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Acoustic doorway states in layered porous silicon structures
Superlattices and Microstructures 121 (2018) 16–22
Contents lists available at ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Acoustic doorway states in layered porous silicon structures Z. Lazcanoa,∗, Y. García-Floresa, A. Díaz-de-Andaa, O. Mezaa, G. Monsivaisb, J. Arriagaa a b
T
Instituto de Física, Benemérita Universidad Autónoma de Puebla, Mexico Instituto de Física, Universidad Nacional Autónoma de México, Mexico
A R T IC LE I N F O
ABS TRA CT
Keywords: Doorway states Porous silicon Acoustics
We report the experimental observation of doorway states phenomenon for acoustic waves in porous silicon structures. We designed and fabricated a structure capable to exhibit the acoustic doorway states phenomenon around 1 GHZ. The structure consists of two porous silicon layers, one with low porosity and another with higher porosity and smaller thickness, followed by a much more thicker layer of crystalline silicon. We study the doorway state phenomenon in the frequency and in the time domain. To do this, we measured the acoustic transmission spectra in the frequency domain, and by using the Fourier transform, we obtain the transmission in the time domain. The transmission spectrum of the fabricated sample shows acoustic doorway states in frequencies around 1 GHz. The experimental results are in good agreement with the transfer matrix calculations.
1. Introduction The manifestation of the doorway state phenomenon, studied in nuclear physics through giant resonances [1,2], have been recently discussed in several systems, which include: atomic and molecular systems [3–5], clusters [6], quantum dots [7] and fullerenes [8,9]. Despite being originally studied in quantum mechanical systems, the doorway states phenomenon has also been observed in classical wave systems: microwave resonators [10], resonance state coupled to the background of many chaotic states [11], optical resonators [12], in the seismic response of sedimentary basins [13,14] and, more recently, in elastic systems [15–17]. All these cases consider two coupled structures with different spectral densities, i.e., one of these systems has a low density of states usually with a simple structure, whereas the other shows a high-density spectrum, formed by a “sea” of usually more complicated states. The low-density states system, act as doorway states whenever it is coupled to the sea of states. When the doorway state is excited, its amplitude spreads over the complicated states with a Lorentzian-like shape; and this is called the strength-function phenomenon [18]. In the present work, we demonstrate the theoretical and experimental evidence of acoustic doorway states (ADS). We propose simple multilayer structures based on porous silicon (PS) to observe the acoustic analogue of this phenomenon in the transmission spectra in Gigahertz range. Theoretical evidence of the temporal behaviour of ADS in such structures is also presented. A schematic representation of the structure considered in this work is shown in Fig. 1. The first silicon layer, A, has a low porosity pA and thickness dA , this layer contains a low density of states, i.e., the resonant states of this layer provide the ADS. The second layer, B, only used to couple the first layer to the third, has a porosity pB and a thickness dB , were pA < pB and dA > > dB . The sea of states is provided by the third layer C, in this case, the substrate. Layer C is crystalline silicon and its thickness dC is much larger than dA .
∗
Corresponding author. E-mail address: [email protected] (Z. Lazcano).
https://doi.org/10.1016/j.spmi.2018.07.018 Received 23 May 2018; Received in revised form 7 July 2018; Accepted 10 July 2018 Available online 17 July 2018 0749-6036/ © 2018 Elsevier Ltd. All rights reserved.
Superlattices and Microstructures 121 (2018) 16–22
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Fig. 1. Schematic representation of the structures considered in this work. Layers A, B and C have thicknesses dA , dB and dC , respectively, with dC > > dA .
