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1
Nonlinear nano-electromechanical lattices for high-frequency, tunable
stress propagation
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)
Active manipulation of mechanical waves at high frequencies opens opportunities in heat
management
1,2
, radio-frequency (RF) signal processing
3-5
, and quantum technologies
6-10
.
Nanoelectromechanical systems (NEMS) are appropriate platforms for developing these
technologies, offering energy transducibility between different physical domains, for
example, converting optical or electrical signals into mechanical vibrations and vice-
versa. Existing NEMS platforms, however, are mostly linear, passive, and not dynamically
controllable. Here, we report the realization of active manipulation of frequency band
dispersion in one-dimensional (1D) nonlinear nanoelectromechanical lattices (NEML) in
the RF domain (10-30 MHz). Our NEML is comprised of a periodic arrangement of
mechanically coupled free-standing nano-membranes, with circular clamped boundaries.
This design forms a flexural phononic crystals with a well-defined band gaps, 1.8 MHz
wide. The application a DC gate voltage creates voltage-dependent on-site potentials,
which can significantly shift the frequency bands of the device. Dynamic modulation of
2
the voltage triggers nonlinear effects, which induce the formation of phononic band gaps
in the acoustic branch. These devices could be used in tunable filters, ultrasonic delay lines
and transducers for implantable medical devices.
Architectured materials, such as photonic, plasmonic, and phononic metamaterials, control
wave propagation with specifically designed geometry of their building blocks. For application
in devices, it is desirable to be able to tune their operational frequencies on demand. Effective
tuning mechanisms exploit mechanical compliance to alter the architecture geometry, including
structure and/or periodicity
11-15
,
but this approach is limited to the use of soft materials. Other
tuning mechanisms involve the change of intrinsic properties of constituent materials, e.g., in
photonic crystal this is achieved by changing the dielectric constant with heating
16
and in
plasmonic metamaterials by changing the charge carrier density or fermi level of materials
17-20
.
Phononic metamaterials present an advantage over their optical and electronic counterparts, in
that they can more easily access nonlinear regimes for tunability. In the small amplitude domain,
elements in phononic crystals can be strongly coupled, enabling controlled energy transfer in
space, like in phonon waveguides
21,22
. At larger amplitudes, mechanical deformations can
access nonlinear regimes that allow energy transfer across different energy states, or vibration
modes, allowing, for example, phonon cavity dynamics
23-25
and synchronization
26
.
Macroscopic closed-packed granular crystals exploit the nonlinear contact mechanics between
two spheres, to change the effective coupling strengths. The application of a small static pre-
compression, can be used to demonstrate tunable band gaps
27
, wave velocity
28
, and wave
localization
29
. In other examples, magnetic repulsive forces have been used to control the on-
site potential of locally resonant units, thereby realizing phononic transistors
30
and
programmable metamaterials
31
. However, due to scaling constraints such tunability could not
be demonstrated at device relevant frequencies, in the ultrasonic domain.
3
Nanoelectromechanical system (NEMS) with voltage induced frequency tuning
32,33
offer a
practical platform to miniaturize phononic devices, making megahertz and gigahertz
transduction accessible with conventional fabrication methods. The application of a DC voltage
to a single resonator demonstrated the ability to trigger different nonlinearities, depending on
the thickness to vacuum gap ratio. The DC voltage increased the resonant frequency by tension-
induced stiffening
32,33
or decreased it by electrostatic softening of the resonators
34
(Supplementary Fig. S1). Two or three coupled-NEMSs with such tunability have shown
interesting nonlinear dynamic phenomena including cooling and amplification
25
. Recently,
NEMS based phononic crystals consisting of suspended GaAs/AlGaAs heterostructures
21,22
have been suggested as RF waveguides. They demonstrated Bragg phononic band gaps, slow
phonons, and defect-mediated nonlinear dynamics. Although promising, these devices are still
non tunable, meaning that the phononic properties are pre-determined during fabrication. In
addition, the defect-mediated dynamic control has a narrow bandwidth and low modulation
efficiency, transmitting substantial energy to the stop band. In this study, we realize a tunable
phononic crystal that harnesses DC/AC electrostatic forces exerted on the membranes, to
control wave transmission over a broad frequency range.
