大弹性常数差二维声子晶体带隙计算中的集中质量法

3­(,P,410073)(2004419;2004512),,.,.,,.,,.:,,PACC:4320,4335,0260,8160H3(:51307).­.E-mail:[e-mail]1110,,[1].,,.,[27]..,,..,,.(5)[8,9],,.(PWE)[5,6,1012](MST)[1316](FDTD)[17,18](VM)[9].PWE,,,,,,.MST,,,.FDTDP,,,,.(LM),[19]..[20],LM,.54320053100023290P2005P54(03)P1247206ACTAPHYSICASINICAVol.54,No.3,March,2005n2005Chin.Phys.Soc.©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.,.,.,LM,PWE,.,LM.,LM..211,z,BA,z,a.2(a),(b)4x-y,x-yz,xyxy=+20+2000xyxy,(1)xzyz=00xzyz.(2)xy={x,y,xy}T;xy={x,y,xy}T;z=1{xz,yz}T;z={xz,yz}T,xy=Dxyxy;z=Dzz,(3)(x,y,xy,xz,yz),(x,y,xy,xz,yz),.31LMLM,.2(a),NN,,,p842154©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.q,,p,q.2(b)4,b=aPN.z,zwr(r=i,j,k,l,i,j,k,l),,z:w=Niwi+Njwj+Nkwk+Nlwl,(4)Nr=14b2b4+xrbxb4+yrby,,z=Bz,(5)B=14b2xr4+xryrb2y,yr4+xryrb2xT,=9w9x,9w9yT,z=[wi,wj,wk,wl]T.(3),z=Dzz=DzBz.(6)[21],Fz=Kzz,(7)Fz={Fi,Fj,Fk,Fl}Ti,j,k,lzz,Kz=[krs],r,s=i,j,k,li,j,k,l,Kz=kunitBTDzBtdxdy=[krs],(8)tz,.krs=xrxsb21+43yrysb2+yrysb21+43xrxsb2.(9),4,4,rmr92wr9t2={Fz}l=Ns=0{kz}rsws,(10)mrr,wrrz.,,.Bloch[22],wr=Crei[qxxr+qyyr-t],(11)qx,qy,,Crmrz.(11)(10),(-{kz}rrPmr-2)Cr=sr{kz}rsCsei[qx(xs-xr)+qy(ys-yr)-t]Pmr.(12):Cs(xs,ys)=Cs(xs+na,ys+na),(13)(13)(12),NN[Sz(q)-2I]C=0.(14)x2y[Sxy(q)-2I]A=0,(15)q={qx,qy}.(14)(15),q,,,().,,(14)(15)4q[22].341I,3,a=20mm,r=8mm,A,B.:A=11600kgm-3,B=1180kgm-3,A=41231010Pa,B=4143109Pa,A=11491010Pa,B=1159109Pa.494213:©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.4I3,a=20mm,r=8mm,A,B.---LMx2yz,PWEx2yz,,LM400,PWE441.,.42915381546155115kHz.5I.,.LM,,,LM.51II[8,9]().,..LM,I,B[8,9],B=1300kgm-3,B=6105Pa,B=4104Pa.6,LM625,PWE1225.,,.5LMPWEIM1,M2,3M44Mz.LM,---PWE.PWE,LM.,(5),PWEGibbs,,LM.6II3,a=20mm,r=8mm,A,B47II.8.57:LMIII,PWEIII.II052154©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.7LMPWEIIM1,M2M3,46Mz.5,LMPWE,.LM625(25164s),.,1225(375183s),PWE.II(5),PWE,.,LM,,PWEGibbs,.,8,,LMPWE,LM,.61,LM[19],.8LMIIIMz,---PWE.,PWE,:11LM,PWEGibbs,,LM,.,.21LM,,,.31PWEMST,LM,.41LM,,.15213:©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.[1]JohnsonSGandJoannopoulosJD2002PhotonicCrystalsTheRoadfromTheorytoPractice(Dordrecht:KluwerAcademic)[2]SigalasMMandEconomouEN1992J.SoundVib.158377[3]SigalasMMandEconomouEN1993SolidStateCommun.86141[4]EconomouENandSigalasMM1993Phys.Rev.B4813434[5]KushwahaMSetal1993Phys.Rev.Lett.712022[6]KushwahaMSetal1994Phys.Rev.B492313[7]Martinez-SalaRetal1995Nature378241[8]LiuZYetal2000Science2891734[9]GoffauxCandSÀnchez-DehesaJ2003Phy.Rev.B67144301[10]WuFGandLiuYY2002ActaPhys.Sin.511434(inChinese)[2002511434][11]WuFGetal2002J.Phys.D:Appl.Phys.35162[12]VasseurJOetal1998J.Phys.:Condens.Mat.106051[13]KafesakiMetal2000Phys.Rev.Lett.846050[14]KafesakiMandEconomouEN1999Phys.Rev.B6011993[15]LiuZYetal2000Phys.Rev.B622446[16]ChenYYandYeZ2001Phys.Rev.E6436616[17]SigalasMandGarcVaN2000J.Appl.Phys.873122[18]WangGetal2003ActaPhys.Sin.521943(inChinese)[2003521943][19]WenJHetal2004ActaPhys.Sin.533384(inChinese)[2004533384][20]JensenJS2003J.SoundVib.2661053[21]HandLNandFinchJD1998AnalyticalMechanics(Cambridge:CambridgeUniversityPress)[22]AshcroftNW1987SolidStatePhysics(InternationalThomsonPublishing)Lumped-massapproachforthecalculationofbandstructureoftwo-dimensionalphononiccrystalswithhighcontrastofelasticconstant3WangGang­WenJi-HongLiuYao-ZongYuDian-LongWenXi-Sen(InstituteofMechatronicalEngineering,andPBGResearchCenter,NationalUniversityofDefenseTechnology,Changsha410073,China)(Received19April2004;revisedmanuscriptreceived12May2004)AbstractWitheachunitcellreplacedbyasystemoffinitefreedomsofmotion,two-dimensionalphononiccrystalscanbesimplifiedtoaninfinitediscreteperiodicsystem.Therefore,theelasticwavebandstructuresofthetwo-dimensionalphononiccrystalscanbecalculatedwithastraightforwardlumped-massapproach,whosecomputationalcostismuchlowerthanthewell-knownplanewaveexpansion(PWE)method.Thenumericalresultsofthetwomethodsareinreasonableagreements.Asthewell-knownGibbsoscillationsinthePWEcanbeeliminatedwiththelumped-massmethod,thisnewapproachisinsensitivetothesharpvariationofelasticconstantsontheinterfacesinsidethephononiccrystals.Furthermore,thelumped-massmethodcanalsobeusedtocalculatethebandstructuresoftwo-dimensionalphononiccrystalswitharbitraryunitshapeseasily.Keywords:phononiccrystal,phononicband-gap,lumped-massmethodPACC:4320,4335,0260,8160H3ProjectsupportedbytheStateKeyDevelopmentProgramforBasicResearchofChina(GrantNo.51307).­Correspondingauthor.E-mail:[e-mail]252154©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.