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Acoustic analog of monolayer graphene and edge states
Physics Letters A 375 (2011) 3533–3536
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Acoustic analog of monolayer graphene and edge states Wei Zhong a , Xiangdong Zhang a,b,∗ a b
Department of Physics, Beijing Normal University, Beijing 100875, China School of Physics, Beijing Institute of Technology, Beijing 100081, China
a r t i c l e
i n f o
Article history: Received 13 July 2011 Received in revised form 13 August 2011 Accepted 13 August 2011 Available online 22 August 2011 Communicated by V.M. Agranovich Keywords: Acoustic structure Monolayer graphene Edge states
a b s t r a c t Acoustic analog of monolayer graphene has been designed by using silicone rubber spheres of honeycomb lattices embedded in water. The dispersion of the structure has been studied theoretically using the rigorous multiple-scattering method. The energy spectra with the Dirac point have been verified and zigzag edge states have been found in ribbons of the structure, which are analogous to the electronic ones in graphene nanoribbons. The guided modes along the zigzag edge excited by a point source have been numerically demonstrated. The open cavity and “Z” type edge waveguide with 60◦ corners have also been realized by using such edge states. © 2011 Elsevier B.V. All rights reserved.
In recent years, there has been a great deal of interest in studying the physical properties of graphene due to the successful fabrication experimented by Novoselov et al. [1]. Graphene is a monolayer of carbon atoms densely packed in a honeycomb lattice, which can be viewed as either an individual atomic plane pulled out of bulk graphite or unrolled single-wall carbon nanotubes [2,3]. In graphene, the energy bands can be described at low energy by a two-dimensional Dirac equation centered on hexagonal corners (Dirac points) of the honeycomb lattice Brillouin zone. The quasiparticle excitations around the Dirac point obey linear Dirac-like energy dispersion. The presence of such Dirac-like quasiparticles is expected to lead to a number of unusual electronic properties in graphene [2,3]. Analogous to the above electron systems, in some two-dimensional (2D) photonic crystals (PCs) with triangular or honeycomb lattices, the band gap may become vanishingly small at corners of the Brillouin Zone, where two bands touch as a pair of cones [4]. Such a conical singularity is referred to as the Dirac point similar to the case of electron graphene [1–3]. Many interesting phenomena in optics relevant to photonic Dirac cone have been demonstrated [4–9]. In recent works, similar phenomena have been observed experimentally for acoustic waves in 2D sonic crystals [10]. Recently, another photonic analog of graphene, namely, honeycomb array of metallic nanospheres, has been proposed and analyzed theoretically [11]. Particle plasmon resonances in the nanoparticles act as if localized orbitals in carbon atom. The tightbinding picture is thus reasonably adapted to this system, and
*
Corresponding author at: Department of Physics, Beijing Normal University, Beijing 100875, China. Tel.: +86 10 58805153; fax: +86 10 58808026 12. E-mail address: [email protected] (X. Zhang). 0375-9601/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2011.08.027
nearly flat bands are found in the zigzag edge for both dipole and quadrupole modes. The problem is whether or not the similar acoustic analog of monolayer graphene consisting of elastic spherical particles arranged in honeycomb lattices can be constructed to observe the Dirac point in energy spectra and zigzag edge states in ribbons of the structure? Motivated by such a problem, in this work we explore the possibility to construct such a material. We consider a honeycomb monolayer consisting of silicone rubber spheres embedded in water as shown in Fig. 1(a). The black dots in the figure represent the silicone rubber spheres. The inter-distance between adjacent √ spheres in the monolayer is a0 = a/ 3, where a is the lattice constant. The Brillouin Zone (BZ) of the structure is shown in Fig. 1(b). In such a homogeneous elastic medium, the lattice displacement (r ) satisfies the elastic wave equation vector u
