crystals Article Comparison of Brillouin Light Scattering and Density of States in a Supported Layer: Analytical and Experimental Study Ossama El Abouti 1 , John Cuffe 2,3 , El Houssaine El Boudouti 4, * , Clivia M. Sotomayor Torres 2,5 , Emigdio Chavez-Angel 2 , Bahram Djafari-Rouhani 6 and Francesc Alzina 2 1 Faculté des Sciences et Techniques, Al Hoceima, Université Abdelmalek Essadi, Tétouan 93000, Morocco 2 Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and BIST, Campus UAB, Bellaterra, 08193 Barcelona, Spain 3 Department of Physics, University College Cork, College Road, T12 K8AF Cork, Ireland 4 LPMR, Département de Physique, Faculté des Sciences, Université Mohammed I, Oujda 60000, Morocco 5 ICREA, Pg. Lluis Companys 23, 08010 Barcelona, Spain 6 Département de Physique, IEMN, Université de Lille, 59655 Lille, France * Correspondence: e.elboudouti@ump.ac.ma Abstract: We provide a detailed analytical calculation of the Brillouin light scattering (BLS) intensity of a layer on a substrate, taking into account both photoelastic and moving boundary (ripple effect) mechanisms, and give a comparison between BLS intensity and density of states (DOS) to determine the dispersion curves of longitudinal guided modes in the supported layer. In particular, in the case where the mismatch between the elastic parameters of the substrate and the adsorbed layer is high, such as in a PMMA layer on a Si substrate, we derive closed-form expressions of BLS and DOS and demonstrate a simple relationship between these two quantities. A very good agreement between experimental and theoretical BLS spectra was found and compared to theoretical DOS spectra. In particular, we show Citation: El Abouti, O.; Cuffe, J.; El that while the peaks in the DOS present a uniform behavior, the BLS spectra follows a sine cardinal Boudouti, E.H.; Sotomayor Torres, C.M.; Chavez-Angel, E.; (sinc) function shape around a given frequency fixed by the chosen laser wavelength. The theoretical Djafari-Rouhani, B.; Alzina, F. calculation is performed within the framework of the Green’s function approach. Comparison of Brillouin Light Scattering and Density of States in a Keywords: Brillouin light scattering; density of states; supported layer; Green’s function Supported Layer: Analytical and Experimental Study. Crystals 2022, 12, 1212. https://doi.org/10.3390/ cryst12091212 1. Introduction Academic Editor: Bernard Bonello Physical properties of thin supported layers are of specific interest, as they are of- ten different from those of bulk materials, mainly the transport properties [1]. These Received: 31 July 2022 properties are of fundamental importance for the development of high-performance de- Accepted: 26 August 2022 vices. The propagation of surface acoustic waves in supported layers has been extensively Published: 28 August 2022 studied [1–13]. On the other hand, the so-called longitudinal guided modes have also at- Publisher’s Note: MDPI stays neutral tracted significant attention [14–19]. Experimental evidence of the existence of these modes with regard to jurisdictional claims in has been obtained for one supported layer deposited on a substrate such as Mo/Si [6], published maps and institutional affil- Si (amorphous)/Si(crystalline) [7], SiO2 /GaAs [8], ZnSe/GaAs [14], WC/Si [10] and re- iations. cently in heterostructure semiconductor/topological insulator Si/Bi2 Te3 [13]. Generation of modes in more complicated structures such as interconnected Al stripes and pillars deposited on a Si substrate has been reported [20]. In this regard, polymer layers are attractive model systems to investigate phonon Copyright: © 2022 by the authors. confinement due to their elasticity, transparency and inexpensive fabrication in contrast to Licensee MDPI, Basel, Switzerland. rigid layers. The acoustic properties of supported polymer layers are of key importance in This article is an open access article distributed under the terms and advancing Nanoimprint Lithography and other polymer-coating-based technologies [21]. conditions of the Creative Commons As well as being a convenient system for the study of phonon confinement, the acoustic Attribution (CC BY) license (https:// properties of nanoscale layers of polymers are important for the fabrication of nanostruc- creativecommons.org/licenses/by/ tures, for a range of applications including micro/nano-electronics, nanophotonics and 4.0/). nanofluidics. An example of a study on the acoustic properties of polymer layers, ranging Crystals 2022, 12, 1212. https://doi.org/10.3390/cryst12091212 https://www.mdpi.com/journal/crystals Crystals 2022, 12, 1212 2 of 15 in thickness from 40 nm to 3 µm, was performed by Gomopoulos et al. [22,23]. By employ- ing BLS, they measured the longitudinal modulus in supported thin polymer films in the direction normal to the film surface. Furthermore, the manipulation of elastic wave propagation given by combining materials of significantly different acoustic impedance in a periodic structure has been investigated both experimentally and theoretically [24–30]. Potential applications of these periodic structures include the enhancement of interaction between acoustic and visible light waves, which can enable coherent phonon generation [31–34], concurrent modulation of light and sound [35–37], acoustic diodes [38] and sensors [39,40]. Recently, the study of hybrid superlattices consisting of alternating silica and poly(methylmethacrylate) (SiO2 - PMMA) studied by non-destructive BLS has advanced our knowledge of phononic wave propagation [41–43]. 