High-Q Slow Sound Mode in a Phononic Fishbone Nanobeam Using an Acoustic Potential Well Cavity

Abstract

Phononic crystals and phononic metamaterials are popular structures for manipulating acoustic waves with artificially arranged units that have different elastic constants. These structures are also used in acousto-optic coupling and optomechanical structures. In such research, a 1-D nanobeam containing a cavity region sandwiched by two mirror regions is one of the most common designs. However, searching bandgaps for suitable operation modes and the need for the mirror region are limitations in the device design. Therefore, we introduce the slow sound mode as the operating acoustic mode and use an acoustic potential well to further trap the phonons in the cavity. Three types of structures are introduced to investigate the effect of the potential well. The products of the mode frequencies and the quality factors of the modes are used to demonstrate the performance of the structures. The displacement field and the strain field show the concentrated slow sound modes of the potential wells and produce high quality factors.

1. Introduction

Phononic crystals (PnCs) [1,2] and phononic metamaterials (PMs) [3,4,5] are artificial structures with special elastic characteristics. By periodically arranging materials with different elastic parameters, PnCs and PMs show great performance in manipulating acoustics waves. One of the special effects of these structures is the acoustic band gap (ABG), which can be induced by specially designed geometric structures. The ABGs can perfectly block the acoustic waves in a frequency range and stop them propagating through the structures in several periods [6,7,8,9,10,11], and they can also provide crack arrest and fracture resistance to the PMs, which have been studied in a recent study [12]. If ABGs exist in PnC or PM structures, they can be found in the phonon-band structures, which occur at a range of frequencies with no eigenvalue roots. Another special effect that can be found in the phonon-band structure is the slow sound (SS) mode [13,14,15,16,17]. In the phonon-band structure of PnCs, the group velocity of a specific mode can be calculated from the first-order derivative of the mode frequency and wave vector, which are the vertical axis and the horizontal axis of the band structure. Some modes are nearly flat in their band structure, meaning the group velocities of these modes are near zero. For example, Nan Wang et al. designed a Bloch-mode micromechanical resonator by fabricating defects in a 2-D silicon PnC slab in 2012 [15]. The radii of the air holes drilled in several periods in the center of the slab were changed to create defects in the PnC. The band structure showed a nearly flat band appearing in the ABG due to the defects. Slow sound modes that can provide a high quality factor (Q) via low energy loss have been studied. On the other hand, a local resonator structure can also produce a nearly flat band in the band structure. Yabin Jin et al. used several hollow pillars that formed a whispering-gallery mode [18]. The mode induced by the line defect in a square lattice PnC presents a narrow pass-band with a high quality factor. Even in recent studies, slow sound modes have been studied in several ways. In 2020, Si-Yuan Yu et al. studied slow surface acoustic waves via lattice optimization [16]. The slow acoustic mode was studied in different types of lattice arrangement, and a technological means of realizing the slow on-chip mode was demonstrated. This shows the potential of the SS mode for designing acoustic devices.

In these studies, the devices usually require high quality factors. Lots of high-Q devices have been studied in the last few decades [15,18,19,20,21,22,23]. For example, Chiu et al. built a fishbone phoxonic resonator by simultaneously inducing phononic BG and photonic BG to trap both acoustic waves and optical waves in the cavity [24]. The phoxonic structure produces high-Q acoustic modes and high-Q optic modes, and has a high acousto-optic coupling coefficient. In recent years, Kassa-Baghdouche et al. presented two sensor devices formed using the point defect in a planar photonic crystal. The high quality factors and small mode volumes provide great sensitivity for both liquid sensing [22] and gas sensing [23]. In these devices, phononic BG or photonic BG is required. With the existence of the forbidden frequency gap, the cavity modes can be formed by the defects, and the mode volumes are affected by the shape of the defects. However, a well-working ABG structure needs at three to five periods of additional structures as the wall of the corresponding waves (acoustic waves or optics waves).