(λ + 2μ)∇(∇ · u ) − μ∇ × ∇ × u + ρω2 u = 0,
(1)
where ρ is the mass density and λ, μ are the Lamé coefficients of the medium. In spherical coordinates, the general solution of Eq. (1) can be expressed as [12–14]
JLlm (r ) = 1 ∇ jl (ql r )Y lm (ˆr ) , ql
JMlm (r ) = jl (qt r ) X lm (ˆr ), JNlm (r ) = 1 ∇ × jl (qt r ) X lm (ˆr ), qt
and
Llm (r ) = H
1 ql
∇ hl (ql r )Y lm (ˆr ) ,
(2)
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W. Zhong, X. Zhang / Physics Letters A 375 (2011) 3533–3536
Mlm (r ) = hl (qt r ) X lm (ˆr ), H Nlm (r ) = 1 ∇ × hl (qt r ) X lm (ˆr ), H (3) qt √ where ql = ω/cl , cl = ((λ + 2μ))/ρ is the speed of the longitudi√ nal wave, and qt = ω/ct , ct = μ/ρ is the speed of the transverse wave. Here jl (x) is the spherical Bessel function and hl (x) is the lm (ˆr ) is a vector spherspherical Hankel function of the first kind, X ical harmonic. The monolayer is assumed in the x– y plane and the relative location of the jth sphere in the unit cell is δ j (δ j is the two-dimensional vector). If we consider a plane longitudinal wave of angular frequency ω with the displacement vector (r , t ) = Re[u(r ) exp(−i ωt )] incident on the system, the total scatu tered wave can be given by using the Bloch theorem [12–14]:
sc (r ) = u
N ∞ l 1 j =1 l=0 m=−l
×
ql
b jlm ∇
n )hl (ql rnj )Y lm (ˆrnj ) , exp(ik · R
(4)
n R
n + δ j ), k is the Bloch vector and R n represents where rnj = r − ( R a two-dimensional (Bravais) lattice vector; b jlm are the scattered coefficients of the jth sphere in the unit cell, which are determined by the incident plane wave and the scattered wave from all the other spheres in the system. The wave scattered from all the other spheres can be expanded into a series of incident vector spherical waves around the j th sphere as
j sc (r ) = u
l ∞ 1 l=0 m=−l
ql
bj lm ∇ × jl (ql rnj )Y lm (ˆrnj ) .
(5)
Here the coefficients bj lm in Eq. (5) are to be determined by the following relation:
bjlm =
n
Ω jlm, j l m bj l m ,
(6)
j =1 l m
where Ω jlm, j l m is the free-space propagator, which the explicit expression can be obtained from the previous papers [12–14]. The key to calculate the propagator of the system is the problem of lattice sum, namely “structure constants”. Here Ewald’s treatment of lattice sums has been used [15]. If the external incident field is expanded in vector spherical waves and the expansion coefficients are characterized by a jlm , we have the Rayleigh identities
b jlm = T jlm
N
Ω jlm, j l m b j l m + a jlm ,
(7)
j =1 l m
where T jlm are the elements of the scattering matrix by the single isotropic sphere, which can be obtained analytically [13]. This is the basic equation for the present multiple-scattering system. The normal modes of the system may be obtained by solving the following secular equation in the absence of an external incident wave:
det δ j j δll δmm − Ω jlm, j l m T jlm, j l m = 0.
(8)
l m
Here T jlm, jl m = T jlm δll δmm for isotropic sphere. Based on such an equation, the dispersion of the acoustic analog of monolayer graphene can be obtained through the numerical calculations. In the calculations, the radius of the silicone rubber sphere is taken as r = 0.2a. The relevant parameters are chosen as follows: mass density ρ = 1.3 × 103 kg/m3 , longitudinal and transverse
Fig. 1. (a) Schematic structure of a 2D honeycomb lattice, a1 and a2 are the primitive vectors. (b) The first Brillouin Zone, b1 and b2 are the reciprocal vectors. (c) and (d) show the elastic band structures of the honeycomb lattice for silicone rubber spheres embedded in water with different frequency regions. The dash lines are the light line, c 0 is the longitudinal velocity of wave in water and the sphere radius is taken as r = 0.2a.
