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Feature: Phononic crystals

It is fascinating to think of the abstract beauty of crys-tals, with countless atoms occupying precise positionson a lattice and giving rise to perfect order and high lev-els of symmetry. Indeed, we need look no further thanthe brilliant appearance and extraordinary propertiesof many precious gems to witness the consequences ofthese precise atomic arrangements.

This picture of crystals is very appealing, but it is notstrictly correct. Even in an ideal crystal that is free fromdefects, the atoms are never static – they are alwaysmoving randomly around their equilibrium positions.Scientists have long assumed that it is impossible tocontrol this random thermal motion. However, a novelclass of artificially structured materials known as“phononic crystals” might make such control possible.

Phononic crystals make use of the fundamentalproperties of waves, such as scattering and interfer-ence, to create “band gaps” – ranges of wavelength orfrequency within which waves cannot propagatethrough the structure. This phenomenon is well knownin physics: the electrons in a semiconductor can onlyoccupy certain energy bands, while photonic crystalsonly allow light in certain frequency ranges to travelthrough them.

The band gap in a photonic crystal is caused by a peri-odic variation in the refractive index of an artificiallystructured material. In a phononic crystal the densityand/or elastic constants of the structure change peri-odically. This changes the speed of sound in the crys-tal, which, in turn, leads to the formation of a phononicband gap.

But why mention waves at all when talking about

random atomic motions? The reason is that the atomsin a solid cannot move independently of each otherbecause they are connected by chemical bonds. Whenan atom is displaced from its equilibrium position, itexerts a force on its neighbours, which causes them tomove. These atoms then cause their neighbours tomove, and the end result is the creation of a “phonon”– a special wave of lattice distortion that propagatesthrough the solid.

Where do band gaps come from?Acoustic waves differ from light waves in several ways.Acoustic waves are mechanical, which means that theycannot travel through a vacuum, whereas light wavesare electromagnetic and can travel through a vacuum.In general, mechanical waves passing through a gas ora liquid are known as acoustic waves, while those pass-ing through a solid are called elastic waves.

There are other important differences betweenmechanical and electromagnetic waves. Whereas alight wave can have two independent polarizations, anelastic wave in a homogeneous solid has three inde-pendent polarizations: two of these are transverse(shear waves) and one is longitudinal (a compressionwave). However, since shear waves are not supportedin liquids and gases, an acoustic wave has just onelongitudinal polarization.

The propagation of mechanical waves in a mediumis usually described by a dispersion relation that relatesthe frequency, !, and wave vector, k, of the propagatingwave. The dispersion relation for waves travelling in ahomogeneous medium is very simple: ! = c • k, wherec is the velocity of sound in the medium. However, thedispersion relations for materials that are not homo-geneous – such as phononic crystals (figure 1) – aremore complicated.

But why are certain waves not allowed to propagatein phononic crystals? To get an intuitive understandingof how band gaps form, consider a 1D crystal composedof alternating layers of two different materials. At everyinterface an incoming wave transfers part of its energyinto secondary, reflected waves, which then interferewith each other.

If this interference is constructive, all the energy ofthe original wave is reflected back and the wave cannotpropagate through the crystal. On the other hand, if theinterference is destructive, then all energy of the origi-nal wave is transmitted through the crystal. Therefore,constructive interference of the secondary waves results

Phononic crystals are novel materials that offer exceptional control over phonons, soundand other mechanical waves. Taras Gorishnyy, Martin Maldovan, Chaitanya Ullal andEdwin Thomas report on a revolution in acoustics

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Taras Gorishnyy,Martin Maldovan,Chaitanya Ullal andEdwin Thomas are inthe Department ofMaterials Scienceand Engineering,MassachusettsInstitute ofTechnology,Cambridge, US, e-mail elt@mit.edu

Sound ideas

! When a wave passes through a periodic structure, interference leads to theformation of “band gaps” that prevent waves with certain frequencies travellingthrough the structure

! Band gaps have been observed for electron waves in semiconductors,electromagnetic waves in photonic crystals and sound waves in phononic crystals

! The periodic variation in the density and/or speed of sound that is needed tomake a phononic crystal can be achieved by making “air holes” in an otherwisesolid structure

! The phenomenon of “negative refraction” – which can be exploited to makesuperlenses that can beat the diffraction limit – has been observed with phononic crystals

! Controlling the dispersion relation for phonons both inside and outside the bandgap could lead to breakthroughs in both fundamental research and applications

At a Glance: Phononic crystals

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Physics World December 2005

Diamond structures and phononic band gapsPeriodic structures can block sound waves incertain frequency ranges and directions wheninterference effects lead to the formation ofphononic band gaps. Phononic crystals withdiamond symmetry, like this artificial 3Dstructure, have the largest band gaps.

