Bragg scattering based acoustic topological transition controlled by local resonance

Abstract

Topological metamaterials offer new routes for control of waves, which are widely realized by Bragg scattering and/or local resonance. Understanding of topological transition by the interaction between these mechanisms is strongly desired to extend the design degrees of freedom for intriguing wave phenomena. Here, we demonstrate a phononic metamaterial consisting of C-shaped elements that enables us to investigate interaction between Bragg scattering and local resonance. We show that by adding resonance scattering a topological band gap is opened from a Bragg scattering based Dirac cone, and its band gap is controlled by the resonance frequency of the cavities relative to the Dirac cone frequency. In addition, we show that topological band-gap opening induced by the Bragg scattering can be reversed into an ordinary state or vice versa by judicious inclusion of the local resonance. Scattering cross-section analysis elaborates the combined effect of the two mechanisms on topological states. By employing lossy resonant elements, we further demonstrate a lossy topological insulator capable of one-way sound propagation immune to sharp corners, potentially leading to an energy-harvesting topological insulator. Our results provide a critical understanding of topological phenomena involving coupled scattering mechanisms.

Full text

PHYSICAL REVIEW B 99, 064305 (2019)
Bragg scattering based acoustic topological transition controlled by local resonance
Taehwa Lee*and Hideo Iizuka
Toyota Research Institute of North America, Toyota Motor North America, Ann Arbor, Michigan 48105, USA
(Received 27 November 2018; revised manuscript received 25 January 2019; published 27 February 2019)
Topological metamaterials offer new routes for control of waves, which are widely realized by Bragg scattering
and/or local resonance. Understanding of topological transition by the interaction between these mechanisms is
strongly desired to extend the design degrees of freedom for intriguing wave phenomena. Here, we demonstrate
a phononic metamaterial consisting of C-shaped elements that enables us to investigate interaction between
Bragg scattering and local resonance. We show that by adding resonance scattering a topological band gap is
opened from a Bragg scattering based Dirac cone, and its band gap is controlled by the resonance frequency
of the cavities relative to the Dirac cone frequency. In addition, we show that topological band-gap opening
induced by the Bragg scattering can be reversed into an ordinary state or vice versa by judicious inclusion of
the local resonance. Scattering cross-section analysis elaborates the combined effect of the two mechanisms on
topological states. By employing lossy resonant elements, we further demonstrate a lossy topological insulator
capable of one-way sound propagation immune to sharp corners, potentially leading to an energy-harvesting
topological insulator. Our results provide a critical understanding of topological phenomena involving coupled
scattering mechanisms.
DOI: 10.1103/PhysRevB.99.064305
I. INTRODUCTION
Topological insulators have drawn much attention, since
their topological invariant enables robust one-way wave
propagation immune to perturbations such as defects
and disorders [1–3]. After the first demonstration of the
acoustic version of topological insulators using rotating flows
[4,5], flow-free acoustic topological insulators were quickly
introduced by realizing pseudospin states [6–8]. Since then,
there have been extensive research efforts devoted to seeking
different types of topological phononic crystals [9–19]. In
these works, a topological phase transition was enabled by
carefully controlling symmetry breaking in two-dimensional
(2D) phononic crystals. Beyond the 2D topological insulators,
the research on topological acoustics has evolved into
higher-order topological insulators (quadrupolar) [20] and
other dimensionalities, e.g., three-dimensional (3D) [21]
and one-dimensional [22,23]. In addition, to prove its
practical merits, backscattering-free edge propagation has
been combined with additional intriguing functionalities such
as acoustic antenna [24], programmable coding [25], bilayer
[26], multiband [27], acoustic delay line [28,29], tunable
strain [30,31], and on-chip [32].
Similar to conventional phononic crystals, the scattering
mechanisms employed in these acoustic topological insulators
include Bragg scattering and local resonance. Bragg scattering
has been commonly used in acoustic topological insulators,
since it requires a simple design process. In addition, local
resonance has shown advantages by enabling subwavelength
control of waves [33–35]. An interesting feature in topolog-
ical insulators may be demonstrated by implementing other
*taehwa.lee@toyota.com
scattering mechanisms (e.g., inertial amplification [36]), or
by simultaneously using more than two mechanisms. Espe-
cially, combination of these mechanisms is advantageous in
increasing the degrees of freedom as a synthetic dimension
necessary for higher-order topological modes (beyond the
octupole moment) [37]. In a system with combined mecha-
nisms, interaction between the constituent scattering mecha-
nisms is important, requiring a platform that allows multiple
mechanisms. Recently, it has been demonstrated that one
platform can encompass three scattering mechanisms by en-
gineering spiral scatterers [13]. However, in this system, each
mechanism was independently employed. There have been
no systematic studies on interaction between the scattering
mechanisms in acoustic topological insulators, as previous
works are focused on one of the scattering mechanisms
[9–19].