velocities cl = 22.87 m/s and ct = 5.54 m/s for the silicone rubber [16], and ρ = 1.0 × 103 kg/m3 and cl = 1.49 × 103 m/s for water. The calculated results for different frequency regions are plotted in Fig. 1(c) and (d). The dashed lines indicate the light line. We only consider the modes under the light line, namely k > ω/c 0 , because these modes extend inside the two-dimensional monolayer and decay outside the layer. The Dirac cones at the K point can be found in the energy spectra of Fig. 1(c) and (d), which corresponds to the frequency ωa/2π c 0 = 0.1123 and 0.1515015, respectively. The existence of Dirac dispersions in such a case is not only a result of the symmetry of the honeycomb structure, it also comes from the locally elastic resonance of silicone rubber spheres [16], which is similar to the plasmonic metal spheres for the photonic cases [11]. The difference between them is that only one longitudinal mode exists in the present case. This will lead to unique zigzag edge states in ribbons with finite size. An important consequence of the Dirac spectrum in graphene electronics is the existence of peculiar edge states for graphene ribbons with finite widths [8,9]. Two types of graphene nanoribbons, namely, zigzag and armchair ribbons, are usually considered. Now, we consider a ribbon of the acoustic structure as shown in Fig. 2(a), which the structure parameters are identical with those in Fig. 1. The upper and lower sides are the zigzag edges, and the left and right sides are the armchair edges. The arrows in Fig. 2(a) indicate the translational (periodic) directions of the ribbon. Fig. 2(b) and (c) show the band structures (blue lines) of the ribbon with 10 zigzag chains, which corresponds to the Dirac spectra in Fig. 1(c) and (d), respectively. The light gray regions are the projected bands for the 2D honeycomb lattice along the direction of the zigzag edge. In the gaps of the projected bands, some edge modes can be observed, which are analogous to the electron edge states in the zigzag graphene ribbons. The present edge states are twofold degenerate within the region of 2π /3 < ka < π , while one resonance mode appears for ka < π . Such a feature is different from those of plasmonic honeycomb lattices in Ref. [11], which the dipole and quadrupole modes are only considered. In contrast, the present results in Fig. 2(b) and (c) are rigorous, which the contribution of multipoles is included. The phenomenon still originates from the local resonance of the silicone rubber spheres in water for the longitudinal mode.
W. Zhong, X. Zhang / Physics Letters A 375 (2011) 3533–3536
Fig. 2. (Color online.) (a) A sketch of the ribbon with 10 zigzag chains. The upper and lower sides are the zigzag edges, the right and left sides are the armchair edges. The arrows indicate the translational (periodic) directions of the ribbon. (b) and (c) show the band structures (blue line) of the ribbon with 10 zigzag chains for two frequency regions. The light gray regions are the projected band diagrams of the honeycomb lattice along the zigzag edge direction. The dash lines are the light line and c 0 is the longitudinal wave velocity in water. The system parameters are the same as in Fig. 1.
In order to verify the edge states certainly happen, we perform a numerical simulation and see whether or not the wave propagates along the zigzag edge of the ribbon when a point source is put near the edge. The calculations are still performed by the multiple-scattering method. The results are plotted in Fig. 3. Here the honeycomb monolayer consists of 456 rubber spheres arranged in water. The lower side of the sample is only taken as the zigzag edge, the upper edge is not. And the right and left sides are taken as the armchair edge as shown in Fig. 3. The point source is arranged at (10.5a, −4.04a, 0) outside the spheres. Fig. 3(a) shows the distribution of displacement field within z = 0 plane of the ribbon for the S 1 state marked in Fig. 2(c) (ωa/2π c 0 = 0.1515015). It can be seen clearly that the field is strongly localized at the zigzag edge sites. The similar phenomena can also be found for S 2 state marked in Fig. 2(c), which the result is plotted