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-TiO2 [44] and PMMA-BaTiO3 [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. Another method that enables revealing the surface acoustic waves’ existence in multilayers is the so-called picosecond ultrasonics based on the ultrafast pump–probe technique [46]. Recently, this method has been used to generate and detect surface acoustic waves on a structure consisting of nanoscale Al lines on a SiO2 layer deposited on Si substrate [47] and to probe the mechanical properties of single vegetal cells [48]. Among different mathematical approaches, the Green’s function formalism [1,28] is particularly convenient for studying the spectral properties of these types of excitations; in particular, it enables the calculation of the total or local DOS in which the resonant (leaky) modes appear as well-defined peaks. Let us recall that the vibrational DOS is often used for the integrated function over the phonon wavevector q in condensed matter physics such as the heat capacity and thermal conductivity [49]. Several works have reported the extraction of the DOS from measurable quantities such as: the nuclear resonant scattering to derive the phonon DOS [50], the differential conductance in scanning tunneling spectroscopy to obtain electron charge density [51] and the conductivity correction in semiconductors to deduce the Fermi energy [52]. Additionally, several works have used the DOS spectra to analyze the different modes obtained from the experimental BLS measurements in layered media [10,14,43,53–55]. In the case of oblique incidence (in-plane propagation) and for an opaque layer, the scattering occurs at the free surface of the layer and it was shown that the BLS spectra is proportional to the local DOS evaluated at the surface [10,28]. For a normal incidence (out-of-plane propagation), it was shown that the experimental BLS spectra follows a sine cardinal (sinc) function [14,22,23,55,56]. However, to our knowledge, there is no analytical comparison between the total DOS and the BLS spectra in the latter case. In this work we give an analytical and experimental demonstration of the comparison between BLS and DOS to determine the dispersion curves of pure longitudinal acoustic modes propagating perpendicular to a thin layer deposited on the substrate. In the case where the mismatch between the elastic parameters of the substrate and the adsorbed layer is high such as PMMA layer on Si substrate, we obtain closed-form expressions of BLS and DOS and demonstrate a simple relationship between these two quantities. A very good agreement between experimental and theoretical BLS spectra is found and compared to theoretical DOS spectra. In particular, we show that while the peaks in the DOS present a uniform behavior, the BLS spectra follow a sinc function shape around a given frequency fixed by the chosen laser wavelength. Another result of the paper is an exact relationship between DOS and the delay time of the reflection coefficient which can be measured in an acoustic experiment. present a uniform behavior, the BLS spectra follow a sinc function shape around a given frequency fixed by the chosen laser wavelength. Another result of the paper is an exact relationship between DOS and the delay time of the reflection coefficient which can be measured in an acoustic experiment. Crystals 2022, 12, 1212 This paper is organized as follows: in Section 2 we present the analytical3 calculations of 15 of the BLS intensity and DOS. These calculations are obtained using the Green’s function method [28]. Section 3 presents the experimental and numerical results concerning the BLS This and paper DOS spectra of PMMA is organized layerindeposited as follows: onpresent Section 2 we Si substrate. Sectioncalculations the analytical 4 summarizes the principal results of this work. of the BLS intensity and DOS. These calculations are obtained using the Green’s function method [28]. Section 3 presents the experimental and numerical results concerning the 2. BLSAnalytical Results and DOS spectra of PMMA layer deposited on Si substrate. Section 4 summarizes the principal results of this work. 2.1. Scattering Intensity in a Supported Layer We consider 2. Analytical (Figure 1) a structure formed out of one adsorbed layer (labeled i = 1) of Results thickness d depositedinon 2.1. Scattering Intensity a Supported Layer a homogeneous substrate (labeled i = 2) along the 𝑥3 axis. Al the interfaces We consider are(Figure taken1)toabe parallel structure to theout formed 𝑥1 of − one𝑥2 plane. adsorbed At layer normal incidence (labeled i = 1) and for of thickness d deposited on a homogeneous substrate (labeled isotropic media, there is a decoupling between longitudinal and transverse3waves i = 2) along the x axis. in the All the interfaces are taken to be parallel media and the former can be treated separately. to the x − 1 Eachx 2 plane. At normal medium within the stack incidence and is defined for isotropic media, there is a decoupling between longitudinal and by its mass density 𝜌𝑖 , its optical index 𝑛𝑖 , longitudinal sound velocity 𝑣𝑖 , and photoelastictransverse waves in the media and the former can be treated separately. Each medium within the stack is constant 𝑝𝑖 . The corresponding elastic constants and acoustic impedances are given by defined by its mass density ρi , its optical index ni , longitudinal sound velocity vi , and 𝑖 𝐶11 = 𝜌𝑖 𝑣𝑖2 and photoelastic 𝑍𝑖 = 𝜌p𝑖.𝑣𝑖The constant respectively (i = 1, 2). The dielectric constants of the vacuum corresponding elastic constants and acoustic impedances i supported layer i and substrate 2 are 𝜀 are given by C11 = ρi vi and Zi = ρi vi respectively0 𝜀1 and 𝜀2 , (i, =respectively. Since we 1, 2). The dielectric are interested constants of in longitudinal phonons propagating the vacuum, supported alongare layer and substrate theε 0normal , ε 1 andtoε 2the interfaces 𝑥Since , respectively. 