applied sciences

Article Elastic Characterization of Transparent and Opaque Films, Multilayers and Acoustic Resonators by Surface Brillouin Scattering: A Review

Giovanni Carlotti ID Dipartimento di Fisica e Geologia, Università di Perugia, 06123 Perugia, Italy; [email protected]; Tel.: +39-0755-852-767

Received: 17 December 2017; Accepted: 5 January 2018; Published: 16 January 2018

Abstract: There is currently a renewed interest in the development of experimental methods to achieve the elastic characterization of thin films, multilayers and acoustic resonators operating in the GHz range of frequencies. The potentialities of surface Brillouin light scattering (surf-BLS) for this aim are reviewed in this paper, addressing the various situations that may occur for the different types of structures. In particular, the experimental methodology and the amount of information that can be obtained depending on the transparency or opacity of the film material, as well as on the ratio between the film thickness and the light wavelength, are discussed. A generalization to the case of multilayered samples is also provided, together with an outlook on the capability of the recently developed micro-focused scanning version of the surf-BLS technique, which opens new opportunities for the imaging of the spatial profile of the acoustic field in acoustic resonators and in artificially patterned metamaterials, such as phononic crystals.

Keywords: acousto-optics at the micro- and nanoscale; photon scattering by phonons; elastic constants of films and multilayers

1. Introduction The problem of a detailed knowledge of the elastic constants of thin film materials has attracted much attention in recent years, because of the growing importance of single- and multi-layered structures in advanced applications such as in devices for information and communication technology (ICT). For instance, current filters and duplexers in mobile communication devices exploit either bulk or surface acoustic wave (BAW or SAW) resonators operating at several GHz [1–4], whose design and optimization require a detailed knowledge of the elastic properties of the constituent layers. In fact, as illustrated in Figure1, free-standing bulk acoustic wave resonators (FBAR) and solidly mounted resonators (SMR) consist of a stack of several materials, including the active resonator (usually a film of AlN, ScAlN or ZnO), metallic layers for electrical electrodes (Ti, Au, etc.) and other materials for temperature compensation (SiO2) or for Bragg mirrors (SiO2 and W). The thicknesses of the above layers range from a few tens of nanometers up to a few microns. It follows that it is crucial to dispose of advanced techniques and methods to provide information about the elastic constants of the constituent layered materials and to image the spatial profile of the acoustic field, both inside and outside the resonator. In this paper, I will review the capability of the surface Brillouin light scattering technique (surf-BLS) to the above aims and in particular to successfully achieve a selective determination of the elastic constants of various types of films and multilayers. This technique is based on the inelastic scattering of photons by thermal phonons, naturally present in any material at room temperature, through the modulation of the optical constants of the medium, which is induced in transparent media via the elasto-optic effect [5,6]. However, in the case of opaque media, either metals or semiconductors,

Appl. Sci. 2018, 8, 124; doi:10.3390/app8010124 www.mdpi.com/journal/applsci Appl. Sci. 2018, 8, 124 2 of 19 Appl. Sci. 2018, 8, 124 2 of 19

the rippling of the free surface, caused by the presence of thermal phonons, gives a substantial the rippling of the free surface, caused by the presence of thermal phonons, gives a substantial contribution to the light scattering [7]. In fact, the periodic modulation of the optical constant (either in contribution to the light scattering [7]. In fact, the periodic modulation of the optical constant (either the interior or at the surface of a medium) associated to a propagating acoustic wave results in a moving in the interior or at the surface of a medium) associated to a propagating acoustic wave results in grating so that Brillouin scattering can be understood in very simple terms as a Doppler shift of the a moving grating so that Brillouin scattering can be understood in very simple terms as a Doppler diffracted light. From the quanto-mechanical point of view, however, one considers a photon-phonon shift of the diffracted light. From the quanto-mechanical point of view, however, one considers a collision in which the frequency and the momentum are conserved through phonon creation or photon-phonon collision in which the frequency and the momentum are conserved through phonon annihilation, corresponding to either Stokes or anti-Stokes processes, respectively. It follows that in creation or annihilation, corresponding to either Stokes or anti-Stokes processes, respectively. It follows a typical Brillouin light scattering experiment, the probed phonons have wavelength comparable to that in a typical Brillouin light scattering experiment, the probed phonons have wavelength comparable that of light sand this implies that their frequencies are in the range between about 10 and 150 GHz. to that of light sand this implies that their frequencies are in the range between about 10 and 150 GHz. The cross section of the scattering process is relatively low, so that in order to extract the tiny inelastic The cross section of the scattering process is relatively low, so that in order to extract the tiny inelastic component of light from the elastically scattered contribution, a high-resolution spectrometer is component of light from the elastically scattered contribution, a high-resolution spectrometer is required. To this aim, the best combination of high resolution and good throughput is achieved using required. To this aim, the best combination of high resolution and good throughput is achieved using a multipass Fabry-Perot interferometer. a multipass Fabry-Perot interferometer.