2. Theoretical model In a layered solid, the acoustic waves can be either longitudinal or transversal. Here, we only consider longitudinal waves propagating along the perpendicular direction of the layers. This propagation is described by the wave equation [19],
∂2uj (z , t ) ∂z 2
−
1 ∂2uj (z , t ) = 0, ∂t 2 v j2
(1)
where uj (z , t ) is the atomic displacement and vj is the sound speed in layer j, given by vj = (λ + 2μ) j / ρj , being ρj the mass density, and λ and μ the Lamé parameters. The mass density ρj is a function of the porosity and is described by ρj = ρ0 (1 − Pj ), being ρ0 = 2.330 g / cm3 the mass density of bulk silicon and Pj the porosity in the j − th layer. The acoustic velocity dependence on porosity is given empirically by Ref. [20],
v = v0 (1 − P )k
(2)
being v0 the longitudinal velocity of sound in bulk silicon along the (100) crystallographic direction and k ≥ 0.5 is a constant. In general, the parameter k depends on PS morphology which in turn depends on the doping level of the Si substrate [20,21]. We have used v0 = 8.44 km/s and the parameter k = 0.56 from Ref. [22]. The solution of equation (1) can be written in the form of harmonic functions as,
u (zj , t ) = [A+j exp (ikj zj ) + A−j exp (−ikj zj )] exp (−iωt )
(3)
where ω = vj kj and ω = 2πf being f the frequency in s−1. After imposing the boundary conditions, at each interface, relating to the continuity of atomic displacement and stress, the propagation can be solved conveniently by using the transfer matrix method [23]. The transfer matrix of the complete system relates the displacement vector at the beginning of the structure with that at the end. With this formalism, the reflectivity and transmission of the structure can be easily calculated. ADS can be observed in the transmission spectrum (in frequency domain). Here, we present calculations and measurements of acoustic transmission as a function of frequency to demonstrate the existence of acoustic doorway states and the strength function in GHz range. To demonstrate the doorway state mechanism in the time domain, we calculated the time dependent acoustic transmission of both, theoretical and experimental results by using the inverse fast Fourier transform (IFFT) algorithm. 3. Numerical results As mentioned before, the doorway-state phenomenon occurs when two systems, one with a low density of states is coupled with another of large density. In our case, layer A and layer C play these roles respectively, and they are coupled through layer B. To have a deeply understanding of the doorway-state phenomenon, we consider slightly different structures described by slightly different parameters of the system appearing in Fig. 1. The direction of propagation, z-axis, is perpendicular to the layers and it is indicated by an arrow in Fig. 1. The number of states in each layer is determined by the thickness, the porosity and the velocity of propagation of the mechanical wave. As it is expected, the layer B has a very low density of states. We determine the appropriate thicknesses and porosities so that the states of layer B (which is used only to couple layers A and C) do not interfere with those of layers A and C, or do not appear in the range of frequencies considered here, i.e. from 0.7 to 1.5 GHz. In Fig. 2 we show the typical behaviour of the acoustic transmission for the system depicted in Fig. 1. The solid line corresponds to the acoustic transmission of the complete system, while in dashed line we have superimposed the transmission spectrum of layer A only. The maxima of the transmission appear at 0.75, 1.12 and 1.49 GHz, and they can be fitted to Lorentzian-like functions. From Fig. 2 we can clearly see that the transmission of the complete system is modulated around the frequencies provided by layer A, which provides the doorway states. Meanwhile, the fine oscillations are due to the dense part of the system, i.e., correspond to the high-density spectrum of layer C. This response is equivalent to the strength function analyzed in the quantum-mechanical case [24]. If the thickness or the porosity of the layer with ADS (layer A) is changed, the width of the Lorentzian changes, in addition, the maxima of such Lorentzians move in frequency. In other words, the influence of the doorway-states on the sea of states extends depending on the width or the porosity of layer A, besides that, the central frequency of the modes moves in frequency. We show the analysis of this behaviour in Figs. 3 and 4. For this purpose, we calculate acoustic transmission spectra and adjust a Lorentzian function to the peak around 1.12 GHz for a set of systems with pB = 80%, dB = 0.7μm , dC = (525 − dA) μm varying thickness (dA ) and 17
Superlattices and Microstructures 121 (2018) 16–22
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Fig. 2. Acoustic transmission for the system of Fig. 1 (solid line): with pA = 20%, dA = 10μm , pB = 80%, dB = 0.7μm and dC = 514.3μm . The dashed line corresponds to the acoustic transmission for the layer A with the same parameters.
Fig. 3. Central frequency fD (left axis, dots) and Lorentzian width ΓD (right axis, squares) as a function of the thickness of layer A (dA ). The thickness changes from 8 to 11 μm, keeping the other parameters constant: pA = 20% , pB = 80%, dB = 0.7μ m and dC = 525 − dA μ m. Solid lines correspond to a numerical fitting of the form ΓD = 0.79/ dA and fD = 11.18/ dA , respectively.
Fig. 4. Central frequency fD (to the left axis, dots) and Lorentzian width Γ (right axis, squares) as a function of the porosity of layer A ( pA ). The porosity changes from 2 to 20%, keeping constant the other parameters: dA = 10μ m, pB = 80%, dB = 0.7μ m and dC = 514.3μ m. Solid lines correspond to a numerical fitting of the form fD = −7.4x10−3dA + 1.27 and ΓD = 7.6x10−4dA + 0.064 , respectively.
