stress loads.
An alternate derivation of this formula is given in Ref. S
4.
B. Change in dimensions (Geometric Effect in Table I)
Due to the boundary conditions in a doubly
-clamped beam, after an isotropic in
-plane stress is
applied to the beam the strain map that develops is:
(S9)
Assuming that the Young’s modulus of the material remains unchanged during the application
of stress, the relative change in the resonant frequency would be:
(S10)
That, using (S9) with (S10), yields:
(S11)
which is the expression found in Table I in the main manuscript. In the case of doubly
-clamped beams
and the typical geometries that are used
, the
stress effect
described previously is much larger
than this
geometric effect
.
2
2
6.459 /
/
RR
h
fE
L
ωπ
ρ
=
=
R
ωωω
∆=−
(
)
(
)
(
)
(
)
1
11
0;
1
;
1
TT
ss
xx
yy
zz
Eh
Eh
νσ
ν ν νσ
εε ν
ε
ν
+−
==
+=
Δ
1Δ Δ Δ 1ΔV Δ Δ 3 1 3
22
2
2
222
xx
yy
zz
R
tL
tL
tL VtL
ωρ
εεε
ωρ
=− +− = +− ≈− + +
(
)(
)
(
)
1
12 1
Δ
1
T
s
R
Eh
ν ν νσ
ω
ων
++ −
1
L
h

>>


Supporting Information:
“S
tress-Induced Variations in the Stiffness of
Micro
- and Nano
-C antilever Beams
R. B. Kara
balin, L. G. Villanueva, M. H. Matheny, J. E. Sader, M. L. Roukes
SI-5
II. (Unphysical) Axial Force M
odel for Cantilever Beams
It has been widely assumed that application of surface stress to a cantilever beam induces an
axial force along the beam length. This so
-called “axial force model” has been shown to be unphysical
and in violation of Newton’s 3
rd
law
S2,S5,S6
. For completeness, however, we reproduce the resulting
formula derived from this model
S7
. This allows comparison and assessment with measurements
performed in this study.
In the limit of small surface stress loads, the axial force model has be
en reported under various
forms
S7-12
. Since the underlying model is unphysical, we refrain from any discussion on the merits of
each form and simply report the most common formula that has been claimed to yield good agreement
with measurements
S7
:
.
(S12)
Comparing this result with the (physically correct) model for doubly
-clamped beams in Eq.
(S8), we observe that the axial force model predicts a larger shift in frequency in cantilever beams than
in doubly
-clamped beams, by a factor of:
.
(S13)
Since
, the axial force model predicts that cantilever beams are much more sensitive to
surface stress changes than doubly
-clamped beams. This is not observed in the controlled
measurements of doubly
-clamped and cantilever beams reported in this study.
22
2
12
1.2
TT
ss
R
LL
LL
Eh b
h
Eh b
h
σσ
ω
ωπ


=




(
)
8
1
RR
cant
beam
L
b
ωω
ωω
ν
∆∆
=
/1
Lb
>>
Supporting Information:
“S
tress-Induced Variations in the Stiffness of
Micro
- and Nano
-C antilever Beams
R. B. Kara
balin, L. G. Villanueva, M. H. Matheny, J. E. Sader, M. L. Roukes
SI-6
III. Piezoelectric loads in doubly
-clamped beams
The piezoelectric effect couples the mechanical and electrical degrees of freedom of a material
by the following relation
(S14)
where (
ε
) and (
σ
) are the mechanical strain and stress vectors, respectively;
[
C
] is the compliance
matrix of the material, (
E
) is the electric field vector and [
d
] is the piezoelectric matrix, which is
directly responsible for the mechanical/electrical coupling. For the piezoelectric material considered in
this study, the piezoele
ctric matrix is of the form (with axis 1, 2 and 3 pointing along the beam,
transversal in
-plane and out
-of-plane respectively):
.
(S15)
The application of an external voltage between two electrodes contacting the piezoelectric
material (as in our exp
erimental case) leads to the generation of an electric field and, consequently,
deformation of the material. Following a similar approach to that taken in Section I of the SI, we first
calculate the deformation of the unrestrained piezoelectric material. T
his allows the material to expand
or contract freely and therefore the in
-plane strains
ε
xx
along the beam and
ε
yy
perpendicular to the beam
and out
-of-plane strain
ε
zz
are given by:
,
(S16)
where
V
is the applied voltage and
is the thickness of the piezoelectric layer.
However, the composite beam structure used in the present devices contains other materials that
are not piezoelectric. Those materials impose some restrictions on the total deformation, which will be
a combina
tion of net elongation and bending. As a first approximation, bending does not contribute to
the stiffness of the beam
S2
and therefore we can limit our analysis to the net (average) strain. This net
strain is given by Eq. (S17) when the Poisson ratios of t
he different materials are identical (which is an
excellent approximation in the present case):
( )
[ ]
( )
[ ]
( )
··
Cd
εσ
=
+
E
[ ]
31
31
31
31
33
45
45
00
00
00
00
0 00
d
d
ddd
d
d
d




