of 17
1
Experimental realization of on
-
chip
topological nanoelectromechanical
metamaterials
Authors:
Jinwoong Cha
1
,2
,
Kun
W
oo Kim
3
,
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
3
Korea Institute for
Advanced Study, Seoul, Republic of Korea
*
Correspondence to: Prof. Chiara D
a
raio (daraio@caltech.edu)
T
opological mechanical metamaterials
translate
condensed
matter
phenomena,
like non
-
reciprocity and robustness to defects,
in
to
classical platforms
1,2
. At small
scales, t
opological nano
electro
mechanical metamaterials
(NEMM)
can
enable
the
realization of
on
-
chip
acoustic
components
,
like unidirectional waveguides and
compact
delay
-
lines
for
mobile
devices.
Here, we
report the experimental realization
of
NEMM
phononic topological insulators,
consisting of two
-
dimensional arrays of
free
-
standing silicon nitride (SiN) nanomembranes
that
operate
at
high frequencies
(10
-
20 MHz).
We experimentally demonstrate
the presence of edge states
,
by
characterizing their
localization and
Dirac cone
-
like
frequency dispersion.
Our
topological waveguides
also
exhibit
robustness to waveguide distortion
s
and
2
pseudo
spin
-
dependent transport
.
The suggested devices
open wide opportunities
to
develop
functional
acoustic
systems
for high
-
frequency
signal processing
application
s
.
Wave
-
guiding through a stable
physical channel is strongly desired for reliable
information transport
in on
-
chip devices
.
H
owever, energy transport in h
igh
-
frequency
mechanical systems, for e
xample
based on
microscale
phononic devices,
is
particularly
sensitive to defects and sharp turns because of
back
-
scattering and losses
3
.
T
wo
-
dimensional
topological insulators
, first described
as quantum spin hall insulators (QSHE) in
condensed
matter
4
-
7
,
demonstrated
robust
ness
and spin
-
dependent energy transport along
m
aterials’
boundaries
and
interfaces
.
Translating these properties in th
e classical domain offers
opportunity for scaling the size of acoustic components to on
-
chip device levels.
T
opological phenomena
ha
ve
been shown
in
various
architecture
d
materials
like
photonic
8
-
11
,
acoustic
12,13
, and
mechanical
1,2,14
-
18
metamaterials.
Especially, photonic systems
have recently demonstrated
the use of topological effects
for lasing
19
-
21
and quantum
interface
s
22
. However, acoustic
and
mechanical
topological systems
have so far
been realized
only in
large
-
scale
systems
,
like arrays of
pendula
1
, gyroscopic lattices
2
, and arrays of
steel
rods
12
,
laser
-
cut plates
18
which require external driving systems.
T
o fulfil their potential in
device applications, mechanical topological systems need to be scaled to on
-
chip level for high
-
frequency tr
ansport.
Nanoelectromechanical systems
23,24
(NEMS)
can
be
employed to
build
on
-
chip
topological acoustic devices
, thanks to
their
ability to transduce
electrical
signal into
mechanical motion
, which is essential
in
applications
.
NEMS systems with
few
degree
s
of
freedom have already demonstrated quantum
-
analogous phenomena
,
like cooling and
amplification
25
, and
Rabi
-
oscillation
26,27
.
One
-
dimensional
(1
-
D)
nanoel
ectromechanical
lattices (NEML) are a different class of NEMS devices used to study
lattice dynamics
for
3
example, in
waveguiding
28,29
,
and energy focusing
30
.
Recently,
1
-
D NEML
s
made of SiN
nanomembranes
ha
ve
demonstrated
active
manipulation
of phononic dispersion, leveraging
electrostatic softening effects and nonlinear
ity
31
.
Here, we
fabricate and test
high
-
frequency phononic topological insulators in
engineered
,
two
-
dimensional
,
nano
-
electromechanical
metamaterials
.
W
e employ
an
extended
honeycomb
lattice, which contains six
lattice site
s in a unit cell
, satisfying
C
6
symmetry
(Supplementary information)
. Th
is
lattice exploits Brillouin
zone
-
folding
to demonstrate
a
double
-
Dirac cone at
point of the Brillouin zone
(BZ)
. Th
e
zone
-
folding method
has been
recently
used in
various topological
photonic
9,10
,
acoustic
12,13
and elastic
16
systems, by
introducing the concept of pseudospins
that
satisfy Kramers theorem
7
.
BZ folding
allows to
realize
a
pseudo
-
time
reversal
symmetry
invariant system, where
a time
-
reversal
operator is
defined from
the symmetry
(
C
6
)
of the lattice.
