# Topological rainbow trapping and broadband piezoelectric energy harvesting
of acoustic waves in gradient phononic crystals with coupled interfaces

Xiao-Lei Tang  Tian-Xue Ma<EMAIL_ADDRESS>Miso Kim  Yue-Sheng Wang
<EMAIL_ADDRESS>Department of Mechanics, School of Mechanical Engineering,
Tianjin University, Tianjin 300350, PR China  Department of Mechanics, School
of Physical Science and Engineering, Beijing Jiaotong University, Beijing
100044, PR China  School of Advanced Materials Science and Engineering,
Sungkyunkwan University (SKKU), Suwon 16419, Republic of Korea  SKKU Institute
of Energy Science and Technology (SIEST), Sungkyunkwan University (SKKU),
Suwon 16419, Republic of Korea

###### Abstract

Topological phononic crystals (PCs) offer a novel approach to control the
behavior of acoustic or elastic waves. In this paper, we design the gradient
PC structures with coupled interfaces to realize topological rainbow trapping
and broadband acoustic energy harvesting. Based on the geometric symmetry of
PC unit cells, we construct the coupled topological interfaces by amalgamating
two PCs with distinct topological phases. Also, the gradient modulation of
structural parameters along the coupled interfaces is employed for the
generation of rainbow trapping. Subsequently, a polyvinylidene fluoride (PVDF)
film is mounted along the coupled interfaces of the gradient PC structures,
facilitating the conversion of acoustic energy into electrical energy. The
numerical and experimental results show that the acoustic waves at different
frequencies stop and are amplified at different positions of the coupled
interfaces. Compared with the harvester in a bare structure, the topological
PC energy harvester significantly enhances the output power at different
excited frequencies. In the experiments, the maximum amplification ratio of
the output power reaches 91. Besides, it is demonstrated that the topological
rainbow trapping of acoustic waves exhibits robustness against random
structural disorders. Notably, the coupled interfaces have broadband and
multi-mode features, which are promising for other applications such as
selective filtering and sensing enhancement.

###### keywords:

phononic crystal, topological state, acoustic wave, piezoelectric energy
harvesting, rainbow trapping

##  1 Introduction

Acoustic energy harvesting is emerging as a popular approach for powering
small electronic devices. Various transduction mechanisms, such as
piezoelectric [1, 2, 3, 4, 5], electromagnetic [6, 7, 8] and triboelectric [9,
10, 11, 12] approaches, have been used to convert acoustic energy into
electrical energy. Though sounds are pervasive in daily life their energy is
typically characterized by low power density, which limits the efficiency of
acoustic energy harvesting. To address the aforementioned issue, sounds need
to be concentrated and localized at specified positions. Previous studies have
reported several acoustic energy harvesters employing Helmholtz resonators
[13] and 1/4-wavelength resonators [14, 15]. Liu $et$ $al$. [16] achieved the
acoustic energy harvesting by coupling a piezoelectric patch into the
Helmholtz resonator. Yuan $et$ $al$. [17] proposed an acoustic triboelectric
nanogenerator based on a 1/4-wavelength resonator system. The proposed system
was capable of generating a power output of 4.33 mW under the 100 dB sound
pressure level excitation.

Recently, with the studies of phononic crystals (PCs) in diverse disciplines,
interest and attempts have also been devoted to their possibility for wave
energy harvesting. PCs are artificially designed materials with peculiar wave
properties, including negative refraction [18], negative modulus [19] and band
gap [20]. Owing to their unique abilities to control acoustic or elastic
waves, PCs have been extensively studied by the scientific community for sound
transmission [21], cloaking [22] and energy harvesting [23, 24, 25, 26]. When
a point defect is introduced into a perfect PC, the point defect state of the
PC induces an acoustic localization effect. That is, the acoustic waves are
concentrated in and around the point defect, and hence, the pressure amplitude
is remarkably improved. The acoustic localization effect in PC structures can
be harnessed for energy harvesting applications [27, 28, 29]. Ma $et$ $al$.
[30] combined the PC and the Helmholtz resonator to improve the sound energy
density via wave localization and amplification. In addition, PC lenses [31,
32] are a distinct type of gradient PC structures for the control of wave
propagation, in which the effective refractive index of each unit cell is
spatially distributed according to specific profiles. Owing to their wave
focusing ability, various PC lenses have been constructed for the
piezoelectric energy harvesting of elastic [33, 34] or acoustic waves [35].
Kim $et$ $al$. [36] integrated the Helmholtz resonance mechanism with a PC
lens to achieve high power output for sounds lower than 1 kHz. Lee $et$ $al$.
[37] proposed a gradient PC energy harvester comprising unconventional unit
cells derived from machine learning.