In recent years, optomechanical devices and phoxonic crystals have become popular research topics [25,26,27]. In these studies, the 1-D nanobeam is a common structure for high-Q design [28,29,30]. Most phoxonic nanobeams or optomechanic nanobeams are designed to simultaneously guide the photons and phonons and trap them in a cavity. The nanobeam will include a cavity region sandwiched by two mirror regions, and the acoustic band gap and the optic band gap designed in the mirror region can trap the corresponding mode in the cavity region and induce acousto-optic coupling [24,31,32,33,34]. Such modes in nanobeam structures can be measured by another fiber near the nanobeam. In 2022, Daniel et al. studied this system and measured the nanobeam signal through an additional light coupling input using a fiber placed at a very close position [35]. In these studies, some common points can be found: 1. Lots of periods are needed to form the whole structure due to the mirror region on the two sides of the cavity, which has at least five periods on each side. 2. The cavity regions are usually designed with a gradient-varying geometric parameter to reduce the difference between the region of the mirror and the cavity, and the length of the cavity becomes very long. Both of the points cause an increase in the total length of the nanobeam. In acousto-optic coupling structures, the nanobeams are usually considered to be fabricated as suspended beams. Therefore, a long suspended length of the beam will increase the difficulty of fabrication.

From the point of view of numerical analysis, although the frequency bandgap is one of the main characteristics in these periodic structures, a bandgap with the desired frequency or an adequate frequency width only exists within specific structures with designed geometric parameters. Therefore, designing a phoxonic crystal that simultaneously has both a phononic bandgap and a photonic bandgap is very valuable, but hard. The SS mode and the band gaps are found in the band structure. The SS mode can be found easier than the band gap. The SS mode can be created in several ways. The wave velocities may not be so close to zero in all the SS modes found in band structures, and a relatively slow velocity can provide a large enough effect for wave manipulation. Therefore, choosing the SS mode as the operation mode can provide more possibilities for acoustic structure design.

To form a cavity without using the bandgap characteristic, we consider an acoustic potential well (APW) as the box of the cavity. In classical quantum physics, the electron energy levels can be counted and illustrated as an electron band structure for a certain material or structure. An electron potential well can be built by combining different materials or different geometric designs that have different electron potential energy to trap the electrons [36,37,38]. In acoustics, the energy of the lattice vibration in crystals, induced by a propagating elastic wave, can be quantized. The quantum of energy is the phonon, just like the photons in electromagnetic waves. The energy of the elastic waves can then be defined as proportional to the angular frequency ω of the phonons [39]. The frequencies of the SS modes in APW structures can be checked and calculated using the phononic band structure.

In this study, we aimed to design a phononic nanobeam cavity without a mirror region. We present an SS-mode structure based on a classical fishbone structure, which is commonly used for acoustic–optical coupling. We designed a gradient-varying structure to build an APW. With the APW, the SS mode can be trapped in the center of the fishbone nanobeam like a cavity without the need for the mirror region. Just like the traditional potential well devices, we can wisely combine several fishbone units that have different geometric parameters, the same SS mode, and different frequencies to form an APW structure. The cavity-like mode is formed by the APW in the fishbone nanobeam, and we confirmed that it can effectively increase the quality factor. The structure designed by the APW can save the space occupied by the mirror regions located beside the cavity and provide another choice for minifying the device size.

2. Structures

A fishbone nanobeam structure is formed by periodically arranging several half air cylinders on two sides of the nanobeam. Figure 1a illustrates the unit cell of the fishbone and the geometry parameters. As shown in Figure 1a, the thickness (H) of the suspended nanobeam is 220 nm; w = 440 nm (width). Each unit cell has a lattice constant a = 440 nm, the radii of the half air cylinders are defined as r = 0.32a, and an additional width, dw, is the varying parameter. Such values of the geometric parameters are chosen to keep the wavelength of the sound waves within the micrometer level to further use them in the acousto-optic coupling. Structures with these parameters have been studied in our past studies [24,40,41]. Figure 1b shows the band structure of a standard unit cell (i.e., dw = 0) with one mode marked in red circles. The band structure was calculated via the finite element method (FEM) using the solid mechanics physics model in COMSOL Multiphysics. In band structures, the vertical axes usually show the eigenfrequencies of the modes, and the horizontal axes are the wave vectors (k). Each point in the band structure is calculated by the eigenvalue solver at a specific k vector. According to the definitions of the axes in band structures, the first-order derivative of each line in the picture should then be proportional to the group velocity of the corresponding mode. It can be seen that the first-order derivative of the red line in Figure 1b is almost zero, meaning that it is an SS mode with near-zero velocity in this structure, and we use this SS mode as the operation mode in this study.