In our systems, the
topol
ogical phase
is controlled
by the coupling strength
,
t
, between
unit cell
s within the extended honeycomb
super
cell
and
between adjacent cells,
t
’ (Fig.
S
1
and S2
)
.
According to a classification method for
topological phonons
15
,
extended honeycomb
lattices are a
class
A
I
I
topological
insulators
, characterized by
a
Z
2
topological
invariant with
1
TT
UU
. H
ere
,
T
U
is
the
time
-
reversal operator
,
which is anti
-
unitary
(Supplementary
information)
.
To calculate the topological invariant, we consider
spin
-
Chern numbers from
BHZ model
6
near the
point
,
as
pseudospins are not fully
preserved
in entire BZ.
The spin
-
Chern numbers
C
±
= 0 for
t > t’, C
±
= ± 1 for
t
<
t’
confirms the pseudospin
-
dependent edge
states (Supplementary information). The
Z
2
topological invariants
are
= (
C
+
-
C
-
)/2 (mod 2) =
0 for
t > t’
,
and
=
1 for
t
<
t’
,
and
support different topological phases.
The difference in the
coupling
s
,
t
and
t
’,
lead
s
to
a
band gap opening at
point
, with
a
dipole
-
to
-
quadrupole
vibrational
band inversion
for a non
-
trivial lattice (
t
>
t’
)
(Supplementary information)
.
If a
4
domain wall
is
formed between the two
topologically d
istinct phases
, gapless
topological
edge
states emerge.
We realize these to
pological properties in our NEMM
by
periodically
arranging
etch
holes
, with 500 nm diameter, in
an extended honeycomb lattice
. The etch holes enable
a
buffered oxide
etchant
(BOE)
to
partially
remove
the
sacrificial thermal oxide layer and release
the
SiN
suspended membranes
(Fig. 1a).
We engineer the topological phase
s
of the lattice
,
by
c
hanging
the distance
between etch holes
,
w
. This strategy allows us to control the lattice
couplings
,
t
and
t’
,
from
the overlaps between the circular etching paths from the neighboring
etch holes (
Extended Data Fig. 1
). Recent reports
28
-
31
on 1
-
D NEML have exploited
the
isotropic nature of HF etching
for
device
fabrication.
Our NEMM
consists of a periodic array
of free
-
standing SiN nanomembrane
forming a flexural phononic crystal
.
The average thickness
of the nano
-
membranes is ~79 nm.
The average vacuum gap distance b
etween the SiN layer
and the highly doped silicon
substrate i
s
~
147
nm. These
values are estimated considering the
partial etching
rate
of the SiN
in the BOE
etchant
(~0.3 nm/min).
We perform finite element
simulations
using
COMSOL©
,
to
numerically compute
frequency dispersion curves
for a unit cell with a lattice parameter
,
a
= 18
m
(Extended
Data Fig. 2)
. We
vary the distance between two
neighboring
holes,
w
,
from 5.5
m to 6.5
m
(Fig. 1b to 1e).
For a unit cell with
w =
6.0
m
=
a /
3, a double Dirac
-
cone is present
around 14.55 MHz
at
the
point
of the BZ.
(Fig.
1c
). The frequency dispersion
curves
typical
l
y
start
from
around
12 MHz,
because of the presence of clamped boundaries.
The
frequency dispersion curves for
w =
5.5
m and 6.5
m
show
the emergence of
~
1.8 MHz
-
wide band gaps at
the
point, ranging from 14 MHz to 15.8 MHz.
T
he lattice with
w
=
5.5
m
exhibits
two additional
band gaps below and above the center
band gap
around 15
MHz (Fig. 1b), while
the lattice with
w
=
6.5
m
(Fig. 1d) do
es
not. The four vibrational
5
modes,
p
x
, p
y
, d
xy
,
and
d
x
2
-
y
2
, at
the
point
are degenerate
at the Dirac point
for the lattice
with
w
=
6
m
(Fig. 1c and 1e)
.
The
four
degenerate modes are split
into two separate
degenerate modes,
for
w
<
6
m and
w
>
6
m
(Fig. 1e)
,
opening a band gap
. The band
inversion
between the dipole vibrational modes (
p
x
, p
y
)
and the quadrupole ones (
d
xy
,
d
x
2
-
y
2
)
appears at
the
point
for
w
>
6
m
, which support
s
the topological non
-
triviality of the
lattice
.
