Ultrasonics 44 (2006) e1259–e1263 www.elsevier.com/locate/ultras
Velocity of a SAW propagating in a 2D phononic crystal B. Bonello *, C. Charles, F. Ganot Institut des NanoSciences de Paris, CNRS (UMR 7588), Universite´s Pierre et Marie Curie et Denis Diderot, 140, rue de Lourmel, 75015 Paris, France Available online 5 June 2006
Abstract We have studied the propagation of a surface acoustic waves (SAW), in a structure constituted by a 2D phononic ﬁlm (a few micrometers thick and having lattice constants of a few hundreds of micrometers in the two directions of the propagation plane) deposited onto a homogeneous semi-inﬁnite substrate. First, we have calculated the dispersion relations of the acoustic modes by using a plane waves expansion method. We found that the surface branch exhibits both the folding eﬀect and a band gap for the propagation along some particular directions. This is a very interesting result which demonstrates that the eﬀects related to the existence of the band gap (sound velocity dispersion, diﬀraction, refraction, ultrasound tunneling, etc.) can all appear, even if the thickness of the phononic ﬁlm is much less than the penetration depth of the SAW. Then, we used an all-optical technique to monitor the spectral content of the SAW propagating along the CX direction in the reduced Brillouin zone. We show that a wave with frequency in the stop band, is destructively diﬀracted after it propagates through less than ten periods. Finally, we report on measurements of the Rayleigh wave phase velocity and we show that the transit time is independent of the distance traveled inside the phononic crystal, suggesting that tunneling trough the sample is involved. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Phononic crystal; Surface acoustic wave; Laser ultrasonics
1. Introduction Phononic crystals are artiﬁcial structures made of inclusions of a material A, periodically embedded in a host matrix of material B. It is now well established, both theoretically and experimentally, that frequency band gaps over which the propagation of elastic waves is forbidden, exist in these systems, even if the mismatch of the acoustic impedance for the materials A and B is small. Most of the works devoted to the subject to date deal with the propagation of bulk waves in ﬂuid/solid or ﬂuid/ﬂuid systems, for which only longitudinal acoustic modes are relevant. Despite the great importance of surface acoustic waves in modern technology, one must recognize that very few was known about their interaction with a phononic crystal until the pioneering work of Tanaka and Tamura . These authors have *
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calculated the dispersion relations for a 2D phononic crystal constituted by a square array of AlAs cylinders, periodically embedded in a GaAs host matrix. They have found that the surface-phonon branch presents a stop band for waves with a wave vector perpendicular to the cylinders. In that case the gap does not extend throughout the ﬁrst Brillouin zone; it only exists in speciﬁc directions of propagation. However, the cylinders were considered as inﬁnitely long, which in some experimental circumstances is far from the actual situation. This is the reason why we have adopted another approach. Indeed, we have studied how a surface acoustic wave interacts with an heterostructure consisting in a 2D phononic ﬁlm (a few micrometers thick and having a lattice constant of a few hundreds of micrometers along both the directions of the propagation plane) deposited onto a homogeneous semi-inﬁnite substrate. In that case, the SAW propagates partly in the phononic ﬁlm and partly in the underlying substrate and there is to date no evidence that the formation of a band gap can arise in such a
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heterostructure. In the present article, we try to give a theoretical and an experimental answer to this question. Original phenomena, such as focusing or sound tunneling through the heterostructure, have already been observed in 3D phononic crystals [2,3]. Both these phenomena are related to the anisotropy or to the large sound velocity dispersion for frequencies outside, but close to the band gap. Getting a deeper understanding of these eﬀects requires the prior knowledge of how the periodicity aﬀects the wave transport. This can be done experimentally by measuring the components of the displacement vector at any place inside the crystal. Although feasible, this is very diﬃcult in a 3D phononic crystal with microscopic dimensions. Actually, the phase information can be easier recorded in a 2D system. To this end, we used a laser ultrasonic technique, based on the optical excitation and detection of surface acoustic pulses, to measure the transit time of Rayleigh waves propagating in a phononic ﬁlm patterned onto a semi-inﬁnite substrate. We brieﬂy describe the main features of this technique below, and we report on our measurements which demonstrate that the transit time of the acoustic pulse is independent of the distance traveled inside the phononic crystal. 