Effect Of Point Defect On Band Gap Of Two - Dimensional Phononic Crystal

Phononic crystal is a kind of artificial periodic composite materials. Acoustic orelastic wave propagation will also form a band structure in the structure of periodicelastic medium. The band gap which appears between the band called as phononicband gap. If the frequency of elastic waves appears in the frequency band gap, theelastic wave propagation will be suppressed, bound. Because of the band gapproperties of phononic crystals, photonic crystal has very wide application prospect.According to different needs, the different phonon crystals will be manufactured bychanging the topology of the phonon crystals and the filling rate.Defect states are easier to produce the band gaps than perfect states, when thephonon crystals are artificially manufactured defects. At the present stage, researchon defect states is mostly theoretical study on point, line defect. In addition, study onpoint defect is few. In this paper, the band gap of two-dimensional phononic crystalof different point defects structure is studied theoretically. The band gaps of differentstructures were calculated by the plane-wave expansion method such as the squarecolumn iron in epoxy resin, the square column copper in epoxy resin and thecylindrical iron in epoxy resin. The band gaps of the various structures werecompared. The main results are as follows:(1)Compare with the largest band gap width of defect-free, it was found that thelargest band gap width of defect states is larger than that of the defect-free crystal.Especially, the largest band gap width (△a/2ct=1.39) of crystal point defect with(11) direction next-nearest-neighbor coupling is about5times as large as that of thefree defect for the structure of2D square lattices of iron cylinders with square crosssection periodically inserted into epoxy resin; The largest band gap width (△a/2ct=1.929) of crystal point defect with (11) direction next-nearest-neighborcoupling is about7.65times as large as that of the free defect for the structure of2Dsquare lattices of copper cylinders with square cross section periodically inserted intoepoxy resin; The largest band gap width (△a/2ct=0.546) of crystal point defect with (10) direction next-nearest-neighbor coupling is about2.3times as large as thatof the free defect for the structure of2D square lattices of iron cylinders periodicallyinserted into epoxy resin.(2) The relative width of the minimum band gap of defect-free and defect statesare compared. When the filling fraction is F=0.4, the largest relative widths of theminimum band gap (△g=0.171429) is in the crystal point defect states with (10)direction coupling for the structure of2D square lattices of iron cylinders with squarecross section periodically inserted into epoxy resin; When the filling fraction is F=0.7,the largest relative widths of the minimum band gap (△g=1.644226) is in thecrystal point defect states with (10) direction coupling for the structure of2D squarelattices of copper cylinders with square cross section periodically inserted into epoxyresin; When the filling fraction is F=0.6, the largest relative widths of the minimumband gap (△g=1.85348) is in the crystal point defect states with (11) directionnext-nearest-neighbor coupling for the structure of2D square lattices of ironcylinders periodically inserted into epoxy resin.(3) Compare with the band gap number of defect-free, it was found that the bandgap number of defect states is more than that of the defect-free crystal. Especially,when the filling fraction is0.9, the band gap number of crystal point defect with (10)direction coupling reaches a maximum value of15for the structure of2D squarelattices of iron cylinders with square cross section periodically inserted into epoxyresin; when the filling fraction is0.3, the band gap number of crystal point defectwith (10) direction next-nearest-neighbor coupling reaches a maximum value of18for the structure of2D square lattices of copper cylinders with square cross sectionperiodically inserted into epoxy resin; when the filling fraction is0.5, the band gapnumber of crystal point defect with (11) direction next-nearest-neighbor couplingreaches a maximum value of16for the structure of2D square lattices of ironcylinders periodically inserted into epoxy resin.（4）The structure of square column Fe-epoxy and square column Cu-epoxy isthe same in two-dimensional phononic crystal. The density of the scatterer and thesubstrate in square column Cu-epoxy is larger than that of square column Fe-epoxy. The largest band gap width、the relative width of the minimum band gap and the bandgap number of square column Cu-epoxy are larger than those of square columnFe-epoxy in two-dimensional phononic crystal. This also proves that it is easy toproduce the band gaps when the density of the scatterer and the substrate is larger.（5）The materials of square column Fe-epoxy and cylindrical Fe-epoxy are thesame in two-dimensional phononic crystal. The largest band gap width of squarecolumn Fe-epoxy is larger than that of cylindrical Fe-epoxy in two-dimensionalphononic crystal. However, the relative width of the minimum band gap and the bandgap number of cylindrical Fe-epoxy are larger than those of square column Fe-epoxyin two-dimensional phononic crystal.These conclusions can provide a theoretical reference for practical applicationsof phononic crystals.