FigureFigure 1.1. SchematicSchematic drawingsdrawings ofof free-standingfree-standing ((leftleft)) andand solidlysolidly mountedmounted ((rightright)) bulkbulk acoustic-waveacoustic-wave resonators,resonators, namednamed free-standingfree-standing bulkbulk acousticacoustic wavewave resonatorsresonators (FBAR)(FBAR) andand solidlysolidly mountedmounted resonatorsresonators (SMR),(SMR), respectively, that are are currently currently used used in in filters filters and and duplexers duplexers for for mobile mobile communication communication devices. devices.

MeasurementMeasurement of the whole set ofof constantsconstants is usuallyusually outout ofof thethe reachreach ofof conventionalconventional staticstatic techniques,techniques, suchsuch asas micromechanicalmicromechanical testtest oror nanoindentation,nanoindentation, whichwhich cancan onlyonly give information about specificspecific elasticelastic modulimoduli [[8].8]. On the otherother hand,hand, useuse ofof acousticacoustic techniquestechniques basedbased onon surfacesurface acousticacoustic waveswaves (SAW)(SAW) presents presents technical technical difficulties difficulties connected connected with with the the fabrication fabrication of the of acousticalthe acoustical delay delay line. Inline. addition, In addition, one has one to has consider to consider that in that the in usual the usual frequency frequency range ofrange hundreds of hundreds of MHz, of theMHz, large the values large ofvalues the acoustic of the acoustic wavelength, wavelength, compared compared to the film to thickness,the film thickness, require a require careful considerationa careful consideration of the effect of ofthe the effect substrate of the onsubstrate the propagation on the propagation of the SAW of [th9].e TheseSAW [9]. limitations These limitations can be overcome can be overcome using surf-BLS, using becausesurf-BLS, it because has specific it has advantages specific advantages over conventional over conventional techniques. techniques. In particular, In particular, - it does not require any external generation of acoustic waves (since it probes thermal acoustic - it does not require any external generation of acoustic waves (since it probes thermal acoustic phonons naturally present into the medium); phonons naturally present into the medium); - it is a local probe, since the investigated portion of the sample is defined by the dimension of the - focusedit is a local laser probe, spot and since this the permits investigated to operate portion a scan of of the the sample elastic properties is defined of by inhomogeneous the dimension specimens;of the focused laser spot and this permits to operate a scan of the elastic properties of - itinhomogeneous is suited to analyze specimens; short wavelength and high-frequency phonons, up to several GHz, such - asit isthose suited of tocurrent analyze and short forthcoming wavelength ICT and devices high-frequency operating at phonons, microwave up to frequencies. several GHz, such as those of current and forthcoming ICT devices operating at microwave frequencies. The latter characteristic descends from the fact that, thanks to the wavevector conservation in the photon-phononThe latter characteristic interaction, descends the wavelength from the of fact th that,e revealed thanks elastic to the waves wavevector is of the conservation same order inof themagnitude photon-phonon of that of interaction, light. It is theinteresting wavelength to notice of the that revealed most of elastic the experimental waves is of the investigations same order ofof magnitudelayered structures, of that ofsuch light. as Itsupported is interesting films to and notice supe thatrlattices, most ofhave the been experimental carried out investigations in the eighties of layeredand early structures, nineties, i.e., such just as after supported the availability films and of superlattices, the high-contrast have Tandem been carried Fabry-Perot out in the Interferometer. eighties and Later on, around the turning of the millennium, thanks to the growing importance of the rising fields of spintronics and magnonics, most of the research groups that had been active in the application of Appl. Sci. 2018, 8, 124 3 of 19 early nineties, i.e., just after the availability of the high-contrast Tandem Fabry-Perot Interferometer. Later on, around the turning of the millennium, thanks to the growing importance of the rising fields of spintronics and magnonics, most of the research groups that had been active in the application of surf-BLS to acoustic phonons [such as the group of B. Hillebrands in Kaiserslautern (Germany), that of M. Grimsditch in Argonne (USA), and the groups of F. Nizzoli in Ferrara and mine in Perugia (Italy)] switched their main interest to the study of magnetic films and multilayers by detection of spin waves. In some sense, this caused a slowing down in the transfer of expertise and in the education of young researchers in the application of surf-BLS to the elastic characterization of layered samples. The result is that, according to my experience, there is currently a gap between the needs of the industrial manufacturers of advanced acoustic components (such as filters and duplexers) for mobile telecommunication devices and the availability of experts in the application of surf-BLS to their needs. The main aim of this paper is just to provide a contribution to fill this gap. The structure of the paper is as follows. In Section2, I recall the physical mechanism that is at the origin of the experiments, focusing on the interaction between light and acoustic phonons and discussing in details the quantitative relations that hold in the backscattering geometry that is the most popular for the study of opaque media. Section3 is devoted to the presentation of the situations that may occur when applying the surf-BLS technique to the elastic characterization of different types of films. In particular, the experimental methodology that apply to specific case studies are discussed together with the discussion of the amount of information that can be extracted, depending on the thickness of the film and on its transparency. A short presentation of the limitations of the technique and a generalization to the case of multilayered samples is also provided. In the last Section, the recent development of the micro-focused, scanning, version of the surf-BLS technique are shortly discussed, addressing the new opportunities that are at hands for the imaging of the spatial profile of the acoustic field in acoustic resonators and phononic crystals, with a deeply submicrometric resolution.

