# Comparison of Brillouin Light Scattering and Density of States in a Supported Layer: Analytical and Experimental Study

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## Abstract

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## 1. Introduction

_{2}/GaAs [8], ZnSe/GaAs [14], WC/Si [10] and recently in heterostructure semiconductor/topological insulator Si/Bi

_{2}Te

_{3}[13]. Generation of modes in more complicated structures such as interconnected Al stripes and pillars deposited on a Si substrate has been reported [20].

_{2}-PMMA) studied by non-destructive BLS has advanced our knowledge of phononic wave propagation [41,42,43]. The large mismatch between the physical properties of polymer layers is the key parameter in determining the phononic properties of these periodic nanocomposites, as the width of the phonon band gaps depends on the difference between the acoustic impedances of the constituents. More recently, the direction-dependent elastic and electromagnetic wave propagation has been studied by some of us experimentally and theoretically in supported films of PMMA-TiO

_{2}[44] and PMMA-BaTiO

_{3}[45] multilayers with a periodicity of about 100–140 nm. In these studies, full theoretical description of the phononic density of states (DOS) recorded by BLS was derived.

_{2}layer deposited on Si substrate [47] and to probe the mechanical properties of single vegetal cells [48].

## 2. Analytical Results

#### 2.1. Scattering Intensity in a Supported Layer

_{i}of this medium: $\delta {\epsilon}_{i}$ $={\epsilon}_{i}^{2}{p}_{i}\frac{\partial {u}_{i}\left({x}_{3}\right)}{\partial {x}_{3}}$ (${p}_{i}$stands for the photoelastic constant${p}_{12}$in layer $i$ ($i=1,2$)). Then, in the presence of an incident electric field${E}_{i}\left({x}_{3}\right)$, the strain due to displacement field ${u}_{i}\left({x}_{3}\right)$induces a polarization given by [57]

#### 2.2. Density of States and Reflection Delay Time

#### 2.3. Particular Case of a Soft Layer on a Hard Substrate

## 3. Numerical and Experimental Results

#### 3.1. Experimental Setup

#### 3.2. Results and Discussion

## 4. Conclusions

## Supplementary Materials

_{3}axis is normal to the surface.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Effect of the Moving Interface Mechanism on BLS Spectra

**Figure A1.**Theoretical Brillouin light scattering spectra displayed at $\mathrm{q}=0.036{\text{}\mathrm{nm}}^{-1}$ for a series of supported PMMA films with different thicknesses deposited on Si substrate, with (continuous curves) and without (dashed curves) taking into account the deformation of the interfaces.