18
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porosity ( pA ) of layer A. First, we consider different values of dA from 8 to 11 μm keeping pA = 20%, and the results appear in Fig. 3. In Fig. 3 we observe that if we increase the thickness of the layer with the doorway states then the central frequency fD of the fitted Lorentzian shifts to lower frequencies, whereas the width ΓD of the Lorentzian function decreases. The solid lines correspond to fitted hyperbolic functions of the form 1/ dA . A similar analysis was done for the case when pA varies from 2 to 20 % and keeping dA = 10μm constant, and the corresponding results appear in Fig. 4. In this case, the central frequency fD of the fitted Lorentzian shifts to lower frequencies when the porosity of the layer of doorway states increases, while the width ΓD of the Lorentzian increases. The solid lines correspond to a linear fitting for the central frequency fD , and the width ΓD of the Lorentzian function of the form fD = −7.4x10−3dA + 1.27 and ΓD = 7.6x10−4dA + 0.064 , respectively. From the results of Figs. 3 and 4, we conclude that we can choose the frequency in order to obtain the doorway state mechanism and modulate the amplitude of the sea of states around the doorway state. In addition, it is possible to increase or decrease the frequency range of influence of a given doorway state by changing the porosity or the thickness of the layer with ADS. 4. Experimental details Porous silicon structures were electrochemically etched into boron-doped (100)-oriented Si substrates with a resistivity of 0.001–0.002 Ω⋅ cm. Room-temperature anodization was performed using a 3:1 solution of HF (48%) and ethanol (99.98%). The etching current densities used to grow layers A and B were 2.3 mA/cm2 for 4500 s and 102.8 mA/cm2 for 7 s, respectively. Layer C correspond to the remaining crystalline silicon below the porous layers A and B. The acoustic transmission experiments presented 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 operation bandwidth of ∼800 MHz. To couple the transducers to the sample 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 are incident normally into the samples. It is important to remark that the transducers were connected to the VNA ports and a calibration of the complete system, including the coupling liquid, was done before to measure the transmission spectra of the samples. This calibration guarantee that the measured signal corresponds only to the sample. The experimental setup used to measure the transmission spectrum appears in Ref. [25]. Transmission spectrum of the sample was measured between 0.7 and 1.5 GHz, corresponding to the interval frequency response of the transducers. 5. Results and discussion Fig. 5 shows SEM images of the cross section of the fabricated structure. Thicknesses dA and dB are 10.6 μm and 0.6 μm, respectively. This leads a thickness of 513.8 μm for layer C. The porosities pA = 0.50 and pB = 0.62 , were determined from the analysis of the Fabry-Perot interference fringes of the optical reflectance spectra by fitting experimental measurements and comparing them with theoretical simulations for single layers fabricated previously. Fig. 6(a) shows the simulated acoustic transmission. This spectrum was obtained using the transfer matrix method described before, with In-Ga eutectic as the coupling liquid between the transducers and the sample. The parameters used for simulation were: dA = 10.6μm , dB = 0.6μm , pA = 0.50 and pB = 0.62 . We can see that three doorway state modes appear in the frequency range showed in the figure. The corresponding frequencies for each mode are: 0.74, 0.99 and 1.25 GHz. We observe that the amplitude of the acoustic transmission is modulated around the frequencies of the doorway states, i.e. the strength function phenomenon is clearly observed. Experimental measurement of the acoustic transmission appears in Fig. 6(b). We can observe that the doorway frequencies appear at: 0.72, 1.00 and 1.25 GHz, very similar to the theoretical simulation. However, although the shape of acoustic transmission and the position in frequency of doorway modes are in excellent agreement, the amplitude in both cases (theoretical and experimental) are quite different. This difference can be attributed to the losses in the experimental set-up, since the theoretical model corresponds to an ideal system, and it does not consider any kind of loss in the wave propagation. Fig. 6(c) and (d) show theoretical and experimental spectra around one of the ADS. The Lorentzian fitted curve the theoretical transmission is depicted in dashed line in Fig. 6(c) and has the form,
T=
C , (f − fD )2 + (ΓD /2)2
(4)
centered at fD = 1.25 GHz, with corresponding ΓD = 87.8 MHz width. In Fig. 6(d) can be seen that the behaviour of the experimental transmission matches very well to a Lorentzian function with the same central frequency ( fD ) and width (ΓD ). The response of the system shows the strength function phenomenon with a spreading width which is given by ΓD . This broadening is named Natural Line. In the case of atoms and molecules, the Natural broadening effect is due to the phonon or photon relaxation of an excited state to its ground state. In this way, ΓD is related to the temporal dynamics of the systems. Our system can be modelled as a classical damped harmonic oscillator, where ΓD is the damping constant [27]. On the other hand, the time required by the waves to travel through the layers A, B and C are, respectively: τA = dA/ vA , τB = dB / vB and τC = dC / v0 . Then, the total time required by the waves to travel along the complete sample from one transducer to the other is: τT = τA + τB + τC . We have chosen τA > τB and τC > > τA . For the sample analyzed here, τA = 1.9 ns, τB = 0.1 ns and τC = 60.9 ns, giving τT = 62.9 ns. In order to analyze the behaviour of ADS in time domain, we calculated the IFFT of transmission spectra. Fig. 7(a) and (b) shows 19
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Fig. 5. Cross-sectional SEM images of the fabricated structure for (a) 1,500× and (b) 5,000×. Labels A, B and C correspond to each layer. In the bottom image the thicknesses of the layers A and B are clearly observed.