To
study the
topological p
roperties of
our
NEM
M
s
, we
first
fabricate a straight
topological edge waveguide
(Fig. 2a and 2b)
,
formed at the interface of the topologically
trivial (
w
= 5.5
m
, Fig. 2c
) and non
-
trivial (
w
= 6.5
m
, Fig. 2d
) lattices.
Topological
edge states do not exist at free
-
boundaries of our systems, owing to the lack of
C
6
symmetry.
The number of unit cells of each phase is approximately 200
,
so that the edge waveguide
has 20 supercells with 18
m o
ne
-
dimensional lattice spacing.
To c
haracterize the edge
states, we
excite
the flexural motion of the membrane
s
by applying a dynamic electrostatic
force
,
F
(
V
DC
+
V
AC
)
2
,
to the
excitation
electrode.
Here,
V
DC
and
V
AC
are DC and AC
voltages,
which are simulteneously applied between
th
e
excitation electrode
and the
grounded substrate
(Fig. 2a
).
We perform measurements using
a home
-
built Michelson
interferometer with a balanced homodyne detection scheme (Met
h
ods). To obtain the
dispersion curves of the edge states, we measure the frequency responses
of 20 sites along
the edge waveguide
,
by spatially scanning the
measurement points
(yellow
strip
, Fig.
2a
)
with a 18
m step size
(Extended Data Fig. 3)
. The Dirac
-
like edge state frequency
disperison
curves, isolated from
the
bulk dispersion,
are present in the frequency range
between
14.1 MHz
and
15.8
MHz
, showing a good agreement with the numerical
dispersion curves (Fig. 2f).
We
also
observe
a
defect mode at the crossing point of the edge
state dispersion curves (Fig. 2e).
This stem
s
from a point
defect mode
from the boundary
6
near the excitation region
. The broken
C
6
symmetry at the interfaces introduce
s
a
small
band gap in the middle of
the
edge state dispersions
(Fig. 2b)
. Despite the presence of the
band gap, the defect mode is allowed to transmit
non
-
negligible
energy to the end of the
waveguide owing to the long decay length of the evanescent mode.
We also characterize the localization o
f the edge states
,
by scanning the
measurement point
across the waveguide (yellow dashed
-
line AB in Fig.
2a
)
also
with
an
18
m step size. The edge states are strongly localized
(Fig. 2h)
within
± 36
m distance
from the
interface
(Fig. 2g
).
Beyond this range, the frequency responses
(Fig. 2
g
) start to
show
clear
band gaps with
similar
positions and
magnitude
to
the numerical
dispersions
,
shown in Fig1b and 1d
. The trivial lattice side presents three band gaps (
Fig. 2
g
, left
) and
the non
-
trivi
al lattice side shows only one topological band gap (Fig.
2
g
, right
), as predicted
in the numeric
a
l frequency dispersion (Fig. 1b to 1
d
)
.
The
frequency responses
show
evidence of
different
topological
phases
in
the two lattices
,
w
= 5.5 and 6.5
m,
confirming
that the waveguiding effect is
topological.
One remarkable feature of topological edge modes is the
ir
robustness to
waveguide
imperfections
,
like sharp corners
.
To
study
this,
we
additionaly
fabricate
a
z
ig
-
zag
wavegui
de with two sharp corners
,
with
60
o
angle
s
(Fig. 3a). We perform
steady
-
state and
transient
response
measurements
,
by scanning
the laser spot
along the waveguide with 18
m scanning steps. The
frequency disper
s
i
on
from the steady state measurements
confirms
the presence of the topological edge states (
Extended
Data
Fig. 4
).
T
o validate th
e back
-
scattering immunity
, we perform transient response measurements with propagating pulses,
because the
steady
-
state
responses include
signals from
boundary
-
scattered waves.
For
transient measurements, we apply a DC (
V
DC
) and a
chirped
(
V
P
) voltages to the excitation
electrode
(Fig. 3a)
.
A pulse with 14.25 MHz center
-
frequency and 0.2 MHz bandwidth
,
propages
at 77 m/s group velocity. The space
-
time evolutions of the pulse
show
a stable
7
energy transport
with negligible
backscattering from the two sharp corners (Fig. 3b). Th
e
s
e
result
s demonstrate
robustness of the waveguides.