=






31
33
;
xx
yy
zz
PZE
PZE
VV
dd
hh
εε
ε
==
−=
PZE
h
Supporting Information:
“S
tress-Induced Variations in the Stiffness of
Micro
- and Nano
-C antilever Beams
R. B. Kara
balin, L. G. Villanueva, M. H. Matheny, J. E. Sader, M. L. Roukes
SI-7
(S17)
where
is the Young’s modulus of the piezoelectric material,
h
tot
is the total thickness of the beam,
and the summation in the denominator extends t
o all four layers of the composite structure, with
and
being the Young’s modulus and thickness of every layer, respectively. The brackets in equation
(S17)
,
, mean average over the cross section of the beam (
y-z
plane).
Following the approach of Section I of the SI, we initially consider the case of a doubly
-
clamped beam. The solution above corresponds to Subproblem (2) in Section I. Subproblem (2) then
requires a displacement load in the
x
-direction to match the required
clamped boundary conditions at
the beam ends, as discussed above. This axial displacement induces the following axial load:
.
(S18)
Equation (S18) can now be directly compared to Eq. (S3) and the derivation performed in Section
I is valid here. We can thus write an expression equivalent to (S8), in this case for a piezoelectric beam:
,
(S19)
where
<EI>
is the effective flexural rigidity of the beam. If the Young’s moduli of the different
materials in the composite structure are approximately the same, Eq. (S19) immediately leads to:
,
(S20)
which is only dependent on total thickness
h
tot
. Comparing
Equations (S8) and (S20), it is evident that
the effects of an applied voltage are equivalent to the effects of an applied surface stress, as required.
This leads to the following relation connecting an equivalent surface stress to a given applied voltage
:
.
(S21)
Note that in the calculation of the change in frequency for a doubly
-clamped beam, we have
neglected any differential change in dimensions, since this is much smaller than the contribution given
by Eq. (S20) (we assume that any beam holds the condition
L>h
tot
).
31
33
;
PZE
xx
yy
zz
i i
tot
E
V
dV
d
Eh
h
εε
ε
=
=−=
PZE
E
i
E
i
h
ii
ε
31
axial
PZE
F
d E
Vb
=
2
31
0.0123
PZE
R
E
d V
bL
EI
ω
ω
=
2
31
0.1475
R
tot
tot
dV
L
hh
ω
ω

=


(
)
31
1
T
s
E
dV
σ
ν
=
Supporting Information:
“S
tress-Induced Variations in the Stiffness of
Micro
- and Nano
-C antilever Beams
R. B. Kara
balin, L. G. Villanueva, M. H. Matheny, J. E. Sader, M. L. Roukes
SI-8
IV.
Cantilever
beams
A. Isotropic material –
surface stress load
We begin by considering a cantilever plate composed of an isotropic material under a surface
stress load – this effect is due to surface stress change; see ref. S
2.
The
in
-plane stress
effect has been analyzed
previously and yields
S2
:
,
(S22)
where numerical coefficient
is estimated via FEM and accounts for 3D effects of the
stress distribution close to the clamp
. A detail exposition of the derivation of this formula and its
physical features is given in Ref. S4
.
The
geometric
effect is given by the differential change in dimensions of the cantilever beam,
due to the surface stress load. This tunes the stiffness and the resonant frequency of the cantilever, and
yields (considering
):
,
(S23)
In the limit as
(commensurate with the assumptions of thin plate theory
S2
), the in-
plane stress contribution in Eq. (S22) dominates the geometric effect in Eq. (S23), for
large and fixed
aspect ratio
. The in-
plane stress effect remains prevalent for thin beams under the condition:
,
(S24)
whereas for thicker beams, the geometric effect will dominate.
These results are gathered in Table I of the main manuscript.
( )
2
1-
()
T
s
R
bb
Eh
L
h
νσ
ω
φν
ω
  
=
  
  
0. 2
)
04
(
φ νν
≈−
xx
yy
εε
=
13
2
22
(1 2 )
T
s
zz
xx
R
hL
hL
Eh
ωρ
ε
ν
ε
ωρ
σ
+
∆ ∆∆ ∆
=− +− = −=
/0
hb
/
Lb
b
h
<
b
h
critical
5
1 + 2
ν
ν
1
ν
( )
L
b
Supporting Information:
“S
tress-Induced Variations in the Stiffness of
Micro
- and Nano
-C antilever Beams
R. B. Kara
balin, L. G. Villanueva, M. H. Matheny, J. E. Sader, M. L. Roukes
SI-9
B. Piezoelectric material –
voltage load
For the piezoelectric beams used in this study, the geometric effect can also be calculated:
.
(S27)
The changes in the Young’s modulus and Poisson ratio of the materials are neglected. For the
piezoelectric material used here,
, and hence Eq. (S27) becomes:
.
(S28)
Calculation of the stress effect in the devices considered is complicated by the mul
tilayer
structure of the piezoelectric cantilevers, and their non-
isotropic material properties. Three
-dimensional
FEM analysis is thus used to obtain the combined contributions of stress and geometric effects,
complementing Eq. (S28). This total effect is
plotted in Fig. 3. We find that the geometric effect
dominates the results.
33
31
1 33
=2
22
2
zz
xx
Rt
tot
ot
hL
V
h
h
L
dd
h
ωρ
εε
ωρ