3 , thewe BLS areis envis- aged in a in interested backscattering longitudinal phononsgeometry with bothalong propagating incident (𝑘𝑖 ) and the normal scattered to the interfaces(𝑘𝑠 x) 3wavevectors , the forming an angleinαa with BLS is envisaged the 𝑥3 axis. backscattering geometry with both Additionally, weincident consider (k i )the andso-called scatteredPP (k s )configu- wavevectors ration with both,forming thean angle α electric incident x3 axis. with thefield Additionally, 𝐸𝑖 and the scattered we consider electricthe so-called field 𝐸𝑠 , polarized PP configuration with both, the incident electric field E and the along the 𝑥2 direction (transverse electric polarization). Let us also notice that the i scattered electric field Es , magni- polarized along the x2 direction (transverse electric polarization). Let us also notice that the tudes of the scattered and incident wavevectors are very close to each other 𝑘𝑖 ≈ 𝑘𝑠 . In- magnitudes of the scattered and incident wavevectors are very close to each other k i ≈ k s . ′ deed, the phonon frequency Indeed, the phonon frequencyisismuch much lower than that of the light, so the incident 𝜔 and lower than that of the′ light,′ so the incident ωi0 and 𝑖 ′ scattered 𝜔 0 𝑠 scattered ω photon photon frequencies are almost frequencies are almost identical ω0 ≈ 𝜔 identical ω𝑖0 . ≈ 𝜔𝑠 . s i s Figure Figure 1.1.Schematic Schematicrepresentation representation of one of one layer layer i = 1)𝑖 of=width (labeled (labeled 1) ofdwidth d deposited deposited on a substrate on a substrate (labeledi 𝑖==2 2). (labeled Thex3𝑥axis ). The 3 axis is is normal normal to to the the surface. surface. k i 𝑘 and 𝑖 and k s are𝑘the 𝑠 are the incidentincident and and scattered scattered wavevectors in vacuum. The electromagnetic fields are polarized along the x2 axis. α and θ are the incident and refracted angles respectively. Crystals 2022, 12, 1212 4 of 15 There are two different mechanisms contributing to the scattered light intensity, namely the photoelastic and interface motion effects. The former mechanism can be described as an acoustic modulation of the dielectric constant in the bulk of each medium by means of ∂ui ( x3 ) the photoelastic constant pi of this medium: δε i = ε2i pi ∂x (pi stands for the photoelastic 3 constant p12 in layer i (i = 1, 2)). Then, in the presence of an incident electric field Ei ( x3 ), the strain due to displacement field ui ( x3 ) induces a polarization given by [57] ∂ui ( x3 ) P( x3 ) = δε i Ei ( x3 ) = ε2i pi Ei ( x3 ) (1) ∂x3 (layer ) The contribution of the scattered electric field Es of the layer at any point x30 is obtained by the following equation [57] Zd (layer ) ∂u1 ( x3 ) x30 , ω 0 ε21 p1 E1 ( x3 ) g0 x3 , x30 , ω 0 dx3   Es = (2) ∂x3 0 where g0 ( x3 , x30 , ω 0 ) is the electromagnetic Green’s function between the point x3 where the polarization is located and any point x30 where the scattered field is observed (here the vacuum medium, Figure 1), and ω 0 is the electromagnetic angular frequency. (sub) Similarly, we can calculate the contribution of the scattered field Es coming from the substrate as follows [57] Z0 (sub) ∂u2 ( x3 ) x30 , ω 0 ε22 p2 E2 ( x3 ) g0 x3 , x30 , ω 0 dx3   Es = (3) ∂x3 −L where L is the depth over which light is supposed to interact with phonons in the substrate. The second mechanism, called “moving interface” or “ripple” effect, results from the difference between the dielectric constants of two adjacent media i and j, since the displacement at their interface changes the dielectric constants of a thin layer of material from ε i to ε j and vice versa. Consequently, it is proportional to the interface displacement in contrast to the photoelastic effect, which depends on the strain. It can be written as [57]: (ripple) x30 , ω 0 = (ε 0 − ε 1 )u1 (d) E1 (d) g0 d, x30 , ω 0 + (ε 1 − ε 2 )u1 (0) E1 (0) g0 0, x30 , ω 0 (4)    Es The scattered intensity of the structure is then given as the square of the sum of all (layer ) (sub) (ripple) scattered electric fields Es , Es and Es , namely (layer ) (sub) (ripple) 2 I = Es + Es + Es (5) To calculate the above scattered fields (Equations (2)–(4)), we need the spatial distribu- tion of the displacement field and the incident electric field, as well as the electromagnetic Green function g0 ( x3 , x30 , ω 0 ) of the multilayer structure. The details of these calculations are given in the Supplementary Materials. Let us notice that the expressions of the displace- ment field and the incident electric field are very easy to calculate and the utilization of the Green function approach in this work is mainly motivated by the knowledge that the function g0 ( x3 , x30 , ω 0 ) will later allow us to calculate the vibrational density of states. For instance, the displacement field distribution can be obtained by launching an incident wave from the substrate (see also Supplementary Materials) and calculating its reflection at the boundaries x3 = 0 and x3 = d. In this way, we obtain the implicit expressions of the displacement fields in the layer (0 ≤ x3 ≤ d) and substrate (x3 ≤ 0), respectively, as follows h d d i u1 ( x3 ) = C A(ω )e− jk1 ( 2 − x3 ) + B(ω )e jk1 ( 2 − x3 ) (6) and u2 ( x3 ) = C (e− jk2 x3 + re+ jk2 x3 ) (7) Crystals 2022, 12, 1212 5 of 15 where k i = ω/vi is the wavevector in medium i, vi is the longitudinal velocity of phonons in medium i (i = 1, 2) and ω is their elastic angular frequency. C is the constant of integration of the displacement field and A(ω ) and B(ω ) are the amplitudes of the incident and reflected elastic waves in layer 1 containing all the information on the mechanical properties of the system. Their explicit expressions are given in Supplementary Materials. The parameter r corresponds to the acoustic reflected wave in the substrate; its expression (see Supplementary Materials) is given by     Z1 sin ωd v − jZ2 cos ωd v r=−  1  1 (8) ωd ωd Z1 sin v + jZ2 cos v 1 1 It is worth noticing that the displacement field throughout the structure must be normal- ized. Indeed, the constant C in Equations (6) and (7) is obtained by the normalization condition of the displacement field throughout the system (see Supplementary Materials), namely Z0 Zd   2 2 2 2 1 ω ρ2 |u2 ( x3 )| dx3 + ω ρ1 |u1 ( x3 )| dx3 = n(ω ) + ћω (9) 2 −L 0 where n(ω ) is the Bose–Einstein population factor and  h̄ is the reduced  Planck constant (h/2π). It should be noted that for high temperature n(ω ) + 21 ћω ≈ k B T, where k B is the Boltzmann constant. Similarly, the implicit expressions of the electric fields in