The displacement field and the strain field are displayed in Figure 1c,d, respectively. From the displacement field shown in Figure 1c, we can see that the vibrations are mainly located on the wings of the fishbone. The waving wings induce strong strain in the center of the fishbone, as shown in Figure 1d. Due to the EM wave modes in the fishbone structures usually existing in the center of the fishbone, the strain distributions in acoustic mode will bring some benefits for acousto-optic coupling in the future. The distributions of the field in Figure 1c,d can also be used to validate the SS mode in the following sections. The out-of-plane vibration of the wings and the concentrated strain in the center of the fishbone are the characteristics of the SS mode. According to the vibrations of the wings, we can introduce the manipulation of the SS mode frequency via the parameter dw.

Figure 1. Sketches of (a) the fishbone unit cell with the geometric parameters and (b) the band structure with the SS mode marked by red circles. (c) The out-of-plane displacement field and (d) the strain field of the SS mode in (b). (e) A sketch of three types of the APW fishbone structures and (f) their APW display as blue circles (type-I), orange triangles (type-II), and black crosses (type-III).

Figure 1. Sketches of (a) the fishbone unit cell with the geometric parameters and (b) the band structure with the SS mode marked by red circles. (c) The out-of-plane displacement field and (d) the strain field of the SS mode in (b). (e) A sketch of three types of the APW fishbone structures and (f) their APW display as blue circles (type-I), orange triangles (type-II), and black crosses (type-III).

For quantized acoustic waves, the phonon energy for an acoustic wave with an angular frequency ω can be defined as [39]:

$$E=\left(n+\frac{1}{2}\right)\hslash \omega$$

(1)

where the acoustic mode is excited to quantum number n, meaning there are n phonons occupying the mode, $\hslash$ is the reduced Planck constant, and the term $\frac{1}{2}\hslash \omega$ is the zero-point energy of the acoustic mode. The formula tells us that we can use the frequency of the acoustic mode to represent the energy due to the proportional relationship between E and ω. Therefore, we can build an APW structure via continuity of varying mode frequencies—i.e., continuity of mode energy. As shown in Figure 1e, we arranged the unit cells with different widths (w + dw) to build fishbone structures. The eigenfrequencies of the SS modes in each unit with different lengths of wings are listed in

Table 1. The eigenfrequencies of the SS mode in the fishbone unit cells with different values of dw.

a = 440 nm	dw
w = 440 nm	0 w	0.1 w	0.2 w	0.3 w	0.4 w	0.5 w
Eigenfrequency (GHz)	7.465	6.618	5.913	5.331	4.852	4.454

. Based on the eigenfrequencies in Table 1, we designed three types of APW structures. The width of the units in the center of each fishbone in Figure 1e was the widest, and they gradually became narrower from the center toward the sides of the fishbone.

We can classify three types of devices by the width of the center unit and the gradient of the width. We can first define the index for each unit in the nanobeam. By defining the index of the central unit as 0, the units on the right-hand side can be defined from 1 to the number of units, such as 3 in the type-I and type-III structures and 5 in the type-II structure; additionally, the units on the left-hand side are defined from −1 to the number of units as a negative: −3 in the type-I and type-III structures and −5 in the type-II structure. The type-I structure in Figure 1e has a $\mathrm{w}+\mathrm{d}\mathrm{w}=1.3 \mathrm{w}$ center width, meaning that $\mathrm{d}\mathrm{w}=0.3 \mathrm{w}$, and the width of the units becomes narrower by a gradient of $-0.1 \mathrm{w}/\mathrm{unit}$ from the center to the two sides of the fishbone, finally reaching w and making contact with the normal nanobeam. The actual widths are illustrated on each unit on the right-hand side in Figure 1e, and the widths on the left-hand side mirror those on the right-hand side. According to the frequencies in Table 1 and the unit indices, we can plot the APW of the type-I structure as a function of the unit index. In Figure 1f, the blue line with circle markers shows the potential well of the type-I device. The central unit with an index number of 0, which has the widest width and the lowest mode frequency, defines the depth of the APW in this structure. The width of the APW will be decided by the gradient of the width, which controls the number of unit cells on the two sides of the central unit, as shown in Figure 1f.