Our NEML consists of Si-rich LPCVD silicon nitride (SiN
x
) membranes (10
m in diameter)
that form a phononic crystal with periodic, curved boundaries (Fig. 1a). SiN
x
, an insulator, is
used to electrically separate the excitation and the tuning electrodes. Neighboring membranes
are overlapped to create mechanical coupling (Fig. 1b and 1c). We construct membrane arrays
with variable numbers of repeating units (Supplementary Fig. S2). The periodicity of the lattice
is chosen to be
a =
7
m owing to the expected frequency dispersion, with prominent acoustic
and optical branches separated by a well-defined phononic band gap (Fig. 1d). To characterize
the devices, we measure the frequency response of NEML with and without tuning electrodes
using a laser interferometer (see Methods). Flexural motion of the membranes is triggered by
4
simultaneously applying DC and AC voltages (Fig. 1a). As shown in Figure 1d, the spectra of
NEML without and with electrodes present clear stop bands below 12 MHz and 11.5 MHz,
respectively. The NEML without electrodes has a band gap from 17 MHz to 19.8 MHz, while
the NEML with electrodes has a gap from 16 MHz to 17.8 MHz, confirming that the gold
electrodes do not significantly change the operating frequencies. Passband spectra show
discrete peaks due to the finite number of unit cells (N = 120). We perform frequency response
measurements with varying number of unit cells (N = 30, 60, 90, and 120) and observed well-
defined phononic band gaps even at the lowest unit cell numbers (Supplementary Fig. S3). The
average quality factor (Q-factor) of the device with (without) electrodes is 1700 (2800) and it
depends on the surface and the intrinsic material loss due to the evaporated gold
35
. For
applications in quantum state transfer that requires low loss, the Q-factor can be enhanced by
replacing the gold electrodes with graphene
36
, increasing the residual stress of the film, and
operating in cryogenic environments.
To inform our experiments, we perform numerical simulations using Comsol
Multiphysics© with Floquet periodic boundary condition, considering geometric nonlinearity.
The experimental dispersion curves, obtained from fast-Fourier transformation of the spatially
scanned data, capture the phononic behavior of our NEML (Supplementary Fig. S4). The
experimental results show good agreement with numerical simulations (Fig. 2a-d) at gating
voltages (
V
T
) ranging from 0 V to 22 V, capturing correctly the location and the size of pass
and stop bands. Flat band regions exist near the band edges, at
q
=0 and
/a, suggesting slow,
near zero group velocities. The locally clamped boundaries, acting as grounding springs, cause
the acoustic band edge at
q
=0 to start from finite frequencies. To systematically investigate the
dependence of the transmission properties on the applied gate voltage, we measure the
frequency response of the last resonator (located on the opposite boundary from the excitation)
while varying the gate voltage from 0 V to 25 V (Fig. 2f). We note a broadening of the acoustic
5
branch from 4.5 MHz at
V
T
= 0 V to 6 MHz at
V
T
=25 V and a decreases of the band gap size
from 1.8 MHz at
V
T
= 0 V to 1.3 MHz at
V
T
=25 V.
To explain the experimental observations, we developed an analytical reduced order model
of the finite samples based on discrete interconnected rigid plates (Supplementary Fig. S5).
This model clarifies the interplay of the experimental parameters and their effect on the
dispersion response of the NEML (Supplementary Fig. S6 and S7). The presence of a gate
voltage below the membrane acts as an on-site potential that tunes the transmission. An increase
of the electrostatic force applied by the tuning electrodes softens the on-site potentials and
consequently down-shifts the frequency bands. In addition, the presence of clamped boundaries
causes the development of an axial tension along the lattice. In the discrete model, this tension
decreases the contrast of the rotational couplings, reducing band gap size. The boundary effect
is not captured by the finite element model that assumes periodic boundary conditions (see
Methods). The experimentally measured variations of the center frequency and the band size
match well the numerical data (Fig. 2f and 2h) below
V
T
=23 V. We characterize the Q-factor
as a function of the gating voltage of the NEML and show that it decreases quadratically due to
dielectric losses (Fig. 2i).