The discovery of topological insulators has opened up unprecedented ways for
the control of electromagnetic [38], acoustic [39, 40, 41] and elastic [42,
43, 44] waves. Topologically protected states can be achieved by assembling
the materials with different topological phases according to the bulk-edge
correspondence. In contrast to conventional wave states, topological interface
states (TISs) exhibit excellent robustness against structural defects or
imperfections. For instance, despite defects or sharp bands, the TISs can
propagate smoothly along the interface under topological protection. In the
last decade, acoustic topological insulators have been employed for
unidirectional transmission [45, 46], logic operation[47, 48] and acoustic
tweezers [49, 50, 51]. Besides, the rainbow trapping phenomenon of TISs was
reported, in which the TISs at different frequencies are split and stop in
different locations [52, 53]. Moreover, the unique wave properties of
topological PCs make them a potential candidate for acoustic energy
harvesting. Fan $et$ $al$. [54] proposed a one-dimensional (1D) PC tube with
TISs to harvest the energy of acoustic waves. Zhao $et$ $al$. [55]
theoretically and experimentally demonstrated subwavelength acoustic energy
harvesting by using the TISs in 1D Helmholtz resonator arrays. Tang $et$ $al$.
[56] numerically and experimentally investigated the topological rainbow
trapping and energy amplification of acoustic waves in a gradient PC
structure. Li $et$ $al$. [57] designed a novel acoustic energy harvesting
device based on the TIS of a multi-resonant PC, where the device robustness
was improved by the TIS and the operating frequency was lowered by the use of
multi-resonant cavities.

However, the piezoelectric energy harvesting of acoustic waves with broadband
features and excellent robustness is rarely reported. Notably, in the regime
of electromagnetic waves, Elshahat $et$ $al$. [58] observed the rainbow
trapping of TISs in the coupled topological interfaces of gradient photonic
crystal structures. Interestingly, it was found that the coupling of TISs has
broadband and multi-mode features. Here, inspired by the above-mentioned
study, we propose a gradient PC structure with coupled interfaces to realize
broadband topological rainbow trapping and piezoelectric energy harvesting of
acoustic waves. The paper is organized as follows. In Section 2, the
dispersion relations and topological properties of 2D PCs, as well as the
group velocities of TISs, are investigated. Section 3 presents the numerical
and experimental results of topological rainbow trapping and sound pressure
amplification in designed PC structures with coupled interfaces. Section 4
reports the experimental validation of acoustic energy harvesting. Finally,
the conclusions of this paper are provided in Section 5.

##  2 Band diagram analysis

Figure 1: (a) Schematic diagram of the 2D square-latticed PC, in which the
polymeric scatterers (green areas) are placed periodically in the air
background (light cyan area). (b) Four choices of the PC unit cells, where
different inversion centers are marked by red points. (c) Dispersion relations
of the PC unit cells, where the band gap is highlighted by gray color. (d)
Pressure distributions of the first bulk band at the high symmetry points (X
and Y) and the corresponding Zak phases ($\theta_{x}^{Zak},\theta_{y}^{Zak}$).

We construct a 2D square-latticed PC by embedding polymeric scatterers (green
areas) in an air background (light cyan area), as sketched in Fig. 1(a). The
lattice constant $a$ and the radius of circular scatterers $r$ can describe
the geometric configuration of the 2D PC. Thanks to the significant contrast
in acoustic impedance between polymer and air, the polymeric scatterers are
considered as acoustically rigid bodies. The mass density and the sound speed
of air are $\rho=1.21$ $\rm{kg/m^{3}}$ and $c=343$ m/s, respectively.
Throughout this paper, we utilize the finite element method with COMSOL
Multiphysics for numerical simulations. Noticeably, this 2D PC has four
inversion centers, which are marked as A, B, C and D, respectively [see Fig.
1(a)]. According to the crystal periodicity, four types of PC unit cells can
be obtained, as illustrated in Fig. 1(b). The band diagram of the 2D PC is
shown in Fig. 1(c), where the lattice constant is $a$ = 45 mm and the
scatterer radius is $r$ = 0.45$a$. It can be seen from Figs. 1(b) and 1(c)
that despite four different choices of unit cells, the band diagrams for the
PC are identical.

In this work, the 2D PC is protected by the inversion symmetry, restricting
the Zak phase to two values: 0 or $\pi$. By studying the symmetry properties
of pressure field ($p$) distribution at the high symmetry points (X and Y) in
the Brillouin zone [see Fig. 1(d)], the corresponding
($\theta_{x}^{Zak},\theta_{y}^{Zak}$) of four unit cells can be obtained [59].
Considering the mirror-symmetry of the pressure profile at point $\Gamma$, if
the pressure profile at the high symmetry points (i.e., X and Y) is
symmetric$/$antisymmetric (S$/$A), the Zak phase $\theta_{j}^{Zak}$ of the
first band is 0 (trivial phase)$/$$\pi$ (non-trivial phase) in the
corresponding direction. As depicted in Fig. 1(d), the pressure fields ($p$)
at X(Y) of the four unit cells are A/A, S/A, A/S and S/S, respectively, and
hence the corresponding 2D Zak phases ($\theta_{x}^{Zak},\theta_{y}^{Zak}$)
are ($\pi$, $\pi$), (0, $\pi$), ($\pi$, 0) and (0, 0), respectively. In
addition, the Zak phase of the first band of the 2D PC along the $j$ ($j=x,y$)
direction is defined as $\theta_{j}^{Zak}$ =$\int\rm$ $dk_{x}dk_{y}$
Tr$[A_{j}(k_{x},k_{y})]$, where
$A_{j}(k_{x},k_{y})=\left\langle\psi\left|i\partial
k_{j}\right|\psi\right\rangle$ is the Berry connection, $\psi$ denotes the
periodic part of the Bloch pressure eigenfunction, and the integration is
performed over the first Brillouin zone [60, 61, 62]. The numerically
calculated Berry connections of different 2D PC unit cells are provided in the
Supplementary Material. When two PCs with different topological phases (i.e.,
Zak phases) are placed adjacently, the combined PC structure will generate
topologically protected states at the interface (i.e., TISs). For an acoustic
TIS, the acoustic waves propagate steadily along the interface of the PC
structure without backscattering.