Based on the same gradient as the type-I structure, the type-II structure increases the dw of the central unit, making w + dw = 1.5 w, and therefore, more units are needed to reduce the width to w. We can obtain an APW structure with a deeper well depth than the type-I structure. The APW of the type-II structure is plotted by the triangular markers on the orange line in Figure 1f. However, due to the same gradient of the unit width, the additional units in the type-II structure also make the width of the well wider than that of the type-I structure. In contrast, to determine how the depth and the width of the APW affect the quality factor of the SS mode, we built the type-III structure.

The type-III structure has the same dw as the type-II structure, but a larger gradient of width by $-0.2 \mathrm{w}/\mathrm{unit}$. In this way, the type-III structure takes the same number of units as the type-I structure to reach w and connect with the normal nanobeam, which means the type-III structure has the same width as the APW of the type-I structure. Meanwhile, the same dw in the central unit gives the type-III structure the same APW depth as the type-II one. The APW of the type-III structure is plotted as a black line with x markers in Figure 1f. By comparing the type-III and type-I structures, we can find the effects of the well depth, and the effects of the well width can be found by comparing the type-III and type-II structures.

3. Results

For a better understanding of how the APW affects the SS mode in fishbone structures, we used standard fishbones as the control group, which have the same number of units as the corresponding type of the APW structure, and meanwhile, every unit has the same width w in the whole fishbone. Figure 2 illustrates sketches of the type-I structure with a 1.3 w central width and the standard fishbone, and their frequency spectra. The standard fishbone structure with eight periods (seven complete units and two half units on the two sides of the central units) is shown in Figure 2a. Each unit in Figure 2a is a standard unit cell with dw = 0.

The frequency spectrum of this fishbone is plotted in Figure 2b, and was calculated using the finite element method (FEM). The material of the fishbone nanobeams was Si with an anisotropic $6\times 6$ elastic matrix, for which, $C_{11}=C_{22}=C_{33}=165.7\times {10}^9, C_{12}=C_{13}=C_{23}=63.9\times {10}^9, \mathrm{and} C_{44}=C_{55}=C_{66}=79.56\times {10}^9$. The five complete units on the outside of the two sides of the nanobeam were treated as the connecting parts between the waveguides and the fishbone structure. The bottom boundaries of three of the furthest outside units were set as fixed, and simulated as the parts that contact the substrates. In contrast, the rest of the fishbone structure was treated as suspended and vibrated in the free space. The acoustic wave came from the left side and propagated to the right side, through a boundary load set on the top boundary of the fourth complete unit cell on the left side, as shown in Figure 2a, and the loss factor conditions were set in the complete units to avoid reflective waves.

Figure 2. Sketches of the type-I fishbone and the standard fishbone (the control group): (a) A sketch of the standard fishbone. (b) The frequency spectrum of the standard fishbone. The SS mode is marked by a red arrow, and the insets are the displacement field (left) and the strain field (right). (c) A sketch of the type-I APW fishbone. (d) The frequency spectrum of the type-I fishbone. The SS mode is marked by a red arrow, and the insets are the displacement field (left) and the strain field (right).

Figure 2. Sketches of the type-I fishbone and the standard fishbone (the control group): (a) A sketch of the standard fishbone. (b) The frequency spectrum of the standard fishbone. The SS mode is marked by a red arrow, and the insets are the displacement field (left) and the strain field (right). (c) A sketch of the type-I APW fishbone. (d) The frequency spectrum of the type-I fishbone. The SS mode is marked by a red arrow, and the insets are the displacement field (left) and the strain field (right).

We took the logarithm of the displacement field of the complete fishbone units as a function of the frequency of the acoustic wave, as illustrated in Figure 2b. The insets in Figure 2b are the out-of-plane displacement field (left) and the strain field (right) of the peak marked by the red arrow. The deforming shape in the displacement field shows vertical vibration perpendicular to the fishbone, and the wings have the maximum displacement, which is similar to those in Figure 1c. The strain field also shows a distribution similar to that of Figure 1d. According to the distributions in the two fields, we can confirm that it is the SS mode in the standard fishbone. The distribution of the fields shows that this SS mode at 7.471 GHz exists in a large area of the fishbone and in almost the whole structure.