2. Brillouin Scattering from Thin Films and Opaque Media: Historical Background and Interaction Mechanism As introduced in the previous paragraph, the surf-BLS technique is presently the most powerful tool to access the elastic characteristics of thin films and layered structures in the GHz range of is frequencies that are of growing importance in nowadays ICT devices. The physical principle that stands behind this technique was initially argued, almost a hundred years ago, by the pioneering and independent works of Leon Brillouin and Leonid Mandelstam [10,11] concerning the scattering of light by acoustic waves in liquids and transparent media. Extension to thin films and metallic or semiconductor solids (i.e., opaque to visible light) were achieved in the seventies thanks to both the advent of lasers and the construction of high-contrast Fabry-Perot interferometers [12–14]. John R. Sandercock developed an original type of these machines in the so-called “tandem” configuration during the seventies, while working in the research division of the RCA company in Zurich, but a few years later he established his own company (formerly named “JRS Scientific Instruments” and now “The Table Stable”) [15], which launched into the market the multi-pass tandem Fabry-Perot interferometer (TFP). After more than thirty years from its commercialization, this kind of interferometer is still without alternatives on the market for the detection of either phonons or magnons from thin films and opaque solids. Several upgrades of the setup have been performed during the last two decades to achieve better performance in terms of contrast, resolution and signal-to-noise ratio. Let us now recall some relevant characteristics of the physical mechanism lying below the surf-BLS from acoustic waves. As stated in the Introduction, it is based on the coupling between the electric field of a monochromatic incoming light beam and the periodic variation of the dielectric constant of the medium associated with acoustic waves. The modulation of the dielectric constant of the medium, induced by the acoustic wave, can be described according to the following equation:

∆εαβ = kαβγδ Sγδ (1) Appl. Sci. 2018, 8, 124 4 of 19

where Sγδ is the strain field associated with the wave and kαβγδ is the elastooptic tensor. Following the derivation in Ref. [16], the scattered electric field Es at large distance from the scattering volume V is proportional to the following integral:

−i(Ki−Ks)·r y V ∆εαβ Eie dV (2) where Ei is the electric field of the incident light, Ki and Ks are the wavevectors of the incident and scattered light. The above integral leads directly to the conservation of momentum, because for a given −iQ×r acoustic excitation of wavevector Q, ∆εαβ is proportional to e so that, if the interaction volume V is large with respect to the acoustic wavelength, the integral becomes a delta function δ(Ki − Ks ∓ Q) and this means that the exchanged wavevector must be conserved. As for the intensity of the inelastic scattering from a bulk acoustic wave in an isotropic medium at temperature T, the scattering cross section (scattered power divided by the solid angle), for an unitary interaction volume and incident power, can be expressed as [17]:

dP k Tπ2 = B |g|2 (3) dΩ 4$v2λ4 where $ is the mass density, λ is the light wavelength and v the acoustic velocity. Interestingly, g is a vector that is determined by the values of the contracted elastooptic coefficients kαβ [18] according to:

1   gα = (k11 − k12)(eαQ × Ei + Qαe × Ei) + 2k12δαβEβQ × e (4) QEs where e is the unit polarization vector of the acoustic wave. It results that the scattered intensity depends on the values of the elastooptic coefficients and on the polarization direction of the incoming and scattered light beams, as well as of the acoustic wave. However, one can derive the following simple guiding rule. Longitudinal acoustic waves give origin only to polarized scattering, i.e., the light polarization of the scattered photons is the same of that of incoming photons. In particular, if the incoming and scattered beams are both polarized perpendicular to the plane of incidence, s-s 2 2 scattering, one has that the cross section is proportional to |g| = 4k12, while for p-p scattering one has 2 | |2 = ( − ) 2 ψ + g 4 k11 k12 cos 2 k12cosψ , where ψ is the scattering angle, formed by Ki and Ks. Instead, shear horizontal (transverse) waves that are polarized perpendicular to the scattering plane, give origin only to depolarized scattering (p-s or s-p scattering), and the cross section is 2 2 ψ proportional to |g| = (k11 − k12) cos 2 . From the brief description above, it follows that the polarization of the light scattered by acoustic waves can be either parallel or perpendicular to that of the incident light and this fact is of great help for the experimentalist in order to select the scattering from specific acoustic modes, as we will show in the following. Moreover, as anticipated in the Introduction, there is another interaction mechanism that contributes to the light scattering and is particularly relevant in opaque samples: the rippling of the free surface, or of the interface of a thin film, induced by the acoustic phonons. It contributes to polarized scattering, only, but the relative importance of the elastooptic and the ripple mechanism depends on the specific medium under investigation, with a prevalence of the former (latter) in the case of transparent (opaque) media, as we will see in the following [6,7]. From the experimental point of view, the most simple and popular surf-BLS interaction geometry, suitable for thin films and opaque samples, is that represented in of Figure2, called backscattering geometry, where the scattering angle ψ = π and the same lens (typically a camera objective with f number comprised between 2 and 4 and focal length of 35 or 50 mm) is used for focusing the laser beam on the sample and for collecting the light that is back-scattered within a solid angle. A discussion of the influence of the f number of the collecting lens on the measurement accuracy can be found in Refs. [19,20]. Two classes of phonons contribute to the scattering in this geometry. First, through the Appl. Sci. 2018, 8, 124 5 of 19 conservation of the momentum component parallel to the surface, one can reveal acoustic waves propagating parallel to the free surface (“surface-like” waves) with an acoustic wavenumber QS = 2ki sin(θ) with ki = 2π/λ, with λ the optical wavelength (typically 514.5 or 532 nm) and θ the angle of incidence of light. Moreover, when the film thickness is much larger than the wavelength of light, “bulk-like” acoustic waves with wavevector QB = 2 nki enter the scattering process, with n the refractive index of the film. Due to optical refraction, these bulk waves propagate within the film material at an angle α = sin−1 (sin θ/n), as seen in Figure2. Appl. Sci. 2018, 8, 124 5 of 19