∆ ∆∆ ∆
− +− = − =


−+
33
31
2
dd
= −
31
4
R
tot
d
h
V
ω
ω
=
Supporting Information:
“S
tress-Induced Variations in the Stiffness of
Micro
- and Nano
-C antilever Beams
R. B. Kara
balin, L. G. Villanueva, M. H. Matheny, J. E. Sader, M. L. Roukes
SI-10
V. Measurement of NEMS Device Response
We employ an optical interferometric detection scheme to sense the mechanical motion of our
NEMS devices
S13
. This detection mechanism is used
due to ease of implementation, high mechanical
responsivity to out
-of-plane beam displacements and negligible RF background noise.
Figure S2 |
Schematic of the optical interferometric detection
setup used for sensing the motion of NEMS devices
A schem
atic of the experimental setup is shown in Fig. S2. The devices are mounted in a
custom
-built room temperature vacuum chamber fitted with a quartz optical window for interrogation.
An optical cavity is created between a two
-mirror system formed by the bott
om of the substrate and the
top surface of the NEMS device. The light source consists of a regular helium neon laser (λ = 632 nm)
followed by a beam expander and an attenuator. After the original laser beam passes the beam splitter
oriented at 45
o
, one com
ponent is focused onto the NEMS device with a lens of 0.15 numerical
aperture, resulting in a spot size ~10–
20 μm. Interference between laser beams reflected from the top
surface of the mechanical resonator and bottom substrate is detected using a high-
ban
dwidth low
-noise
photodetector. As a result, the resonator motion modulates the light intensity in proportion to the
magnitude of the mechanical displacement.
Supporting Information:
“S
tress-Induced Variations in the Stiffness of
Micro
- and Nano
-C antilever Beams
R. B. Kara
balin, L. G. Villanueva, M. H. Matheny, J. E. Sader, M. L. Roukes
SI-11
An AC voltage between top and bottom electrodes of the NEMS device actuates the out
-of-
plane mot
ion, while a DC bias produces the required external stress studied in this work. Both voltages
are combined using a bias
-tee. A vector network analyzer
is used to measure the resonance spectral
response of the resulting signal. To perform the frequency shi
ft experiments, the network analyzer
operates in a continuous wave (CW) regime and is controlled by an external computer that is used to
provide phase locked loop (PLL) operation.
A slight asymmetry between tunability slopes is observed in doubly clamped beams. We
attribute this to an electrostatic effect from the substrate that affects the resonant frequency as
2
.
When fitting the experimental data to a second
-degree polynomial, the fit is more accurate but the
linear slope, nevertheless, remains the same (relative change smaller than 1%). In subsequent
experiments, we used a wafer with a thick sacrificial silicon
oxide layer underneath the seed aluminum
nitride layer and this reduced by two orders of magnitude the coefficient of second order in the
polynomial. In this case, again, the linear slope remains the same. We therefore believe that the
conclusions drawn i
n the current paper are not affected by the observed small symmetry
-breaking.
Fig. 3 in the main manuscript also shows some apparent
asymmetry and, in addition, an
apparent
discrepancy of the scaling
with
1/
2
. In order to clarify this issue, we replot Fi
g. 3
from the
main manuscript as Fig. S
3, in this case
with error bars
. The error bars for the experimental data are
calculated based o
n the measured Allan Deviation of 5·
10
-6
, representing the 95% confidence bounds.
The error bars for the FEM data corresp
ond to the typical expected accuracy of FEM simulations,
accounting for
geometrical
and material uncertainties
and convergence of the frequency shift; see
Section VI
-A. The discrepancy between theoretical prediction and measurement slopes is on the order
of ~5
-15%. This is a reasonable agreement considering the shown
experimental error
of ~10%,
especially visible on scaled plot shown in Fig S3(a).
Therefore, t
he slight asymmetry
and discrepancy
with the scaling appear both to be negligible, when careful ana
lysis of measurement errors is taken into
account.
Supporting Information:
“S
tress-Induced Variations in the Stiffness of
Micro
- and Nano
-C antilever Beams
R. B. Kara
balin, L. G. Villanueva, M. H. Matheny, J. E. Sader, M. L. Roukes
SI-12
Figure S3
|
(a)
Relative
Δ푓
/
and
(b)
absolute
Δ푓
frequency shift for three cantilevers
(Fig. 3
in the
main manuscript)
with error bars. The experimental uncertainties
show 95% confidence bounds and
they
are calculated based on the observed Allan Deviation of 5·
10
-6
– a measure of cant
ilevers’
frequency
fluctuations
. The error bars for FEM data are based on estimated geometrical and material
uncertainties and convergence of the frequency shift (see Section VI
-A).