Acoustic analog of monolayer graphene and edge states
Wei Zhong a, Xiangdong Zhang a,b,∗a Department of Physics, Beijing Normal University, Beijing 100875, Chinab School of Physics, Beijing Institute of Technology, Beijing 100081, China
a r t i c l e i n f o a b s t r a c t
Article history:Received 13 July 2011Received in revised form 13 August 2011Accepted 13 August 2011Available online 22 August 2011Communicated by V.M. Agranovich
Keywords:Acoustic structureMonolayer grapheneEdge states
Acoustic analog of monolayer graphene has been designed by using silicone rubber spheres of honeycomblattices embedded in water. The dispersion of the structure has been studied theoretically using therigorous multiple-scattering method. The energy spectra with the Dirac point have been verified andzigzag edge states have been found in ribbons of the structure, which are analogous to the electronicones in graphene nanoribbons. The guided modes along the zigzag edge excited by a point source havebeen numerically demonstrated. The open cavity and “Z” type edge waveguide with 60◦ corners havealso been realized by using such edge states.
In recent years, there has been a great deal of interest instudying the physical properties of graphene due to the success-ful fabrication experimented by Novoselov et al. [1]. Graphene is amonolayer of carbon atoms densely packed in a honeycomb lattice,which can be viewed as either an individual atomic plane pulledout of bulk graphite or unrolled single-wall carbon nanotubes [2,3].In graphene, the energy bands can be described at low energy bya two-dimensional Dirac equation centered on hexagonal corners(Dirac points) of the honeycomb lattice Brillouin zone. The quasi-particle excitations around the Dirac point obey linear Dirac-likeenergy dispersion. The presence of such Dirac-like quasiparticles isexpected to lead to a number of unusual electronic properties ingraphene [2,3].
Analogous to the above electron systems, in some two-dimen-sional (2D) photonic crystals (PCs) with triangular or honeycomblattices, the band gap may become vanishingly small at corners ofthe Brillouin Zone, where two bands touch as a pair of cones [4].Such a conical singularity is referred to as the Dirac point similarto the case of electron graphene [1–3]. Many interesting phenom-ena in optics relevant to photonic Dirac cone have been demon-strated [4–9]. In recent works, similar phenomena have been ob-served experimentally for acoustic waves in 2D sonic crystals [10].
Recently, another photonic analog of graphene, namely, hon-eycomb array of metallic nanospheres, has been proposed andanalyzed theoretically [11]. Particle plasmon resonances in thenanoparticles act as if localized orbitals in carbon atom. The tight-binding picture is thus reasonably adapted to this system, and
* Corresponding author at: Department of Physics, Beijing Normal University, Bei-jing 100875, China. Tel.: +86 10 58805153; fax: +86 10 58808026 12.
nearly flat bands are found in the zigzag edge for both dipoleand quadrupole modes. The problem is whether or not the similaracoustic analog of monolayer graphene consisting of elastic spher-ical particles arranged in honeycomb lattices can be constructed toobserve the Dirac point in energy spectra and zigzag edge states inribbons of the structure?
Motivated by such a problem, in this work we explore the pos-sibility to construct such a material. We consider a honeycombmonolayer consisting of silicone rubber spheres embedded in wa-ter as shown in Fig. 1(a). The black dots in the figure representthe silicone rubber spheres. The inter-distance between adjacentspheres in the monolayer is a0 = a/
√3, where a is the lattice con-
stant. The Brillouin Zone (BZ) of the structure is shown in Fig. 1(b).In such a homogeneous elastic medium, the lattice displacementvector �u(�r) satisfies the elastic wave equation
(λ + 2μ)∇(∇ · �u) − μ∇ × ∇ × �u + ρω2�u = 0, (1)
where ρ is the mass density and λ, μ are the Lamé coefficientsof the medium. In spherical coordinates, the general solution ofEq. (1) can be expressed as [12–14]
3534 W. Zhong, X. Zhang / Physics Letters A 375 (2011) 3533–3536
�HMlm(�r) = hl(qtr) �Xlm(r̂),
�HNlm(�r) = 1
qt∇ × hl(qtr) �Xlm(r̂), (3)
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 thespherical Hankel function of the first kind, �Xlm(r̂) is a vector spher-ical harmonic. The monolayer is assumed in the x–y plane andthe relative location of the jth sphere in the unit cell is �δ j (�δ jis the two-dimensional vector). If we consider a plane longitu-dinal wave of angular frequency ω with the displacement vector�u(r, t) = Re[�u(r)exp(−iωt)] incident on the system, the total scat-tered wave can be given by using the Bloch theorem [12–14]:
�usc(�r) =N∑
j=1