Figure 2c,d show the type-I structure and its frequency spectrum. According to the inset in Figure 2d, the displacement field (left) and the strain field (right) can help us confirm the mode marked by the red arrow is the SS mode. It can be noticed that due to the APW, the frequency of the SS mode dropped to 5.744 GHz, which is very close to the bottom of the well (see dw = 0.3 in Table 1). From the distribution of the acoustic fields, we also can see that the SS mode is concentrated in the central units, which clearly shows the effect of the APW trapping the phonon effectively in the designed fishbone structure.

For resonators, the product of the frequency and the quality factor (fQ) is one of the most important parameters. Based on the characteristic of the SS mode, even the standard fishbone can provide a good fQ value in GHz.

Table 2. The Q value and the mode volume of the SS mode in the three types of fishbone structure.

	Type-I		Type-II		Type-III
	Standard	Potential Well	Standard	Potential Well	Potential Well
Frequency (GHz)	7.471	5.744	7.473	4.852	5.137
fQ value	$1.14\times {10}^{13}$	$2.73\times {10}^{14}$	$1.36\times {10}^{10}$	$6.12\times {10}^{12}$	$2.16\times {10}^{15}$
Mode volume	$4.02\times {10}^{-20}$	$3.49\times {10}^{-20}$	$6.05\times {10}^{-20}$	$4.56\times {10}^{-20}$	$4.03\times {10}^{-20}$

lists the fQ value of the fishbone structures. The fQ value of the SS mode in the standard fishbone shown in Figure 2b was $1.14\times {10}^{13}$. Benefitting from the APW, the SS mode in the type-I fishbone shows a cavity-like distribution and has a much larger fQ value of $2.73\times {10}^{14}$. The other important parameter is the mode volume, which can give a quantified view of the concentrated SS modes. As listed in Table 2, the SS mode in the standard fishbone has a mode volume of $4.02\times {10}^{-20}$, and the mode in the type-I fishbone can reduce the mode volume to $3.49\times {10}^{-20}$. This result tells us that the APW structure can not only concentrate the phonons forming a cavity, but also increase the quality factor of the corresponding modes exactly.

We then further investigated the effect of the depth of the APW. In Figure 3, the type-II structure with a central width of 1.5 w and the standard fishbone with 12 periods (11 complete units and 2 half units on the two sides of the central units) are illustrated. Using the same simulation settings as for the type-I structure in Figure 2, the SS mode of the standard fishbone structure was detected and is marked by a red arrow in Figure 3b at 7.473 GHz. The insets in Figure 3b are the out-of-plane displacement field (left) and the strain field (right) of the peak marked by the red arrow. The deforming shape in the displacement field shows vertical vibration perpendicular to the fishbone, and the wings have the maximum displacement; they are similar to those in Figure 1c. The strain field also shows a distribution similar to that in Figure 1d. According to the distributions in the two fields, we can confirm that it is the SS mode in this standard fishbone. The SS mode in a standard fishbone exists in a large area across almost the whole structure; the fQ value can be calculated as $1.36\times {10}^{10}$; and the mode volume is $6.05\times {10}^{-20}$. In contrast, the type-II APW structure has an SS mode marked by a red arrow at 4.852 GHz, which is close to the bottom of the well (see dw = 0.5 in Table 1). Similarly to the type-I structure, the inset in Figure 3d also shows distributions of the cavity-like SS mode. The displacement field and the strain field display that the acoustic mode only exists in three of the central units. The fQ value of the SS mode in the type-II APW is $6.12\times {10}^{12}$, and the mode volume is $4.56\times {10}^{-20}$.

We can see that the fQ values of the type-II structure and the standard structure are lower than those of type-I, and the mode volumes of the type-II fishbone are bigger than those of the type-I structure. This can be attributed to the longer length of the fishbone structure in the type-II structure, which will cause more energy loss in the propagation. Regarding the contrast between structures with and without the APW, although the type-II structures have lower fQ values than the type-I structures, the increase in the fQ value in the type-II structure is much larger than in the type-I structure. The improvement in the fQ values in the type-I and type-II structures demonstrates that a greater depth of the well will provide better improvement for the specific mode.

