of 24
Letters
https://doi.org/10.1038/s41565-018-0252-6
Electrical tuning of elastic wave propagation in
nanomechanical lattices at MHz frequencies
Jinwoong Cha
1,2
and Chiara Daraio
2
*
1
Department of Mechanical and Process Engineering, ETH Zurich, Zurich, Switzerland.
2
Engineering and Applied Science, California Institute of Technology,
Pasadena, CA, USA. *e-mail: daraio@caltech.edu
SUPPLEMENTARY INFORMATION
In the format provided by the authors and unedited.
NaturE NaNotECHNology
|
www.nature.com/naturenanotechnology
1
Supplementary Information for
Electrical
tuning of elastic wave propagation in
nanomechanical lattices at
MHz frequenc
ies
Authors:
Jinwoong Cha
1,2
, Chiara Daraio
2
*
Affiliations:
1
Department of Mechanical and Process Engineering, ETH Zurich,
Switzerland
2
Engineering and Applied Science, California Institute of Technology, Pasadena, CA, USA
*Correspondence to: Prof. Chiara Daraio (daraio@caltech.edu)
1.
Tuning mechanisms
Here, w
e qualitatively discuss the frequency tuning mechanism of a
single resonator. We
assume the dynamics of a thin circular plate resonator can be described by the Von
-
Karman plate
theory [S1]
, relevant for thin plates. The equation of motion
for
a single resonator
,
can be
then
written as
:
2
4
2
()
e
W
h
D
W
h
W
F
t
σ
,
(S1)
where
, h, W
, and
D
are the density, the plate thickness, the displacement field, and bending
rigidity
,
D=Eh
3
/12(1
2
)
, respectively
.
σ
is an in
-
plane stress tensor
,
which contains the
contributions from the intrinsic residual stress
,
0
σ
,
and the stress
induced
by the plate
’s
deformation
,
induced
σ
.
F
e
=
0
V
2
/2(d
W)
2
is the electrostatic force per unit area applied to the plate,
where
V
is the
applied voltage and
d
is the
vacuum gap. Considering an axisymmetric deformation,
the equation of motion can be reduced to
:
2
4
2
2
2
4
2
2
e
W
W
W
F
W
F
h
D
F
t
r
r
r r
r r
r
.
(S2)
2
Here,
F
is an in
-
plane stress function that satisfies the following compatibility equation
:
42
42
1
F
W
W
Eh r
r
r r
.
(S3)
We can express the displacement
W
(
r
,t
) as the summation
of
a dynamic displacement
W
D
(
r
,t
)
and a static displacement
W
0
(
r
).
Provided that the
dynamic displacement is much smaller than the
static displacement when subject to an applied DC voltage,
the electrostatic force can be
approximated as
:
2
0
2
00
2
(1
)
2(
)
D
e
V
W
F
d W
d W
.
(S4)
Therefore, the governing equation can be decomposed into a static part and a dynamic part.
4
2
2
2
0
0
0
0
4
2
2
2
0
2(
)
W
W
W
V
FF
D
r
r
r r
r r
r
d W
with
2
4
00
42
1
WW
F
Eh r
r
r r
(S5)
2
2
4
2
2
0
2
4
2
2
3
0
0
()
D
D
D
D
D
V
W
W
W
W
FF
h
D
W
t
r
r
r r
r r
r
d W
(S6)
If we assume that the dynamic displacement is too small
to induce an in
-
plane stres
s, the in
-
plane stress function
F
can be assumed to be just a function of the static displacement field
0
()
W
r
.
Consequently, the static and the dynamic parts can be decoupled as in Equation
s
S5 and S6.
First,
we can calculate t
he static deflection
,
0
()
W
r
,
and the in
-
plane stress function
,
F
,
under any applied
DC voltages. Then, the corresponding eigenfrequency can be found from the dynamic part
applying
appropriate boundary
conditions. The third
and
fourth
terms in Eq. S6 denote the tension induced
stiffening, while the
fifth term in Eq
.
S6 denote
s
the electrostatic softening.