in Fig. 3(b). Comparing them, we find that the localized properties for the S 2 state are weaker than those of the S 1 state. This is because the S 1 state is determined by the single scattering resonance of rubber spheres (the resonance frequency of single sphere is ωa/2π c 0 = 0.151500977). So, such a localized edge state is robust against disorder. These phenomena are in contrast to the case in the band region. For comparison, the corresponding distribution of displacement field at ωa/2π c 0 = 0.1515744 (in the band region) is shown in Fig. 3(c). The propagating properties of the field in whole monolayer are observed clearly. In the previous investigations, the people have pointed out that some electronic or optical devices can be designed by using edge states in the graphene ribbons [8,9]. The advantage of such devices is robust against the disorder. In fact, we can also use the present edge states in elastic structures to design acoustic devices. For example, Fig. 4(a) shows “Z” type edge waveguide with 60◦ corners. When it is excited by a point source at ωa/2π c0 = 0.1515015, a clear guided mode along the zigzag edge is observed. Fig. 4(b) shows a hexagonal cavity consisting of silicone rubber spheres in a honeycomb lattice bounded by zigzag edges. The fields corresponding to the edge mode are strongly localized on the edge spheres of the structure under the excitation of the point source. This means that we can control the transport or
3535
Fig. 3. (Color online.) (a) Distribution of the displacement field within z = 0 plane at one edge of the zigzag ribbon excited by a point source at ωa/2π c 0 = 0.1515015, corresponding to the S 1 state marked in Fig. 2(c). The point source is at (10.5a, −4.04a, 0.0) (marked with red fork in the figure). (b) The corresponding result at ωa/2π c 0 = 0.1515044, corresponding to the S 2 state marked in Fig. 2(c). (c) The corresponding result at ωa/2π c 0 = 0.1515744 in the band region. The other parameters are the same as those in Fig. 2.
Fig. 4. (Color online.) (a) Distribution of the displacement field within z = 0 plane along “Z” type edge waveguide with 60◦ corners excited by a point source at ωa/2π c 0 = 0.1515015. The position of the point source is marked with red fork in the figure. (b) The corresponding distribution of the displacement field for a hexagonal cavity excited by a point source. The other parameters are the same as those in Fig. 3.
transfer of the acoustic wave on the edges of a finite-size sample in free space. The above discussions are only for the theoretical results. If we want to observe these phenomena experimentally, the corresponding samples have to be fabricated. In order to fix the positions of the spheres, the best method is to attach them to the wall. We can find some solid materials such as Teflon, which the mass density
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W. Zhong, X. Zhang / Physics Letters A 375 (2011) 3533–3536
and velocities of acoustic wave in such a material are near to those in the water [17]. If we take such a material as substrate, the effect of the substrate on the energy spectra of the monolayer acoustic graphene is small. All phenomena disclosed above still exist when the substrate is introduced. In summary, we have designed a honeycomb lattice by using silicone rubber spheres embedded in water. We have calculated the dispersion of such a structure by using the rigorous multiplescattering method. The energy spectra with the Dirac point have been verified and zigzag edge states have been found in ribbons of the structure, which are analogous to the electronic ones in graphene nanoribbons. The guided modes along the zigzag edge excited by the point source have been numerically demonstrated. The open cavity and “Z” type edge waveguide with 60◦ corners have also been realized by using such edge states. Our findings provide a new way for controlling the transport of elastic waves similar to the graphene for electrons and thereby open up the possibility for developing new acoustic devices. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 10825416), the National Key Basic Research Special Foundation of China under Grant 2007CB613205.