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Feature: Phononic crystals

in the creation of a band gap, while destructive inter-ference leads to the formation of propagation bands.

The condition for constructive interference is simplythat the path differences between the interfering wavesmust be equal to an integer multiple of their wave-length, ". Since the path difference is determined bythe lattice parameter of the crystal, a, it is easy to seethat constructive interference occurs when the latticeparameter is comparable to the wavelength. And sincefrequency is inversely proportional to wavelength, thefrequency at the centre of the band gap, !g, is alsoinversely proportional to the lattice parameter:!g ~ 1/" ~ 1/a. As a result, we can create a band gap atany frequency or wavelength we choose by simplychanging the size of the unit cell. The width of the bandgap is directly related to the ratio of the densities andsound velocities in the different layers: the larger theratio, the wider the gap.

Moreover, the position and width of the band gapdepend on the direction of the waves because the pathdifference depends on the angle of incidence. Somephononic crystals form band gaps for waves propagat-ing in any direction – these are known as absolute orcomplete band gaps. Other materials possess partialband gaps that only stop waves travelling in certaindirections. It is easy to see that a 1D crystal does nothave an absolute band gap because its mechanicalproperties only vary in one direction: waves travellingat right angles to this direction will not be reflected, sothere will not be a band gap in this direction.

Symmetry and phononic band gapsHow do we design a phononic crystal to have a com-plete band gap? It is clear from our 1D example thatthe density and sound velocity need to vary in all threedirections of space. However, very few 3D periodic

structures will form a complete phononic band gap. Infact, it is still quite difficult to determine the structureswith large absolute band gaps.

For electromagnetic waves, which only have twoindependent (transverse) polarizations, sinusoidalmodulations of the dielectric constant along certaindirections create photonic crystals with absolute gapsfor three different, highly symmetric lattices: simplecubic, body-centred cubic and face-centred cubic. Thediamond structure, which is face-centred cubic, pos-sesses the champion photonic band gap, i.e. the largestband gap for a given dielectric constant.

Mechanical waves can have both longitudinal andtransverse components in a solid, although only longi-tudinal waves are allowed in fluids. As a result, if wewant to create a complete phononic band gap, we mustdesign structures that have band gaps for both longitu-dinal and transverse waves in the same frequencyregion. This could be harder than designing structuresfor photonic crystals because electromagnetic wavesonly have transverse modes.

The search for structures with complete phononicband gaps began in 1992 with theoretical work byMichael Sigalas and Eleftherios Economou while theywere both at Iowa State University in the US. Theyshowed that structures that consist of a periodic 3Dlattice of identical high-density spheres placed withina low-density host material gave rise to phononic bandgaps. These structures can either be solid–solid orliquid–liquid.

Despite the fact that elastic waves propagate at twodifferent speeds within solids while acoustic wavestravel at a single speed in fluids, Sigalas and Economoupredicted that complete phononic band gaps shouldexist in both cases. A few months later they showed thatan infinite 2D array of high-density parallel cylindersembedded in a low-density host material should alsopossess a complete band gap in two dimensions. Un-aware of this work, Manvir Kushwaha of the Univer-sidad Autónoma de Puebla and co-workers elsewherein Mexico and in France reported the existence ofphononic band gaps for polarized elastic waves in 2Delastic systems in 1993.

The existence of structures with complete phononicband gaps has obvious applications. For instance, aphononic crystal will reflect incoming sound waves withfrequencies within the gap and can therefore be usedas an acoustic isolator. Moreover, the introduction ofdefects within the structure allows sound waves withfrequencies in the band gap to be trapped near a point-like defect (figure 2), or guided along linear defects.

Phononic crystals and soundSound manipulation is perhaps the most obvious appli-cation of phononic crystals. Sound is immensely valu-able in our daily lives for communication, informationtransfer or simply for its aesthetic value as exhibited inmusic and rhythms. For human hearing, sound is basi-cally made up of acoustic waves with frequenciesroughly between 20 Hz and 20 kHz, or wavelengthsranging from metres to several tens of centimetres.Therefore, if we assemble periodic structures with lat-tice constants in this range, we can expect them toinhibit sound and act as sonic mirrors.