Figure 2: (a) Schematic diagram of the PC super-cell with two coupled
interfaces, in which one unit cell of PC2 is sandwiched between ten unit cells
of PC3. (b) Band diagram of the PC super-cell, where three different TISs
(marked by green, yellow and red dots, respectively) emerge inside the band
gap. The black dots denote the bulk modes. (c) Pressure distributions of the
three TISs at the Brillouin zone boundary $k_{x}=\pi/a$.

Next, we combine two kinds of PC unit cells with distinct topological phases
($\theta_{x}^{Zak}=0,\theta_{y}^{Zak}=\pi$) to form a super-cell, as
illustrated in Fig. 2(a). One unit cell of PC2 is sandwiched between ten unit
cells of PC3, forming two coupled topological interfaces. The radius of
scatterers along the coupled interfaces (i.e., scatterers in PC2) is denoted
as $r_{\textrm{i}}$. Compared with the conventional topological interface, the
coupled topological interfaces have the superiority of multi-mode and
broadband, which are beneficial to multi-mode rainbow trapping and broadband
energy harvesting. The TISs in the conventional topological interfaces are
discussed in the Supplementary Material. Moreover, the coupled interface
region provides sufficient space to host the piezoelectric structures or
materials for energy transduction. The band diagram of the PC super-cell is
shown in Fig. 2(b), where the black dots denote the bulk states, and the red,
yellow and green dots denote the TISs within the band gap. The pressure
distributions of the three TISs at $k_{x}=\pi/a$ are plotted in Fig. 2(c).
TISs 1, 2 and 3 exhibit S, S and A patterns with respect to the mirror plane
(marked by black dashed line). To facilitate the experiment, TISs 1 and 2 are
considered for the realization of rainbow trapping.

Figure 3:  Dispersion curves of TISs 1 (a) and 2 (b) for different radii of
scatterers along the coupled interfaces. Group velocity curves of TISs 1 (c)
and 2(d) for different radii of scatterers along the coupled interfaces.

It should be noted that in Figs. 3(a) and 3(b), the zero group velocity of
TISs 1 and 2 appears at the Brillouin zone boundary $k_{x}$ = $\pi/a$. The
radius of scatterers along the coupled interfaces $r_{\textrm{i}}$ are changed
from $0.04a\sim 0.182a$, and $0.1a\sim 0.28a$ for TISs 1 and 2, respectively.
With the increase of $r_{\textrm{i}}$, the dispersion curve of TIS 1 (TIS 2)
shifts to the lower (higher) frequency region. According to the definition of
group velocity $v_{\textrm{g}}$ ($v_{\textrm{g}}$ =
$\partial$$\omega$/$\partial$$k$ with $\omega$ and $k$ being the angular
frequency and wave number, respectively), it can be inferred that the group
velocity of the TISs is affected by the geometry of the scatterers along the
interfaces. The group velocity $v_{\textrm{g}}$ of TISs 1 and 2 as a function
of $r_{\textrm{i}}$ is shown in in Figs. 3(c) and 3(d), respectively. As
$r_{\textrm{i}}$ increases, the cut-off frequencies (i.e.,
$v_{\textrm{g}}$$\rightarrow$ 0) of TISs 1 and 2 shift to lower and higher
frequencies, respectively. As a result, once the acoustic waves enter into the
PCs with zero group velocity, they cannot propagate any more and hence be
trapped.

##  3 Topological rainbow trapping

Figure 4:  Schematic diagrams of PC samples I (a) and II (b). Topological
rainbow trapping in PC samples I (c) and II (d) at different frequencies. (e)
Schematic diagram of the PC samples with structural disorders. Topological
rainbow trapping in perturbed PC samples I (f) and II (g) at different
frequencies.

In this section, two configurations of gradient PC structures (with the size
of $20\times 11$ unit cells) are constructed to observe topological rainbow
trapping, where PC structures I and II correspond to TISs 1 and 2,
respectively, as schematically shown in Figs. 4(a) and 4(b). The radius of
scatterers along the coupled interfaces $r_{\textrm{i}}$ increases linearly
along the $+x$ direction. The expression of $r_{\textrm{i}}$ is described as
$r_{\textrm{i}}$ = $r_{0}+(n-1)\delta_{r}$, where $\delta_{r}$ is the step
variation in $r_{\textrm{i}}$ and $n$ ($n=1,2,...,19$) denotes the position
along the interfaces. The configuration of PC structure I (II) is given as
follows: $r_{0}$ = $0.04a$ and $\delta_{r}$ = $0.00789a$ ($r_{0}$ = $0.1a$ and
$\delta_{r}$ = $0.01a$). It is known from Section 2 that if the radius of
scatterers along the coupled interfaces $r_{\textrm{i}}$ decreases
monotonously, the position of zero group velocity of TISs at different
frequencies varies correspondingly. As a result, the gradient PC structures
with coupled interfaces generate an intriguing phenomenon, i.e., topological
rainbow trapping. In the numerical simulations, a point monopole source
(marked by a star) is used to generate acoustic waves and placed on the left
side of the PCs. In addition, the perfectly matched layers (PMLs) are utilized
to reduce the acoustic wave reflection at the outer boundaries. Furthermore,
in order to prove the robustness of topological rainbow trapping, the
structural disorders are introduced into the scatterers of PC structures I and
II in the following three ways [see Fig. 4(e)]. The rotation angle $\varphi$
(highlighted by blue color) and the shape (highlighted by green color) are
perturbed, as well as and the absence of some scatterers is introduced, where
$\varphi$ changes randomly from $-$$$ to $None$.