the layer (0 ≤ x3 ≤ d) and substrate (x3 ≤ 0) can be written, respectively, (see Supplementary Materials) as 0 d 0 d E1 ( x3 ) = A0 ω 0 e− jk1 ( 2 − x3 ) + B0 ω 0 e+ jk1 ( 2 − x3 )   (10) 0 E2 ( x3 ) = t0 e− jk2 x3 (11) where k0i= ω0 n i /c = 2πni /λ is the wavevector of the electric field in each medium (i = 1, 2). λ is the wavelength of the incident light in vacuum. The amplitudes A0 (ω ) and B0 (ω ) of the incident and reflected electromagnetic waves in layer 1 contain all the information on the optical properties of the system. Their explicit expressions are given in Supplementary Materials. The Green’s function between any source point in the layer (0 ≤ x3 ≤ d) and the observation point in vacuum ( x30 ≥ d) is given by (see Supplementary Materials) 0 0 0 d 0 d n o g0 x3 , x30 , ω 0 = e jk0 ( x3 −d) A0 ω 0 e− jk1 ( 2 − x3 ) + B0 ω 0 e+ jk1 ( 2 − x3 )    (12) where k00 = ω 0 /c = 2π/λ is the wavevector of the electric field in vacuum. In the same way, the Green’s function between any source point in the substrate (x3 ≤ 0) and the observation point in vacuum ( x30 ≥ d) is given by (see Supplementary Materials) − jt0 jk00 (x30 −d) − jk02 x3 g0 x3 , x30 , ω 0 =  e e (13) 2k00 By replacing Equations (6), (10) and (12) in Equation (2), we obtain ( (layer ) 0 0 k1 k01 Es ( x30 ) = ε21 p1 d e jk0 (x3 −d) 2 [ A ( ω ) A 0 ( ω 0 )2 sin(k1 d) sin(k01 d)   (k1 +2k01 )d + B(ω ) B0 (ω 0 )2 ]sinc 2   (14) k1 d −2[ A(ω ) A0 (ω 0 ) B0 (ω 0 ) + B(ω ) B0 (ω 0 ) A0 (ω 0 )]sinc 2  ) (k1 −2k01 )d h i 2 2 + A(ω ) B0 (ω 0 ) + B(ω ) A0 (ω 0 ) sinc 2 Crystals 2022, 12, 1212 6 of 15 Due to the shape of the sinc function, the three terms in the RHS of Equation (14) take significant values only around k1 = vω = −2k01 = −4πn1 /λ (anti-Stokes contribution), 1 k1 = vω = 0 (elastic scattering) and k1 = vω = 2k01 = 4πn1 /λ (Stokes contribution). We can 1 1 notice that the Stokes term in Equation (14) gives a peak around the Brillouin frequency of the layer f 1 = 2n1 v1 /λ. Similarly, by replacing Equations (7), (11) and (13) in Equation (3), we can calculate the contribution of the scattered field coming from the substrate as follows 2 k2 k02 t0 (k2 + 2k02 ) L (k2 − 2k02 ) L      (sub) 0 0 x30 = ε22 p2 e jk0 ( x3 −d) L  Es 2 sinc + r sinc (15) sin(k2 L) sin k02 L 2 2 where k2 = ω/v2 , k02 = ω 0 n2 /c = 2πn2 /λ and k00 = ω 0 /c = 2π/λ. We can notice that the Stokes term in Equation (15) gives a peak around the Brillouin frequency of the substrate f 2 = 2n2 v2 /λ. Finally, From Equations (6), (10) and (12), one can derive the moving interface (ripple) contribution (Equation (4)) as  h d d ih d d i2  jk − jk 0 0 jk 0 0 − jk (ε 0 − ε 1 ) A(ω )e 1 2 + B(ω )e 1 2 A (ω )e 1 2 + B (ω )e 1 2      (ripple) 0 jk00 ( x30 −d)  Es x3 = Ce (16) j 2     − 0 (ε − ε 2 )(1 + r )t 0   2k0 1 2.2. Density of States and Reflection Delay Time The density of states (DOS) of the whole system is an interesting quantity as it enables us to deduce the distribution and the weight of the different modes. In the case of the supported layer, we can calculate the variation of DOS (∆n(ω )) between the supported layer and the two constituting materials (layer and substrate) taken separately. This quantity is given by [1] 1 d ∆n(ω ) = [ Arg(g(0, 0))] (17) π dω where g(0, 0) is the acoustic Green’s function at the interface x3 = 0 (see Supplementary Materials). From the expression of g(0, 0), one can easily derive 1 Z1 Z2 ∆n(ω ) =  2    2 (18) π Z1 sin ωd v + Z2 cos ωd v 1 1 This variation of DOS can be related to the acoustic reflection delay time defined by d τ( ω ) = [ Arg(r )] (19) dω From the expression of r (Equations (8) and (18)), one can further conveniently derive an expression for the acoustic reflection delay time as 2Z1 Z2 τ( ω ) =  2    2 (20) Z1 sin ωd v + Z2 cos ωd v 1 1 Equations (18) and (20) provide the relationship between the variation of DOS and the reflection delay time τ as: τ(ω ) = 2π∆n(ω ) (21) 2.3. Particular Case of a Soft Layer on a Hard Substrate In order to explain the behavior of the experimental BLS spectra, we have simplified the theoretical expression of the light scattered intensity of the system (layer/substrate) which is defined as the sum of both layer and substrate contributions. In order to examine each contribution, it is convenient to evaluate their amplitudes separately. However, several assumptions can be made for a soft layer on a hard substrate, i.e., (i) the main intensity Crystals 2022, 12, 1212 7 of 15 contribution comes from the surface layer as phonons are confined essentially in the top layer. (ii) The light wave in the substrate is attenuated due to imaginary parts of the complex refractive index of the substrate (see Section 3.2). (iii) The substrate contribution in the scattering intensity falls at a higher frequency far from those of the layer modes (see Section 3.2). (iv) The photoelastic constant of the substrate (here silicon) is very small in comparison with the one of the surface layer (PMMA) (see Table 1). With the above assumptions, the main contributions to the BLS intensity (Equation (5)) come from the (layer ) photoelastic contribution of the layer (Es , Equation (2) or Equation (14)) and the moving (ripple) interface contribution (Es , Equation (4) or Equation (16)). We shall see below that the latter contribution only contributes non-negligibly if the layer is very thin, namely below (1) 300 nm. Otherwise, the only remaining contribution comes from Es , whose Stokes term can be written from Equation (14) as:  ! (1) 0  h 0 0 2 0  i 0 2 k1 − 2k01 d 0 0 Es x3 ∝ A(ω ) B ω + B(ω ) A ω e − k 0 ( x3 − d )  sinc (22) 2 Table 1. Physical quantities of the PMMA layer and Si substrate used in the theoretical calculations. Mass density (ρ), sound velocity (v), photoelastic constant ( p12 ) and refractive index (n). Material p (Kg/m3 ) v (m/s) p12 n PMMA 1150 2778 0.3 1.4932 Si 2335 8431 0.01 3.5 − 0.26i From the expressions of A(ω ), B(ω ), A0 (ω 0 ) and B0 (ω 0 ) (Equations (S10), (S11), (S23) and (S24)), one can get the following simple expression "   (1) 0 (k1 +2k01 )d Es ( x3 ) ∝ n1 + n2 sin 2 2  2 !  # sinc (k1 −2k10 )d (23) 2 (k1 +2k01 )d 0 0   e − k 0 ( x3 − d ) − j2n1 n2 cos 2   Z2 cos ωd v − jZ 1 sin ωd v 1 1 where the main modes of the surface layer are observed around k1 = 2k01 . We can see that the optical modulation of the refractive index in the system will affect only the amplitude in Equation (23) through the term between brackets, which means that the main behavior of the scattering amplitude follows a sinc function. Indeed, around k1 = 2k01 , the intensity can be written very simply as    sinc2 vω − q d/2 1 I∝   2   2 (24) ωd Z2 cos v 1 + Z1 sin ωd v 1 where q = 2k01 = 4πn1 /λ is the analog of scattering wavevector in layer 1. From Equations (17) and (24), one can see that the scattered intensity can be con- sidered  as a density   of states modulated by the square modulus of the sinc function sin c2 ω v1 − q d/2 , namely    ω I ∝ ∆n(ω ) sinc2 − q d/2 (25) v1 3. Numerical and Experimental Results 3.1. Experimental Setup The polymer chosen for the investigation was PMMA 75K300 from MicroChem Corp. Layers with thickness ranging from 250 nm to 10 µm were fabricated by spin-coating for 30 s with angular velocities ranging from 500 to 3000 revolutions per minute on top of a Si substrate. The spectra were taken with a multipass (3+3) Tandem Fabry–Perot Crystals 2022, 12, 1212 8 of 15 Interferometer from JRS Scientific Instruments. The λ = 514.5 nm line of an Ar gas-ion laser was used for the incident radiation, at a mirror spacing of 6 mm and scanning amplitude of 490 nm, resulting in a free spectral range (FSR) of ± 23.8 GHz. The backscattering angle was set to 8 degrees [58]. The finite angle prevented reflected light from entering the collection objective, and subsequently the spectrometer. This angle causes an increase in the effective cavity length of d/cosθ. However, due to Snell’s law, the angle inside the polymer is reduced to 5.3 degrees, which causes an overestimation of the thickness of the polymer layer of less than half a percent. As the polymer is assumed to be isotropic, the change in angle does not affect the acoustic velocity. The laser power was kept as small as possible to avoid any heating effect, the glass transition temperature of PMMA was taken around 110 ◦ C, the polarization of incident and reflected beam was PP with respect to the sample plane, the crystallographic orientation of the Si substrate was (100), the resistivity or carrier concentration of the Si substrate was 11.5 ohm·cm, p doped boron and the FPI finesse for FSR = 23.8 GHz was set to 100. 3.2. Results and Discussion The Brillouin spectra for the PMMA layers of different thicknesses are shown in Figure 2a by full curves. When the thickness of the film was reduced to below one microme- ter, a triplet structure was observed around q = 0.036 nm−1 (i.e., f 1 = qv1 /2π = 15.9 GHz). Indeed, multiple reflections of an acoustic excitation from the film surface and interface with the substrate give rise to the equally spaced modes yielding to standing waves. Fur- thermore, a sufficiently large elastic impedance Z = ρv mismatch (involving two physical quantities, density $ and longitudinal sound velocity v) is required for these standing waves observations and results in the quantification of their frequencies. At normal incidence, the eigenmodes   (standing modes) of the soft layer in contact with a hard substrate are given by cos ωd v1 = 0, i.e., f m = (2m + 1)v1 /4d (m is an integer). One can notice that for films with further increasing thicknesses, the spacing between the equidistant longitudinal acoustic modes ∆ f m = v1 /2d decreases, giving rise to a single mode (for thick layer d = 10 µm) falling at the frequency of the longitudinal bulk-like acoustic mode in the polymer, namely f 1 = v1 q/2π ' 16 GHz. Indeed, for a thick layer, the frequencies of the modes become very close to each other and the sinc function becomes a delta function, with the result that only the phonon wavevector respecting the conservation of the momentum q = k i − k s is observed, as illustrated in Figure 2a for d = 10 µm. The intensity of the peak coming from the longitudinal bulk-like acoustic mode in the substrate falls at a high Brillouin frequency f 2 = 2n2 v2 /λ ' 114 GHz, and therefore does not interfere with the spectra coming from the layer observed around 16 GHz. We have reproduced the theoretical results corresponding to the experimental spectra in Figure 2a represented by open circles using Equations (6), (11), (14) and (15). The elastic, optic and photoelastic parameters of both PMMA and Si materials used in the theoretical calculations are listed in Table 1 [41,59,60]. Good agreement between the theoretical curves and the experimental data was found. The theoretical spectra had to be convoluted with the instrumental broadening function (Gaussian with Γ ≈ 0.53 GHz) to match the experimental spectra in Figure 2a. It is worth noticing that the contribution coming from the motion of the interfaces (Equation (14)) is negligible, especially for thick layers as it is explained in Figure A1 in the Appendix A. Figure 2b shows the displacement field versus the space position for the mode at f = 16 GHz and d = 490 nm. One can see that the elastic field is well confined to the layer and almost vanishes in the substrate. Figure 2c shows the behavior of the transmission of the optical wave as a function of the wavelength. The transmission amplitude (evaluated at the top of the substrate x3 = 0) shows Fabry–Perot oscillations with an amplitude of around 0.75 at λ = 514.5 nm. The variation of the electric field in the x3 direction is plotted in Figure 2d for λ = 514.5 nm and d = 490 nm. One can notice a sharp decrease of the amplitude of the field as a function of the penetration depth which results from the absorption coefficient in Si (see Table 1). These results demonstrate the weak contribution ystals 2022, 12, x FOR PEER REVIEW 9 o Crystals 2022, 12, 1212 9 of 15 of the