in solid lines the square of the Fourier transform for theoretical and experimental acoustic transmission, respectively. The first peak correspond to τT ≈ 63 ns, for both cases. The next peaks correspond to resonances inside the sample at the interfaces between the layers. The positions of the peaks in both spectra are in good agreement, which means that the measured signal comes only from the sample. The exponential decay of the time response comes from the spreading width phenomenon, i.e. the Lorentzian envelope with ΓD width. Dashed lines in Fig. 7(a) and (b) correspond to decaying curves of the form exp[-(t − τT )/τD ] for t > τT . From the theoretical time resolved transmission (Fig. 7(a)), we obtain a decay time, τD equal to 1.81 ns, meanwhile, for the experimental transmission (Fig. 7(b)), this value is equal to τD = 1.84 ns. The spreading width ΓD = 1/2πτD coincides very well with that obtained from the square of the Fourier transform of curves appearing in Fig. 7, with a decay time τD obtained from the experimental and theoretical results. The lifetime is an important factor, as it determines the time available for the doorway to interact with or diffuse in its environment, and hence the information available from its phonon emission [26]. A possible application of our porous system is in the manufacture of fast-pulse generator transducer. Typically the transducer (ZnO for sample) is coated with a material to protect it from the environment. This coated material must allow the transmission of a pulse and in the same way not to influence the subsequent pulses. According to the exponential nature of the doorway intensity, at time t = 1.84 ns, the intensity decreases 36.8%. Analogously, at time t = 5 τD = 9.2 ns the intensity decreases less than 0.7%. In this way, our SP system could be used in the manufacture of fastpulse generator transducer at 1.25 GHz with Δt > 9.2 ns. 6. Conclusions By considering very simple structures based on porous silicon, we have successfully demonstrated both theoretically and experimentally the acoustic doorway states phenomenon. We analyzed the behaviour of the acoustic transmission spectra in both frequency and time domain. We have demonstrated that the strength function phenomenon is observed with a spreading width given by ΓD = 1/2πτD , where τD corresponds to the time decay of the square of IFFT of acoustic transmission. Theoretical and experimental results are in good agreement. From our theoretical results we conclude that it is possible to choose the central frequency where the doorway state mechanism occurs and modulate the amplitude of the sea of states around the doorway state by changing the porosity or the thickness of the layer with ADS. 20
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Fig. 6. Acoustic transmission spectra for the sample described in the text. (a) Theoretical transfer matrix calculations and (b) experimental measurement. (c) and (d) correspond to an amplificated view of the doorway states around 1.25 GHz. Dashed lines are the corresponding adjust to Lorentzian functions given by equation (4) with fD = 1.25 GHz and ΓD = 87.8 MHz, respectively.
Fig. 7. Square of IFFT of acoustic transmission of Fig. 6(a) and (b). Dashed lines correspond to exponential decays of the form I = I0 exp[−(t − τT )/ τD], with (a) τD = 1.81 ns and (b) τD = 1.84 ns.
Acknowledgements This work has been partially supported by PRODEP-SEP under project No. 511-6/17-8017. Project MELO-EXC17-G and 100522989-VIEP2018 BUAP. Authors thank R. Silva for SEM images. References [1] [2] [3] [4] [5]
M. Goldhaber, E. Teller, Phys. Rev. 74 (1948) 354. G.E. Brown, Unified Theory of Nuclear Models (North Holland, Amsterdam), (1964). I. Kawata, H. Kono, Y. Fujimura, A.D. Bandrauk, Phys. Rev. A 62 (2000) 031401(R). R. Curík, C.H. Greene, Phys. Rev. Lett. 98 (2007) 173201. A.L.D. Kilcoyne, A. Aguilar, A. Muller, S. Schippers, C. Cisneros, G. Alna Washi, N.B. Aryal, K.K. Baral, D.A. Esteves, C.M. Thomas, R.A. Phaneuf, Phys. Rev. Lett. 105 (2010) 213001.