## References

- Dobrzynski, L.; El Boudouti, E.H.; Akjouj, A.; Pennec, Y.; Al-Wahsh, H.; Lévêque, G.; Djafari-Rouhani, B. Phononics, 1st ed.; Elsevier: Amsterdam, The Netherlands, 2017. [Google Scholar]
- Carlotti, G. Elastic Characterization of Transparent and Opaque Films, Multilayers and Acoustic Resonators by Surface Brillouin Scattering: A Review. Appl. Sci.
**2018**, 8, 124. [Google Scholar] [CrossRef] - Bottani, C.E.; Fioretto, D. Brillouin scattering of phonons in complex materials. Adv. Phys. X
**2018**, 3. [Google Scholar] [CrossRef] - El Boudouti, E.H.; Djafari-Rouhani, B. Acoustic waves in finite superlattices. Phys. Rev. B
**1994**, 49, 4586–4592. [Google Scholar] [CrossRef] - Alami, M.; El Boudouti, E.; Djafari-Rouhani, B.; El Hassouani, Y.; Talbi, A. Surface acoustic waves in one-dimensional piezoelectric-metallic phononic crystal: Effect of a cap layer. Ultrasonics
**2018**, 90, 80–97. [Google Scholar] [CrossRef] [PubMed] - Bell, J.A.; Zanoni, R.; Seaton, C.T.; Stegeman, G.I.; Makous, J.; Falco, C.M. Elastic constants of, and Stonely waves in, molybdenum films measured by Brillouin scattering. Appl. Phys. Lett.
**1988**, 52, 610–612. [Google Scholar] [CrossRef] - Zhang, X.; Comins, J.D.; Every, A.G.; Stoddart, P.R.; Pang, W.; Derry, T.E. Surface Brillouin scattering study of the surface excitations in amorphous silicon layers produced by ion bombardment. Phys. Rev. B
**1998**, 58, 13677–13685. [Google Scholar] [CrossRef] - Zhang, X.; Manghnani, M.H.; Every, A.G. Evidence for a shear horizontal resonance in supported thin films. Phys. Rev. B
**2000**, 62, R2271–R2274. [Google Scholar] [CrossRef] - Lai, K.; Finkelstein-Shapiro, D.; Lehmann, S.; Devos, A.; Mante, P.-A. Fano resonance between Stokes and anti-Stokes Brillouin scattering. Phys. Rev. Res.
**2021**, 3, L032010. [Google Scholar] [CrossRef] - Wittkowski, T.; Distler, G.; Jung, K.; Hillebrands, B.; Comins, J.D. General methods for the determination of the stiffness tensor and mass density of thin films using Brillouin light scattering: Study of tungsten carbide films. Phys. Rev. B
**2004**, 69, 205401. [Google Scholar] [CrossRef] - Izadi, A.; Sinha, M.; Papson, C.; Roccabianca, S.; Anthony, R. Mechanical behavior of SiNC layers on PDMS: Effects of layer thickness, PDMS modulus, and SiNC surface functionality. RSC Adv.
**2020**, 10, 39087–39091. [Google Scholar] [CrossRef] - Alonso-Redondo, E.; Belliard, L.; Rolle, K.; Graczykowski, B.; Tremel, W.; Djafari-Rouhani, B.; Fytas, G. Robustness of elastic properties in polymer nanocomposite films examined over the full volume fraction range. Sci. Rep.
**2018**, 8, 16986. [Google Scholar] [CrossRef] [PubMed] - Trzaskowska, A.; Mielcarek, S.; Wiesner, M.; Lombardi, F.; Mroz, B. Dispersion of the surface phonons in semiconductor/topological insulator Si/Bi2Te3 heterostructure studied by high resolution Brillouin spectroscopy. Ultrasonics
**2021**, 117, 106526. [Google Scholar] [CrossRef] [PubMed] - Zhang, X.; Bandhu, R.S.; Sooryakumar, R.; Jonker, B.T. High-frequency standing longitudinal acoustic resonances in supported thin films. Phys. Rev. B
**2003**, 67. [Google Scholar] [CrossRef] - Groenen, J.; Poinsotte, F.; Zwick, A.; Torres, C.M.S.; Prunnila, M.; Ahopelto, J. Inelastic light scattering by longitudinal acoustic phonons in thin silicon layers: From membranes to silicon-on-insulator structures. Phys. Rev. B
**2008**, 77, 045420. [Google Scholar] [CrossRef] - Hartschuh, R.; Ding, Y.; Roh, J.H.; Kisliuk, A.; Sokolov, A.P.; Soles, C.L.; Jones, R.L.; Hu, T.J.; Wu, W.L.; Mahorowala, A.P. Brillouin scattering studies of polymeric nanostructures. J. Polym. Sci. Part B Polym. Phys.