In this paper, we study interaction between the Bragg
scattering and local resonance. To independently control the
effect of each mechanism in phononic crystals, we use C-
shaped elements that are constructed by circular rods with
inside circular cavities and narrow slits. We find that the local
resonance induced by the inner cavities and slits leads to band-
gap opening and the topological phase transition in Bragg
scattering. The size of the band-gap opening is strongly in-
fluenced by the resonance frequency relative to the frequency
where the double Dirac cone appears. In addition, topological
states are determined by the orientation of the slits as well
as the resonance frequency. By employing scattering cross-
section analysis, we investigate the effect of the resonance
on the Bragg scattering induced topological phase transitions.
Lastly, we show that the lossy topological insulator, which
is constructed by including losses into the resonant elements,
exhibits robust one-way transmission.
2469-9950/2019/99(6)/064305(11) 064305-1 ©2019 American Physical Society

TAEHWA LEE AND HIDEO IIZUKA PHYSICAL REVIEW B 99, 064305 (2019)
FIG. 1. (a) Conventional topological phase transitions with respect to the double Dirac cone for R/a=R0/a0=0.3928 with the lattice
constant, a, and the radius, R. The topological transitions occur via the change of the hexagon (varying afor the constant R=R0)ordifferent
radii (R) for the constant a=a0. (b) Resonant cavity-induced Bragg scattering topological phase transitions, depending on the slit orientation
(ϕ) and the cavity radius (r) of the resonant element: (i) from the Dirac cone (R=R0,a=a0) to the ordinary (ϕ=0◦) or topological
(ϕ=180◦) state, (ii) from the ordinary (R=R0,a<a0) to topological state (ϕ=180◦), and (iii) from the topological (R<R0,a=a0)to
ordinary state (ϕ=0◦).
II. DESIGN OF THE C-SHAPED SCATTERER
To study interaction of local resonance with Bragg scat-
tering, we first consider a honeycomb lattice consisting of
circular rods (the radius Rand the lattice constant a)inair,
which features a Bragg scattering induced double Dirac cone
with a fourfold degeneracy for R/a=0.3928. Such a double
Dirac cone occurs as a feature of phononic crystals character-
ized by C6vpoint-group symmetry. We have chosen a honey-
comb lattice widely implemented for topological insulators,
although a triangular lattice of ring-type scatterers can show
the double Dirac cone [9]. The double Dirac cone with four
degenerate modes creates pseudo-spin-up and -down states
by hybridizing these modes. The specific R/afor the double
Dirac cone is obtained from simulation (see Appendix A)
using a hard boundary condition on the scatterers (the same
R/afor the double Dirac cone was reported for steel rods
and air in Ref. [7]). Owing to a symmetry inversion in the
reciprocal space, this phononic crystal results in energy-band
inversion and thereby topological phase transition from an
ordinary state to a topological state. Here, the ordinary state
(i.e., a topologically trivial state) is characterized by two dipo-
lar modes in the lower-frequency bands and two quadrupolar
modes in the upper-frequency bands (i.e., zero-spin Chern
number), whereas the topological state is characterized by two
quadrupolar modes in the lower bands and two dipolar modes
in the upper bands (non-zero-spin Chern number). As summa-
rized in Fig. 1(a), it is well known that the topological phase
transition is readily realized by either controlling the filling
ratio (R/a)[7] or changing the size of the hexagon (expanded
or shrunk) [38]. Then, we add local resonance into the non-
resonant crystals by integrating resonant elements consisting
of a cavity and a narrow slit, as shown in Fig. 1(b).Forthe
C-shaped elements, the cavity (area S) and slit (width hand
length ls) constitute a Helmholtz resonator, the resonance fre-
quency ( fr) of which is approximately given by fr=c
2πh
Sls.
For independent control of each contribution of local res-
onance and Bragg scattering, we assume that these reso-
nant elements only result in additional scattering without
changing the Bragg scattering owing to the negligible rigid-
ity change and the small slit width (hλ, wavelength).
In Fig. 1(b), we show that adding the resonant elements
to the gapless phononic crystal [case i in the figure, the
double Dirac cone] leads to band-gap openings (two twofold
degeneracies) with an ordinary or topology state depending
on the radius of the cavity and the orientation of the slit
(defined as ϕ): the ordinary (ϕ=0◦) and topological (ϕ=
180◦) states. For the orientations (ϕ=0◦and 180◦)ofthe
resonant elements, C6vpoint-group symmetry is preserved.
