By taking advantage of the Crenigacestat possibility to modulate the elastic properties of PS layers, and considering that it is possible to create localized modes by introducing a defect layer with different acoustic properties into a periodic structure, in this paper, we investigate the propagation of longitudinal acoustic waves in multilayer structures based on PS, that exhibit resonant cavity modes in frequencies of gigahertz (GHz), consisting of defect layers intentionally introduced in periodic structures. The design and material parameters that allow to create these localized acoustic modes is discussed,
and experimental results of the measured acoustic transmission in PS samples fabricated by electrochemical Ralimetinib solubility dmso etching are presented. Methods Theoretical models The multilayer PS structures studied here have thicknesses in micrometer range and the procedure used to fabricate
them creates mesoporous silicon with an average pore diameter of 20 to 50 nm. On the other hand, in our experiments, the typical longitudinal wavelengths excited throughout the samples are 3 to 7 μm depending on porosity. Accordingly, each of the individual layers in the structures is assumed to be homogeneous. The longitudinal acoustic wave equation in the continuum limit for a solid inhomogeneous along the z direction (but homogeneous along the x and y directions) is given by [23], (1) where ρ j is the mass density, and u(z,t) is the atomic displacement. Here, j is an index identifying each layer. The limits ATM Kinase Inhibitor price of the elastic
continuum description of wave propagation in ordered media depends on the dimensions of the system compared with the wavelength. When the dimensions approach nanometer-length scales, atomistic treatments using first principles or semi-empirical methods may become necessary [24]. However, in our case, the thicknesses of the layers are in the micrometer range and each layer can be considered as a homogeneous layer; thus, the model described before is assumed valid. Tau-protein kinase In a solid, the acoustic waves can be longitudinal or transversal. In this letter, only longitudinal waves propagating through PS are considered because in our experiments, the waves are coupled to the samples through a liquid at normal incidence. The mass density ρ is a function of the porosity and is described by ρ=ρ 0(1−P) where ρ 0=2.330 g/cm 3 is the density of bulk silicon and P the porosity. The acoustic velocity dependence on porosity is given empirically by v L =v L0(1−P) k , being v L0 the longitudinal velocity of sound in bulk silicon along the (100) crystallographic direction and k≥0.5 is a constant [25–28]. In general, the parameter k depends on PS morphology which in turn depends on the doping level of the Si substrate [25, 26].