2. Theory We used a plane waves expansion method to calculate the dispersion relations of bulk and surface acoustic waves propagating in a phononic ﬁlm deposited onto an homogeneous half-space. The phononic crystal is a square array of cylinders (diameter d, material A) periodically embedded in a thin ﬁlm (material B). The axis of the cylinders are perpendicular to the interface between the phononic crystal and the substrate (material C). Both the height of the cylinders and the thickness of the ﬁlm are denoted h; the repeat distance of the cylinders along the two directions of the free surface is denoted a. The wave vector lies in the plane perpendicular to the cylinder axis. We have restricted the calculation to the case h 6 a which the most closely describes the actual experimental situation. To calculate the dispersion relations one must solve the equation of motion for the displacement vector u(r, t) both in the phononic crystal layer and in the substrate, assuming stress-free boundary conditions at the surface and the continuities of both the displacements and the normal components of stresses across the interface. The equation of motion in the structured ﬁlm is: o2 ui ðr; tÞ o oum ðr; tÞ qðx; yÞ ¼ C ðx; yÞ ði ¼ 1; 2; 3Þ; ijmn ot2 oxj oxn ð1Þ where both the density q and the elastic constants Cijmn are position dependant. A similar equation holds in the substrate, but in this medium the density and the elastic constants do not depend on the coordinates. One can then take advantage of the spatial periodicity in the upper layer
Fig. 1. Dispersion relations for a phononic ﬁlm deposited onto a silicon substrate, in the CX direction. The phononic ﬁlm is constituted by cylindrical iron inclusion in a copper matrix (f = 0.45). The thickness of the ﬁlm is h = a/2. The results are given in terms of the normalized frequency, deﬁned as ma/cT, where cT is the transverse sound velocity in copper (cT = 2012 m s1). Inset: folding and stop band for the surface branch.
and solve Eq. (1) by expanding u, q, and Cijmn in the Fourier domain, with respect to a ﬁnite number of 2D reciprocal-lattice vectors. Note that the method is well adapted to the calculation of the dispersion relations for materials A and B showing a small impedance mismatch. Indeed, in that case, a satisfactory convergence is obtained with a relatively small number of reciprocal-lattice vectors (we used generally 25 vectors in the simulations shown below). We present in Fig. 1 the result of the numerical simulations for the propagation along the CX direction, in a heterostructure made of iron cylinders (material A) embedded in a copper background (material B) deposited onto a silicon semi-inﬁnite substrate. The thickness of the phononic layer is h = a/2 and the ﬁlling fraction, deﬁned as f = pd2/4a2, is 0.45 for which the gap takes a maximum value. The calculation accounts for the anisotropy in the three materials. The slowest branch in Fig. 1 corresponds to the surface wave mode (the others branches will be described elsewhere). It is very interesting to notice that this branch exhibits both the folding eﬀect and a stop band gap at the zone-edge X point (see inset in Fig. 1). This important result demonstrates for the ﬁrst time that the physical eﬀects associated with the periodicity are all likely to occur in these composite structures, even if the thickness h of the phononic ﬁlm is less than the repetition distance a. In the region far from the stop band, the sound velocity is non-dispersive and the phononic ﬁlm can be regarded as an eﬀective medium. We deduced from the slope at low frequency, the velocity of the surface mode in that region, to be about 2631 m s1. This is more than 5% faster than the velocity of a Rayleigh wave propagating at the surface of a phononic crystal in which qA,B A;B and C ijmn are constant along the cylinder axis (i.e., h is
B. Bonello et al. / Ultrasonics 44 (2006) e1259–e1263