Moreover, the resonance can change topological states in
phononic crystals with bulk band gaps: the high-filling ratio
crystal [case ii in the figure] from an ordinary to a topological
state by resonance (ϕ=180◦), and the expanded crystal [case
iii in the figure] from a topological to an ordinary state
by ϕ=0◦.
III. RESULTS AND DISCUSSIONS
A. Topological phase transition in C-shaped phononic crystals
Figure 2(a) shows the band structures of the phononic
crystals [case i in Fig. 1(b)]. The phononic crystal with no
resonant elements (R=15 mm, a=38.2 mm, and R/a=
0.3928) leads to the double Dirac cone at a frequency of
fD=5100 Hz, where four conical bands touch at a point. We
find that reduced Dirac cone frequency ( fDa) in this paper is
the same as that used in Ref. [7], i.e., fDa=193.8m/s, indi-
cating that the Dirac cone frequency for different acan be esti-
mated by fD=193.8(m/s)/aif a hard boundary condition is
imposed on scatterers in air. The addition of circular cavities
with different radii and orientations (radius r=6mm,ϕ=
0◦;r=4mm,ϕ=180◦), respectively, results in ordinary and
topological states having similar band-gap sizes (∼400 Hz).
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FIG. 2. (a) Band structures of the honeycomb phononic crystals for the ordinary state (left), double Dirac cone (middle), and topological
state (right). For the ordinary state, the radius (r) and orientation (ϕ)aregivenbyr=6mmandϕ=0◦, while for the topological state r=
4mmandϕ=180◦. The topological band inversion occurs (dtype, quadrupolar; ptype, dipolar). The horizontal dashed line indicates the
frequency ( fD) at which the double Dirac cone occurs. (b) Pressure fields for the ordinary (left panel) and topological (right panel) states. The
d-type (quadrupolar) bulk modes correspond to dxy and dx2−y2, while the p-type (dipolar) bulk modes correspond to pxand py[px:x−/y+,
py:x+/y−,dx2−y2:x+/y+,anddxy :x−/y−with x(y)±the even or odd symmetry along the xor yaxis].
The topological state is confirmed with the band inversion,
showing d-type (quadrupolar) bands at lower frequencies and
p-type (dipolar) bands at higher frequencies (±1 Chern num-
bers), whereas the ordinary state exhibits lower p-type bands
and upper d-type bands (zero Chern number). The circular
cavities along with the identical slit size (h=ls=1mm)
have resonance frequencies of 2000 Hz (r=6 mm) and 4200
Hz (r=4 mm), much smaller than the frequency ( fD=
5100 Hz). This indicates that these resonant elements serve
as off-resonance scatterers, but they significantly influence
topological phases. One interesting feature is that such local
resonance dominantly affects only either of the two band types
while the other band type remains similar in its corresponding
frequency at (close to fD=5100 Hz). Specifically, for the
ordinary state, the frequencies of the quadrupolar states (d
type) are only increased by the local resonance, whereas for
the topological states only the dipolar states (ptype) con-
siderably experience their frequency shifts. Such orientation
dependence of the topological phases is explained by the
pressure field inside the cavities, as shown in Fig. 2(b). Here,
the subscripts of pand drepresent the symmetry of the
pressure field [px:x−/y+,py:x+/y−,dx2−y2:x+/y+, and
dxy :x−/y−with x(y)±the even or odd symmetry along the
xor yaxis]. For the ordinary state (ϕ=0◦), the dipolar field
profiles are negligibly influenced by the resonant elements,
showing low pressures in the cavities (green). Notably, the
slits of the cavities are aligned to the nodal line of the pressure
field. In contrast, the quadrupolar states have relatively high
pressure in the cavities away from the nodal lines. We find
that the topological states (ϕ=180◦) show strong coupling
of the cavities with the dipolar pressure fields, thus causing
the dipolar (p-type) bands to blueshift.
For much stronger interaction between local resonance and
Bragg scattering, it is natural to consider the resonance fre-
quency ( fr) close to fD. To control the resonance frequency,
we vary the radius (r) of the resonant cavity in the phononic
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TAEHWA LEE AND HIDEO IIZUKA PHYSICAL REVIEW B 99, 064305 (2019)
FIG. 3. Interaction between Bragg scattering and local resonance with varying the radius (r) of the circular cavity. (a) Band structure for
the crystal with no resonance (r=0mmandR=15 mm), showing the double Dirac cone. (b) Topological band structures of the phononic
crystals (ϕ=180◦)forr=5mmandr=4 mm , the resonance frequencies ( fr) of which are smaller than the Dirac cone frequency ( fD), i.e.,
fr<fD. The red shade and purple shade indicate band gaps and resonance-induced flat bands, respectively. The dipolar (quadrupolar) bands
are indicated as pwith red circle symbols (dwith blue circle symbols) at the point. The frequency of the lower flat band, labeled as fwith
purple circle symbols, corresponds to the resonance frequency. (c) Ordinary band structures of the phononic crystals (ϕ=180◦)forr=3.1
and 2.8 mm ( fr>fD). (d) Simulated scattering cross section of isolated resonant scatterers, illustrated in the inset. (e, f) Phase plots (top two,
pyand dx2−y2) and acoustic intensity field (bottom, fat resonance) of the topological (r=4 mm) (e) and ordinary ( r=3.1 mm) (f) states.