Here, we can see a simple summary: First, the frequency of the operation mode can be shifted down to near the frequency of the well bottom, meaning that the APW can also be used to change the operation frequency. Second, comparing the results between type-I and type-II, the length of the 1-D device in the propagation direction could cause a reduction in the quality factor due to the propagation loss. However, the well with a deeper depth could provide a larger improvement in the quality factor. Third, the results of the APW structures and the standard fishbone structure show the ability of the APW to form a cavity-like resonator and increase the quality factor effectively.

Therefore, we further designed the type-III APW structure to confirm the effects of the well depth and the device size. The width of the central unit cell of the type-III structure was 1.5 w, the same as the width of the type-II structure. The gradient of the unit width increased to $-0.2 \mathrm{w}/\mathrm{unit}$ from the center to the edge to give the type-III structure the same number of units as the type-I structure. In Figure 4a, the frequency spectrum of the type-III structure is illustrated. The SS mode is pointed out by a red arrow, as well, and the displacement field and the strain field are shown in Figure 4b,c, respectively. It can be noticed that although the bottom frequency of the APW in type-III is as deep as that in the type-II structure, due to the larger gradient of width in type-III, the frequency of the SS mode in Figure 4a is 5.137 GHz, which is a little higher than the SS mode in Figure 3d. From the fields in Figure 4b,c, we can see that The distributions in Figure 4b,c are “the further concentrated in a smaller area” ones. The fQ value of the SS mode in type-III reaches $2.16\times {10}^{15}$, which is even higher than the type-I structure.

Finally, we studied the fQ value and the mode volume as functions of the central unit width and the width gradient, which represent the well depth and the well width, respectively. Figure 5a shows the fQ value and the mode volume of the fishbone structures with a width gradient of $-0.1 \mathrm{w}/\mathrm{unit}$ varying with the central width. The magnitudes of the fQ values marked by the blue line and circle markers have a logarithmic form for better comparison, and the mode volumes are marked by the triangle markers on the orange line. The tendency of the fQ value on the blue line is inversely proportional to the central width. This tendency confirms the results in type-I and type-II structures. Due to the fixed width gradient, the larger central width means a longer total length. Therefore, the mode volume increases with the central width growth.

In Figure 5b, the fQ values and the mode volumes are displayed as functions of the width gradient. The central width was fixed to 1.1 w for a larger gradient investigation. The fQ value on the blue line shows a tendency to be proportional to the width gradient while the gradient is smaller than $-0.3 \mathrm{w}/\mathrm{unit}$. The fQ values of the structures with a gradient larger than $-0.4 \mathrm{w}/\mathrm{unit}$ become smaller due to the drastic variation in the width. The mode volume marked by an orange line shows a steep reduction in the width gradient. Based on the results in Figure 5, a larger gradient of width can effectively reduce the mode volume and increase the quality factor. A large width gradient will need a larger central width, which may reduce the quality factor, and too large a width gradient will further reduce the quality factor of the SS mode.

4. Conclusions

In this study, we designed three types of SS mode fishbone structures based on the APW. The effects of the APW were investigated and used to form cavity-like SS mode structures without using the mirror region used in traditional fishbone structures. The product of the mode frequency and the quality factor of the mode (fQ), and the mode volume, were used to demonstrate the performance of the structures. The SS mode operating in a standard fishbone has a rather high fQ value, showing the potential of the SS mode in resonators. By manipulating the length of the wings in the fishbone unit cells, the APW can further improve the quality factor and reduce the mode volume of the SS modes.

The frequency of the bottom of the well can affect the mode frequency, and the gradient of the width can cause an additional shift in frequency. The depth of the APW confirmed that a greater depth can induce a higher fQ value. However, the length of the fishbone may reduce the quality factor due to the propagation loss. Therefore, the gradient of the width of the fishbone units should be chosen wisely to obtain a high quality factor at a suitable operation frequency. The fQ value and the mode volume depend on the central width and width gradient investigated. We presented a different perspective for designing the acoustic fishbone. The APW fishbone structure can provide high-quality-factor SS modes with tunable frequency. Not requiring the mirror region is advantageous and gives the APW structure stronger potential for designing AO devices.