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M.S. Hussein, V. Kharchenko, L.F. Canto, R. Donangelo, Ann. Phys. (N.Y.) 284 (2000) 178. L.O. Baksmaty, C. Yannouleas, U. Landman, Phys. Rev. Lett. 101 (2008) 136803. T. Laarmann, I. Shchatsinin, A. Stalmashonak, M. Boyle, N. Zhavoronkov, J. Handt, R. Schmidt, C.P. Schulz, I.V. Hertel, Phys. Rev. Lett. 98 (2007) 058302. I.V. Hertel, I. Shchatsinin, T. Laarmann, N. Zhavoronkov, H.H. Ritze, C.P. Schulz, Phys. Rev. Lett. 102 (2009) 023003. S. Aberg, T. Guhr, M. Miski-Oglu, A. Richter, Phys. Rev. Lett. 100 (2008) 204101. D.V. Savin, Acta Phys. Pol. A 132 (2017) 1688. A. Díaz-de-Anda, K. Volke-Sepúlveda, J. Flores, C. Sánchez-Pérez, L. Gutiérrez, Am. J. Phys. 83 (2015) 1012–1018. J. Flores, O. Novaro, T.H. Seligman, Nature 326 (1987) 783. J.A. Franco-Villafane, J. Flores, J.L. Mateos, R.A. Méndez-Sánchez, O. Novaro, T.H. Seligman, EPL 94 (2011) 30005. A. Morales, A. Díaz-de-Anda, J. Flores, L. Gutiérrez, R.A. Méndez-Sánchez, G. Monsivais, P. Mora, EPL 99 (2012) 54002. J.C. Torres-Guzmán, A. Díaz-de-Anda, J. Flores, G. Monsivais, L. Gutiérrez, A. Morales, EPL 114 (2016) 54001. J.A. Otero, G. Monsivais, A. Morales, L. Gutiérrez, A. Díaz-de-Anda, J. Flores, J. Acoust. Soc. Am. 142 (2017) 646. F.S. Levin, H. Feshbach, Reaction Dynamics, Gordon and Breach, New York, 1973, p. 209. L.D. Landau, E.M. Lifshitz, Theory of Elasticity, third ed., Elsevier, 2005. G.N. Aliev, B. Goller, D. Kovalev, P.A. Snow, Appl. Phys. Lett. 96 (2010) 124101. (a) R.J.M. Fonseca, J.M. Saurel, A. Foucaran, J. Camassel, E. Massone, T. Taliercio, J. Mater. Sci. 30 (1995) 35; (b) R.J.M. Da Fonseca, J.M. Saurel, A. Foucaran, E. Massone, T. Talierco, J. Camassel, Thin Solid Films 255 (1995) 155. Z. Lazcano, G.N. Aliev, J. Arriaga, J. Appl. Phys. 115 (2014) 154505. C.L. Mitsas, D.I. Siapkas, Appl. Opt. 34 (1995) 1678. V.V. Sokolov, V. Zelevinsky, Phys. Rev. C 56 (1997) 311. Z. Lazcano, J. Arriaga, Appl. Phys. Lett. 105 (2014) 231901. J.R. Lakowicz, Introduction to Fluorescence: Chapter Principles of Fluorescence Spectroscopy, third ed., Springer, Boston, MA, 2006. H.M. Nussenzveig, Causality and Dispersion Relations, Academic Press, 1972.
22
Contents lists available at ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Acoustic doorway states in layered porous silicon structures Z. Lazcanoa,∗, Y. García-Floresa, A. Díaz-de-Andaa, O. Mezaa, G. Monsivaisb, J. Arriagaa a b
T
Instituto de Física, Benemérita Universidad Autónoma de Puebla, Mexico Instituto de Física, Universidad Nacional Autónoma de México, Mexico
A R T IC LE I N F O
ABS TRA CT
Keywords: Doorway states Porous silicon Acoustics
We report the experimental observation of doorway states phenomenon for acoustic waves in porous silicon structures. We designed and fabricated a structure capable to exhibit the acoustic doorway states phenomenon around 1 GHZ. The structure consists of two porous silicon layers, one with low porosity and another with higher porosity and smaller thickness, followed by a much more thicker layer of crystalline silicon. We study the doorway state phenomenon in the frequency and in the time domain. To do this, we measured the acoustic transmission spectra in the frequency domain, and by using the Fourier transform, we obtain the transmission in the time domain. The transmission spectrum of the fabricated sample shows acoustic doorway states in frequencies around 1 GHz. The experimental results are in good agreement with the transfer matrix calculations.
1. Introduction The manifestation of the doorway state phenomenon, studied in nuclear physics through giant resonances [1,2], have been recently discussed in several systems, which include: atomic and molecular systems [3–5], clusters [6], quantum dots [7] and fullerenes [8,9]. Despite being originally studied in quantum mechanical systems, the doorway states phenomenon has also been observed in classical wave systems: microwave resonators [10], resonance state coupled to the background of many chaotic states [11], optical resonators [12], in the seismic response of sedimentary basins [13,14] and, more recently, in elastic systems [15–17]. All these cases consider two coupled structures with different spectral densities, i.e., one of these systems has a low density of states usually with a simple structure, whereas the other shows a high-density spectrum, formed by a “sea” of usually more complicated states. The low-density states system, act as doorway states whenever it is coupled to the sea of states. When the doorway state is excited, its amplitude spreads over the complicated states with a Lorentzian-like shape; and this is called the strength-function phenomenon [18]. In the present work, we demonstrate the theoretical and experimental evidence of acoustic doorway states (ADS). We propose simple multilayer structures based on porous silicon (PS) to observe the acoustic analogue of this phenomenon in the transmission spectra in Gigahertz range. Theoretical evidence of the temporal behaviour of ADS in such structures is also presented. A schematic representation of the structure considered in this work is shown in Fig. 1. The first silicon layer, A, has a low porosity pA and thickness dA , this layer contains a low density of states, i.e., the resonant states of this layer provide the ADS. The second layer, B, only used to couple the first layer to the third, has a porosity pB and a thickness dB , were pA < pB and dA > > dB . The sea of states is provided by the third layer C, in this case, the substrate. Layer C is crystalline silicon and its thickness dC is much larger than dA .