The tunability of the dispersion response means that the transmissible frequencies can be
dynamically selected, but also that the group velocity can be controlled. This is evident from
the transient responses of the lattice at
V
T
= 0 V and 24 V (Fig. 3a and 3b). We send a chirped
pulse with frequencies between 8 MHz ~ 22 MHz through the lattice, and measure the responses
of the end unit. The measured response contains several displacement pulses, which originate
from boundary reflections. To analyze the dispersive behaviors at single frequencies, we apply
a Burtterworth filter with 200 kHz bandwidth to the broadband responses. We then plot the
envelope of the filtered pulses in time domain, at each frequency (Fig. 3d and 3f). The time-of-
flight of the wave near the band edges is much longer than that in the middle of pass bands,
6
confirming the strong dispersion near the band edges. The experimental group velocity is
obtained calculating the velocity of the traveling pulses (
v
s
) at each frequency, using
v
s
=2Na/
t
(Figure 3c and 3e)
.
Here,
N
=120 is the number of unit cells,
a
is the periodicity and
t
is the
time-of-flight. The factor 2 accounts for reflections. The experimental group velocities show
excellent agreement with the numerical group velocities. The small discrepancy found in the
group velocities at
V
T
= 24 V (Fig. 3e) originates from boundary effects. The envelope of waves
near the acoustic band edge (14.5 MHz) broadens due to dispersion. Propagation losses are
deduced from the amplitude decay of two neighboring pulses at 13.5 MHz and are found to be
around 25% in voltages or 44% in energy per 1.68 mm travelled, which is equivalent to 2.6
dB/mm. At 12 MHz, the pulse speed at
V
T
= 24 V is about twice than the pulse speed at
V
T
= 0
V, demonstrating group velocity tuning. This observation is important for signal processing
applications, e.g., in tunable delays line.
Metamaterials under dynamic environments have shown many interesting features, e.g.,
non-reciprocity
37
and non-equilibrium phenomena
38
. We investigate the behavior of our system
under dynamic, AC tuning. In addition to the AC voltage, we apply a 5V DC voltage to the
tuning electrode to increase the modulation amplitude. This DC voltage is small enough not to
induce significant softening effects. Since the tuning electrode simultaneously excites the entire
lattice, we expect their modulation to predominantly affect the lowest modes in the phononic
band. We note that the AC tuning triggers nonlinear dynamic phenomena in the NEML leading
to a classical hysteretic behavior (Fig. 4a). The resonance curves show a hardening behavior,
due to tension induced nonlinearity. Beyond the bifurcation point, at modulation frequencies
(
p
) above 11.3 MHz, a small band gap emerges in the acoustic branches (Fig. 4b and 4c). The
gap size increases with increasing modulation frequency, changing its position. The shift of
band gap originates from the up-shift of the entire dispersion curves, due to stronger tension
induced at higher modulation frequencies. The presence of the modulation-induced band gap is
7
also evident in the real space scanning data obtained at
p
=11.5 MHz, with 2
a
=

m scanning
step (Fig. 4d). Analysis of the corresponding dispersion curve confirms the effective
“dimerization” of the unit cell, with a gap at the new band edge
q
=
/2
a
(Fig. 4e), at a center
frequency around 12.7 MHz. Accordingly, the nonlinear second harmonics of the modulation
frequency lies at
q
=
/2
a
in the dispersion relation and the modes’ amplitudes at
q
=
/2
a
have
2
a
periodicity (Supplementary Fig. S8 to S10).
Here, we demonstrate the static and dynamic control of nanoelectromechanical lattices by
electrostatic forces. Their frequency dispersion can be tuned over a wide range by applying DC
gate voltage. The DC tuning can be used to shift the band edges, reduce the width of the band
gap, and control the group velocity. Dynamic AC modulation of the gate shows a new
mechanism for band gap formation, induced by nonlinear resonances. This NEML contributes
to the development of miniaturized high-frequency components, such as tunable filters and
delay lines for signal processing applications, with smaller foot-print compared to the
electromagnetic counterparts.
Acknowledgements
We acknowledge partial support for this project from NSF EFRI Award No. 1741565. We thank
Emre Togan at ETH Zurich for his advice on interferometers.
Author contributions
J.C. and C.D. conceived the idea of the research. J.C. designed and fabricated the samples.
J.C. built the experimental setups and performed the experiments. J.C. developed the
analytical models and performed the numerical simulations. J.C. and C.D. wrote the
manuscript.
