Localization of acoustic modes in periodic porous silicon structures
© Lazcano et al.; licensee Springer. 2014
Received: 12 May 2014
Accepted: 4 July 2014
Published: 21 August 2014
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© Lazcano et al.; licensee Springer. 2014
Received: 12 May 2014
Accepted: 4 July 2014
Published: 21 August 2014
The propagation of longitudinal acoustic waves in multilayer structures based on porous silicon and the experimental measurement of acoustic transmission for the structures in the gigahertz range are reported and studied theoretically. The considered structures exhibit band gaps in the transmission spectrum and these are localized modes inside the band gap, coming from defect layers introduced in periodic systems. The frequency at which the acoustic resonances appear can be tuned by changing the porosity and/or thickness of the defect layer.
The study of acoustic and elastic wave propagation in phononic crystals (PCs) [1–3] have been studied theoretically  and experimentally  in recent years. In analogy with the photonic band gap materials, emphasis in phononic crystals has been on achieving large acoustic band gaps within which propagation of sound is forbidden.
The interest about phononic band gap materials is in both, pure and applied physics. Fundamental physics has a special interest concerned with the localization phenomena of sound and vibrations in PCs. Researchers have prospected numerous applications based on cavity structures built around PCs, such as wave filters, waveguides, and splitters [6–9]. Furthermore, it is possible to design cavities for coherent (single-wavelength) phonon generation and control, to attain phonon amplification and ‘lasing’ in the called ‘saser’, one of the most important potential applications [10–12].
Periodic solid-state structures exhibit transmission stop bands for waves at certain frequencies. By placing one or more defects into a perfect phononic crystal, acoustic cavities are created inside the system. The presence of these defects, produces localization of elastic or acoustic modes inside the phononic band gap. These localized modes are the acoustic analog of donor or acceptor states produced inside the band gap of semiconductors. In analogy with electronic systems, one can consider these acoustic states to control the sound propagation through the structure.
If a defect is introduced into a periodic structure, the translational symmetry is broken and highly localized defect modes within the band gaps are created [6, 8, 13, 14]. Point, linear, and planar defect states have been theoretically investigated in one-dimensional (1D), two-dimensional (2D), and three-dimensional (3D) phononic crystals [3, 15, 16]. In 1D structures, a microcavity can be a spacer layer of thickness λ/2 enclosed by two Bragg reflectors . In 2002, Trigo et al. proposed phonon cavities in structures consisting of two semiconductor superlattices enclosing a spacer layer, showing that acoustical phonons can be confined in such layered structures if the spacer thickness is an integer multiple of the acoustic half-wavelength at the center of one of the superlattice-folded minigaps. These acoustic cavities are semiconductor multilayers in the nanometer scale and are fabricated by molecular beam epitaxy (MBE), which is a sophisticated and expensive technique that requires ultra-high vacuum system and a very tight control on the growth parameters, and modulate the thicknesses is easier than to modulate the elastic properties of the layers. Contrasting, porous silicon (PS) multilayer fabrication is relatively easy and considerably less costly, besides that this material allows to modulate both the thicknesses and the elastic properties of each layer.
PS is known as a versatile material with applications in light emission, sensing, and photonic devices . The possibility of producing acoustic band gaps in PS was proposed in 2005 , and detailed calculations of predicted bandwidths were subsequently published . Recently, experimental results of Brillouin light scattering suggested the existence of zone-folded phonons and phononic band gaps in PS multilayers . G. N. Aliev et al. in 2010 reported experimental measurements of transmission for longitudinal acoustic waves and observation of hypersonic band gaps in a periodic PS structure with an acoustic microcavity consisting of a spacer layer of thickness λ/2 .
By taking advantage of the 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 etching are presented.
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 μ m depending on porosity. Accordingly, each of the individual layers in the structures is assumed to be homogeneous.
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 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 . 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.
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=ρ0(1−P) where ρ0=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 =vL 0(1−P) k , being vL 0 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].
If we consider a structure formed by N layers, the total transfer matrix representing the structure is obtained by multiplying, in the appropriate order, a series of N transfer matrices, each one given by a matrix of the type appearing in Equation 6. The obtained matrix relates the displacement vector at the beginning of the structure with that at the end, and represents a 2 × 2 set of equations that can be fully solved.
With the above formalism, one can derive the acoustic eigenenergies and eigenvectors. The reflectivity and transmission can also be calculated as the square modulus of and , respectively, by imposing the boundary conditions and for a wave traveling from right to left. Here 0 and N label the first and last layer of the structure, respectively.
Attenuation can be included by taking the wave vector k j complex, such that K j =k j −α i , where α i is attenuation coefficient. The form of the attenuation coefficient depends on the physical process causing loss and we assume that the Akhiezer model is dominant in a semiconducting material. This gives α=η ω2/2ρ v3, where η is the viscosity. However, it is known that introducing acoustic attenuation into the model leads to important effects as the shrinking of gaps, only for frequencies higher than 180 GHz ; therefore, no absorptive behavior is considered in our model since no important effects are obtained if they are included.