crystals of ϕ=180◦[Fig. 2(a)] while the slit size remains the
same. The scattering cross-section spectra for different rin
Fig. 3(d) show that the resonance frequency increases with
reducing r. Starting from the band structure for r=0mm
showing the double Dirac cone [Fig. 3(a)], we observe an
increase in the topological band gap from 200 to 400 Hz by
reducing r(5 →4 mm) and thereby the frequency difference
(fD−fr),asshowninFig.3(b). In the band structure, such
a resonance-frequency shift is also confirmed with resonance-
created bands, which are highlighted as the purple shade. It
is interesting to note that the local resonance significantly
influences the topological phase transition despite its small
scattering cross section at frequencies around fD. Note that
although the scattering analysis of an isolated resonant scat-
terer provides an insight into understanding of the topological
phase transitions its scattering characteristics are different
from those of an array of resonant scatterers (each strongly
coupled with the others).
When the cavity radius (r) is further reduced, the resonance
frequency ( fr) is increased, exceeding fD, i.e., fr>fD,as
shown in Figs. 3(c) and 3(d).InFig.3(c), we find that the band
gaps for r=3.1 and 2.8 mm are relatively small (∼50 Hz)
compared to those for fr<fD. Interestingly, for fr>fDwe
observe band inversion into the ordinary state (i.e., d-type
bands at higher frequencies). This feature is explained with
the pressure phase of the cavities, as shown in Figs. 3(e)
and 3(f). Depending on sign ( fr−fD), the phase polarity of
the resonant cavity is drastically changed. For the topological
crystals ( fr<fD), the polarity of the inner cavities is out of
phase in the dipolar field [Fig. 3(e)], whereas for the ordinary
crystal ( fr>fD) the polarity in the dipolar field is reversed to
be in phase [Fig. 3(f)]. This indicates that the polarity of the
cavity determines the effect of the resonance scattering on the
topological phase transition.
The topological transition depending on sign ( fr−fD)for
ϕ=180◦is similarly observed for ϕ=0◦. Figures 4(b) and
4(c) summarize the frequencies of p- and d-type bands at 
with respect to the cavity radius (r) for the phononic crystals
of ϕ=0◦and 180◦, respectively. The resonance frequency
for different ris shown in Fig. 4(a). For both ϕ=0◦and
180◦, the band gaps are similarly increased by decreasing
rdown to 4 mm, i.e., increasing the resonance frequency.
Moreover, it is noted that for the same rthe band gaps
of ϕ=0◦are much larger than those of ϕ=180◦. Such a
difference in the band-gap size between the orientations arises
from the coupling between the resonators that depends on
the orientation. For r=3.5 mm, the resonance frequency is
similar to fD, i.e., fr≈fD. In this case, the local resonance
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FIG. 4. Band-gap size and topological phase transition. (a) Res-
onance frequency ( fr) with respect to the radius (r) of the circular
resonant cavity. The horizontal dashed line indicates the frequency
(fD) for the double Dirac cone of the nonresonant phononic crystal
(r=0). The gray shade region indicates that the radii lead to fr∼
=fD.
(b, c) Band-gap sizes as a function of rfor the two phononic crystals,
ϕ=0◦(b) and ϕ=180◦(c). p-type (red line) and d-type (blue line)
bands are labeled as pand d.
appears within the band gap induced by the Bragg scattering
(see Appendix B).
B. Scattering cross-section analysis
As the topological phase transition is induced by the in-
teraction between the local resonance and Bragg scattering,
we investigate such interaction from a scattering cross-section
perspective. Although the band structures of our phononic
crystals are determined by scattering from an ensemble of
the C-shaped scatterers, the scattering cross-section analysis
of an individual scatterer is useful because it allows us to
identify the contribution of each scattering mechanism. To see
each one’s contribution, the C-shaped scatterer is decomposed
into the direct and resonant components, as illustrated in
Fig. 5(a). The direct component (i.e., geometrical scattering)
corresponds to a rigid rod, while the resonant component is
modeled as a resonant cavity embedded in a semi-infinite
reflector. Such decomposition is valid for the slit widths much
smaller than both the wavelength and the outer radius ((R).