The frequency responses of the two PC structures with different gradient
interfaces are calculated. First, for structure I, the acoustic pressure
fields at four selected frequencies of 3250, 3350, 3450 and 3550 Hz are shown
in Fig. 4(c). As expected, the position where the acoustic energy is
concentrated along the coupled interfaces (marked by the red line) is
dependent on the frequency of incident waves. The acoustic waves are well-
guided along the coupled interfaces, and the waves at lower frequencies travel
further than those at higher frequencies. Interestingly, due to the effect of
zero group velocity, the acoustic waves at different frequencies are trapped
in different spatial locations. The topological rainbow trapping caused by the
gradient interfaces is observed. Then, for structure II, the pressure profiles
with the incident waves at 4400, 4500, 4600 and 4700 Hz are presented in Fig.
4(d). Apparently, the topological rainbow trapping is also achieved in PC
structure II. In comparison, the results for the perturbed PC structures are
plotted in Figs. 4(f) and 4(g). The results for the topological rainbow
trapping with and without disorders match well, demonstrating its robustness
against structural disorders.

Figure 5: (a) Photographs of the 3D-printed PC samples (I and II). (b)
Photograph of the experimental setup, where the inset shows the enlarged view
of the drilled holes for inserting the microphone (red wireframe) and the PC
dimensions (green wireframe). Numerically calculated and experimentally
measured pressure distributions in the perfect PC samples and the bare
structure: (c) sample I and (d) sample II.

For the purpose of experimental validation, we fabricate the polymeric PCs via
the 3D-printing technique. The photos of the fabricated PC samples are given
in Fig. 5(a). The gradient PC samples consist of $20\times 11$ unit cells and
the radius of scatterers on the coupled interfaces $r_{\textrm{i}}$ varies
linearly along the $+x$ axis, which are the same as the numerical simulations
[see Figs. 4(a) and 4(b)]. The experimental setup is shown in Fig. 5(b). To
satisfy the 2D approximation, the PC samples are placed in a planar waveguide
with 35 mm height, which are sandwiched by the PMMA and PE plates at the top
and bottom, respectively. To accurately measure the acoustic field, we drill
19 holes in the upper PMMA plate of the planar waveguide to accommodate the
microphone (Brüel $\&$ Kjær, Type 4939-A-011). Thus, the microphone can be
inserted into these holes to facilitate the detection of acoustic pressure
along the PC interfaces. The signals over the considered frequency range are
generated by a generator module (Brüel $\&$ Kjær, Type 3160-A-042) and then
are amplified by a power amplifier (Krohn-Hite, Model 7500). The amplified
signals are played by a loudspeaker, which is placed on the left side of the
interfaces of the fabricated PC structures. Additionally, a bare structure
(planer waveguide without PCs) is also considered for comparison.

Figures 5(c) and 5(d) show the numerical and experimental results of
topological rainbow trapping for PC structures I and II as a function of
frequency, respectively. Herein, we will discuss the results of PC structure I
in detail. Obviously, the acoustic waves at lower frequencies (e.g., 3385 Hz)
can pass through the entire interface of the gradient PC structure. In
contrast, the acoustic waves at higher frequencies (e.g., 3700 Hz) can only
travel a short distance before being stopped owing to the rainbow trapping
effect. The measured and calculated results match well, although the location
of the high pressure region in the experiment changes slightly from the
numerical prediction. On the other hand, the rainbow trapping of PC structure
II is also verified experimentally, as shown in Fig. 5(d). Importantly,
compared with the sound pressure in the bare structure, the topological
rainbow trapping in the designed PC structures has a significant amplification
[see Figs. 5(c) and 5(d)].

Figure 6:  Photographs of PC samples I (a) and II (b) with several structural
disorders, where the scatterers are randomly modulated in rotation angle
(highlighted by blue color), shape (highlighted by green color) and absence.
Numerically calculated and experimentally measured pressure distributions in
the disordered PC samples and the bare structure: (c) sample I and (d) sample
II.

Furthermore, to demonstrate the robustness of topological rainbow trapping and
acoustic energy amplification, several structural disorders are introduced
into PC structures I and II, as shown in Figs. 6(a) and 6(b). The introduced
disorders are consistent with the numerical simulations in Fig. 4(c). Figures
6(c) and 6(d) show the calculated and measured pressure distributions in the
corresponding frequency ranges for the two disordered PC structures. The
numerically calculated and experimentally measured results for topological
rainbow trapping are in good agreement. There is a slight deviation at the
high pressure position due to experimental errors. Comparing Figs. 5 and 6, we
infer that the rainbow trapping of the TISs remains despite the introduction
of different types of structural disorders.

