Existence Of Absolute Band Gaps Research Articles

(Invited) Phonon Dispersion Engineering and Thermal Transport in Si Membranes

Present silicon technology provides single crystal films and membranes with thicknesses on the order of 10 nm and below. Along the years, the thermal studies on such structures have shown a reduction of the thermal conductivity consistent with the decrease of the characteristic size. Although the lowering of thermal conductivity is detrimental for heat dissipation processes in nanoelectronic devices, it becomes advantageous to increase the figure of merit (ZT) of Si and turning it as a promising material for thermoelectric applications. It is widely accepted that, for submicrometer thicknesses and down to 20 nm, the reduction of the in-plane thermal conductivity is mainly determined by the shortening of the phonon mean free path due to the diffusive scattering of phonons at the boundaries. In this case an analytical model that adopts the phonon bulk properties and includes the boundary scattering by a conductivity reduction factor is shown to match the experimental trend of systematical reduction of the thermal conductivity as the thickness of the film or membrane decreases. In this talk we will report on the modification of the phonon properties in Si membranes, such as the dispersion relation, and its dependence on the thickness. The thermal conductivity in Si results from the cumulative contribution of the transport of phonons with a broad range of wavevectors and mainly from long-wavevector phonons. Thus, heat transport at room temperature can be influenced by the reduction of the membrane thickness as the dispersion relation of the short-wave-phonons starts to be affected by emerging new vibrational modes arising from the boundary conditions at the membrane surface. Our measurements of the thermal conductivity in ultra-thin sub-20 nm Si membranes have demonstrated this extreme and, in fact, we have been able to disentangle the effects of phonon confinement from those related to surface morphology and chemical composition. Finally, we will discuss the prospects of employing the increased modification of phonon dispersion relation achieved in phononic crystals in controlling heat transport. While some applications require the existence of absolute band gaps, the approach to reduce the thermal conductivity is more linked to the decrease of phonon’s group velocity and phonon-phonon interactions in an extended range of frequencies.

Phononic crystals and manipulation of sound

AbstractThe acoustic properties of phononic crystals are presented, using numerical simulations. In a first part we study some possible applications of these composite materials. In particular, we show that defect modes (cavities, wave guides, stubs...) inserted inside the 2D periodic structure may lead to the possibility of manipulating the propagation of sound and realizing selective frequency filters and efficient devices for wavelength demultiplexing. We also report on the vibration modes of a 2D plate, i.e. a phononic crystal of finite thickness along the axis of the inclusions. We investigate the existence of absolute band gaps in the band structure of the plate, either free or deposited on a substrate. This should open new perspectives in the field of high‐frequency radiofrequency devices. Finally, we present the possibility of devicing sonic insulators for frequencies of the order of kHz with relatively small thicknesses of the phononic crystal sample as compared to the wavelength. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Design and characterization of stop-band filters using PZT layer on silicon substrate phononic crystals

Phononic crystals are periodic structures exhibiting absolute band gaps i.e. frequency bands in which the propagation of elastic waves is forbidden in all directions. Filtering is then a possible application of phononic crystals. Recently, the existence of absolute band gaps has also been theoretically demonstrated for guided elastic waves in a piezoelectric plate on a substrate [J. Vasseur et al, J. Appl. Phys, 101, 114904 (2007)], which is a geometry of interest for possible co-integration on silicon chip. The 2D phononic crystal was constituted by a square arrangement of cylindrical holes in a PZT layer deposited on a silicon substrate. In this communication, the realization of a stop-band filter constituted by a periodically patterned PZT layer, polarized along thickness, on silicon substrate and interdigitated electrodes (IDE) for emission/reception of guided elastic waves, is investigated. The filter characteristics are theoretically evaluated by using finite element simulations: dispersion curves of patterned PZT layer are computed for various pattern geometries to obtain the absolute band gap. Complete structure is then modeled, with appropriate IDE to propagate a guided mode in the piezoelectric layer. Finally, filtering capability of the structure is evaluated. Work supported by STMicroelectronics (Nano2008 program of French ministry of industry).