∗
Corresponding author. E-mail address: [email protected] (Z. Lazcano).
https://doi.org/10.1016/j.spmi.2018.07.018 Received 23 May 2018; Received in revised form 7 July 2018; Accepted 10 July 2018 Available online 17 July 2018 0749-6036/ © 2018 Elsevier Ltd. All rights reserved.
Superlattices and Microstructures 121 (2018) 16–22
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Fig. 1. Schematic representation of the structures considered in this work. Layers A, B and C have thicknesses dA , dB and dC , respectively, with dC > > dA .
2. Theoretical model In a layered solid, the acoustic waves can be either longitudinal or transversal. Here, we only consider longitudinal waves propagating along the perpendicular direction of the layers. This propagation is described by the wave equation [19],
∂2uj (z , t ) ∂z 2
−
1 ∂2uj (z , t ) = 0, ∂t 2 v j2
(1)
where uj (z , t ) is the atomic displacement and vj is the sound speed in layer j, given by vj = (λ + 2μ) j / ρj , being ρj the mass density, and λ and μ the Lamé parameters. The mass density ρj is a function of the porosity and is described by ρj = ρ0 (1 − Pj ), being ρ0 = 2.330 g / cm3 the mass density of bulk silicon and Pj the porosity in the j − th layer. The acoustic velocity dependence on porosity is given empirically by Ref. [20],
v = v0 (1 − P )k
(2)
being v0 the longitudinal velocity of sound in bulk silicon along the (100) crystallographic direction and k ≥ 0.5 is a constant. In general, the parameter k depends on PS morphology which in turn depends on the doping level of the Si substrate [20,21]. We have used v0 = 8.44 km/s and the parameter k = 0.56 from Ref. [22]. The solution of equation (1) can be written in the form of harmonic functions as,
u (zj , t ) = [A+j exp (ikj zj ) + A−j exp (−ikj zj )] exp (−iωt )
(3)
where ω = vj kj and ω = 2πf being f the frequency in s−1. After imposing the boundary conditions, at each interface, relating to the continuity of atomic displacement and stress, the propagation can be solved conveniently by using the transfer matrix method [23]. The transfer matrix of the complete system relates the displacement vector at the beginning of the structure with that at the end. With this formalism, the reflectivity and transmission of the structure can be easily calculated. ADS can be observed in the transmission spectrum (in frequency domain). Here, we present calculations and measurements of acoustic transmission as a function of frequency to demonstrate the existence of acoustic doorway states and the strength function in GHz range. To demonstrate the doorway state mechanism in the time domain, we calculated the time dependent acoustic transmission of both, theoretical and experimental results by using the inverse fast Fourier transform (IFFT) algorithm. 3. Numerical results As mentioned before, the doorway-state phenomenon occurs when two systems, one with a low density of states is coupled with another of large density. In our case, layer A and layer C play these roles respectively, and they are coupled through layer B. To have a deeply understanding of the doorway-state phenomenon, we consider slightly different structures described by slightly different parameters of the system appearing in Fig. 1. The direction of propagation, z-axis, is perpendicular to the layers and it is indicated by an arrow in Fig. 1. The number of states in each layer is determined by the thickness, the porosity and the velocity of propagation of the mechanical wave. As it is expected, the layer B has a very low density of states. We determine the appropriate thicknesses and porosities so that the states of layer B (which is used only to couple layers A and C) do not interfere with those of layers A and C, or do not appear in the range of frequencies considered here, i.e. from 0.7 to 1.5 GHz. In Fig. 2 we show the typical behaviour of the acoustic transmission for the system depicted in Fig. 1. The solid line corresponds to the acoustic transmission of the complete system, while in dashed line we have superimposed the transmission spectrum of layer A only. The maxima of the transmission appear at 0.75, 1.12 and 1.49 GHz, and they can be fitted to Lorentzian-like functions. From Fig. 2 we can clearly see that the transmission of the complete system is modulated around the frequencies provided by layer A, which provides the doorway states. Meanwhile, the fine oscillations are due to the dense part of the system, i.e., correspond to the high-density spectrum of layer C. This response is equivalent to the strength function analyzed in the quantum-mechanical case [24]. If the thickness or the porosity of the layer with ADS (layer A) is changed, the width of the Lorentzian changes, in addition, the maxima of such Lorentzians move in frequency. In other words, the influence of the doorway-states on the sea of states extends depending on the width or the porosity of layer A, besides that, the central frequency of the modes moves in frequency. We show the analysis of this behaviour in Figs. 3 and 4. For this purpose, we calculate acoustic transmission spectra and adjust a Lorentzian function to the peak around 1.12 GHz for a set of systems with pB = 80%, dB = 0.7μm , dC = (525 − dA) μm varying thickness (dA ) and 17
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Fig. 2. Acoustic transmission for the system of Fig. 1 (solid line): with pA = 20%, dA = 10μm , pB = 80%, dB = 0.7μm and dC = 514.3μm . The dashed line corresponds to the acoustic transmission for the layer A with the same parameters.