inﬁnite), namely 2488 m s1. This might be related to the eﬀect of the silicon substrate. 3. Experiment We used a non-contact technique to experimentally validate the predictions of the preceding theory. Acoustic pulses were generated by focusing a light pulse issued from a frequency-doubled (532 nm) Q-switched Nd:YAG, at the surface of the sample, through a cylindrical lens. The line shaped spot was about 5 mm long and 100 lm across. Each optical pulse had an energy of about 50 lJ and a duration of 10 ns. These optical features, combined with the lateral dimensions of the spot, ensure that the laser ﬂuence was well below the ablation threshold and therefore, that the acoustic generation was in the thermoelastic regime and linear. The geometry is well adapted to the generation of a SAW pulse propagating at the Rayleigh velocity in the direction perpendicular to the line source. To conclude this brief outline, it should be noted that the frequency spectrum of the Rayleigh acoustic mode is determined by several factors including the duration of the light pulse, the size of the laser spot and the acoustic phase velocity. In our experiments, and depending on the sample under study, the maximum frequency in the spectrum was limited to 10–15 MHz. The acoustic waves were analyzed at a variable distance from the acoustic source, using a Michelson interferometer in which the light source was an He–Ne laser. One beam of the interferometer was focused on the sample (acting as one of the mirrors of the interferometer) through a achromat to a spot size of 15 lm, whereas the reference beam was reﬂected by an actively stabilized mirror. The interference pattern was collected by a high-speed photodiode and then digitized at 100 Ms s1 by a digital oscilloscope, triggered at 10 Hz by the light pulse issued from the Nd:YAG source. The temporal resolution of our system is only limited by the ratio of the spot size to the velocity of the surface wave. This latter quantity being always more than 2 km s1 in the samples that we have investigated, we estimate the temporal resolution to be better than 10 ns. Both the cylindrical lens and the sample were mounted on translation stages in such a way that the probe beam could be scanned across the sample with a precision of about 1 lm. Note that this noncontact technique allowed us to measure the displacement ﬁeld at any point at the surface of the sample and to resolve hence ﬁne details of the interaction of the acoustic waves with the phononic crystal. Note also that the interferometric method is only sensitive to the component of the displacements normal to the surface but not to the in-plane components. In a phononic crystal, the expected eﬀects related to the periodicity are all the more large than the mismatch in the elastic properties of materials A and B is large. To unambiguously demonstrate the formation of a stop band, we prepared a set of samples in which the material making the inclusions is air. Moreover, to avoid the Sezawa modes to be excited by the illumination, we have prepared samples
in which materials B and C are identical. They was constituted by a squared lattice of 200 cylindrical holes drilled in a semi-inﬁnite copper substrate (20 rows of 10 holes each). The repetition period was a = 1 mm and, depending on the sample, the depth of the holes was h 15–20 lm. The ﬁlling fraction was either f = 0.45 or f = 0.25. To discriminate the coherent eﬀects associated to the periodicity from the ones simply due to the incoherent diﬀusion of the waves by the inclusions, we prepared also a sample constituted by 200 cylindrical holes randomly drilled in a semi-inﬁnite copper substrate. The depth of the holes was h = 17 lm and the ﬁlling fraction f = 0.45 (for this sample, f is deﬁned as the ratio of the total cross-section of the holes to the whole surface covered by the phononic crystal). All the results described below were obtained by exciting the surface acoustic waves outside the phononic crystal and probing the acoustic pulse at diﬀerent places in between two consecutive inclusions (see inset in Fig. 3). 4. Results and discussion We show in Fig. 2a the normal component of the displacement recorded behind the sixth spatial period, corresponding to a distance of about 8 mm from the excitation spot. We show for comparison the signal after the waves have propagated over the same distance at the surface of a non-patterned copper sample (Fig. 2b). It is clear from the diﬀerence between the two signals (Fig. 2c) that the acoustic pulse undergoes a noticeable alteration while propagating within the phononic crystal, although the air inclusions merely aﬀect a very thin region underneath the free surface. Another interesting feature is the fact that the leading part of the pulse seems less altered by the interaction with the phononic crystal than the tail is, suggesting that the eﬀect strongly depends on the frequency. This can be conﬁrmed by computing the Fourier spectra of
Fig. 2. Normal component of the displacement recorded at 8 mm apart from the excitation, (a) after the surface acoustic waves have traveled through six elementary cells of the phononic crystal. The ﬁlling fraction is f = 0.45 and the wave vector is along the direction x (CX in the Brillouin zone); (b) propagation over 8 mm at the surface of a copper sample. Curve (c) corresponds to the diﬀerence (b)–(c), scaled by a factor 5 for clarity.