Absolute band gaps and waveguiding in free standing and supported phononic crystal slabs

Using the finite element method (FEM), we investigate the existence of absolute band gaps and localized modes associated with a guide in thin films of phononic crystals. Two different structures based on two-dimensional (2D) phononic crystals are considered, namely a free standing plate and a plate deposited on a silicon substrate. The 2D phononic crystal is constituted by a square array of cylindrical holes drilled in an active piezoelectric PZT5A matrix. We demonstrate the existence of absolute band gap in the band structure of the phononic crystal plate and, then, the possibility of guided modes inside a linear defect created by removing one row of air holes. In the case of the supported plate, we show the existence of an absolute forbidden band in the plate modes when the thickness of the substrate significantly exceeds the plate thickness.

Waveguiding in supported phononic crystal plates

We investigate, with the help of the finite element method, the existence of absolute band gaps in the band structure of a free-standing phononic crystal plate and of a phononic crystal slab deposited on a substrate. The two-dimensional phononic crystal is constituted by a square array of holes drilled in an active piezoelectric (PZT5A or AlN) matrix. For both matrix materials, an absolute band gap occurs in the band structure of the free-standing plate provided the thickness of the plate is on the order of magnitude of the lattice parameter. When the plate is deposited on a Si substrate, the absolute band gap still remains when the matrix of the phononic crystal is made of PZT5A. The AlN phononic crystal plate losses its gap when supported by the Si substrate. In the case of the PZT5A matrix, we also study the possibility of localized modes associated with a linear defect created by removing one row of air holes in the deposited phononic crystal plate.

Waveguiding in two-dimensional piezoelectric phononic crystal plates

We investigate the possibility of designing phononic crystal-based devices for telecommunication applications using materials commonly employed in microfabrication. We focus our attention on a phononic crystal made of a square array of cylindrical holes drilled in an active piezoelectric PZT5A matrix. Two different structures are considered, namely, a freestanding phononic crystal plate and a plate deposited on a silicon substrate. The geometrical characteristics of the phononic crystal plates (lattice parameter and thickness) were chosen to ensure the existence of an absolute band gap around 1.5GHz; a common frequency in radio frequency telecommunications. Computations of the dispersion curves of these active structures were conducted with the help of the finite element method. We demonstrate the existence of absolute band gaps in the band structure of the phononic crystal plates and, then, the possibility of guided modes inside a linear defect created by removing one row of air holes in the phononic crystal. In the case of the supported phononic crystal plates, we show the existence of an absolute forbidden band in the plate modes when the thickness of the substrate significantly exceeds the plate thickness. We discuss the conditions to realize waveguiding through a linear defect inside the supported plate. The present work provides evidences that phononic crystal properties can be integrated with existing silicon based microdevice technology.

Rayleigh-wave attenuation by a semi-infinite two-dimensional elastic-band-gap crystal

In this paper, we report experiments on the scattering of surface-elastic waves by a periodic array of cylindrical holes. The experiments were performed in a marble quarry by drilling cylindrical holes in two different configurations: honeycomb and triangular lattices. The attenuation spectra of the surface waves show the existence of absolute band gaps for elastic waves in these semi-infinite two-dimensional crystals. Results are compared with theoretical calculations based on a scalar-wave approach. The scaling property of the underlying theory has led us to explore the possible application of the results obtained to the attenuation of surface waves in seismic movements. @S0163-1829~99!07419-6#

Acoustic band gaps in fibre composite materials of boron nitride structure

We present elastic band-structure results for a new geometry of two-dimensional phononic crystals: the boron nitride- (BN-) like structure. This array is constituted of two kinds of infinite elastic parallel cylinder located at the vertices of a regular hexagon and surrounded by an elastic background. This geometry includes both the triangular and graphite structures as particular cases. The inclusions and matrix are either both fluids or both solids, the constituent materials being water and mercury, and carbon (or tungsten) and epoxy. We discuss the evolution of the band structure, and especially the existence of absolute band gaps, as a function of the ratio between the radii of the two cylinders in the BN geometry. We also discuss the existence of these gaps in relation to the physical parameters of the materials involved, and compare the results with those for square and triangular structures.