**2004**, 42, 1106–1113. [Google Scholar] [CrossRef] - Link, A.; Sooryakumar, R.; Bandhu, R.S.; Antonelli, G.A. Brillouin light scattering studies of the mechanical properties of ultrathin low-k dielectric films. J. Appl. Phys.
**2006**, 100, 013507. [Google Scholar] [CrossRef] - Nakamura, N.; Ogi, H.; Hirao, M. Stable elasticity of epitaxial Cu thin films on Si. Phys. Rev. B
**2008**, 77, 245416. [Google Scholar] [CrossRef] - Zerdali, M.; Hamzaoui, S.; Teherani, F.; Rogers, D. Growth of ZnO thin film on SiO2/Si substrate by pulsed laser deposition and study of their physical properties. Mater. Lett.
**2006**, 60, 504–508. [Google Scholar] [CrossRef] - Trzaskowska, A.; Hakonen, P.; Wiesner, M.; Mielcarek, S. Generation of a mode in phononic crystal based on 1D/2D structures. Ultrasonics
**2020**, 106, 106146. [Google Scholar] [CrossRef] - Razeeb, K.M.; Dalton, E.; Cross, G.; Robinson, A. Present and future thermal interface materials for electronic devices. Int. Mater. Rev.
**2017**, 63, 1–21. [Google Scholar] [CrossRef] - Gomopoulos, N.; Cheng, W.; Efremov, M.; Nealey, P.F.; Fytas, G. Out-of-Plane Longitudinal Elastic Modulus of Supported Polymer Thin Films. Macromolecules
**2009**, 42, 7164–7167. [Google Scholar] [CrossRef] - Gomopoulos, N.; Saini, G.; Efremov, M.; Nealey, P.F.; Nelson, K.; Fytas, G. Nondestructive Probing of Mechanical Anisotropy in Polyimide Films on Nanoscale. Macromolecules
**2010**, 43, 1551–1555. [Google Scholar] [CrossRef] - Parsons, L.C.; Andrews, G.T. Observation of hypersonic phononic crystal effects in porous silicon superlattices. Appl. Phys. Lett.
**2009**, 95, 241909. [Google Scholar] [CrossRef] - Walker, P.M.; Sharp, J.S.; Akimov, A.V.; Kent, A.J. Coherent elastic waves in a one-dimensional polymer hypersonic crystal. Appl. Phys. Lett.
**2010**, 97, 073106. [Google Scholar] [CrossRef] - Aliev, G.N.; Goller, B.; Kovalev, D.; Snow, P.A. Hypersonic acoustic mirrors and microcavities in porous silicon. Appl. Phys. Lett.
**2010**, 96, 124101. [Google Scholar] [CrossRef] - Hesami, M.; Gueddida, A.; Gomopoulos, N.; Dehsari, H.S.; Asadi, K.; Rudykh, S.; Butt, H.-J.; Djafari-Rouhani, B.; Fytas, G. Elastic wave propagation in smooth and wrinkled stratified polymer films. Nanotechnology
**2018**, 30, 045709. [Google Scholar] [CrossRef] - El Boudouti, E.H.; Djafari-Rouhani, B.; Akjouj, A.; Dobrzynski, L. Acoustic waves in solid and fluid layered materials. Surf. Sci. Rep.
**2009**, 64, 471–594. [Google Scholar] [CrossRef] - Pennec, Y.; Vasseur, J.O.; Djafari-Rouhani, B.; Dobrzyński, L.; Deymier, P.A. Two-dimensional phononic crystals: Examples and applications. Surf. Sci. Rep.
**2010**, 65, 229–291. [Google Scholar] [CrossRef] - Quotane, I.; Amrani, M.; Ghouila-Houri, C.; El Boudouti, E.H.; Krutyansky, L.; Piwakowski, B.; Pernod, P.; Talbi, A.; Djafari-Rouhani, B. A Biosensor Based on Bound States in the Continuum and Fano Resonances in a Solid–Liquid–Solid Triple Layer. Crystals
**2022**, 12, 707. [Google Scholar] [CrossRef] - Ezzahri, Y.; Grauby, S.; Rampnoux, J.M.; Michel, H.; Pernot, G.; Claeys, W.; Dilhaire, S.; Rossignol, C.; Zeng, G.; Shakouri, A. Coherent phonons in Si/SiGe superlattices. Phys. Rev. B
**2007**, 75, 195309. [Google Scholar] [CrossRef] - Belliard, L.; Huynh, A.; Perrin, B.; Michel, A.; Abadias, G.; Jaouen, C. Elastic properties and phonon generation in Mo/Si superlattices. Phys. Rev. B
**2009**, 80, 155424. [Google Scholar] [CrossRef] - Kimura, N.D.L.; Fainstein, A.; Huynh, A.; Perrin, B.; Jusserand, B.; Miard, A.; Lemaitre, A. Coherent Generation of Acoustic Phonons in an Optical Microcavity. Phys. Rev. Lett.