8
Competing Financial Interests
Nothing to report.
Methods
Sample fabrication
The fabrication of the nanoelectromechanical lattices begins with low pressure chemical
vapor deposition (LPCVD) of 100 nm low-stressed Si-rich silicon nitride (SiN
x
) on both sides
of a SiO
2
(150 nm)/Si(525
m) wafer. The sacrificial thermal oxide under the SiN
x
layer defines
the vacuum gap distance when the device is released in the HF etching step. The wafer is then
cleaned with acetone and iso-proylalchol (IPA). We then spin-coat 495K A4 and 950K A2
Poly-methly-methacrylate (PMMA). The electrode patterns are transferred using an electron-
beam (e-beam) lithography, and the PMMA resists are developed in a MIBK and IPA mixture.
45 nm-gold and 5 nm-chromium layers for e-beam alignment markers, excitation and tuning
electrodes are then deposited using an e-beam evaporator. Then, the samples are immersed in
acetone for 2 hours to lift of the remaining PMMA resists covered by the metals. For the next
e-beam exposure, we spin-coat 500 nm ZEP520 e-beam resist to serve as an etch mask for the
following reactive ion etching process. A second e-beam lithography is then performed to
pattern the arrays of the etch holes with 500 nm diameter and the exposed resist is developed
in a ZED-N50 developer. We employ an inductively coupled plasma (ICP) dry etching to drill
the holes in the SiN
x
device layer. The remaining resist is dissolved in a N-Methyl-2-
pyrrolidone (NMP) based solvent. The wafer is then scribed into 3 mm
3 mm dies and the
dies are immersed in Buffered Oxide (BOE) solution to etch the sacrificial SiO
2
layer through
the etch holes. After the etching process, we use a critical point dryer to avoid the adhesion of
the released free-standing membranes to the substrates and obtain perfectly isotropic circular
9
membranes. The Si-rich SiN
x
is hardly etched by buffered HF (etch rate of ~0.3 nm/min).
Furthermore, the LPCVD process leads to minimal disorder and structural defects, ensuring
uniform adhesion to the sacrificial layer, and low residual stress (~100 MPa).
Experiments
The measurements of the mechanical motion of the membranes are performed using a home-
built optical interferometer, at room temperature and a vacuum pressure P < 10
-6
mbar. The
interferometer is a phase-locked Michelson interferometer that employs a balanced homodyne
detection scheme. The phase-lock is enabled by moving a reference mirror mounted on a PID-
controlled piezoelectric actuator at 1.5 kHz. The motion of the membranes is electrostatically
excited by simultaneously applying DC and AC voltages through a bias tee (Mini-circuits,
ZFBT-6GW+). A 633 nm laser light with 20
W input power is incident on the sample. The
reflected light interferes with the light from the reference mirror. The intensity of the interfered
light is measured using the balanced photodetector. Its RF output signal is connected to high
frequency lock-in amplifier (Zurich instrument, UHFLI). The position of the laser spot was
monitored with a CMOS camera and the measurement position was controlled by a computer-
controlled XY-linear stage supporting the vacuum chamber. Details of the method can be found
in the Supplementary Information.
Numerical Simulations
All the numerical simulations to calculate the phononic frequency dispersion are performed
using the finite element method (FEM) via COMSOL. We employ the module pre-stressed
eigenfrequency analysis considering geometric nonlinearities, to reflect the effect of membrane
deflection and stresses induced by the DC gate voltage. The DC voltage also introduces an
effective softening potential and we implement this effect by assigning a negative elastic
10
foundation. To model the elastic foundation, we use an effective gap distance
d
eff
= 180 nm
(Supplementary Note 3). The parameters used for silicon nitride films are 3000 kg/m
3
density,
290 GPa Young’s modulus, and 0.27 Poisson ratio. For the properties of the electrodes made
of 45 nm gold and 5 nm chrome, we calculate the geometric averages of the properties and
obtain 18100 kg/m
3
density, 98 GPa Young’s modulus, and 0.417 Poisson ratio. We impose
anisotropic in-plane residual stresses 35 MPa for the direction perpendicular to the lattice and
125 MPa for the parallel direction, to match the experimental dispersion. The contributions of
both residual stress components are discussed in the Supplementary Information.
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