Photonic band structures of two-dimensional systems fabricated from rods of a cubic polar crystal

By the use of the plane-wave method, we calculate the photonic band structure for electromagnetic waves of E and H polarization propagating in a system consisting of an infinite array of identical, infinitely long, parallel cylinders of circular cross section, embedded in vacuum, whose intersections with a perpendicular plane form a square or triangular lattice. The cylinders are fabricated from GaAs, which represents a cubic, polar crystal material, containing two atoms in a primitive unit cell, for which the dielectric function has the form \ensuremath{\epsilon}(\ensuremath{\omega})=${\mathrm{\ensuremath{\epsilon}}}_{\mathrm{\ensuremath{\infty}}}$ (${\mathrm{\ensuremath{\omega}}}_{\mathrm{L}}^{2g}$-${\mathrm{\ensuremath{\omega}}}^{2}$ )/(${\mathrm{\ensuremath{\omega}}}_{\mathrm{T}}^{2g}$-${\mathrm{\ensuremath{\omega}}}^{2}$ ), where ${\mathrm{\ensuremath{\epsilon}}}_{\mathrm{\ensuremath{\infty}}}$ is the optical frequency dielectric constant, while ${\mathrm{\ensuremath{\omega}}}_{\mathrm{L}}$ and ${\mathrm{\ensuremath{\omega}}}_{\mathrm{T}}$ are the frequencies of the longitudinal optical and transverse optical vibration modes of infinite wavelength, respectively. For electromagnetic waves of both polarizations the problem of obtaining the photonic band structure is reduced to the solution of a generalized eigenvalue problem. In comparison with the dispersion curves of electromagnetic waves in vacuum, the photonic band structures are the most significantly affected by the particular form of the dielectric function we have assumed for frequencies in the vicinity of the polariton gap ${\mathrm{\ensuremath{\omega}}}_{\mathrm{T}}$ ${\mathrm{\ensuremath{\omega}}}_{\mathrm{L}}$ . For small filling fractions of the GaAs cylinders the photonic band structure for each polarization, except within the frequency range of the polariton gap, is a slightly perturbed version of the dispersion curves of electromagnetic waves in vacuum. For higher values of the filling fraction the dispersion curves deviate substantially from the dispersion curves of electromagnetic waves in vacuum within a broader frequency range, and in the case of E polarization, the photonic band structures reveal the existence of absolute band gaps. As the most striking feature, the calculated photonic band structures yield additional, nearly dispersionless bands that for waves of both polarizations occur at and below ${\mathrm{\ensuremath{\omega}}}_{\mathrm{T}}$ and in the case of H polarization occur also within the frequency range ${\mathrm{\ensuremath{\omega}}}_{\mathrm{T}}$ ${\mathrm{\ensuremath{\omega}}}_{\mathrm{L}}$ where \ensuremath{\epsilon}(\ensuremath{\omega}) is negative. The results for H polarization and small filling fractions of the GaAs cylinders are reproduced by an approach in which the zeros of a determinant are sought. A possible origin of the flat bands obtained in this polarization by both approaches is discussed.

Hexagonal photonic-band-gap structures.

We investigate a class of two-dimensional hexagonal structures that possess photonic band gaps. This class includes the previously studied triangular structure as a particular case. By symmetry analysis, we obtain a clear insight into the gap opening and we show how the photonic band gaps are affected by the characteristics of materials, according to whether the dielectric constant of the cylinders is larger or smaller than the background one. The photonic band structures present numerous gaps, and we study the evolution of these gaps as functions of the diameter and nature of cylinders. Our results demonstrate the existence of absolute band gaps for many configurations, largest gaps being obtained for triangular and graphite structures. \textcopyright{} 1996 The American Physical Society.