##  4 Experimental validation of acoustic energy harvesting

Figure 7:  (a) Diagram of the experimental setup for piezoelectric energy
harvesting. (b) Schematic of the piezoelectric energy harvesting system.
Measured voltage outputs in the time (c) and frequency (d) domains,
respectively. (e) Measured voltage outputs in the time domain for different
resistance loads.

The experimental setup of the piezoelectric energy harvesting of acoustic
waves based on the topological rainbow trapping effect in gradient PCs is
plotted in Fig. 7(a). To convert the acoustic energy into the electricity, a
PVDF film is placed at the interfaces of the gradient PC structures [see the
inset of Fig. 7(a)]. The circuit of the piezoelectric energy harvesting system
is schematically illustrated in Fig. 7(b). The acoustic signals are produced
using a signal generator (Tektronix AFG3102C) and then amplified through a
power amplifier (Krohn-Hite, Model 7500). The voltage output across the
resistance is measured with an oscilloscope (Tektronix, MDO3014). To validate
the performance of the PC energy harvester, a continuous sine-wave signal is
employed, with the frequencies corresponding to the topological rainbow
trapping.

The measured voltage by the piezoelectric film in topological PC structure I
at 3675 Hz as a function of time is shown in Fig. 7(c). Moreover, the electric
output of the bare structure (without PCs) is measured as the reference. The
corresponding voltages in the frequency domain are also plotted in Fig. 7(d).
Clearly, the output voltage of the topological PC energy harvester is much
higher than that of the reference energy harvester. This is ascribed to the
enhancement of acoustic pressure based on topological rainbow trapping. It is
noteworthy that the load resistance significantly impacts upon both the
voltage and power outputs of the PC piezoelectric energy harvester. Figure
7(e) illustrates the voltage outputs measured from the PC energy harvester
under various external load resistances. It is apparent that as the load
resistance increases, the voltage amplitude increases as well.

Figure 8: Measured voltage (a, b) and power (c, d) outputs as a function of
the resistance load at selected frequencies in PC structures I and II, where
the dots and circles denote the outputs obtained from the PC and bare
structures, respectively. The results of the bare structure are visually
enlarged in the insets of (b) and (d).

To verify that the proposed PC structures based on the topological rainbow
trapping can achieve broadband energy harvesting of acoustic waves, the
piezoelectric films are placed at different positions along the structural
interfaces, as shown in Fig. 7(a). Because the wave characteristics of the
topological rainbow trapping in the spatial and frequency domains, as well as
its robustness against to structural disorders have been discussed in Section
3 (see Figs. 5 and 6), we will focus on the performance of piezoelectric
energy harvesting. Five frequencies (for PC structure I) from 3450 to 3650 Hz
and six frequencies (for PC structure II) from 4450 to 4700 Hz are considered
in the energy harvesting experiments. Accordingly, only five positions for
Structure I (six positions for Structure II) along the interfaces are employed
to host the piezoelectric films. Notably, due to the restriction on the
position of piezoelectric films, the excited frequencies are generally not the
perfect ones for the energy harvester. The voltage outputs generated by the PC
and reference energy harvesters as a function of the load resistance are
plotted in Figs. 8(a) and 8(c). It can be seen that as the resistance
increases the measured voltage rises monotonically, but its rate of increase
gradually degrades and gets saturated at a high resistance. Importantly,
compared to the reference structure, the output voltage of the topological PC
structures is increased by a factor of 5 $\thicksim$ 7 times for structure I
(i.e., TIS 1) and that of 7 $\thicksim$ 9 times for structure II (i.e., TIS
2), achieving the effect of acoustic energy harvesting at all considered
frequencies. The power generated by the piezoelectric harvester can be
calculated according to Ohm’s law $P$ = $U^{2}$/$R$, where $P$, $U$ and $R$
denote the power, voltage and load resistance, respectively. The power output
rises with the increase of load resistance until it reaches the maximum values
and then starts to decline, as shown in Figs. 8(b) and 8(d). The maximum power
generated by the gradient PC structures is about 3.5 $\mu$W with an external
load resistance of 14 k$\Omega$ at 4550 Hz [see Fig. 8(d)]. Herein, the
amplification ratio is defined as the ratio of the power outputs obtained from
the PC and reference bare structures. Remarkably, energy harvesting can be
achieved from 3300 to 3800 Hz (from 4350 to 4900 Hz) in topological PC
structure I (II). The amplification ratios are $20\sim 50$ and $40\sim 91$ for
the proposed PC harvesters I and II, respectively.

##  5 Conclusion

In this study, we propose the gradient PC structures with coupled interfaces
for topological rainbow trapping and acoustic energy harvesting. The proposed
2D PCs are constructed by polymeric scatterers in the air background,
exhibiting a wide band gap with different Zak phases. Thanks to the geometric
symmetry, the Zak phase can be tuned by different choices of PC unit cells.
The TISs emerge at two coupled interfaces between two PCs with different Zak
phases. The results indicate that the acoustic waves of different frequencies
separate, stop and are amplified at different positions in the coupled
interfaces. In addition, the topological rainbow trapping is robust to
randomly structural disorders. To verify the energy harvesting effect, the
piezoelectric film is attached at the coupled interfaces of gradient PCs as
the piezoelectric energy harvester. The proposed PC structure exhibits the
capacity of broadband energy harvesting. Compared with the bare structure, the
maximum output power of the topological PC energy harvester is significantly
amplified by 91 times. It is noted that the design of coupled interfaces
possess the characteristics of multi-mode and broad bandwidth, providing wide
application prospects for energy harvesting, sensing and selective filtering.