Such subwavelength slits do not lead to scattering from the
direct pathway (i.e., geometrical scattering), but they affect
resonant scattering. For the resonant component, the incident
wave is parallel to the semi-infinite reflector to prevent the di-
rect scattering by the reflector. In contrast, the circular surface
of the C-shaped scatterer dominantly contributes to the direct
scattering and the effect of the slits on the direct scattering
is negligible, because the slit width is much smaller than the
outer radius (R) of the scatterer. Thus, the direct scattering
from the C-shaped scatterer can be modeled by using a solid
rod. To construct an analytical model, the scattered pressure
field by an isolated C-shaped scatterer is expressed by the sum
of the direct and resonant scatterings:
psc =psc,dir +psc,res.(1)
For the small slit width (lsλ), the directly scattered field
is identical to the scattered field by the solid cylinder, which
at a point (r,θ)isgivenby[39]
psc,dir (r,θ)=∞

n=−∞
AnH(1)
n(kr)einθ,(2)
where H(1)
nis the Hankel function of the first kind, kis
the wave number, and An=−einπ/2J
n(kR)
H(1)
n(kR)is given for the
Neumann boundary condition on the cylinder’s surface with
the prime being the derivative. The pressure by the resonance
is expressed with uniform velocity (v0) over the slit width by
[40]
psc,res(r,θ)=−iωρh
4πR
v0G(r,R,θ,θ
0),(3)
where G(r,R,θ,θ
0) is the Green’s function of a
cylinder in a free field, given by G(r,R,θ,θ
0)=
−2
kR ∞
n=−∞
H(1)
n(kr)
H(1)
n(kR)ein(θ−θ0).
The scattering cross section of a scatterer is defined as
the ratio of the scattered power to the incident power. The
scattered power is calculated by integration of Eq. (1) taken
over the spherical surface surrounding the scatterer (see
Appendix Cfor details). Figure 5(a) shows calculated scat-
tering cross sections, which are normalized to the scattering
cross-section limit of a subwavelength resonant scatterer,
i.e., σ0=2λ0/π (λ0being the wavelength at resonance fre-
quency). The numerical results (symbols) show good agree-
ment with the analytical results (solid line) obtained from
Eqs. (2) and (3). We find that the scattering cross section of
the resonant component (blue square) around the resonance is
comparable to that of the direct component (black triangle).
The sum (dashed line) of scattering cross sections of direct
(black triangle) and resonant (blue square) components differs
greatly from that of the C-shape scatterer (red circle). For
frequencies higher than fr, the scattering cross section of
the C-shaped scatterer is decreased due to the destructive
interference, whereas for f<frit is constructively increased.
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TAEHWA LEE AND HIDEO IIZUKA PHYSICAL REVIEW B 99, 064305 (2019)
FIG. 5. (a) Scattering cross sections of isolated scatterers: C-shaped rod (combined), solid rod (direct), and resonant cavity (resonant).
The dimensions are given by r=4mm,R=15 mm, and h=ls=1 mm. For the C-shaped scatterer, the orientation of its slit is aligned to
the direction of the incident wave, whereas for the resonant cavity the slit orientation is perpendicular to the incident wave. The scattering
cross section is normalized to σ0=2λ0/π. The dashed line corresponds to the sum of the scattering cross sections of the solid rod and
resonant cavity. The symbols indicate numerical results (COMSOL Multiphysics), while the solid lines represent analytical results obtained
fromEqs.(2)and(3). (b) Angular dependence of the scattering cross section for the C-shaped rods (the angle αis defined as the angle between
the slit orientation and the line vertical to the incident wave): α=0◦,±45°, and ±90°. The symbols indicate numerical results, while the
solid lines represent analytical results. (c) Scattering cross section of the pairs of the C-shaped rods with different slit orientations: two slits
facing each other (pair I) and with an angle of 60° (pair II). Pair I constitutes the phononic crystal of ϕ=0◦, while pair II is for ϕ=180◦.For
comparison, the scattering cross section of the single C-shaped rod is plotted (black dashed line).
Such interference asymmetrical to frprovides an explanation
of the sign ( fr−fD) dependence of the topological phase
transition, observed in Figs. 3and 4.
For the isolated scatterers, the result in Fig. 5(a) is lim-
ited to the specific angle of incidence with respect to the
orientation of the slit. For the phononic crystals, the slits of the
C-shaped scatterers within the unit cell have different angles
with respect to the incident wave, thus affecting the scattering
cross section. The angular dependence of the scattering cross
section is shown in Fig. 5(b). The scattering cross-section
spectra are identical between angles of α=+α0and −α0.