Each
eigenfrequency
of the plate
changes as a DC voltage is applied. The application of the
voltage
deflects the plate, changing
its
equilibrium configuration
by
inducing a radial tension inside
the plate
. This tension
introduce
s
a softening potential term −
0
V
2
/(
d
W
0
)
3
in the dynamic
representation of the system.
3
Solving analytically Eqs. S5 and S6
is difficult. Thus, we numerically solve the problem using
COMSOL multi
-
physics. Instead of solving the Von
-
Karmann equations, COMSOL allows us to
solve more general plate equations obtained from
the
Mindlin
-
plate theory [S1]. The Mindlin
-
plate
theory con
verges to the Von
-
Karmann theory when the thickness of a plate becomes much smaller
than the radius of the plate. Thus, the in
-
plane inertia of the plate (or displacement) can be neglected.
We perform pre
-
stress eigen
-
frequency analyses for a circular plat
e with clamped boundary
conditions by considering geometric nonlinearity. The radius of the plate is fixed to 5
m
for all
performed simulations and we vary the thickness
,
t
,
and the vacuum gap distance
,
d
,
at different
applied gate voltages.
As shown
in Figure S1, thin plates with 5
nm and 10
nm thickness suspended with 600
nm
vacuum gap present hardening behaviors, increasing the resonance frequency with the applied gate
voltage. In this case, the contribution of the induced tension overcomes the elec
trostatic softening
potential. The contribution from the induced tension decreases
when the
thickness increase
s
,
because the increased bending stiffness reduce
s
the induced deformation
s
. In contrast, thicker
plates with 60
nm and 100
nm thickness suspended
over
a
200
nm vacuum gap present the decrease
of the resonance frequency due to softening effect
s
.
Supplementary Figure 1
. Thickness and vacuum gap dependence of
the
eigenfrequency of the
first normal mode
in
a
suspended,
circular plate.
(
a
)
Schematic of a voltage
-
gated circular plate
with radius
r,
thickness
t
, and vacuum gap
d
.
(
b
)
The resonance frequency of the fundamental plate
mode as a function of the applied DC gate voltage. The radius
r
is fixed to 5
m. Blue solid (dashed)
line:
t
= 5nm,
d
= 600nm (
t
= 10nm,
d
= 600nm). Red solid (dashed) line:
t
= 40nm,
d
= 400nm (
t
= 80nm,
d
= 400nm). Black solid (dashed) line:
t
= 60nm,
d
= 200nm (
t
= 100nm,
d
= 200nm)
.
4
2.
D
ependence
of the dynamic response on the lattice spacing
To
study the
effects of different lattice periodicities on the dynamic response of the system
, we
experimentally measure the frequency responses of lattices without electrodes (Fig. S2). The
frequency response
for a one
-
dimensional lattice with lattice spacing
a
=
7
m
presents a clear band
gap that separates acoustic and optical branches. Due to the clamping boundaries,
which act as a
grounding potential,
the acoustic branch starts at 12 MHz. A decrease of lattice periodicity leads to
a reduction
in
the size of
the
ban
d gap, broadening acoustic and optical branches
,
as seen
for
a
= 6
m
and 5
m
. The starting frequencies of the acoustic branch are 11
MHz for
a
=
6
m
and 10
MHz for
a
=
5
m
. The
band gap between the acoustic and optical branches
up
-
shift
s with
decreasing lattice spacing and the band gap size decreases. This
decrease of the band gap size
originates from the decrease of mass, an increase of bending stiffness and a decrease of the contrast
of the equivalent rotational couplings
G
and
H
(
see
supplementary
section
7
). Further decrease of
the periodicity leads to the
vanishing
of the
band gap in the measurement range
. In addition, below
a 4
m spacing, the
lower
acoustic
band edge
maintains a constant frequency,
around 9.8
MHz.
The increase of the distances between resonance peaks
in Fig. S2a (for
a
=2
m)
is the clear
evidence of the broadening of the acoustic branch, meaning that the strip width becomes uniform.
Supplementary Figure 2
. Dependence of the phononic
dispersion on periodicity.
(
a
) Frequency
responses of lattices with different periodicity ranging from
a
=
2
m to 7
m, measured at the end
unit of each lattice. Here no tuning electrode is deposited. (
b
)
Corresponding optical microscope
pictures to the
frequency spectrums in (
a
).