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1 The origins of band gaps

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A 2D phononic crystal can be made by creating an array of air-filledcylinders in a solid material (insert) so that the speed of sound variesperiodically. The dispersion relations – plots of frequency, !, versuswave vector, k – for different phonons in this structure (blue lines)reveal that wave propagation is not supported for certain range offrequencies (yellow region): this is a phononic band gap. In ahomogeneous material, ! = c • k, where c is the velocity of sound, andthe dispersion relation would appear as a straight line on this graph.The directions with the highest symmetry in this structure are #–X and#–M (see insert).

Phononiccrystals willprovide newcomponentsthat offer thesame level ofcontrol oversound thatmirrors andlenses provideover light

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Feature: Phononic crystals

A great illustration of the sonic properties of a peri-odic structure was provided by Francisco Meseguer andco-workers at the Materials Science Institute of Madridin 1995 when they studied the acoustic characteristicsof a kinematic sculpture by Eusebio Sempere (figure3). This minimalist sculpture consists of a periodicsquare array of hollow steel cylinders.

In addition to being visually appealing, Meseguerand co-workers recognized that the sculpture shouldalso possess a sonic band gap, so they measured theacoustic transmission of the sculpture as a function offrequency and direction. They found that sound trav-elling perpendicular to the axes of the cylinders wasstrongly attenuated at a frequency of 1670 Hz – a resultthat provided the first experimental evidence for theexistence of phononic band gaps in periodic structures(Nature 378 241).

Unfortunately, a structure needs to be several metreswide to create a phononic band gap in the sonic regime.While this might not be a problem for architecturalacoustics, it is impractical for many other devices suchas headphones and speakers. However, if we move tothe ultrasonic regime, the relevant wavelengths aremuch shorter, so the phononic crystals are also muchsmaller (from centimetres down to fractions of mil-limetres). These developments – combined with otheradvances such as negative refraction and superlenses– could have a wide range of applications.

Ultrasound, negative refraction and superlensesOne of the hottest topics in optics over the past fiveyears has been the possibility of making “superlenses”with materials that have a negative refractive index.And just as photonic band gaps were followed byphononic band gaps, recent progress in making super-lenses for electromagnetic waves has prompted effortsto make superlenses for acoustic waves.

In general it is not possible for a lens to produce animage that contains details that are finer than the wave-length of the light being focused. However, superlensesmade of “negative index” materials can overcome thisdiffraction limit. Moreover, these superlenses do noteven have to be shaped like traditional lenses – a flat,thin slab of negative-index material can act as a super-lens, which means that they should, in theory, be mucheasier to fabricate than traditional lenses.

In optics the challenge has been to make materialsthat have a negative refractive index at the appropri-ate wavelength (see Physics World August pp23–24).Recent research suggests that it might be possible tomake acoustic superlenses with phononic crystals.

The speed of light or sound in a medium depends onthe refractive index of that medium. And when light orsound travels from one medium into another with a dif-ferent refractive index, both its speed and directionchange. This is refraction. Most traditional materialsexhibit positive refraction, but some specially designedmaterials can exhibit negative refraction (figure 4).

Negative refraction in phononic crystals is possibledue to multiple scattering of sound waves at the solid–air interfaces. To get an intuitive understanding of neg-ative refraction consider a sound wave moving througha homogeneous medium that strikes a phononic crystalat an angle. We can think of the sound wave as having

two components: one that travels parallel to the sur-face, and one that moves at right angles to it. Negativerefraction will occur if the direction of the parallel com-ponent is reversed while that of the normal wave doesnot change. This is actually possible if the parallel com-ponent is reflected by the phononic crystal while thenormal wave is allowed to propagate.

In 2004 Xiandong Zhang of Beijing Normal Uni-versity and Zhengyou Liu of Wuhan University, bothin China, predicted that negative refraction wouldoccur in 2D hexagonal “fluid–fluid” crystals consistingof water cylinders in a mercury matrix. They used adetailed analysis of a phononic band diagram to deriveconditions for all-angle negative refraction of acousticwaves, and simulated the propagation of a wavethrough a flat slab of negative-refraction material.