∞∑l=0
l∑m=−l
(1
qlb jlm∇
×∑�Rn
exp(i�k · �Rn)hl(qlrnj)Ylm(r̂nj)
), (4)
where �rnj = �r − (�Rn + �δ j), �k is the Bloch vector and �Rn representsa two-dimensional (Bravais) lattice vector; b jlm are the scatteredcoefficients of the jth sphere in the unit cell, which are determinedby the incident plane wave and the scattered wave from all theother spheres in the system. The wave scattered from all the otherspheres can be expanded into a series of incident vector sphericalwaves around the j′th sphere as
�u′j′sc(�r) =
∞∑l=0
l∑m=−l
(1
qlb′
j′lm∇ × jl(qlrnj′)Ylm(r̂nj′)
). (5)
Here the coefficients b′j′lm in Eq. (5) are to be determined by the
following relation:
b′jlm =
n∑j′=1
∑l′m′
Ω jlm, j′l′m′b′j′l′m′ , (6)
where Ω jlm, j′l′m′ is the free-space propagator, which the explicitexpression can be obtained from the previous papers [12–14]. Thekey to calculate the propagator of the system is the problem oflattice sum, namely “structure constants”. Here Ewald’s treatmentof lattice sums has been used [15]. If the external incident field isexpanded in vector spherical waves and the expansion coefficientsare characterized by a jlm , we have the Rayleigh identities
b jlm = T jlm
(N∑
j′=1
∑l′m′
Ω jlm, j′l′m′b j′l′m′ + a jlm
), (7)
where T jlm are the elements of the scattering matrix by the sin-gle isotropic sphere, which can be obtained analytically [13]. Thisis the basic equation for the present multiple-scattering system.The normal modes of the system may be obtained by solving thefollowing secular equation in the absence of an external incidentwave:
det
∣∣∣∣δ j j′δll′δmm′ −∑l′′m′′
Ω jlm, j′′l′′m′′ T jlm, j′l′m′
∣∣∣∣ = 0. (8)
Here T jlm, jl′m′ = T jlmδll′δmm′ for isotropic sphere. Based on suchan equation, the dispersion of the acoustic analog of monolayergraphene can be obtained through the numerical calculations.
In the calculations, the radius of the silicone rubber sphere istaken 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 primitivevectors. (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 rubberspheres embedded in water with different frequency regions. The dash lines are thelight line, c0 is the longitudinal velocity of wave in water and the sphere radius istaken as r = 0.2a.
velocities cl = 22.87 m/s and ct = 5.54 m/s for the silicone rub-ber [16], and ρ = 1.0 × 103 kg/m3 and cl = 1.49 × 103 m/s forwater. The calculated results for different frequency regions areplotted in Fig. 1(c) and (d). The dashed lines indicate the lightline. We only consider the modes under the light line, namelyk > ω/c0, because these modes extend inside the two-dimensionalmonolayer and decay outside the layer. The Dirac cones at the Kpoint can be found in the energy spectra of Fig. 1(c) and (d), whichcorresponds to the frequency ωa/2πc0 = 0.1123 and 0.1515015,respectively. The existence of Dirac dispersions in such a caseis not only a result of the symmetry of the honeycomb struc-ture, it also comes from the locally elastic resonance of siliconerubber spheres [16], which is similar to the plasmonic metalspheres for the photonic cases [11]. The difference between themis that only one longitudinal mode exists in the present case.This will lead to unique zigzag edge states in ribbons with finitesize.
An important consequence of the Dirac spectrum in grapheneelectronics is the existence of peculiar edge states for grapheneribbons with finite widths [8,9]. Two types of graphene nanorib-bons, namely, zigzag and armchair ribbons, are usually considered.Now, we consider a ribbon of the acoustic structure as shown inFig. 2(a), which the structure parameters are identical with thosein Fig. 1. The upper and lower sides are the zigzag edges, andthe left and right sides are the armchair edges. The arrows inFig. 2(a) indicate the translational (periodic) directions of the rib-bon. Fig. 2(b) and (c) show the band structures (blue lines) of theribbon with 10 zigzag chains, which corresponds to the Dirac spec-tra in Fig. 1(c) and (d), respectively. The light gray regions are theprojected bands for the 2D honeycomb lattice along the directionof the zigzag edge. In the gaps of the projected bands, some edgemodes can be observed, which are analogous to the electron edgestates in the zigzag graphene ribbons. The present edge states aretwofold degenerate within the region of 2π/3 < ka < π , while oneresonance mode appears for ka < π . Such a feature is differentfrom those of plasmonic honeycomb lattices in Ref. [11], which thedipole and quadrupole modes are only considered. In contrast, thepresent results in Fig. 2(b) and (c) are rigorous, which the contri-bution of multipoles is included. The phenomenon still originatesfrom the local resonance of the silicone rubber spheres in waterfor the longitudinal mode.