## CRediT authorship contribution statement

Xiao-Lei Tang: Data curation, Formal analysis, Investigation, Methodology,
Validation, Visualization, Writing - original draft; Tian-Xue Ma:
Conceptualization, Supervision, Formal analysis, Validation, Writing - review
& editing; Miso Kim: Funding acquisition, Resources, Writing - review &
editing; Yue-Sheng Wang: Funding acquisition, Supervision, Resources, Writing
- review & editing.

## Declaration of Competing Interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Acknowledgments

We sincerely appreciate Innovative Research Group of NSFC (Grant No.
12021002), National Natural Science Foundation of China (Grant No. 12372087),
and the National Research Foundation of Korea grant funded by the Korea
government (No.RS-2023-00254689) for the financial support to this study.
Xiaolei-Tang is grateful to the support of the China Scholarship Council
(Grant No. 202306250172).

## DATA AVAILABILITY

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## References

  * [1] B. Li, J. H. You, Y.-J. Kim, Low frequency acoustic energy harvesting using PZT piezoelectric plates in a straight tube resonator, Smart Materials and Structures 22 (2013) 055013.
  * [2] J. Choi, I. Jung, C.-Y. Kang, A brief review of sound energy harvesting, Nano Energy 56 (2019) 169–183.
  * [3] H. Alqaleiby, M. Ayyad, M. R. Hajj, S. A. Ragab, L. Zuo, Effects of piezoelectric energy harvesting from a morphing flapping tail on its performance, Applied Energy 353 (2024) 122022.
  * [4] P. Eghbali, D. Younesian, S. Farhangdoust, Enhancement of the low-frequency acoustic energy harvesting with auxetic resonators, Applied Energy 270 (2020) 115217\.
  * [5] M. Yuan, Z. Cao, J. Luo, Z. Pang, Low frequency acoustic energy harvester based on a planar Helmholtz resonator, AIP Advances 8 (2018).
  * [6] F. U. Khan, Izhar, Electromagnetic-based acoustic energy harvester, in: INMIC, 2013, pp. 125–130.
  * [7] F. U. Khan, M. U. Khattak, Contributed review: Recent developments in acoustic energy harvesting for autonomous wireless sensor nodes applications, Review of Scientific Instruments 87 (2016) 021501.
  * [8] F. U. Khan, Electromagnetic based acoustic energy harvester for low power wireless autonomous sensor applications, Sensor Review 38 (2018) 298–310.
  * [9] J. Yang, J. Chen, Y. Liu, W. Yang, Y. Su, Z. L. Wang, Triboelectrification-based organic film nanogenerator for acoustic energy harvesting and self-powered active acoustic sensing, ACS Nano 8 (2014) 2649–2657.
  * [10] X. Fan, J. Chen, J. Yang, P. Bai, Z. Li, Z. L. Wang, Ultrathin, rollable, paper-based triboelectric nanogenerator for acoustic energy harvesting and self-powered sound recording, ACS Nano 9 (2015) 4236–4243.
  * [11] Z. Yu, Y. Zhang, Y. Wang, J. Zheng, Y. Fu, D. Chen, G. Wang, J. Cui, S. Yu, L. Zheng, H. Zhou, D. Li, Integrated piezo-tribo hybrid acoustic-driven nanogenerator based on porous MWCNTs/PVDF-TrFE aerogel bulk with embedded pdms tympanum structure for broadband sound energy harvesting, Nano Energy 97 (2022) 107205.
  * [12] C. Sun, S. Si, J. Liu, Y. Xia, Z. Lin, Q. He, H. Wang, L. Chen, H. Wu, J. Liu, Y. Wu, J. Yang, Flexible, ultra-wideband acoustic device for ultrasound energy harvesting and passive wireless sensing, Nano Energy 112 (2023) 108430\.
  * [13] X. Peng, Y. Wen, P. Li, A. Yang, X. Bai, Enhanced acoustoelectric coupling in acoustic energy harvester using dual Helmholtz resonators, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 60 (2013) 2121–2128.
  * [14] B. Li, A. J. Laviage, J. H. You, Y.-J. Kim, Harvesting low-frequency acoustic energy using quarter-wavelength straight-tube acoustic resonator, Applied Acoustics 74 (2013) 1271–1278.
  * [15] G. Zhu, Y. Zhou, Z. Si, Y. Cheng, F. Wu, H. Wang, Y. Pan, J. Xie, C. Li, A. Chen, R. Wang, J. Sun, A multi-hole resonator enhanced acoustic energy harvester for ultra-high electrical output and machine-learning-assisted intelligent voice sensing, Nano Energy 108 (2023) 108237.
  * [16] F. Liu, A. Phipps, S. Horowitz, K. Ngo, L. Cattafesta, T. Nishida, M. Sheplak, Acoustic energy harvesting using an electromechanical Helmholtz resonator, The Journal of the Acoustical Society of America 123 (2008) 1983–1990.