Furthermore, the position and width of the band gap are critical parameters for devices that reflect or localize the acoustic waves . Band structures of many kinds of periodic phononic crystals have been reported [31–33]. The most commonly studied acoustic band gaps in 1D PCs are the Bragg type, appearing at an angular frequency ω of the order of vL(T)/d (vL(T) refers to the longitudinal (transverse) elastic wave velocity and d is the lattice constant).
In , the authors considered periodic semiconductor structures of GaAs/AlAs to introduce microcavities as spacer layers of thickness λ/2. However, for a 10-period GaAs/AlAs mirror, R B ∼0.880, while R B ∼0.996 if n=20. For a PS structure, a porosity variation of 15 % between the constituent layers of 52 % and 67 % of porosity, leads to R B ∼0.997 for n=6. Thus, by modulating the porosity of the PS structures, very high reflectivity values can be achieved. This is an essential condition to obtain narrow transmission bands into the stop bands corresponding to the cavity modes.
where u(z,f) is the displacement field distribution at each frequency, which is obtained by the transfer matrix method.
Samples were electrochemically etched from boron-doped (100)-oriented Si substrates with a resistivity of 0.007 to 0.013 Ω cm. Room-temperature anodization was performed using a 1:1 solution of HF (40 %) and ethanol (99.98 %). The acoustic transmission measurements reported here were done using a Vector Network Analyzer (VNA). Each sample was placed between two ZnO-based piezoelectric transducers with a central frequency of 1.1 GHz and an operation bandwidth of 500 MHz. The transducers consist of a piezoelectric layer driving waves into a silicon pillar with a thickness of 520 μ m. To couple the transducers to the specimen, In-Ga eutectic was used. The transducer front surface was aligned parallel to the sample surface using two orthogonal microscopes so that the acoustic waves impinge normally into the PS layers. The transducers were connected to the VNA ports and transmission parameters were measured as function of frequency, more details of the experimental set-up can be found in . Dependence of sound velocity with porosity for Si wafers and experimental conditions used here were determined previously and reported in , the values of vL 0 and the parameter k used here are 8.44 km/s and 0.56, respectively.
For the fabrication of PS multilayers, we consider the inclusion of ‘etch stops’ or ‘etch breaks’ where the current is interrupted to stop the etching of the Si wafer in order to prevent any depletion of HF . The introduction of these etching breaks is necessary to obtain layers with constant porosity with depth . Because our samples include very thick layers, with large mismatch porosities between them, the number and length of the etch breaks are important to obtain homogeneous structures. We found that etch breaks of 4 s with a ratio (etch break time)/(etching time) from 3.3 for low porosities (52 %) to 7.3 for high porosities (67 %) are enough to minimize any chirp in the layers.
Thicknesses of layers were measured by optical microscopy, and the layer porosities were determined from optical reflectance spectra by fitting our experimental measurements and comparing them with our theoretical simulations for each sample. The acoustic transmission and field intensity distribution have been modeled using the transfer matrix method described before and taking into account the effect of the sample (PS-Si substrate), transducers (Si pillars), and In-Ga eutectic liquid used to couple the transducers to the sample.
Now, based on the concepts mentioned before about cavities, we will show how the intentional introduction of a defect layer between a pair of mirrors can lead to formation of an acoustic cavity mode within the stop band. For this purpose, we consider two structures: sample 2 and sample 3.
The modeled transmittance of the periodic case (sample 1) and for the two cavity structures (samples 2 and 3), obtained by the TMM, shows a good match with the experimental results. The localized acoustic resonances can be tuned at different frequencies (within the acoustic band gap) by changing the porosity of the defect layer.
Moreover, for commercial acoustic mirrors which are components of solidly mounted resonators and filters , a low-acoustic-impedance material such as SiO 2 is layered with high-impedance materials such as tungsten or molybdenum. Following Equation 8, for the layer pair of molybdenum and silica, where acoustic impedances are 66.2 MRayl and 13.1 MRayl, respectively, the fixed impedance ratio is 5.1, and the same impedance ratio can be obtained using PS layers of 30 % and 75 %, so, by modulating the porosity, very high reflectivity values can be achieved. Besides, by increasing the number of pairs of layers enclosing the defect layer leads to narrowing of the transmission band together with deepening of the stop band.
We have demonstrated theoretically by using the TMM and experimentally by acoustic transmission measured directly, the formation of acoustic cavity modes in GHz frequencies by introduction of defects into periodic structures based on PS. Acoustic resonances can be tuned at different frequencies by changing the porosity of the defect. And we proved that these resonant modes appear due to the localization of the field into the defect. The acoustic mirrors and cavity structures based on PS have a performance which is at least comparable with that devices based on semiconductor superlattices. This study could be useful for the design of acoustic devices, such as highly selective frequency filters with applications in GHz range.
The authors acknowledge CONACyT for support under project No. 167939.
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