W. Zhong, X. Zhang / Physics Letters A 375 (2011) 3533–3536 3535
Fig. 2. (Color online.) (a) A sketch of the ribbon with 10 zigzag chains. The upperand 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 twofrequency regions. The light gray regions are the projected band diagrams of thehoneycomb lattice along the zigzag edge direction. The dash lines are the light lineand c0 is the longitudinal wave velocity in water. The system parameters are thesame as in Fig. 1.
In order to verify the edge states certainly happen, we performa numerical simulation and see whether or not the wave prop-agates along the zigzag edge of the ribbon when a point sourceis put near the edge. The calculations are still performed by themultiple-scattering method. The results are plotted in Fig. 3. Herethe honeycomb monolayer consists of 456 rubber spheres arrangedin water. The lower side of the sample is only taken as the zigzagedge, the upper edge is not. And the right and left sides are takenas the armchair edge as shown in Fig. 3. The point source is ar-ranged at (10.5a,−4.04a,0) outside the spheres. Fig. 3(a) showsthe distribution of displacement field within z = 0 plane of the rib-bon for the S1 state marked in Fig. 2(c) (ωa/2πc0 = 0.1515015).It can be seen clearly that the field is strongly localized at thezigzag edge sites. The similar phenomena can also be found for S2state marked in Fig. 2(c), which the result is plotted in Fig. 3(b).Comparing them, we find that the localized properties for the S2state are weaker than those of the S1 state. This is because the S1state is determined by the single scattering resonance of rubberspheres (the resonance frequency of single sphere is ωa/2πc0 =0.151500977). So, such a localized edge state is robust against dis-order. These phenomena are in contrast to the case in the bandregion. For comparison, the corresponding distribution of displace-ment field at ωa/2πc0 = 0.1515744 (in the band region) is shownin Fig. 3(c). The propagating properties of the field in whole mono-layer are observed clearly.
In the previous investigations, the people have pointed outthat some electronic or optical devices can be designed by us-ing edge states in the graphene ribbons [8,9]. The advantage ofsuch devices is robust against the disorder. In fact, we can alsouse the present edge states in elastic structures to design acous-tic devices. For example, Fig. 4(a) shows “Z” type edge waveg-uide with 60◦ corners. When it is excited by a point source atωa/2πc0 = 0.1515015, a clear guided mode along the zigzag edgeis observed. Fig. 4(b) shows a hexagonal cavity consisting of sil-icone rubber spheres in a honeycomb lattice bounded by zigzagedges. The fields corresponding to the edge mode are strongly lo-calized on the edge spheres of the structure under the excitation ofthe point source. This means that we can control the transport or
Fig. 3. (Color online.) (a) Distribution of the displacement field within z = 0plane at one edge of the zigzag ribbon excited by a point source at ωa/2πc0 =0.1515015, corresponding to the S1 state marked in Fig. 2(c). The point source isat (10.5a,−4.04a,0.0) (marked with red fork in the figure). (b) The correspondingresult at ωa/2πc0 = 0.1515044, corresponding to the S2 state marked in Fig. 2(c).(c) The corresponding result at ωa/2πc0 = 0.1515744 in the band region. The otherparameters are the same as those in Fig. 2.
Fig. 4. (Color online.) (a) Distribution of the displacement field within z = 0 planealong “Z” type edge waveguide with 60◦ corners excited by a point source atωa/2πc0 = 0.1515015. The position of the point source is marked with red fork inthe figure. (b) The corresponding distribution of the displacement field for a hexag-onal cavity excited by a point source. The other parameters are the same as thosein Fig. 3.
transfer of the acoustic wave on the edges of a finite-size samplein free space.
The above discussions are only for the theoretical results. If wewant to observe these phenomena experimentally, the correspond-ing samples have to be fabricated. In order to fix the positions ofthe spheres, the best method is to attach them to the wall. We canfind some solid materials such as Teflon, which the mass density
3536 W. Zhong, X. Zhang / Physics Letters A 375 (2011) 3533–3536
and velocities of acoustic wave in such a material are near to thosein the water [17]. If we take such a material as substrate, the effectof the substrate on the energy spectra of the monolayer acousticgraphene is small. All phenomena disclosed above still exist whenthe substrate is introduced.
In summary, we have designed a honeycomb lattice by usingsilicone rubber spheres embedded in water. We have calculatedthe dispersion of such a structure by using the rigorous multiple-scattering method. The energy spectra with the Dirac point havebeen verified and zigzag edge states have been found in ribbonsof the structure, which are analogous to the electronic ones ingraphene nanoribbons. The guided modes along the zigzag edgeexcited by the point source have been numerically demonstrated.The open cavity and “Z” type edge waveguide with 60◦ cornershave also been realized by using such edge states. Our findingsprovide a new way for controlling the transport of elastic wavessimilar to the graphene for electrons and thereby open up the pos-sibility for developing new acoustic devices.
Acknowledgements
This work was supported by the National Natural Science Foun-dation of China (Grant No. 10825416), the National Key Basic Re-search 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.
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