Meanwhile, Jian Zi and co-workers at Fudan Univer-sity, also in China, demonstrated experimentally asuperlens for liquid surface waves for the first time in2004 (figure 4).

The development of superlenses for acoustic waveswould be a major breakthrough for a variety of ultra-sound techniques.

Hypersound and thermal managementWavelengths in the hypersonic regime are even shorterthan those used for ultrasound. Indeed, hypersonicwavelengths are less than 10 µm, which corresponds tofrequencies higher than 100 MHz. However, thebehaviour of hypersonic phonons is crucial for manyphysical phenomena in materials. For example, theinteraction between electrons and high-frequencyphonons determines the efficiency of spontaneouslight emission in silicon and other semiconductor ma-terials that have an “indirect” electronic band gap.Greater control over the phonons in silicon couldtherefore lead to highly efficient silicon-based light-emitting devices. Making such devices is a major goalfor the optoelectronics industry.

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2 Trapping sound

By removing a single cylinder from a square lattice like that in figure 1 we can localize sound atthe resulting defect. The sound in these images is represented by a displacement field in thedirection normal to the page: the magnitude of the displacement is given by the colour map.The frequency of the sound increases as we go from left to right and top to bottom.

Sound barrierScanning electronmicroscope image ofa 3D periodic epoxy–air structure similar tothe diamond structureon page XX.

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Feature: Phononic crystals

Hypersonic phononic crystals could also have a largeimpact in thermal management. Thermal energy insolids is transported primarily by electrons andphonons. The electronic contribution is important formaterials with a large number of free carriers, such asmetals. On the other hand, the thermal conductivity ofdielectric materials and many semiconductors is deter-mined mainly by the phonons. The presence of aphononic band gap would reduce the flow of phononsand therefore the thermal conductivity of a solid.

This could prove very useful for thermoelectricdevices that directly convert thermal energy into elec-tricity. The figure of merit for a thermoelectric device,which is known as ZT, scales as ZT ~ %/(ke + kph),where % is the electrical conductivity, and ke and kph arethe electronic and phononic heat conductivities,respectively. If we reduce the electronic heat conduc-tivity, ke, we will also reduce the electrical conductiv-ity, %, which means that ZT will not increase. However,by using a phononic band gap to reduce the phononicheat conductivity, kph, it could be possible to greatlyimprove the performance of devices such as Peltierthermoelectric coolers, thermocouples and thermo-

electric energy generators.However, it is not easy to design and fabricate hyper-

sonic crystals. In contrast to sonic and ultrasonic crys-tals, which are macroscopic and can be readily madeusing standard manufacturing techniques, hypersoniccrystals require 3D periodic patterns to be created atthe submicron and nanometre scale. A variety of tech-niques are being explored for the fabrication of suchstructures. For example, some researchers are tryingto build hypersonic crystals from alternating layers oftwo materials using lithography-based techniques fromthe semiconductor industry. Others use two-photonlithography to pattern the inside of photosensitive poly-mers with lasers. Some groups are also investigatingself-assembly techniques.

Each of these methods has a variety of advantagesand drawbacks. Recently another approach – holo-graphic interference lithography – has received muchattention. Since light is inherently periodic, the over-lap of multiple beams of light can result in interestingperiodic patterns. For example, when two laser beamsare brought together, they form a 1D periodic inten-sity pattern, while the interference of three beams

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3 Art meets science

This kinematic sculpture by Eusebio Sempere consists of a periodic array of hollow stainless-steel cylinders, each 2.9 cm in diameter, arrangedon a square 10 & 10 cm lattice. In 1995 researchers at the Materials Science Institute of Madrid showed that the sculpture strongly attenuatedsound waves at certain frequencies, thus providing the first experimental evidence for the existence of phononic band gaps in periodic structures.

Developmentof superlensesfor acousticwaves wouldbe a majorbreakthroughfor a variety ofultrasoundtechniques

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Feature: Phononic crystals

results in 2D patterns and so on.If the laser beams overlap inside a photosensitive

material, it is possible to turn the light intensity patternof the light into a solid structure. At the MassachusettsInstitute of Technology (MIT) we have recently beenable to fabricate various 2D and 3D periodic structuresusing interference lithography and then, working withGeorge Fytas at the Max Planck Institute for PolymerResearch in Mainz, Germany, directly measure theirphononic dispersion relation using Brillouin light scat-tering. Previously, Maya Campbell, Andrew Turber-field and co-workers at Oxford University had usedinterference lithography to make photonic crystals.