With decreasing |α|, the scattering cross section decreases for
f<fr, while it increases for f>fr. Note that for α=0◦
the opposite trend of the spectral characteristic is observed,
compared to α=±90◦.
The quality (Q) factor of the resonators is an important
parameter to characterize the interaction of local resonance
with the Bragg scattering. It is known that the resonance
Qfactor is strongly influenced by radiation impedance [40]
and thereby coupling of resonators. Thus, to study coupling
of the C-shaped resonators, we consider two representative
resonator pairs, each from the two phononic crystals (ϕ=0◦
and 180◦with the same size resonant elements), as illustrated
in Fig. 5(c). For the C-shaped resonator pair from the crystal
of ϕ=0◦(pair I), the slits of the resonator face each other,
while the other resonator pair (pair II) has an angle of 60◦
between the orientations of the two slits. In Fig. 5(c),the
scattering cross section of the coupled resonators (pairs I
and II) shows a broader bandwidth than the single resonator,
having a lower Qfactor. This broadened spectrum by the
coupled resonance can explain why the off-resonance scatter-
ers can effectively interact with the Bragg scattering [see the
band-gap opening by fD−fr∼3000 Hz in Fig. 3(b)]. Note
that pair I has slightly larger scattering cross section than pair
II because of its short distance and thereby strong coupling.
Such a difference between pair I and pair II provides an
explanation for the band-gap size difference between ϕ=0◦
and 180◦[Figs. 4(b) and 4(c)]. This indicates that although the
characteristics of the phononic crystals are fully analyzed by
the ensemble of the constituent scatterers the scattering cross-
section analysis of a few scatterers is useful in understanding
the observed topological phase transitions by local resonance
and Bragg scattering.
C. Lossy topological edge mode
We show robust one-way edge propagation at the boundary
between different topological states. To investigate these edge
modes, we use two different phononic crystals in the inset
of Fig. 2(a). The band gaps of these two crystals are similar
in size, as shown in Fig. 2(a). The dispersion of interface
modes in a supercell composed of these crystals [Fig. 6(a)]is
calculated from simulation (see Appendix A), as in Fig. 6(b).
We obtain two band-gap-crossing edge modes because of the
moderate band-gap size (larger band gap leads to gapped edge
states [34]). Also plotted are the pressure fields around the
interface between the ordinary and topological states at a
frequency of 5200 Hz and wave vectors of ±0.075π/a, and
the red arrows indicate the acoustic intensity vectors (i.e.,
Poynting vectors). We observe the rotational vector fields
resulting from modal hybridization [10], mimicking spin-up
and spin-down states.
Topological invariants are manifested in immunity to de-
fects and disorders, supporting loss-free one-way sound prop-
agation. As in reality loss is sometimes unavoidable, a critical
question that arises is whether topological edge modes are
preserved in systems with non-negligible loss [41,22]. Our
C shape is an ideal platform to investigate loss effects on
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FIG. 6. (a) Supercell (1 ×20) for the topological edge state, composed of ordinary (1 ×10) and topological (1 ×10) unit cells. (b) Band
structure of the supercell. In the bottom panel, the pressure fields around the edge are plotted at wave vectors of ±0.075π/a. The red arrows
indicate the acoustic intensity vectors.
topological phases, since its structure contains narrow air
slits capable of dissipating acoustic energy, which is similar
to acoustic absorbers made of Helmholtz resonators [40,42].
Without end correction, viscous and thermal losses occurring
in the slit (h) are considered by using effective parameters
given as [42]
ρs=ρ01−tanhh
2Gρ
h
2Gρ−1
,(4)
κs=κ01+(γ−1)tanhh
2Gκ
h
2Gκ−1
,(5)
where ρ0is the density and κ0is the bulk modulus, Gρ=
√iωρ0/η and Gκ=√iωPrρ0/η, with γthe specific-heat ratio
of air, ηthe dynamic viscosity, and Pr the Prandtl number
(Pr =η
ρ0αwith αthe thermal diffusivity). As shown in the
schematics of Fig. 7(a), these effective parameters are used
only for the lossy slits, resulting in the complex speed of
sound within the lossy slits (c=√κs/ρs) for the wave equa-
tion (see Appendix A). Equations (4) and (5) are valid for
plane-wave propagation inside the slits (the slit width his
much smaller than the wavelength). The effective parameters
take into account viscous and thermal losses, as they are
expressed by the dynamic viscosity (η) and thermal dif-
fusivity (α).