Fig. 3. Central frequency fD (left axis, dots) and Lorentzian width ΓD (right axis, squares) as a function of the thickness of layer A (dA ). The thickness changes from 8 to 11 μm, keeping the other parameters constant: pA = 20% , pB = 80%, dB = 0.7μ m and dC = 525 − dA μ m. Solid lines correspond to a numerical fitting of the form ΓD = 0.79/ dA and fD = 11.18/ dA , respectively.
Fig. 4. Central frequency fD (to the left axis, dots) and Lorentzian width Γ (right axis, squares) as a function of the porosity of layer A ( pA ). The porosity changes from 2 to 20%, keeping constant the other parameters: dA = 10μ m, pB = 80%, dB = 0.7μ m and dC = 514.3μ m. Solid lines correspond to a numerical fitting of the form fD = −7.4x10−3dA + 1.27 and ΓD = 7.6x10−4dA + 0.064 , respectively.
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porosity ( pA ) of layer A. First, we consider different values of dA from 8 to 11 μm keeping pA = 20%, and the results appear in Fig. 3. In Fig. 3 we observe that if we increase the thickness of the layer with the doorway states then the central frequency fD of the fitted Lorentzian shifts to lower frequencies, whereas the width ΓD of the Lorentzian function decreases. The solid lines correspond to fitted hyperbolic functions of the form 1/ dA . A similar analysis was done for the case when pA varies from 2 to 20 % and keeping dA = 10μm constant, and the corresponding results appear in Fig. 4. In this case, the central frequency fD of the fitted Lorentzian shifts to lower frequencies when the porosity of the layer of doorway states increases, while the width ΓD of the Lorentzian increases. The solid lines correspond to a linear fitting for the central frequency fD , and the width ΓD of the Lorentzian function of the form fD = −7.4x10−3dA + 1.27 and ΓD = 7.6x10−4dA + 0.064 , respectively. From the results of Figs. 3 and 4, we conclude that we can choose the frequency in order to obtain the doorway state mechanism and modulate the amplitude of the sea of states around the doorway state. In addition, it is possible to increase or decrease the frequency range of influence of a given doorway state by changing the porosity or the thickness of the layer with ADS. 4. Experimental details Porous silicon structures were electrochemically etched into boron-doped (100)-oriented Si substrates with a resistivity of 0.001–0.002 Ω⋅ cm. Room-temperature anodization was performed using a 3:1 solution of HF (48%) and ethanol (99.98%). The etching current densities used to grow layers A and B were 2.3 mA/cm2 for 4500 s and 102.8 mA/cm2 for 7 s, respectively. Layer C correspond to the remaining crystalline silicon below the porous layers A and B. The acoustic transmission experiments presented 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 operation bandwidth of ∼800 MHz. To couple the transducers to the sample 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 are incident normally into the samples. It is important to remark that the transducers were connected to the VNA ports and a calibration of the complete system, including the coupling liquid, was done before to measure the transmission spectra of the samples. This calibration guarantee that the measured signal corresponds only to the sample. The experimental setup used to measure the transmission spectrum appears in Ref. [25]. Transmission spectrum of the sample was measured between 0.7 and 1.5 GHz, corresponding to the interval frequency response of the transducers. 5. Results and discussion Fig. 5 shows SEM images of the cross section of the fabricated structure. Thicknesses dA and dB are 10.6 μm and 0.6 μm, respectively. This leads a thickness of 513.8 μm for layer C. The porosities pA = 0.50 and pB = 0.62 , were determined from the analysis of the Fabry-Perot interference fringes of the optical reflectance spectra by fitting experimental measurements and comparing them with theoretical simulations for single layers fabricated previously. Fig. 6(a) shows the simulated acoustic transmission. This spectrum was obtained using the transfer matrix method described before, with In-Ga eutectic as the coupling liquid between the transducers and the sample. The parameters used for simulation were: dA = 10.6μm , dB = 0.6μm , pA = 0.50 and pB = 0.62 . We can see that three doorway state modes appear in the frequency range showed in the figure. The corresponding frequencies for each mode are: 0.74, 0.99 and 1.25 GHz. We observe that the amplitude of the acoustic transmission is modulated around the frequencies of the doorway states, i.e. the strength function phenomenon is clearly observed. Experimental measurement of the acoustic transmission appears in Fig. 6(b). We can observe that the doorway frequencies appear at: 0.72, 1.00 and 1.25 GHz, very similar to the theoretical simulation. However, although the shape of acoustic transmission and the position in frequency of doorway modes are in excellent agreement, the amplitude in both cases (theoretical and experimental) are quite different. This difference can be attributed to the losses in the experimental set-up, since the theoretical model corresponds to an ideal system, and it does not consider any kind of loss in the wave propagation. Fig. 6(c) and (d) show theoretical and experimental spectra around one of the ADS. The Lorentzian fitted curve the theoretical transmission is depicted in dashed line in Fig. 6(c) and has the form,