**2007**, 99, 217405. [Google Scholar] [CrossRef] - Beardsley, R.P.; Akimov, A.; Henini, M.; Kent, A. Coherent Terahertz Sound Amplification and Spectral Line Narrowing in a Stark Ladder Superlattice. Phys. Rev. Lett.
**2010**, 104, 085501. [Google Scholar] [CrossRef] - Lacharmoise, P.; Fainstein, A.; Jusserand, B.; Thierry-Mieg, V. Optical cavity enhancement of light–sound interaction in acoustic phonon cavities. Appl. Phys. Lett.
**2004**, 84, 3274–3276. [Google Scholar] [CrossRef] - Maldovan, M.; Thomas, E.L. Simultaneous localization of photons and phonons in two-dimensional periodic structures. Appl. Phys. Lett.
**2006**, 88, 251907. [Google Scholar] [CrossRef] - Entezar, S.R.; Namdar, A. Localized modes in defective multilayer structures. Phys. Rev. A
**2009**, 80, 013814. [Google Scholar] [CrossRef] - Liang, B.; Guo, X.S.; Tu, J.; Zhang, D.; Cheng, J.C. An acoustic rectifier. Nat. Mater.
**2010**, 9, 989–992. [Google Scholar] [CrossRef] - Vollmer, F.; Arnold, S. Whispering-gallery-mode biosensing: Label-free detection down to single molecules. Nat. Methods
**2008**, 5, 591–596. [Google Scholar] [CrossRef] - Botsis, J.; Humbert, L.; Colpo, F.; Giaccari, P. Embedded fiber Bragg grating sensor for internal strain measurements in polymeric materials. Opt. Lasers Eng.
**2005**, 43, 491–510. [Google Scholar] [CrossRef] - Gomopoulos, N.; Maschke, D.; Koh, C.Y.; Thomas, E.L.; Tremel, W.; Butt, H.-J.; Fytas, G. One-Dimensional Hypersonic Phononic Crystals. Nano Lett.
**2010**, 10, 980–984. [Google Scholar] [CrossRef] - Schneider, D.; Liaqat, F.; El Boudouti, E.H.; El Hassouani, Y.; Djafari-Rouhani, B.; Tremel, W.; Butt, H.-J.; Fytas, G. Engineering the Hypersonic Phononic Band Gap of Hybrid Bragg Stacks. Nano Lett.
**2012**, 12, 3101–3108. [Google Scholar] [CrossRef] [PubMed] - Schneider, D.; Liaqat, F.; El Boudouti, E.H.; El Abouti, O.; Tremel, W.; Butt, H.-J.; Djafari-Rouhani, B.; Fytas, G. Defect-Controlled Hypersound Propagation in Hybrid Superlattices. Phys. Rev. Lett.
**2013**, 111, 164301. [Google Scholar] [CrossRef] [PubMed] - Alonso-Redondo, E.; Gueddida, A.; Huesmann, H.; El Abouti, O.; Tremel, W.; El Boudouti, E.H.; Djafari-Rouhani, B.; Fytas, G. Direction-dependent elastic properties and phononic behavior of PMMA/BaTiO
_{3}nanocomposite thin films. J. Chem. Phys.**2017**, 146, 203325. [Google Scholar] [CrossRef] [PubMed] - Alonso-Redondo, E.; Huesmann, H.; El Boudouti, E.-H.; Tremel, W.; Djafari-Rouhani, B.; Butt, H.-J.; Fytas, G. Phoxonic Hybrid Superlattice. ACS Appl. Mater. Interfaces
**2015**, 7, 12488–12495. [Google Scholar] [CrossRef] - Matsuda, O.; Larciprete, M.C.; Voti, R.L.; Wright, O.B. Fundamentals of picosecond laser ultrasonics. Ultrasonics
**2014**, 56, 3–20. [Google Scholar] [CrossRef] - Colletta, M.; Gachuhi, W.; Gartenstein, S.A.; James, M.M.; Szwed, E.A.; Daly, B.C.; Cui, W.; Antonelli, G.A. Picosecond ultrasonic study of surface acoustic waves on periodically patterned layered nanostructures. Ultrasonics
**2018**, 87, 126–132. [Google Scholar] [CrossRef] - Dehoux, T.; Ghanem, M.A.; Zouani, O.F.; Ducousso, M.; Chigarev, N.; Rossignol, C.; Tsapis, N.; Durrieu, M.-C.; Audoin, B. Probing single-cell mechanics with picosecond ultrasonics. Ultrasonics
**2015**, 56, 160–171. [Google Scholar] [CrossRef] - Ashcroft, N.W.; Mermin, N.D. Solid State Physics, College ed.; Thomson Learning Inc.: Noida, India, 1976. [Google Scholar]
- Sturhahn, W.; Toellner, T.S.; Alp, E.E.; Zhang, X.; Ando, M.; Yoda, Y.; Kikuta, S.; Seto, M.; Kimball, C.W.; Dabrowski, B. Phonon Density of States Measured by Inelastic Nuclear Resonant Scattering. Phys. Rev. Lett.