To demonstrate robust sound propagation, we consider
lossy and lossless cases by using interfaces with two sharp
corners, as shown in Fig. 7(a). The lossy case uses the
effective parameters (i.e., ρsand κs) for the slits and the
intrinsic properties (ρ0and κ0) for the rest of the fluid regions.
The top domain is the ordinary phase, whereas the bottom
domain has a topological phase. We consider a source of
plane acoustic waves (5200 Hz) on the left side, which is
placed on the boundary (the source length of 3a). In Fig. 7(a),
we observe backscattering-free propagation regardless of the
loss. Figure 7(b) shows absolute pressure amplitudes along
the right boundary [output in Fig. 7(a)] for the lossless, small
loss, and high loss cases. The small loss corresponds to the
intrinsic loss in the slit [its actual width h0is used for Eqs. (4)
and (5)], while the high loss uses the effective parameters
obtained using a smaller h=0.5h0for Eqs. (4) and (5). The
pressure amplitudes are normalized to the peak value for the
lossless case. Note that the acoustic amplitude along the right
boundary (output ports) is considerably reduced for the small
(red line) and high (blue line) losses. The acoustic pressure
is confined along the interface between the two domains (i.e.,
the width at half maximum <6λ0), although it extends near the
top and bottom boundaries. The confinement can be enhanced
by employing ordinary and topological domains with larger
bulk band gaps [34]. The investigation on the lossy topological
insulator indicates that topological insulators can demonstrate
robust wave propagation despite their intrinsic losses. In ad-
dition, when the lossy resonators are replaced with energy
conversion resonators, the lossy topological insulator can be
modified into an energy-harvesting topological insulator.
IV. CONCLUSIONS
We have demonstrated a topological phase transition using
honeycomb phononic crystals consisting of C-shaped rods.
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TAEHWA LEE AND HIDEO IIZUKA PHYSICAL REVIEW B 99, 064305 (2019)
FIG. 7. (a) Effect of propagation loss on topological edge propagation. Backscattering-free zigzag edge propagation is demonstrated for
the lossless (left) and lossy (right) cases. The illustration shows that the lossy case uses the effective parameters [ρsand κsfrom Eqs. (4)and
(5)] for the slit region, while the lossless case uses the intrinsic parameters (ρ0and κ0) for the entire fluid domain. Output ports along the right
boundary, used for pressure magnitude, are highlighted by the blue dashed (lossless) and solid (lossy) lines. (b) Normalized pressure magnitude
(|p|) at the output ports for lossless and lossy edge propagation at 5200 Hz. The pressure magnitude is normalized to the peak pressure of the
lossless case.
Our phononic crystals have allowed us to investigate the
interaction between Bragg scattering and local resonance, and
each can be controlled independently in our phononic crystals.
We have shown that the Bragg scattering induced gapless
bands experience topological phase transitions due to local
resonance, and that the band-gap size can be controlled by
the resonance frequency relative to the Dirac cone frequency.
Moreover, we have demonstrated a lossy topological insulator
by adding loss in the resonant elements. In this lossy topo-
logical insulator, backscattering-free edge wave propagation
is found to be preserved.
Our approach based on the interplay between Bragg scat-
tering and local resonance can increase the degree of freedom
for control of topological phases. An experimental verification
of our findings is feasible by using 3D-printed C-shaped
rods with the experimental setup found in Ref. [7], as long
as the fabricated C-shaped rods have sufficient structural
rigidity. In addition, our scatterers using local resonance can
benefit from electrically tunable resonators for active topo-
logical insulators without needing any mechanical moving
parts. Our results provide an understanding of the inter-
action between the scattering mechanisms and possibilities
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FIG. 8. (a) Band structures of the crystals (ϕ=180◦) for resonance frequencies close to the Dirac cone frequency (5100 Hz). To control
the resonance frequency, the radius (r) of the cavity is varied from r=3.8 to 3.3 mm. The purple shade indicates two flat (upper and lower)
horizontal bands related to the local resonance, while the dashed line indicates the Dirac cone frequency. The frequency of the lower flat band
corresponds to the resonance frequency. The dipolar (quadrupolar) bands are highlighted by pwith red circle symbols (dwith blue circle
symbols). For r=3.4 mm, the green arrow indicates the double Dirac cone. (b, c) Acoustic intensity fields and phase plots for the lower flat
bands (b) and upper flat bands (c) at r=3.8mm.
for designing fast-tunable or energy-harvesting acoustic topo-
logical insulators.