Another crucial aspect of topolog
ical ins
ulators is
the unidirectional propagation
for
distinct
pseudospin
mode
s
. To characterize
this
, we fabricate
a
nother
NEMM
with a
spin
-
splitter configuration
consisting of four domain walls
,
which has been
employed in several
previous studies
11,12
(Fig. 4a and 4b). Such geometry allows
using a simpler
pseudo
spin
selective excitation. In this con
figuration,
the propagating direction of
a
pseudospin
state
depends on
the spatial configuration of the two topolo
gical phases
,
w =
5.5 and 6.5
m
(Fig. 4a).
The pseudospin
states are
filt
ered to have a single dominant state in the input port
(yellow arrow in Fig. 4a).
After the signal
passes the input channel
, the
filtered spin state
mainly
propagates
to
output
port
1 and 3
(yellow arrows in Fig. 4a)
. We do not observe
propagation to
port
2 since
th
at
channel
does not preserve the
incident
pseudospin state
in
the
propagation direction
(cyan arrow, Fig. 4a)
.
To systematically investigate such
propagation behaviors,
we send vo
ltage pulses to the excitation electrode and measure
transient responses of
the
propagating pulses (Methods)
. We scan
13 sites (7 sites from
the
input channel and 6 sites from each
output channel) near the
channel
s’
crossing point
(Fig.
4b)
.
Note that
s
teady
-
state frequency
response
at the end of
the
three output ports (Fig. 4a)
exhibit
almost identical
edge state
response
s
due to
boundary scatterings
(Extended Data
Fig. 5).
The pulse we investigate
has
a 15.1 MHz center
-
frequenc
y and 0.5 MHz bandwidth,
which is
enough to cover
the
broad
frequency ranges of edge states.
As expected, t
he
measured signals show substantial energy transport to out
put 1 (Fig. 4c) and 3 (Fig. 4e),
but
not to
output 2
(Fig. 4d), after the crossing point (white arrows in Fig. 4c
to 4e). The
presented results
confirm
that the propagation direction depends on the types of
pseudospins.
The use of such
spin selective excitation and detection methods will enable
compact, mechanical
uni
-
directional devices.
8
The results
we present in this work show
that
nanoelectromechanical metamaterials
can be used
as
platform
s
for
on
-
chip
topological acoustic devices
.
In the future, t
hese
systems
can be employed
for
stable ultrasound and radio frequency signal processing.
With
advanced
nanofabrication
techn
i
ques, more sophisticated
structures
can be realized to
design other types of topological devices
,
for example,
based on
perturbative metamaterials
design
method
s
17,18
. Moreover, frequency tunability
in
nanoelectromechanica
l
resonators
via electrostatic forces
32,33
can be used
for electrically tunable devices
31
and actively
reconfigurable topological channel
11
.
Acknowledgements
We acknowledge partial support for this project
from NSF EFRI Award No. 1741565,
and the
Kavli
Nanoscience Institute at Caltech.
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
measurement
s.
J.C.
performed the
numerical simulations. J.C. and K.K performed theoretical studies.
J.C.
,
K.K.
and C.D. wrote
the manuscript.
Competing Financial Interests
Nothing to report.
9
Method
s
Sample
f
abrication
T
he fabrication process
begin
s
with a pattern transfer by electron beam lithography and
development of
a
PMMA resist in a MIBK:IPA=1:3 solution.
The
excitation electrodes
,
made
of
a
Au (45 nm)/ Cr (5 nm) layer
,
are deposited
on a 100nm
-
LPCVD silicon nitride
(SiN
x
)/140 nm
-
thermal SiO
2
/5
2
5
m highly
-
doped Si wafer
, followed by a lift
-
off process
in acetone.
A
second electron beam lithography
step
, with ZEP 520 e
-
beam resist, is
then
performed to
create the pattern
of etch hole
s (
with 500 nm diameters
)
arranged in the
extended
honeycomb
latti
ces
(
Fig. 1a and Extended Data Fig. 1
).
We use a
n
ICP
-
reactive
ion etch
,
to drill
the holes
on
the
SiN
x
layer
.
After we finish the etching of the holes, we
immerse the samples in a Buffered Oxide Etchant (BOE) solution for ~ 45
-
46 minutes
,
to
partia
lly
etch the thermal SiO
2
underneath the SiN
x
device layer. The etching duration
determin
e
s
the diameter of
the
etching circle
s
,
r
(Extended
Data
Fig.1). Detailed
fabrication methods can be found in Re
f.
31
.
Experiment
s
The flexural motions
of the membranes are measured using a home
-
built optical
interferometer
(HeNe
-
laser, 633 nm wavelength)
with
a
balanced homodyne method.