**1995**, 74, 3832–3835. [Google Scholar] [CrossRef] - Hess, C. Introduction to Scanning Tunneling Spectroscopy of Correlated Materials. In Quantum Materials: Experiments and Theory; Pavarini, E., Koch, E., van den Brink, J., Sawatzky, G., Eds.; Verlag des Forschungszentrum Jülich: Jülich, Germany, 2016. [Google Scholar]
- Neumaier, D.; Turek, M.; Wurstbauer, U.; Vogl, A.; Utz, M.; Wegscheider, W.; Weiss, D. All-Electrical Measurement of the Density of States in (Ga,Mn)As. Phys. Rev. Lett.
**2009**, 103, 087203. [Google Scholar] [CrossRef] - Ghislotti, G.; Bottani, C.E. Brillouin scattering from shear horizontal surface phonons in silicon on insulator structures: Theory and experiment. Phys. Rev. B
**1994**, 50, 12131–12137. [Google Scholar] [CrossRef] - Chirita, M.; Sooryakumar, R.; Xia, H.; Monteiro, O.R.; Brown, I.G. Observation of guided longitudinal acoustic modes in hard supported layers. Phys. Rev. B
**1999**, 60, R5153–R5156. [Google Scholar] [CrossRef] - Zhang, X.; Sooryakumar, R.; Every, A.; Manghnani, M.H. Observation of organ-pipe acoustic excitations in supported thin films. Phys. Rev. B
**2001**, 64, 081402. [Google Scholar] [CrossRef] - Sandercock, J.R. Structure in the Brillouin Spectra of Thin Films. Phys. Rev. Lett.
**1972**, 29, 1735–1738. [Google Scholar] [CrossRef] - Djafari Rouhani, B.; Khourdifi, E.M. Light Scattering in Semiconductor Structures and Superlattices; Lockwood, D.J., Young, J.F., Eds.; Springer: New York, NY, USA, 1991. [Google Scholar]
- Cuffe, J. Phonon-Photon Interactions in Nanostructures. Ph.D. Thesis, CORK, Faculty of Science, National University of Ireland, Galway, Ireland, 2011. [Google Scholar]
- Szczurowski, M.K.; Martynkien, T.; Statkiewicz-Barabach, G.; Urbanczyk, W.; Khan, L.; Webb, D.J. Measurements of stress-optic coefficient in polymer optical fibers. Opt. Lett.
**2010**, 35, 2013–2015. [Google Scholar] [CrossRef] [PubMed] - Donadio, D.; Bernasconi, M.; Tassone, F. Photoelasticity of crystalline and amorphous silica from first principles. Phys. Rev. B
**2003**, 68. [Google Scholar] [CrossRef] - Yang, S.; Page, J.H.; Liu, Z.; Cowan, M.L.; Chan, C.T.; Sheng, P. Ultrasound Tunneling through 3D Phononic Crystals. Phys. Rev. Lett.
**2002**, 88, 104301. [Google Scholar] [CrossRef] - Khattou, S.; Amrani, M.; Mouadili, A.; El Boudouti, E.H.; Talbi, A.; Akjouj, A.; Djafari-Rouhani, B. Comparison of density of states and scattering parameters in coaxial photonic crystals: Theory and experiment. Phys. Rev. B
**2020**, 102, 165310. [Google Scholar] [CrossRef] - Graczykowski, B.; Gueddida, A.; Djafari-Rouhani, B.; Butt, H.-J.; Fytas, G. Brillouin light scattering under one-dimensional confinement: Symmetry and interference self-canceling. Phys. Rev. B
**2019**, 99, 165431. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**Schematic representation of one layer (labeled $i=1$) of width d deposited on a substrate (labeled $i=2$ ). The ${x}_{3}$ axis is normal to the surface. ${k}_{i}$ and ${k}_{s}$ are the incident and scattered wavevectors in vacuum. The electromagnetic fields are polarized along the ${x}_{2}$ axis.$\alpha $ and $\theta $ are the incident and refracted angles respectively.