  * [17] M. Yuan, C. Li, H. Liu, Q. Xu, Y. Xie, A 3D-printed acoustic triboelectric nanogenerator for quarter-wavelength acoustic energy harvesting and self-powered edge sensing, Nano Energy 85 (2021) 105962.
  * [18] R. Zhu, X. Liu, G. Hu, C. Sun, G. Huang, Negative refraction of elastic waves at the deep-subwavelength scale in a single-phase metamaterial, Nature Communications 5 (2014) 5510.
  * [19] Z. Liu, C. T. Chan, P. Sheng, Analytic model of phononic crystals with local resonances, Physical Review B 71 (2005) 014103.
  * [20] C. Goffaux, J. Vigneron, Theoretical study of a tunable phononic band gap system, Physical Review B 64 (2001) 075118.
  * [21] G. Jiang, Y. Liu, Y. Wu, W. Xu, Q. Kong, C. Zhang, Transmission and radiation of acoustic oblique incident through tube arrays based on phononic crystals theory, Applied Acoustics 116 (2017) 117–126.
  * [22] L.-Y. Zheng, Y. Wu, X. Ni, Z.-G. Chen, M.-H. Lu, Y.-F. Chen, Acoustic cloaking by a near-zero-index phononic crystal, Applied Physics Letters 104 (2014) 161904\.
  * [23] T.-X. Ma, Z.-Y. Li, C. Zhang, Y.-S. Wang, Energy harvesting of Rayleigh surface waves by a phononic crystal Luneburg lens, International Journal of Mechanical Sciences 227 (2022) 107435.
  * [24] T.-X. Ma, Q.-S. Fan, Z.-Y. Li, C. Zhang, Y.-S. Wang, Flexural wave energy harvesting by multi-mode elastic metamaterial cavities, Extreme Mechanics Letters 41 (2020) 101073.
  * [25] L. Zhang, T. Tan, Z. Yu, Z. Yan, Topological imbalanced phononic crystal with semi-enclosed defect for high-performance acoustic energy confinement and harvesting, Nano Energy 100 (2022) 107472.
  * [26] Y. Wang, X. Zhu, T. Zhang, S. Bano, H. Pan, L. Qi, Z. Zhang, Y. Yuan, A renewable low-frequency acoustic energy harvesting noise barrier for high-speed railways using a helmholtz resonator and a pvdf film, Applied energy 230 (2018) 52–61.
  * [27] L.-Y. Wu, L.-W. Chen, C.-M. Liu, Acoustic energy harvesting using resonant cavity of a sonic crystal, Applied Physics Letters 95 (2009) 013506.
  * [28] W.-T. Gao, J.-P. Xia, H.-H. Sun, S.-Q. Yuan, Y. Ge, X.-J. Liu, Acoustic energy harvesting for low-frequency airborne sound based on compound Mie resonances, Applied Physics Express 12 (2019) 044002.
  * [29] A. Yang, P. Li, Y. Wen, C. Yang, D. Wang, F. Zhang, J. Zhang, High-Q cross-plate phononic crystal resonator for enhanced acoustic wave localization and energy harvesting, Applied Physics Express 8 (2015) 057101.
  * [30] K. Ma, T. Tan, Z. Yan, F. Liu, W.-H. Liao, W. Zhang, Metamaterial and Helmholtz coupled resonator for high-density acoustic energy harvesting, Nano Energy 82 (2021) 105693.
  * [31] Y. Jin, B. Djafari-Rouhani, D. Torrent, Gradient index phononic crystals and metamaterials, Nanophotonics 8 (2019) 685–701.
  * [32] G. Lee, S.-J. Lee, J. Rho, M. Kim, Acoustic and mechanical metamaterials for energy harvesting and self-powered sensing applications, Materials Today Energy 37 (2023) 101387.
  * [33] F. Akbari-Farahani, S. Ebrahimi-Nejad, From defect mode to topological metamaterials: A state-of-the-art review of phononic crystals & acoustic metamaterials for energy harvesting, Sensors and Actuators A: Physical (2023) 114871\.
  * [34] J. Hyun, W. Choi, M. Kim, Gradient-index phononic crystals for highly dense flexural energy harvesting, Applied Physics Letters 115 (2019) 173901.
  * [35] A. Allam, K. Sabra, A. Erturk, Sound energy harvesting by leveraging a 3D-printed phononic crystal lens, Applied Physics Letters 118 (2021) 103504\.
  * [36] S. Kim, J. Choi, H. M. Seung, I. Jung, K. H. Ryu, H.-C. Song, C.-Y. Kang, M. Kim, Gradient-index phononic crystal and Helmholtz resonator coupled structure for high-performance acoustic energy harvesting, Nano Energy 101 (2022) 107544.
  * [37] S. Lee, W. Choi, J. W. Park, D.-S. Kim, S. Nahm, W. Jeon, G. X. Gu, M. Kim, S. Ryu, Machine learning-enabled development of high performance gradient-index phononic crystals for energy focusing and harvesting, Nano Energy 103 (2022) 107846.
  * [38] L. Lu, J. D. Joannopoulos, M. Soljačić, Topological photonics, Nature Photonics 8 (2014) 821–829.