substrate to the scattered field and justify the assumptions made in deducing the Figureanalytical simple 3 showsexpression the theoretical BLS (Equation spectra (25)). obtained The length from L of the the approximate substrate for which lightexpress (Equation (22)). interacts withOne can issee phonons a than less quite good 1 µm agreement (Figure 2d). between the approximate results a Figure 3 shows the theoretical BLS spectra obtained the experimental data for each value of the layer thickness, from the which approximate expression confirms the validity (Equation (22)). One can see a quite good agreement between the approximate results and the simple approximate expression given in Equation (23). The discrepancy between the experimental data for each value of the layer thickness, which confirms the validity observed of the and simplecalculated approximate BLS in Figure expression 3 is in given due to the(23). Equation simple The analytical discrepancyexpression between (Eq tion the (23)) used where observed the contribution and calculated BLS in Figureof3the substrate is due in the to the simple BLS asexpression analytical well as the rip mechanism (Equationare neglected. (23)) used where the contribution of the substrate in the BLS as well as the ripple mechanism are neglected. d = 10,000 nm d = 1000 nm d = 490 nm d = 340 nm d = 250 nm x3 (nm) x3 (nm) Figure 2. (a)2.Theoretical Figure (a) Theoretical(open (opencircles) andexperimental circles) and experimental (continuous (continuous curves) curves) Brillouin Brillouin light scat light scat- ing for a series of supported PMMA films with different thicknesses deposited on a Si substrate. T tering for a series of supported PMMA films with different thicknesses deposited on a Si substrate. mainThe main spectra spectra appearappear at theat frequency the frequency position of position of the the longitudinal longitudinal acoustic phonon acoustic aroundaround phonon 4𝜋𝑛1 q = 4πn1 = 0.036−1 nm −1 i.e., f = qv /2π = 15.9 GHz. (b) Square modulus of the displacement = 0.036 λ nm i.e., 𝑓1 = 𝑞𝑣1 /2𝜋 = 15.9 GHz. (b) Square modulus of the displacement field v 1 1 𝜆 field versus the space position for the mode at f = 16 GHz and d = 490 nm. (c) Optical transmission sus the space position for the mode at f = 16 GHz and d = 490 nm. (c) Optical transmission throu through the layer versus the wavelength. The circle on the abscissa axis corresponds to 514.5 nm the layer versusused wavelength the in wavelength. the experiment.The(d)circle Squareon the abscissa modulus axis corresponds of the electric field versus the to 514.5 space nm wavelen position. used in the experiment. (d) Square modulus of the electric field versus the space position. d = 10,000 nm d = 1000 nm d = 490 nm ystals 2022, 12, x FOR PEER REVIEW Crystals 2022, 12, 1212 10 of 15 d = 10,000 nm d = 1000 nm d = 490 nm d = 340 nm Crystals 2022, 12, x FOR PEER REVIEW 10 of 16 d = 250 nm 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 experi- ments. Figure 3. Theoretical spectra (open circles) evaluated using the approximate expression (Equation (25)) Figure 3. Theoretical spectra (open circles) evaluated using the approximate for the same thicknesses of PMMA film on Si substrate. Solid curves correspond to experiments. (25))The variation of the density of states (DOS) is calculated using Equation (17) for dif- for the same thicknesses of PMMA film on Si substrate. Solid curves ferent The layer thicknesses variation as shown of the density in Figure of states (DOS) is4a. As predicted calculated from Equation using Equation (17), for (17) for different ments.𝜔𝑑 1 𝑍2 cos ( thicknesses ) = 0 (i.e.,as𝑓shown 𝑚 = (2𝑚 + 1)𝑣1 /4𝑑 ), Δ𝑛(𝜔) reaches a maximum (17),value for cosof ωd ≈2 layer 𝑣1 in Figure 4a. As predicted from Equation v𝜋1 𝑍1 = 0 𝜔𝑑  1 𝑍1 and for sin ( ) = 0 (i.e., 𝑓𝑚 = 𝑚𝑣1 /2𝑑 ), Δ𝑛(𝜔) reaches a1 Zminimum value of ≈ 1 1)v1 /4d), ∆n(ω ) reaches a maximum value of π Z ≈ 2 and for sin v 𝜋 𝑍= 2 ωd = (2mvariation (i.e., f m The 𝑣+ of the density of states (DOS) is calculated 2 0 using E 1 1 0.05. Figure 4 (b) provides the frequencies of the layer modes 1 Z1 (full curves) as a function of (i.e., ferentf = mv layer/2d), ∆n ( ) thicknesses reaches a minimum as shown value of ≈ 0.05. Figure 4b provides the from thein πFigure Z2 of the4a. DOSAs predicted in Fig- from m 1 ω the thickness d. These modes are obtained maxima displayed frequencies of the layer modes (full curves) as a function of the thickness d. These modes are ure 4(a)𝜔𝑑 and they coincide reasonably well with the modes obtained from the quantified cos ( 𝑣 from obtained )𝑓==the 0(2𝑚 (i.e., maxima 𝑓/4𝑑. of the =DOS (2𝑚 + 1)𝑣 displayed /4𝑑),4a Δ𝑛 in Figure and(they𝜔) coincide reaches a maximum reasonably expression + 1)𝑣1𝑚 One can notice a 1 decrease of their frequencies as far as d well with1the𝑚modes𝜔𝑑 obtained from the quantified expression f m = (2m + 1)v1 /4d. One can increases. As a matter of comparison, we also reported the modes obtained from the max- and afor notice sinof(their) frequencies decrease ima of the BLS experimental = 0spectra(i.e., 𝑓𝑚as= as far around 𝑚𝑣 /2𝑑), d increases. As a matterΔ𝑛(of𝜔comparison, ) reaches 16 GHz1 by red circles. We can see that the a minim we also reported the modes𝑣obtained 1 from the maxima of the BLS experimental spectra around 16 separation 0.05. between4b Figure the modes decreases as far as d increases. As demonstrated in Equa- GHz by red circles. We provides can see that the the frequencies separation between of the the modes layer modes decreases as far as (full d curv tion (21), increases. the As DOS can demonstrated be extracted in Equation also from (21), thethe DOS reflection can be delay extracted time, alsowhich from is the a measur- reflection thequantity able thickness d. These [61]. This propertymodes has been are used obtained fromthethe DOSmaxima of the DO delay time, which is a measurable quantity [61].recently to extract This property has been fromrecently used the delay to uremeasurement time extract 4a the and DOS from theyinthe coincide photonic delay timecoaxial reasonably well with cables [62].in photonic measurement coaxial the cablesmodes [62]. obtained f expression 𝑓𝑚 = (2𝑚 + 1)𝑣1 /4𝑑. One can notice a decrease of their f d increases.1000 Asnm a matter of comparison, we also reported the modes maxima of the BLS experimental spectra around 16 GHz by red circl the separation 490 nm between the modes decreases as far as d increases. A DOS (a.u.) Equation (21), the DOS can be extracted also from the reflection del measurable 340 quantity nm [61]. This property has been used recently to ex the delay time measurement in photonic coaxial cables [62]. 