**Figure 2.**(

**a**) Theoretical (open circles) and experimental (continuous curves) Brillouin light scattering for a series of supported PMMA films with different thicknesses deposited on a Si substrate. The main spectra appear at the frequency position of the longitudinal acoustic phonon around$\mathrm{q}=\frac{4\pi {n}_{1}}{\lambda}=0.036{\text{}\mathrm{nm}}^{-1\text{}}$ i.e., ${f}_{1}=q{v}_{1}/2\pi =15.9$ GHz. (

**b**) Square modulus of the displacement field versus the space position for the mode at f = 16 GHz and d = 490 nm. (

**c**) Optical transmission through the layer versus the wavelength. The circle on the abscissa axis corresponds to 514.5 nm wavelength used in the experiment. (

**d**) Square modulus of the electric field versus the space position.

**Figure 3.**Theoretical spectra (open circles) evaluated using the approximate expression (Equation (25)) for the same thicknesses of PMMA film on Si substrate. Solid curves correspond to experiments.

**Figure 4.**(

**a**) Variation of density of states (in arbitrary units) of layer modes for the different layer thicknesses. (

**b**) Quantified frequencies of layer modes as a function of layer thickness. Peak positions of experimental spectra are given as red circles around the frequency 16 GHz.

**Figure 5.**Variation of the DOS (blue curves) and BLS intensity (red curves) as a function of the frequency for the PMMA layer of thickness $d=490$nm.

**Figure 6.**Same as in Figure 5 but for a hypothetical substrate with impedance${Z}_{2}=0.1{Z}_{1}$ (

**a**), ${Z}_{2}=0.5{Z}_{1}$ (

**b**) and ${Z}_{2}=1.5{Z}_{1}$ (

**c**).

**Table 1.**Physical quantities of the PMMA layer and Si substrate used in the theoretical calculations. Mass density ($\rho $), sound velocity ($v$), photoelastic constant (${p}_{12}$) and refractive index ($n$).

Material | p (Kg/m^{3}) | v (m/s) | p_{12} | n |
---|---|---|---|---|

PMMA | 1150 | 2778 | 0.3 | 1.4932 |

Si | 2335 | 8431 | 0.01 | 3.5 − 0.26i |

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**MDPI and ACS Style**

El Abouti, O.; Cuffe, J.; El Boudouti, E.H.; Sotomayor Torres, C.M.; Chavez-Angel, E.; Djafari-Rouhani, B.; Alzina, F.
Comparison of Brillouin Light Scattering and Density of States in a Supported Layer: Analytical and Experimental Study. *Crystals* **2022**, *12*, 1212.
https://doi.org/10.3390/cryst12091212

**AMA Style**

El Abouti O, Cuffe J, El Boudouti EH, Sotomayor Torres CM, Chavez-Angel E, Djafari-Rouhani B, Alzina F.
Comparison of Brillouin Light Scattering and Density of States in a Supported Layer: Analytical and Experimental Study. *Crystals*. 2022; 12(9):1212.
https://doi.org/10.3390/cryst12091212

**Chicago/Turabian Style**

El Abouti, Ossama, John Cuffe, El Houssaine El Boudouti, Clivia M. Sotomayor Torres, Emigdio Chavez-Angel, Bahram Djafari-Rouhani, and Francesc Alzina.
2022. "Comparison of Brillouin Light Scattering and Density of States in a Supported Layer: Analytical and Experimental Study" *Crystals* 12, no. 9: 1212.
https://doi.org/10.3390/cryst12091212