References [1] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I.V. Grigorieva, A.A. Firsov, Science 306 (2004) 666. [2] C.W.J. Beenakker, Rev. Mod. Phys. 80 (2008) 1337. [3] A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, A.K. Geim, Rev. Mod. Phys. 81 (2009) 109. [4] F.D.M. Haldane, S. Raghu, Phys. Rev. Lett. 100 (2008) 013904; S. Raghu, F.D.M. Haldane, Phys. Rev. A 78 (2008) 033834. [5] R.A. Sepkhanov, Ya.B. Bazaliy, C.W.J. Beenakker, Phys. Rev. A 75 (2007) 063813; X.D. Zhang, Phys. Lett. A 372 (2008) 3512. [6] X.D. Zhang, Phys. Rev. Lett. 100 (2008) 113903. [7] T. Ochiai, M. Onoda, Phys. Rev. B 80 (2009) 155103. [8] S.R. Zandbergen, M.J.A. de Dood, Phys. Rev. Lett. 104 (2010) 043903. [9] S. Longhi, Phys. Rev. B 81 (2010) 075102. [10] X.D. Zhang, Z.Y. Liu, Phys. Rev. Lett. 101 (2008) 264303. [11] D. Han, Y. Lai, J. Zi, Z.-Q. Zhang, C.T. Chan, Phys. Rev. Lett. 102 (2009) 123904. [12] I. Psarobas, N. Stefanou, Phys. Rev. B 62 (2000) 278. [13] R. Sainidou, N. Stefanou, I. Psarobas, A. Modinos, Compt. Phys. Commun. 166 (2005) 197. [14] Z. Liu, C.T. Chan, P. Sheng, Phys. Rev. B 62 (2000) 2446. [15] J.B. Pendry, Low Energy Electron Diffraction, Academic Press, London, 1974. [16] Z. Liu, X. Zhang, Y. Mao, Y.Y. Zhu, Z. Yang, C.T. Chan, P. Sheng, Science 289 (2000) 1734. [17] E.Y. Wong, M.L. Thorne, H.N. Nikolov, T.L. Poepping, D.W. Holdsworth, Ultrasound in Medicine and Biology 34 (2008) 1846.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Acoustic analog of monolayer graphene and edge states Wei Zhong a , Xiangdong Zhang a,b,∗ a b
Department of Physics, Beijing Normal University, Beijing 100875, China School of Physics, Beijing Institute of Technology, Beijing 100081, China
a r t i c l e
i n f o
Article history: Received 13 July 2011 Received in revised form 13 August 2011 Accepted 13 August 2011 Available online 22 August 2011 Communicated by V.M. Agranovich Keywords: Acoustic structure Monolayer graphene Edge states
a b s t r a c t Acoustic analog of monolayer graphene has been designed by using silicone rubber spheres of honeycomb lattices embedded in water. The dispersion of the structure has been studied theoretically using the rigorous multiple-scattering method. The energy spectra with the Dirac point have been verified and zigzag edge states have been found in ribbons of the structure, which are analogous to the electronic ones in graphene nanoribbons. The guided modes along the zigzag edge excited by a point source have been numerically demonstrated. The open cavity and “Z” type edge waveguide with 60◦ corners have also been realized by using such edge states. © 2011 Elsevier B.V. All rights reserved.
In recent years, there has been a great deal of interest in studying the physical properties of graphene due to the successful fabrication experimented by Novoselov et al. [1]. Graphene is a monolayer of carbon atoms densely packed in a honeycomb lattice, which can be viewed as either an individual atomic plane pulled out of bulk graphite or unrolled single-wall carbon nanotubes [2,3]. In graphene, the energy bands can be described at low energy by a two-dimensional Dirac equation centered on hexagonal corners (Dirac points) of the honeycomb lattice Brillouin zone. The quasiparticle excitations around the Dirac point obey linear Dirac-like energy dispersion. The presence of such Dirac-like quasiparticles is expected to lead to a number of unusual electronic properties in graphene [2,3]. Analogous to the above electron systems, in some two-dimensional (2D) photonic crystals (PCs) with triangular or honeycomb lattices, the band gap may become vanishingly small at corners of the Brillouin Zone, where two bands touch as a pair of cones [4]. Such a conical singularity is referred to as the Dirac point similar to the case of electron graphene [1–3]. Many interesting phenomena in optics relevant to photonic Dirac cone have been demonstrated [4–9]. In recent works, similar phenomena have been observed experimentally for acoustic waves in 2D sonic crystals [10]. Recently, another photonic analog of graphene, namely, honeycomb array of metallic nanospheres, has been proposed and analyzed theoretically [11]. Particle plasmon resonances in the nanoparticles act as if localized orbitals in carbon atom. The tightbinding picture is thus reasonably adapted to this system, and