APPENDIX A: SIMULATION
All simulations are conducted using a commercial finite-
element method solver, COMSOL Multiphysics 5.3 (pressure
acoustic module). The solver numerically solves the wave
equation in the two-dimensional (xand ycoordinates) fluid
domain, given by
∂2p
∂x2+∂2p
∂y2+ω
c2
p=0,(A1)
where pis the pressure, ωis the frequency, and cis the speed
of sound (c=√κ/ρ with κand ρbeing the bulk modulus
and density). For losses in the narrow slits, the speed of sound
used for the wave equation is modified by using the effective
parameters [Eqs. (4) and (5)]. By assuming a large acoustic
impedance mismatch between the background fluid and rigid
scatterers, sound hard boundary conditions (i.e., Neumann
condition) are applied to the fluid/solid interface, which are
given by
n·∇p=0,(A2)
where nis the normal vector to the boundary, ∇is the vector
differential operator (i.e., ∇=∂
∂xi+∂
∂yjwith iand jbeing
the unit vectors in the xand ycoordinates, respectively), and
the dot symbol denotes the dot product of the two vectors.
To calculate the bulk (edge) band structures, Floquet periodic
boundary conditions are applied to periodic surfaces of the
unit cell (supercell), which are given by
pdst =psrceikF·(rdst −rsrc )(A3)
and
ndst ·∇pdst =nsrc ·∇psrceikF·(rdst −rsrc ),(A4)
where the subscripts (dst and src) denote the destination
boundary and source boundary, respectively, kFis the k
vector for Floquet periodicity, and ris the position vector.
For phononic crystals, continuity boundary conditions, i.e.,
Eqs. (A3) and (A4) with kF=0, are used for the top and
bottom boundaries, and radiation boundary conditions are
imposed on the left and right, which are given by
n·∇p−iω
cp=Qi,(A5)
where Qiis the source term. Here, the source term is zero
except for a boundary for incident plane waves. The incident
plane waves, the propagation direction of which is parallel to
the interface between the ordinary and topological regions, are
used for the input on the left boundary. In this case, Qiis given
by
Qi=n·∇pi−iω
cpi,(A6)
where piis the incident pressure wave expressed by pi=
p0eiω
c(r·ek)with p0the pressure amplitude and ekthe normal-
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TAEHWA LEE AND HIDEO IIZUKA PHYSICAL REVIEW B 99, 064305 (2019)
ized wave direction vector. Here, the length of the excitation
along the left boundary is three times the lattice constant (a).
The material properties of air are given by the density
(ρ0=1.2kg/m3) and the speed of sound (c=343 m/s). For
the lossy phononic crystals, the effective parameters [Eqs. (4)
and (5)] are obtained by using additional air properties, which
are given for room temperature (300 K) by the Prandtl num-
ber (Pr =0.707), dynamic viscosity (μ=1.85 ×10−5Pa s),
specific-heat ratio (γ=1.4), and bulk modulus (κ0=γP0for
the ambient pressure P0=101.325 kPa).
APPENDIX B: BAND STRUCTURE FOR fr=fD
Figure 8(a) shows the band structures for different radii (r).
For comparison, bands with two twofold degenerate modes (p
type and dtype) are indicated by pand d, and two horizontal
bands associated with local resonance are shaded in purple.
These two horizontal bands are verified with strong pressure
field inside the cavities, as seen in Figs. 8(b) and 8(c).We
find that the lower-frequency horizontal band corresponds
to the resonance frequency. By varying r, the resonance
frequency (lower-frequency horizontal band) approaches the
Dirac cone frequency (dashed line). For r=3.4mm, the
resonance frequency meets the Dirac cone frequency. In this
case, the upper-frequency horizontal band is placed within
the band gap of the p-type and d-type bands. Note that the
double Dirac cone occurs at a frequency lower than 5100
Hz (marked with a green arrow). In addition, the band gaps
at M and K are smaller than the one at , indicating that
the resonance frequency can be placed as close to the Dirac
cone frequency as possible, unless the bands related to local
resonance interfere with the p-type and d-type bands.
APPENDIX C: CALCULATION OF THE SCATTERING
CROSS SECTION
The scattering cross section of a scatterer in the 2D domain
is given by the ratio of the scattered power (W/m) to the
incident power (W/m2):
σsc[m]=∫s|psc |2
2ρcds
|pi|2
2ρc
,(C1)
where psc is the scattered pressure and piis the incident pres-
sure, i.e., Eq. (A6). The scattered pressure field is calculated
numerically (COMSOL) or analytically [Eqs. (2) and (3)].
Then, the scattered power (W/m) is calculated by the integral
taken over a circular line (s) of a radius (r0) surrounding
the scatterer. For the numerical calculation for the resonant
configuration, the integration for the scatterer power is taken
over a semicircle, as the resonant cavity is embedded in a
semi-infinite reflector.
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