“Blind” and “deaf” materialsSince the same basic ideas underpin both phononic andphotonic band-gap materials, it seems obvious toexplore the possibility of making materials that exhibitboth types of band gap. Indeed, at MIT two of the pre-sent authors (MM and ELT) have recently designed acrystal that is “blind” to electromagnetic waves withwavelengths of about several hundred nanometres and“deaf” to sound at similar wavelengths within the crys-tal. These crystals consist of a square or triangular 2Darray of air holes in silicon. In addition to having com-plete band gaps for both light and elastic waves, wehave found that they can also simultaneously trapsound and light at defects.

The ability to make structures with both types of bandgap could lead to breakthroughs in the field of acousto-optics. In 1997, for example, Harold de Wijn and co-workers at the University of Utrecht in the Netherlandssuggested that it might be possible to use these struc-tures to generate intense sources of coherent mono-chromatic phonons, which could be known as “phononlasers”. Other applications could include optical coolingin solids and optical frequency-conversion devices.

In 2002 Alex Fanstein and co-workers at the CentroAtómico Bariloche in Argentina and the LPN-CNRSlaboratory at Marcoussis in France measured photon–phonon scattering in 1D periodic structures that con-tained both photonic and phononic band gaps. Sinceall the layers in the phononic structure were only sev-eral nanometres thick, they did not get in the way of thelight. Fanstein and co-workers showed that such doublelocalization of photons and phonons increases the effi-ciency of photon–phonon scattering by five orders ofmagnitude compared with the values for similar 1Dstructures with photonic cavities only.

OutlookThe field of phononic crystals is only about 10 years oldand many important questions are just being raised. Thesearch for the best phononic structure is ongoing, buteven the very definition of the “best phononic structure”still needs to be clarified. What goes on outside the bandgap is often just as important in phononic applications.

Negative refraction, for instance, is made possible bythe unusual properties of the dispersion relation in thepropagation band. And to get light out of silicon withthe help of phonons it is necessary to increase ratherthan decrease the phononic density of states. We musttherefore think in terms of engineering the dispersionrelation for phonons, rather than simply making the

band gap as large as possible.Phononic crystals will provide researchers in acou-

stics and ultrasonics with new components that offerthe same level of control over sound that mirrors andlenses provide over light. But hypersonic phononiccrystals can also be used for fundamental science. Theycould, for example, be used to explore changes to thestatistical distribution of random phonons under equi-librium and non-equilibrium conditions, or to studyhow phononic band gaps affect the thermal propertiesof materials.

It is difficult to predict how the field of phononic crys-tals will develop in the future, but one thing is clear: wewill hear a lot more about these materials in every senseof the word.

More about: Phononic crystalsG Chen et al. 2004 Engineering nanoscale phonon and photontransport for direct energy conversion Superlattices andMicrostructures 35 161T Gorishnyy et al. 2005 Hypersonic phononic crystals Phys. Rev.Lett. 94 115501J Sánchez-Dehesa 2004 Phononic crystals bring sound to afocus Physics World September p23M M Sigalas and E N Economou 1993 Band structure of elastic waves in two-dimensional systems Solid StateCommun. 86 141M Trigo et al. 2002 Confinement of acoustical vibrations in asemiconductor planar phonon cavity Phys. Rev. Lett. 89227402X Zhang and Z Liu 2004 Negative refraction of acoustic waves intwo-dimensional phononic crystals App. Phys. Lett. 85 341

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4 Negative refraction of water waves

Jian Zi and co-workers at Fudan University in China have shown that surface waves experiencenegative refraction when they encounter a slab containing copper cylinders in a square10 & 10 mm lattice (right column: Phys. Rev. E 69 030201). The top row shows experimentaldata, while the middle row shows the results of simulations; the panels in the bottom row areschematic. At a frequency of 4.50 Hz a virtual image is formed inside the slab cylinders,corresponding to normal refraction (left column). At 6.15 Hz a real image is produced as aresult of negative refraction (middle column). At 7.20 Hz an image does not form, but negativerefraction results in a well-collimated wave propagating away from the slab.

point source

4.50 Hz 7.20 Hz6.15 Hz

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