The
measurements are performed at room temperature and a
vacuum pressure P < 10
-
6
mb
ar
. The
optical path length difference between the reference and the sample arms are stabilized
by
actuating
a
reference
mirror. This mirror is
mounted on
a
piezoelectric actuator
which is
controlled by a PID controller.
The
motion of the membranes
is
electr
ostatically excited by
s
imultaneously applying DC and time
-
varying
voltages
through a bias tee (Mini
-
circuit
s,
ZFBT
-
6GW+
)
.
The intensity of the interfered light from the reference mirror and
the
sample is
10
measured
using
a
b
alanced photodetector
,
which is connected to a high
-
frequency lock
-
in
amplifier (Zurich instrument, UHFLI). The measurement position
, monitored via a CMOS
camera,
can be controlled by
moving a vacuum chamber mounted on
a motorized XY linear
stage.
For
the
dispersion curve meas
urements in Figure 2, we measure (
at
steady
-
state)
frequency responses ranging from 10 to 20 MHz of 20 scanned sites along the edge waveguide.
The scanning step is the one
-
dimensional lattice spacing,
a
= 18
m. The lock
-
in amplifier
(Zurich instrument, UH
FLI) allows
measuring
the amplitude responses and the phase
differences between the measured signal and the excitation source. To plot the frequency
dispersion, we perform Fast
-
Fourier
transformation of
the
amplitude
sin(phase) data. The
amplitude only data and the phase
-
considered data are shown in
Extended Data Figure 3a
and 3b.
For transient measurements
in Figure 3 and 4
,
we send a chirped signal (AWG module
in UHFLI) and measure the signal with an oscilloscope (
Tektronix, DPO3034).
As the signal is
invisible for a low excitation amplitude, we first filter the RF
-
output signals from the
photodetector with a passive band
-
pass filter (6
22 MHz bandwidth) and average 512 data n
time
-
domain. For robustness measureme
nts (Fig. 3), we send a pulse containing frequency
content ranging from 14 to 15 MHz, by applying
V
DC
=
15 V and
V
P
= 75 mV to the excitation
electrode. We then perform post
-
signal processing to extract signals of interest
,
by applying
a Burtterworth
filter with 14.25 MHz center
-
frequency and 0.2 MHz bandwidth. For
pseudospin
-
dependent transport measurements,
we use a pulse (14
16 MHz) and applied
V
DC
=
15 V and
V
P
= 22.5 mV.
We then apply a Burtterworth filter with 15.2 MHz center
-
frequency and 0.5
MHz bandwidth.
11
Numerical Simulations
W
e perform finite
-
element
simulations
to calculate the phononic f
requency dispersion
curves
using COMSOL
multiphysics.
We employ
the
pre
-
stressed eigenfrequency analysis
module
in membrane mechanics. We also
consider
geometric
nonlinearity
,
to reflect the effect
s
of
residual stresses
.
The physical properties of SiN
x
use
d
in the simulations are 3000 kg/m
3
density, 290 GPa Young’s modulus, 0.27 Poisson ratio, and 50 MPa isotropic in
-
place
residual stress
.
The
la
ttice parameter,
a
, is chosen to
be 18
m. We calculate
frequency
dispersion
curves for various unit cell geometries with different
w
ranging from 5.5
m to
6.5
m. The center hexagon and the six corners
of each unit cell
are fixed
,
due to the
presence of
un
etched SiO
2
(light grey
regions
in
the
scannin
g microscope images in Fig.
2b
-
d). The radi
i
of
the
etch
ed
circles are set to
r
= 4.9
m
(Fig. 1 and Extended
Data Fig.
2
).
We apply
Bloch periodic conditions
to the six sides of a unit cell
,
(
)
( ) exp(
)
u r
R
u r
iq R
,
via
Floquet periodic
ity in COMSOL
. Here,
r
is a position
within a unit cell,
R
is a lattice translation vector, and
q
is a wave vector.
We
calculate
the
dispersion curves along
the boundary of
the
irreducible Brilluoin zone
M
A
A
K
A
(Fig.
S1
b)
We also
numerically
calculate the frequency dispersion curves of the edge states to
validate the topological behaviors.
As
we are interested in one
-
dimensional dispersion
along the interface, we build a
strip
-
like
super cell with 18
m periodicity
. Each topological
phase (
w =
6.0 ± 0.5
m) span
s
about
±
160
m from the interface in
the direction
perpendicular to the interfac
e.
We calculate the frequency dispersion by applying one
-
dimensional Floquet periodic condition.
12
Data availability.
The data that support the findings of this study are available from the
corresponding author upon reasonable request.