250 nm 1000 nm Figure 4. 4. (a) Variation Variation of of density density of of states states (in (in arbitrary arbitrary units) units) of layer layer modes for the different layer thicknesses. (b) Quantified frequencies thicknesses. (b) Quantified 490 frequencies nm of layer modes as a function of layer of layer modes as a function of layer thickness. thickness. PeakPeak posi- positions DOS (a.u.) tions of experimental spectra are given as red circles around the frequency of experimental spectra are given as red circles around the frequency 16 GHz. 16 GHz. In order to give a comparison between BLS and DOS spectra, we have plotted in Figure 5 the DOS (blue curves) and the theoretical BLS spectra without convolution (red 340 nm curves) for d = 490 nm. One can see clearly that among all the peaks in the DOS, only those falling around 16 GHz are observed in the scattering spectrum as the latter decreases around the main peak as a sinc function. This result confirms our theoretical findings Crystals 2022, 12, 1212 11 of 15 2022, 12, x FOR PEER REVIEW In order to give a comparison between BLS and DOS spectra, we have plotted in Figure 5 the DOS (blue curves) and the theoretical BLS spectra without convolution (red curves) for d = 490 nm. One can see clearly that among all the peaks in the DOS, only those falling around 16 GHz are observed in the scattering spectrum as the latter decreases around the main peak as a sinc function. This result confirms our theoretical findings demonstrated in Equation (25). DOS BLS 0 5 10 15 20 25 Frequency (GHz) Figure 5. Variation of the DOS (blue curves) and BLS intensity (red curves) as a function of the Figure 5. Variation of the DOS (blue curves) and BLS intensity frequency for the PMMA layer of thickness d = 490 nm. frequency for the PMMA layer of thickness 𝑑 = 490 nm. As a matter of completeness, we have also numerically checked the effect of the impedance of the substrate on the shape of the DOS and the theoretical BLS spectra for the same film of thickness d = 490 nm. We have found that when the impedance of the As a matter of completeness, we have also numericall substrate Z2 is lower than that of the PMMA layer Z1 (Z2  Z1 , i.e., Z2 = 0.1Z1 ), then the pedance of the substrate on the shape of the DOS and the modes of the surface layer become very close to the stationary modes of a layer with free surfaces on bothsides (called an unsupported layer or membrane [17,63]). These modes same ωd thickness 𝑑 = 490 nm. We have found that w are given film by sin ofv = 0 (i.e., f m = mv1 /2d) as it is shown in the DOS of Figure 6a 1 strate 𝑍2 is lower displayed by a blue curve. than ofthat The behavior of(redthe the BLS PMMA curve) shows againlayer 𝑍1 (𝑍2 ≪ a sinc function around the main peak at 17 GHz, similar to the case of the PMMA layer with a free surface modes from ofandthe one side surface a blocked surfacelayer from thebecome very5).close other side (Figure towhere In the case the thestation impedance of the substrate and the surface layer are such that 0.5Z1 < Z2 < 1.5Z1 , then surfaces on both sides (called an unsupported layer or me oscillations in the DOS become less deep, especially for Z2 ≈ Z1 , and the behavior of ωdthe shape of a sinc function as it is shown in Figure 6b,c. the BLS spectra does not follow are given by sin ( ) = 0 (i.e., 𝑓 = 𝑚𝑣 /2𝑑) as it is show v1is due to the fact that 𝑚 The reason for this discrepancy the modes are1not well confined in the layer and the displacement field is expanded over the substrate. Therefore, the thin played by a blue curve. The behavior of the BLS (red curve layer behaves as a large layer where only the main mode around 15.5 GHz is observed around the main peak at 17 GHz, similar to the case of the P as it is shown in Figure 6c. Additionally, the frequencies of the resonant modes in the DOS for 0 ≤ Z2 < 0.5Z1 are close to those of the PMMA layer with free surfaces, whereas from one side and a blocked surface from the other side (F the frequencies of the resonant modes in the DOS for 1.5Z1 ≤ Z2 are close to those of the PMMA layer with blocked surface from the bottom side (Figure 4a). impedance of the substrate and the surface layer are such oscillations in the DOS become less deep, especially for 𝑍2 BLS spectra does not follow the shape of a sinc function as (c). The reason for this discrepancy is due to the fact that th in the layer and the displacement field is expanded over the Crystals 2022, 12, x FOR PEER REVIEW Crystals 2022, 12, 1212 12 of 15 (a) DOS BLS 10 12 14 16 18 20 22 24 (b) DOS BLS 10 12 14 16 18 20 22 24 (c) BLS DOS 10 12 14 16 18 20 22 24 Frequency (GHz) Figure Figure 6.6.Same Same as as in in Figure Figure 5 but5 for buta for a hypothetical hypothetical substratesubstrate with impedance with impedance 𝑍2 = 0.1 𝑍 Z2 = 0.1 Z1 (a), Z 0.5= 0.5 2 𝑍 (b) Z (b) and Z = 1.5 Z 1 and 𝑍 2= 1.5 𝑍1 (c). (c). 