*
Corresponding author at: Department of Physics, Beijing Normal University, Beijing 100875, China. Tel.: +86 10 58805153; fax: +86 10 58808026 12. E-mail address: [email protected] (X. Zhang). 0375-9601/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2011.08.027
nearly flat bands are found in the zigzag edge for both dipole and quadrupole modes. The problem is whether or not the similar acoustic analog of monolayer graphene consisting of elastic spherical particles arranged in honeycomb lattices can be constructed to observe the Dirac point in energy spectra and zigzag edge states in ribbons of the structure? Motivated by such a problem, in this work we explore the possibility to construct such a material. We consider a honeycomb monolayer consisting of silicone rubber spheres embedded in water as shown in Fig. 1(a). The black dots in the figure represent the silicone rubber spheres. The inter-distance between adjacent √ spheres in the monolayer is a0 = a/ 3, where a is the lattice constant. The Brillouin Zone (BZ) of the structure is shown in Fig. 1(b). In such a homogeneous elastic medium, the lattice displacement (r ) satisfies the elastic wave equation vector u
(λ + 2μ)∇(∇ · u ) − μ∇ × ∇ × u + ρω2 u = 0,
(1)
where ρ is the mass density and λ, μ are the Lamé coefficients of the medium. In spherical coordinates, the general solution of Eq. (1) can be expressed as [12–14]
JLlm (r ) = 1 ∇ jl (ql r )Y lm (ˆr ) , ql
JMlm (r ) = jl (qt r ) X lm (ˆr ), JNlm (r ) = 1 ∇ × jl (qt r ) X lm (ˆr ), qt
and
Llm (r ) = H
1 ql
∇ hl (ql r )Y lm (ˆr ) ,
(2)
3534
W. Zhong, X. Zhang / Physics Letters A 375 (2011) 3533–3536
Mlm (r ) = hl (qt r ) X lm (ˆr ), H Nlm (r ) = 1 ∇ × hl (qt r ) X lm (ˆr ), H (3) qt √ where ql = ω/cl , cl = ((λ + 2μ))/ρ is the speed of the longitudi√ nal wave, and qt = ω/ct , ct = μ/ρ is the speed of the transverse wave. Here jl (x) is the spherical Bessel function and hl (x) is the lm (ˆr ) is a vector spherspherical Hankel function of the first kind, X ical harmonic. The monolayer is assumed in the x– y plane and the relative location of the jth sphere in the unit cell is δ j (δ j is the two-dimensional vector). If we consider a plane longitudinal wave of angular frequency ω with the displacement vector (r , t ) = Re[u(r ) exp(−i ωt )] incident on the system, the total scatu tered wave can be given by using the Bloch theorem [12–14]:
sc (r ) = u
N ∞ l 1 j =1 l=0 m=−l
×
ql
b jlm ∇
n )hl (ql rnj )Y lm (ˆrnj ) , exp(ik · R
(4)
n R
n + δ j ), k is the Bloch vector and R n represents where rnj = r − ( R a two-dimensional (Bravais) lattice vector; b jlm are the scattered coefficients of the jth sphere in the unit cell, which are determined by the incident plane wave and the scattered wave from all the other spheres in the system. The wave scattered from all the other spheres can be expanded into a series of incident vector spherical waves around the j th sphere as
j sc (r ) = u
l ∞ 1 l=0 m=−l
ql
bj lm ∇ × jl (ql rnj )Y lm (ˆrnj ) .
(5)
Here the coefficients bj lm in Eq. (5) are to be determined by the following relation:
bjlm =
n
Ω jlm, j l m bj l m ,
(6)
j =1 l m
where Ω jlm, j l m is the free-space propagator, which the explicit expression can be obtained from the previous papers [12–14]. The key to calculate the propagator of the system is the problem of lattice sum, namely “structure constants”. Here Ewald’s treatment of lattice sums has been used [15]. If the external incident field is expanded in vector spherical waves and the expansion coefficients are characterized by a jlm , we have the Rayleigh identities
b jlm = T jlm
N
Ω jlm, j l m b j l m + a jlm ,
(7)
j =1 l m
where T jlm are the elements of the scattering matrix by the single isotropic sphere, which can be obtained analytically [13]. This is the basic equation for the present multiple-scattering system. The normal modes of the system may be obtained by solving the following secular equation in the absence of an external incident wave:
det δ j j δll δmm − Ω jlm, j l m T jlm, j l m = 0.