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14
Figure 1
. Unit
cell geometry and topological phase transitions
.
(
a
)
Schemat
ic
of
a
two
-
dimensional
nanoelectromechanical metamaterial
. The grey area
represents the
silicon nitride
nanomembrane
suspended over a highly doped
n
-
type silicon substrate
. The black dot
s, forming
a honeycomb lattice, represent etch holes. The
light
blue hexagons r
eprese
nt the unetched
thermal oxide
, acting as
fixed boundaries
. The unit cell
geometry
(black solid hexagon)
is
shown in
the
right inset
, with relevant parameters
.
An example flexural mode is shown in the
left inset.
The
topological phase
s
are
controlled by changing
w
. (
b
), (
c
), (
d
)
Frequency
dispersion
curves along a boundary of the irreducible Brillouin
-
zone
M
KM
,
when (
b
)
w
= 5.5
m, (
c
), 6.0
m, and (
d
) 6.5
m. The red and green shaded regions correspond to topol
ogical
and non
-
topological band gaps
, respectively
. (
e
) Eigenfrequencies
above and below the
topological band gap
at
the
point
,
as a function of
w
.
Blue (red) dots denote the
eigenfrequencies for
flexural
modes
p
x
and
p
y
(
d
xy
and
d
x
2
-
y
2
). The flexural mode shapes
are
presented for
w
= 5.5
m
(left)
, 6.0
m
(middle)
, and 6.5
m, respectively
(right)
.
15
Figure 2.
Characterization of
topological edge state
s
.
(
a
)
Scanning electron microscope
image
of a straight topological edge waveguide.
The
two
different
topological
phases are shaded
by blue (non
-
trivial) and red (trivial)
false colors
.
F
lexural membrane motions are excited by
simultaneously applying DC and AC voltages (
V
DC
= 2 V,
V
AC
= 20 mV) to the excitation
electrode via a bias tee.
Scale bar,
100
m
.
(
b
)
, (
c
), (
d
) Scanning electron microscope images
of (
b
) edge
region
,
yellow shaded strip
in (
a
), (
c
) trivial lattice with
w
= 5.5
m, the red shaded
area in (
a
), and (
c
)
non
-
trivial lattice with
w
=
6
.5
m, the
blue
shaded area in (
a
). Scale bars
are
10
m.
The red and blue dots in (
b
) denote the lattice points for
w
= 5.5
m, and
w
= 6.5
m, respectively. The blue and red hexagons in (
c
) and (
d
) represent the unit cells for
w
= 5.5
m, and
w
= 6.5
m.
(
e
)
Ex
perimental and (
f
) numerical frequency dispersion curves along the
edge waveguide (from C to D in
a
).
Yellow
and light blue dots in the edge state dispersion in
(
f
) represent propagating waves for up
-
and down
-
pseudospins, respectively.
Time
-
evolution
of the mode shapes at points 1,2,3, and 4 are provided in supplementary video files.
(
g
)
Frequency responses
for
19
different sites along line A
-
B shown in
(
a
)
(middle panel). The left
and right
panels
represent frequency responses at site A and B, respectively. The red and green
shaded regions represent the band gaps. (
h
)
F
lexural modes for point A and B in the
numerical
dispersion shown in
f
.
The width of the strip is 18
m
, identical to the lattice parameter
a
.
16
Figure 3.
Waveguid
e
robustness against imperfections.
(
a
)
S
canning el
ectron microscope
picture of a zig
-
zag
topological edge waveguide.
The red and blue shaded regions
represent
topologically triv
i
al and non
-
trivial lattices, respectively.
The flexural membrane motions are
excited by simultaneously applying DC and
a chirped signal
with frequency content ranging
from 14 MHz to 15 MHz.
The applied voltages are
V
DC
= 15
V,
V
P
=
75
mV, here
V
P
is the
amplitude of the
chirped
signal
.
The orange dots represent
m
e
a
surement points
.
Scale bar is
10
0
m.
(
b
)
Transient
responses
along the edge waveguide
in a space
-
time domain.
A pulse
with 14.23 MHz center
-
frequency and 0.2 MHz bandwidth is
considered
.
The position denotes
the
m
easurement points
in
the scanning direction shown in (
a
)
.
c1 and c2
mark the position of
the
sharp corners.
The solid line
is added to highlight the
trajectory of the
propagating pulses
.
The two dotted lines
mark the expected trajectory of the backscattered signal, if
the propagating
pulse
were
reflected from corner
s
c1 and c2.