1 2 1 4. Conclusions 4. Conclusion The results presented in this paper are based on an analytical calculation of the Brillouin Thelight scattering results and thein presented density of states this paper forbased are a layerondeposited on a substrate. an analytical calculation We have presented first a detailed analytical calculation of the BLS intensity taking into louin accountlight scattering the modulation and the of elastic, density optic of states and photoelastic for a layer constants deposited in the different on a sub media have presented constituting firstThen, the system. a detailed in orderanalytical to explain thecalculation of the behavior of the BLS intensity experimental light taki scattering intensity, we have simplified the theoretical expression of count the modulation of elastic, optic and photoelastic constants in the differ the intensity of the system (layer/substrate) in the case of a soft layer on a hard substrate, which enables constituting the system. Then, in order to explain the behavior of the experim us to give an analytical comparison between BLS and DOS intensities. Theoretical and scattering experimentalintensity, we have results are given simplified in the case of PMMAthe theoretical layer expression of the inten on Si substrate. system (layer/substrate) A very good agreement has inbeen the obtained case of abetween soft layer on a and theoretical hard substrate,BLS experimental which en spectra. In particular, we have shown that while the peaks in the DOS give an analytical comparison between BLS and DOS intensities. Theoretical a present a uniform behavior, the BLS spectra follow a sinc function shape around a given frequency fixed by mental the chosenresults are givenThese laser wavelength. in theresults casealso of PMMA layer remain valid on the when Si substrate. impedance of the A very substrate good is lower thanagreement has been that of the surface obtained layer; however, between a differencetheoretical and experim appears between spectra. In particular, we have shown that while the peaks in the DOStopresent DOS and BLS spectra when the impedances of the substrate and surface layer are close each other. The results presented in this paper can be generalized to multilayered structures. behavior, the BLS spectra follow a sinc function shape around a given frequen This work is in progress. the chosen laser wavelength. These results also remain valid when the imped substrate is lower than that of the surface layer; however, a difference appea DOS and BLS spectra when the impedances of the substrate and surface layer each other. The results presented in this paper can be generalized to multilay Crystals 2022, 12, 1212 13 of 15 Crystals 2022, 12, x FOR PEER REVIEW 13 of 16 Supplementary Materials: The following supporting information can be downloaded at: https: //www.mdpi.com/article/10.3390/cryst12091212/s1, S1: Elastic displacement field calculation; S2: Author field Electric Contributions: Conceptualization, and electromagnetic C.M.S.T. Green’s function and B.D.-R.; calculation; S3: Formal analysis,ofJ.C., Determination the E.C.-A. constantand of F.A.; Investigation, E.H.E.B. and B.D.-R.; Methodology, C.M.S.T.; Software, O.E.A.; Supervision, normalization of the displacement field. Figure S1: Schematic representation of one layer (labeled i = 1) C.M.S.T. of width dand B.D.-R.; deposited onValidation, F.A.; Visualization, a substrate (labeled i = 2). The x3E.C.-A. and F.A.; axis is normal Writing to the surface.– original draft, O.E.A. and J.C.; Writing – review and editing, E.H.E.B. All authors have read and agreed to the Author published version of theConceptualization, Contributions: manuscript. C.M.S.T. and B.D.-R.; Formal analysis, J.C., E.C.-A. and F.A.; Investigation, E.H.E.B. and B.D.-R.; Methodology, C.M.S.T.; Software, O.E.A.; Supervision, Funding: ICN2 is funded by the CERCA program/Generalitat de Catalunya and is supported by the C.M.S.T. and B.D.-R.; Validation, F.A.; Visualization, E.C.-A. and F.A.; Writing—original draft, O.E.A. Severo Ochoa program from Spanish MINECO (Grant No. SEV-2017-0706). F.A., E.C.A. and C.S.T and J.C.; Writing—review and editing, E.H.E.B. All authors have read and agreed to the published acknowledge support from the Spanish MICINN project SIP (PGC2018-101743-B-I00). version of the manuscript. Conflicts of Interest: The authors declare no conflicts of interest. Funding: ICN2 is funded by the CERCA program/Generalitat de Catalunya and is supported by the Severo Ochoa program from Spanish MINECO (Grant No. SEV-2017-0706). F.A., E.C.-A. and C.M.S.T. Appendix support from the Spanish MICINN project SIP (PGC2018-101743-B-I00). acknowledge Conflicts Effect ofofthe Interest: Moving TheInterface authors declare no conflict Mechanism on of interest. BLS Spectra In Figure A1, we give a comparison of the BLS spectra with (continuous curves) and Appendix A. Effect of the Moving Interface Mechanism on BLS Spectra without (dashed curves) taking into account the term associated to the moving interface In Figure or ripple effectA1, we give(14)). (Equation a comparison of the We can see thatBLS bothspectra curveswith (continuous are almost curves) similar and especially without (dashed for thick layers. curves) taking into account the term associated to the moving interface or ripple effect (Equation (14)). We can see that both curves are almost similar especially for thick layers. 30 d = 250 nm d = 340 nm 25 d = 490 nm d = 1000 nm Intensity (a.u.) 20 15 10 5 0 5 10 15 20 25 Frequency (GHz) −1 1 Figure A1. Figure A1. Theoretical Theoretical Brillouin Brillouinlight lightscattering scatteringspectra displayedatatqq==0.036 spectradisplayed 0.036nmnm−for a series for of a series supported PMMA films with different thicknesses deposited on Si substrate, with (continuous of supported PMMA films with different thicknesses deposited on Si substrate, with (continuous curves) and curves) and without without (dashed (dashedcurves) curves)taking takinginto intoaccount accountthe thedeformation deformationof ofthe theinterfaces. interfaces. References References 1. 1. Dobrzynski, L.; Dobrzynski, L.; El Boudouti, Boudouti, E.H.; Akjouj, A.; Pennec, Y.; Al-Wahsh,Al-Wahsh H.; H.; Lévêque, Lévêque, G.; G.; Djafari-Rouhani, Djafari-Rouhani,B. B.Phononics, Phononics, 1st ed.; Elsevier: Amsterdam, Elsevier: Amsterdam, The The Netherlands, Netherlands, 2017. 2017. 2. 2. Carlotti,G. Carlotti, G. 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