  * [39] Y.-F. Chen, Z.-G. Chen, H. Ge, C. He, X. Li, M.-H. Lu, X.-C. Sun, S.-Y. Yu, X. Zhang, Various topological phases and their abnormal effects of topological acoustic metamaterials, Interdisciplinary Materials 2 (2023) 179–230.
  * [40] S. Yin, L. Ye, H. He, M. Ke, Z. Liu, Acoustic valley-locked waveguides in heterostructures of a Square lattice, Physical Review Applied 18 (2022) 054073\.
  * [41] S. Zheng, X. Man, Z.-L. Kong, Z.-K. Lin, G. Duan, N. Chen, D. Yu, J.-H. Jiang, B. Xia, Observation of fractal higher-order topological states in acoustic metamaterials, Science Bulletin 67 (2022) 2069–2075.
  * [42] T.-X. Ma, J. Liu, C. Zhang, Y.-S. Wang, Topological edge and interface states in phoxonic crystal cavity chains, Physical Review A 106 (2022) 043504.
  * [43] Z. Ma, Y. Liu, Y.-X. Xie, Y.-S. Wang, A simple elastic phononic crystal plate with adjustable topological valley transmission paths, Extreme Mechanics Letters 57 (2022) 101910.
  * [44] T.-X. Ma, Q.-S. Fan, C. Zhang, Y.-S. Wang, Flexural wave energy harvesting by the topological interface state of a phononic crystal beam, Extreme Mechanics Letters 50 (2022) 101578.
  * [45] B.-Z. Xia, T.-T. Liu, G.-L. Huang, H.-Q. Dai, J.-R. Jiao, X.-G. Zang, D.-J. Yu, S.-J. Zheng, J. Liu, Topological phononic insulator with robust pseudospin-dependent transport, Physical Review B 96 (2017) 094106.
  * [46] C. He, X. Ni, H. Ge, X.-C. Sun, Y.-B. Chen, M.-H. Lu, X.-P. Liu, Y.-F. Chen, Acoustic topological insulator and robust one-way sound transport, Nature Physics 12 (2016) 1124–1129.
  * [47] Y.-J. Lu, Y. Wang, Y. Ge, S.-Q. Yuan, D. Jia, H.-X. Sun, X.-J. Liu, Multifunctional acoustic logic gates by valley sonic crystals, Applied Physics Letters 121 (2022) 123506.
  * [48] H. Pirie, S. Sadhuka, J. Wang, R. Andrei, J. E. Hoffman, Topological phononic logic, Physical Review Letters 128 (2022) 015501.
  * [49] H. Dai, L. Liu, B. Xia, D. Yu, Experimental realization of topological on-chip acoustic tweezers, Physical Review Applied 15 (2021) 064032.
  * [50] P. Liu, H. Li, Z. Zhou, Y. Pei, Topological acoustic tweezer and pseudo-spin states of acoustic topological insulators, Applied Physics Letters 120 (2022) 222202\.
  * [51] L. Du, G. Hu, Y. Hu, Q. Wang, Acoustic forceps based on focused acoustic vortices with different topological charges, Sensors 23 (2023) 6874.
  * [52] G. J. Chaplain, J. M. De Ponti, G. Aguzzi, A. Colombi, R. V. Craster, Topological rainbow trapping for elastic energy harvesting in graded Su-Schrieffer-Heeger systems, Physical Review Applied 14 (2020) 054035.
  * [53] Z. Tian, C. Shen, J. Li, E. Reit, H. Bachman, J. E. Socolar, S. A. Cummer, T. Jun Huang, Dispersion tuning and route reconfiguration of acoustic waves in valley topological phononic crystals, Nature Communications 11 (2020) 762.
  * [54] L. Fan, Y. He, X.-a. Chen, X. Zhao, Acoustic energy harvesting based on the topological interface mode of 1D phononic crystal tube, Applied Physics Express 13 (2019) 017004.
  * [55] D. Zhao, X. Chen, P. Li, X.-F. Zhu, Subwavelength acoustic energy harvesting via topological interface states in 1D Helmholtz resonator arrays, AIP Advances 11 (2021) 015241.
  * [56] X.-L. Tang, T.-X. Ma, Y.-S. Wang, Topological rainbow trapping and acoustic energy amplification in two-dimensional gradient phononic crystals, Applied Physics Letters 122 (2023) 112201.
  * [57] B. Li, H. Chen, B. Xia, L. Yao, Acoustic energy harvesting based on topological states of multi-resonant phononic crystals, Applied Energy 341 (2023) 121142.
  * [58] S. Elshahat, M. S. M. Esmail, H. Yuan, S. Feng, C. Lu, Broadband multiple topological rainbows, Annalen der Physik 534 (2022) 2200137.
  * [59] M. Xiao, G. Ma, Z. Yang, P. Sheng, Z. Zhang, C. T. Chan, Geometric phase and band inversion in periodic acoustic systems, Nature Physics 11 (2015) 240–244.
  * [60] J. K. Asbóth, L. Oroszlány, A. Pályi, A short course on topological insulators, Lecture Notes in Physics 919 (2016) 166.
  * [61] F. Zangeneh-Nejad, R. Fleury, Topological Fano resonances, Physical Review Letters 122 (2019) 014301.
  * [62] C. Lu, C. Wang, M. Xiao, Z. Zhang, C. T. Chan, Topological rainbow concentrator based on synthetic dimension, Physical Review Letters 126 (2021) 113902.