(8)
l m
Here T jlm, jl m = T jlm δll δmm for isotropic sphere. Based on such an equation, the dispersion of the acoustic analog of monolayer graphene can be obtained through the numerical calculations. In the calculations, the radius of the silicone rubber sphere is taken as r = 0.2a. The relevant parameters are chosen as follows: mass density ρ = 1.3 × 103 kg/m3 , longitudinal and transverse
Fig. 1. (a) Schematic structure of a 2D honeycomb lattice, a1 and a2 are the primitive vectors. (b) The first Brillouin Zone, b1 and b2 are the reciprocal vectors. (c) and (d) show the elastic band structures of the honeycomb lattice for silicone rubber spheres embedded in water with different frequency regions. The dash lines are the light line, c 0 is the longitudinal velocity of wave in water and the sphere radius is taken as r = 0.2a.
velocities cl = 22.87 m/s and ct = 5.54 m/s for the silicone rubber [16], and ρ = 1.0 × 103 kg/m3 and cl = 1.49 × 103 m/s for water. The calculated results for different frequency regions are plotted in Fig. 1(c) and (d). The dashed lines indicate the light line. We only consider the modes under the light line, namely k > ω/c 0 , because these modes extend inside the two-dimensional monolayer and decay outside the layer. The Dirac cones at the K point can be found in the energy spectra of Fig. 1(c) and (d), which corresponds to the frequency ωa/2π c 0 = 0.1123 and 0.1515015, respectively. The existence of Dirac dispersions in such a case is not only a result of the symmetry of the honeycomb structure, it also comes from the locally elastic resonance of silicone rubber spheres [16], which is similar to the plasmonic metal spheres for the photonic cases [11]. The difference between them is that only one longitudinal mode exists in the present case. This will lead to unique zigzag edge states in ribbons with finite size. An important consequence of the Dirac spectrum in graphene electronics is the existence of peculiar edge states for graphene ribbons with finite widths [8,9]. Two types of graphene nanoribbons, namely, zigzag and armchair ribbons, are usually considered. Now, we consider a ribbon of the acoustic structure as shown in Fig. 2(a), which the structure parameters are identical with those in Fig. 1. The upper and lower sides are the zigzag edges, and the left and right sides are the armchair edges. The arrows in Fig. 2(a) indicate the translational (periodic) directions of the ribbon. Fig. 2(b) and (c) show the band structures (blue lines) of the ribbon with 10 zigzag chains, which corresponds to the Dirac spectra in Fig. 1(c) and (d), respectively. The light gray regions are the projected bands for the 2D honeycomb lattice along the direction of the zigzag edge. In the gaps of the projected bands, some edge modes can be observed, which are analogous to the electron edge states in the zigzag graphene ribbons. The present edge states are twofold degenerate within the region of 2π /3 < ka < π , while one resonance mode appears for ka < π . Such a feature is different from those of plasmonic honeycomb lattices in Ref. [11], which the dipole and quadrupole modes are only considered. In contrast, the present results in Fig. 2(b) and (c) are rigorous, which the contribution of multipoles is included. The phenomenon still originates from the local resonance of the silicone rubber spheres in water for the longitudinal mode.
W. Zhong, X. Zhang / Physics Letters A 375 (2011) 3533–3536
Fig. 2. (Color online.) (a) A sketch of the ribbon with 10 zigzag chains. The upper and lower sides are the zigzag edges, the right and left sides are the armchair edges. The arrows indicate the translational (periodic) directions of the ribbon. (b) and (c) show the band structures (blue line) of the ribbon with 10 zigzag chains for two frequency regions. The light gray regions are the projected band diagrams of the honeycomb lattice along the zigzag edge direction. The dash lines are the light line and c 0 is the longitudinal wave velocity in water. The system parameters are the same as in Fig. 1.