B. Bonello et al. / Ultrasonics 44 (2006) e1259–e1263
Fig. 3. Fourier transform of the normal component of the displacement, recorded in three diﬀerent places (same sample as in Fig. 2): away from the phononic crystal, transmission through two and nine elementary cells. The oscillation in the high frequency region results from numerical processing.
the normal component of the displacement at various distances from the excitation. We show the result in Fig. 3. The Fourier spectra displayed in this ﬁgure were recorded, respectively, a few micrometers ahead of the phononic crystal, and after transmission through two and nine elementary cells. The most striking feature in this result is the dip around 2.5 MHz which is more pronounced the further the waves advance in the crystal. It is also interesting to note that the attenuation is almost complete after 5–6 repetition periods, as already observed in other systems [4–7]. At last, one should notice that our experimental technique does not allow to measure accurately the magnitude of the gap since it is only sensitive to the normal component u3, when in fact the displacements could be parallel to the surface at the upper edge of the stop band . Since it is now proved, both theoretically and experimentally, that surface acoustic waves interact with a periodically patterned ﬁlm deposited onto a semi-inﬁnite substrate as they do with a ‘‘bulk’’ phononic crystal, one can legitimately expect that the physical eﬀects related to the stop band can all appear in these systems. Among phenomena of particular interest, abnormal wave transport has already been observed both in photonic [8,9] and in bulk phononic  crystals. Indeed, several authors have reported on transit time through the crystal independent from the sample thickness, demonstrating thus that tunneling is involved. Our experimental technique being very well suited to studying this phenomenon, we have measured the transit time of a surface acoustic pulse against the number of elementary cells in the path. In these experiments, the excitation is at a variable distance ahead of the phononic crystal and the distance between the excitation and the detection spots is strictly maintained at a constant value. Moreover, this geometry allows to make measurements outside the area occupied by the crystal, deﬁning so a reference value for the transit time. We show in Fig. 4 (open triangles) the changes in the transit time, relatively to the reference time, as a function of the number of elementary cells in the path.
Fig. 4. Changes in the transit time as a function of the number of elementary cells in the path (triangles). The repeat distance is a = 1 mm. We show for comparison the changes in the transit time for a sample in which the inclusions are randomly arranged (circles); in that case, the abscissa units are mm. For both curves, the distance between the excitation and the detection is d = 14.4 mm.
These data were derived from the arrival time of the ﬁrst minimum of the surface acoustic pulse (see Fig. 2). It is clear from the observed variations that the surface waves goes faster in the phononic crystal than they do at the surface of the copper substrate. The change in the velocity is manifest after the path contains only three elementary cells. Moreover, it is interesting to notice that the transit time is almost independent (within the experimental errors) of the path length in the crystal. The alteration in the velocity cannot be ascribed to the softening of the surface, resulting from the local lowering of the density. Indeed, it is well known that softening of a solid generally leads to the slowing down of the elastic waves and not to the increase of the velocity, as observed here. Moreover, we show also in Fig. 4 as open circles, the changes in the transit time recorded in a sample where the air inclusions are randomly arranged on the surface (for that curve, the abscissa units are mm). In that case, we have not observed any noticeable alteration of the transit time, conﬁrming that our experimental ﬁndings are correlated to the spatial periodicity of the sample. 5. Conclusions In this work, we have shown both theoretically and experimentally that a stop band and a folding eﬀect can appear for a surface acoustic pulse propagating in a phononic ﬁlm deposited onto a homogeneous substrate, even if the penetration depth of the waves is larger than the thickness of the phononic ﬁlm. We have also observed an abnormal sound velocity in this system, suggesting that tunneling is involved in the wave transport through the phononic ﬁlm. However, tunneling can only occur at frequencies inside the gap; it should be therefore very enlightening to measure the transit time of a surface pulse with a carrier frequency in the stop band. Another physical parameter of great interest is the phase of the acoustic sig-
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