T=
C , (f − fD )2 + (ΓD /2)2
(4)
centered at fD = 1.25 GHz, with corresponding ΓD = 87.8 MHz width. In Fig. 6(d) can be seen that the behaviour of the experimental transmission matches very well to a Lorentzian function with the same central frequency ( fD ) and width (ΓD ). The response of the system shows the strength function phenomenon with a spreading width which is given by ΓD . This broadening is named Natural Line. In the case of atoms and molecules, the Natural broadening effect is due to the phonon or photon relaxation of an excited state to its ground state. In this way, ΓD is related to the temporal dynamics of the systems. Our system can be modelled as a classical damped harmonic oscillator, where ΓD is the damping constant [27]. On the other hand, the time required by the waves to travel through the layers A, B and C are, respectively: τA = dA/ vA , τB = dB / vB and τC = dC / v0 . Then, the total time required by the waves to travel along the complete sample from one transducer to the other is: τT = τA + τB + τC . We have chosen τA > τB and τC > > τA . For the sample analyzed here, τA = 1.9 ns, τB = 0.1 ns and τC = 60.9 ns, giving τT = 62.9 ns. In order to analyze the behaviour of ADS in time domain, we calculated the IFFT of transmission spectra. Fig. 7(a) and (b) shows 19
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Fig. 5. Cross-sectional SEM images of the fabricated structure for (a) 1,500× and (b) 5,000×. Labels A, B and C correspond to each layer. In the bottom image the thicknesses of the layers A and B are clearly observed.
in solid lines the square of the Fourier transform for theoretical and experimental acoustic transmission, respectively. The first peak correspond to τT ≈ 63 ns, for both cases. The next peaks correspond to resonances inside the sample at the interfaces between the layers. The positions of the peaks in both spectra are in good agreement, which means that the measured signal comes only from the sample. The exponential decay of the time response comes from the spreading width phenomenon, i.e. the Lorentzian envelope with ΓD width. Dashed lines in Fig. 7(a) and (b) correspond to decaying curves of the form exp[-(t − τT )/τD ] for t > τT . From the theoretical time resolved transmission (Fig. 7(a)), we obtain a decay time, τD equal to 1.81 ns, meanwhile, for the experimental transmission (Fig. 7(b)), this value is equal to τD = 1.84 ns. The spreading width ΓD = 1/2πτD coincides very well with that obtained from the square of the Fourier transform of curves appearing in Fig. 7, with a decay time τD obtained from the experimental and theoretical results. The lifetime is an important factor, as it determines the time available for the doorway to interact with or diffuse in its environment, and hence the information available from its phonon emission [26]. A possible application of our porous system is in the manufacture of fast-pulse generator transducer. Typically the transducer (ZnO for sample) is coated with a material to protect it from the environment. This coated material must allow the transmission of a pulse and in the same way not to influence the subsequent pulses. According to the exponential nature of the doorway intensity, at time t = 1.84 ns, the intensity decreases 36.8%. Analogously, at time t = 5 τD = 9.2 ns the intensity decreases less than 0.7%. In this way, our SP system could be used in the manufacture of fastpulse generator transducer at 1.25 GHz with Δt > 9.2 ns. 6. Conclusions By considering very simple structures based on porous silicon, we have successfully demonstrated both theoretically and experimentally the acoustic doorway states phenomenon. We analyzed the behaviour of the acoustic transmission spectra in both frequency and time domain. We have demonstrated that the strength function phenomenon is observed with a spreading width given by ΓD = 1/2πτD , where τD corresponds to the time decay of the square of IFFT of acoustic transmission. Theoretical and experimental results are in good agreement. From our theoretical results we conclude that it is possible to choose the central frequency where the doorway state mechanism occurs and modulate the amplitude of the sea of states around the doorway state by changing the porosity or the thickness of the layer with ADS. 20
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Fig. 6. Acoustic transmission spectra for the sample described in the text. (a) Theoretical transfer matrix calculations and (b) experimental measurement. (c) and (d) correspond to an amplificated view of the doorway states around 1.25 GHz. Dashed lines are the corresponding adjust to Lorentzian functions given by equation (4) with fD = 1.25 GHz and ΓD = 87.8 MHz, respectively.
Fig. 7. Square of IFFT of acoustic transmission of Fig. 6(a) and (b). Dashed lines correspond to exponential decays of the form I = I0 exp[−(t − τT )/ τD], with (a) τD = 1.81 ns and (b) τD = 1.84 ns.
Acknowledgements This work has been partially supported by PRODEP-SEP under project No. 511-6/17-8017. Project MELO-EXC17-G and 100522989-VIEP2018 BUAP. Authors thank R. Silva for SEM images. References [1] [2] [3] [4] [5]
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