In order to verify the edge states certainly happen, we perform a numerical simulation and see whether or not the wave propagates along the zigzag edge of the ribbon when a point source is put near the edge. The calculations are still performed by the multiple-scattering method. The results are plotted in Fig. 3. Here the honeycomb monolayer consists of 456 rubber spheres arranged in water. The lower side of the sample is only taken as the zigzag edge, the upper edge is not. And the right and left sides are taken as the armchair edge as shown in Fig. 3. The point source is arranged at (10.5a, −4.04a, 0) outside the spheres. Fig. 3(a) shows the distribution of displacement field within z = 0 plane of the ribbon for the S 1 state marked in Fig. 2(c) (ωa/2π c 0 = 0.1515015). It can be seen clearly that the field is strongly localized at the zigzag edge sites. The similar phenomena can also be found for S 2 state marked in Fig. 2(c), which the result is plotted in Fig. 3(b). Comparing them, we find that the localized properties for the S 2 state are weaker than those of the S 1 state. This is because the S 1 state is determined by the single scattering resonance of rubber spheres (the resonance frequency of single sphere is ωa/2π c 0 = 0.151500977). So, such a localized edge state is robust against disorder. These phenomena are in contrast to the case in the band region. For comparison, the corresponding distribution of displacement field at ωa/2π c 0 = 0.1515744 (in the band region) is shown in Fig. 3(c). The propagating properties of the field in whole monolayer are observed clearly. In the previous investigations, the people have pointed out that some electronic or optical devices can be designed by using edge states in the graphene ribbons [8,9]. The advantage of such devices is robust against the disorder. In fact, we can also use the present edge states in elastic structures to design acoustic devices. For example, Fig. 4(a) shows “Z” type edge waveguide with 60◦ corners. When it is excited by a point source at ωa/2π c0 = 0.1515015, a clear guided mode along the zigzag edge is observed. Fig. 4(b) shows a hexagonal cavity consisting of silicone rubber spheres in a honeycomb lattice bounded by zigzag edges. The fields corresponding to the edge mode are strongly localized on the edge spheres of the structure under the excitation of the point source. This means that we can control the transport or
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Fig. 3. (Color online.) (a) Distribution of the displacement field within z = 0 plane at one edge of the zigzag ribbon excited by a point source at ωa/2π c 0 = 0.1515015, corresponding to the S 1 state marked in Fig. 2(c). The point source is at (10.5a, −4.04a, 0.0) (marked with red fork in the figure). (b) The corresponding result at ωa/2π c 0 = 0.1515044, corresponding to the S 2 state marked in Fig. 2(c). (c) The corresponding result at ωa/2π c 0 = 0.1515744 in the band region. The other parameters are the same as those in Fig. 2.
Fig. 4. (Color online.) (a) Distribution of the displacement field within z = 0 plane along “Z” type edge waveguide with 60◦ corners excited by a point source at ωa/2π c 0 = 0.1515015. The position of the point source is marked with red fork in the figure. (b) The corresponding distribution of the displacement field for a hexagonal cavity excited by a point source. The other parameters are the same as those in Fig. 3.
transfer of the acoustic wave on the edges of a finite-size sample in free space. The above discussions are only for the theoretical results. If we want to observe these phenomena experimentally, the corresponding samples have to be fabricated. In order to fix the positions of the spheres, the best method is to attach them to the wall. We can find some solid materials such as Teflon, which the mass density
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and velocities of acoustic wave in such a material are near to those in the water [17]. If we take such a material as substrate, the effect of the substrate on the energy spectra of the monolayer acoustic graphene is small. All phenomena disclosed above still exist when the substrate is introduced. In summary, we have designed a honeycomb lattice by using silicone rubber spheres embedded in water. We have calculated the dispersion of such a structure by using the rigorous multiplescattering method. The energy spectra with the Dirac point have been verified and zigzag edge states have been found in ribbons of the structure, which are analogous to the electronic ones in graphene nanoribbons. The guided modes along the zigzag edge excited by the point source have been numerically demonstrated. The open cavity and “Z” type edge waveguide with 60◦ corners have also been realized by using such edge states. Our findings provide a new way for controlling the transport of elastic waves similar to the graphene for electrons and thereby open up the possibility for developing new acoustic devices. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 10825416), the National Key Basic Research Special Foundation of China under Grant 2007CB613205.
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