Figure 2. Back-scattering geometry for conventional surface Brillouin light scattering (surf-BLS) Figure 2. Back-scattering geometry for conventional surface Brillouin light scattering (surf-BLS) experiments. The wave vectors of the incident, reflected, transmitted and scattered light are indicated experiments.in green, The while wave those vectors of acoustic of the waves incident, are in reflected, red (surface transmitted waves) or in and blue scattered (bulk waves). light All are are indicated in green,contained while thosein the plane of acoustic of incidence. waves are in red (surface waves) or in blue (bulk waves). All are contained in the plane of incidence. One can therefore directly derive the connection between the frequency shift f of Brillouin peaks and the phase velocity v of the corresponding acoustic mode, as follows: One can therefore directly derive the connection between the frequency shift f of Brillouin peaks λ and the phase velocity v of the correspondingvS = 2π acousticf/QS = f/2sin( mode,θ) as follows: (5) for surface-like modes, and v = 2πf/Q = λf/2sin(θ) (5) S vB = 2πf/QS B = λf/2n (6) for surface-likefor bulk-like modes, modes. and 3. Determination of the Elastic ConstantsvB = of 2π Nanometricf/QB = λf/2n or Micrometric Films (6) for bulk-likeLet modes. us now consider the case of a film with either micrometric or submicrometric thickness (such as those used in acoustic-wave resonators), whose elastic constants are to be determined. We will see 3. Determinationin the following of the that Elastic the kind Constants of acoustic ofmodes Nanometric that can be or detected Micrometric in a surf-BLS Films experiment, and the consequent amount of information that can be achieved, depends on the thickness of the film, its Lettransparency us now consider and the the presence case ofof aa filmsubstrate with where either th micrometrice acoustic velocity or submicrometric is larger or smaller thickness than in (such as thosethe used film. in It acoustic-wavefollows that the above resonators), conditions whose must be elastic carefully constants considered are if toone be targets determined. a complete We will see in theelastic following characterization that the of kind thin films of acoustic in terms modesof the whole that set can of beindependent detected elastic in a surf-BLS constants. experiment,Note and thethat consequent even in the amount case of opaque of information media, where that the can penetration be achieved, depth of depends light can onbe as the small thickness as ten of the film, itsnanometers, transparency the gained and the information presence concerns of a substrate the whole where coherence the depth acoustic of the velocity acoustic iswaves, larger which or smaller is of the order of the light wavelength, i.e., hundreds of nanometers. Therefore, for submicrometric film than inthickness the film. the It influence follows of that substrat thee aboveon the measured conditions frequencies must be is substantial carefully and considered must be properly if one targets a completemodeled elastic if one characterization wants to extract information of thin films about in the terms films. of This the is wholedifferent set from of the independent case of spin elastic constants.waves Note in ferromagnetic that even in films, the casebecause of opaquein that ca media,se the presence where theof a penetrationnon-magnetic depth substrate of lightis can be as smallcompletely as ten non-influent nanometers, [21]. the gained information concerns the whole coherence depth of the acoustic waves, which is of the order of the light wavelength, i.e., hundreds of nanometers. Therefore, 3.1. Transparent Films with Micrometric Thickness: Guided Modes and Selective Determination of the for submicrometricElastic Constants film thickness the influence of substrate on the measured frequencies is substantial and must be properly modeled if one wants to extract information about the films. This is different The most favorable situation occurs when one deals with a single transparent film of micrometric from the case of spin waves in ferromagnetic films, because in that case the presence of a non-magnetic thickness, supported by a reflecting substrate (such as optically polished Si or GaAs wafers). In this substratecase, is completelyit is possible non-influent to obtain, in a [21 relatively]. straightforward way, the whole set of independent effective elastic constants. These are only two for acoustically isotropic films, such as amorphous glasses and soft dielectric films used in electronics, while may increase to three independent constants in the case of epitaxial cubic films, or to five for films of hexagonal (cylindrical) symmetry, such as piezoelectric films with a textured polycrystalline structure. Appl. Sci. 2018, 8, 124 6 of 19

3.1. Transparent Films with Micrometric Thickness: Guided Modes and Selective Determination of the Elastic Constants The most favorable situation occurs when one deals with a single transparent film of micrometric thickness, supported by a reflecting substrate (such as optically polished Si or GaAs wafers). In this case, it is possible to obtain, in a relatively straightforward way, the whole set of independent effective elastic constants. These are only two for acoustically isotropic films, such as amorphous glasses and soft dielectric films used in electronics, while may increase to three independent constants in the case of epitaxial cubic films, or to five for films of hexagonal (cylindrical) symmetry, such as piezoelectric films with a textured polycrystalline structure. Let us now consider the latter (more complex) case that has been considered in previous investigations devoted, for instance, to ZnO [22] or AlN [23,24] films. Using different angles of Appl. Sci. 2018, 8, 124 6 of 19 incidence and a careful polarization analysis of the scattered light, one can detect and identify several guidedLet acoustic us now modesconsider [the25 ],latter such (more as the complex) surface case Rayleigh that has been wave considered (RW), the in shearprevious horizontal mode (SHM),investigations the longitudinal devoted, for mode instance, (LM) to andZnO the[22] longitudinalor AlN [23,24] bulkfilms. (LB)Using mode, different whose angles propagation of direction andincidence polarization and a careful are polarization summarized analysis in of Figurethe scattered3. (Please light, one note can detect that and in someidentify papers several the LM guided acoustic modes [25], such as the surface Rayleigh wave (RW), the shear horizontal mode (SHM), is labeledthe as LGM,longitudinal longitudinal mode (LM) and guided the longitudinal mode, and bulk the (LB) LB mode, is labeled whose propagation as LA, longitudinal direction and acoustic wave). Theirpolarization detection are providessummarized a sufficientlyin Figure 3. (Please large note amount that in ofsome information papers the LM to is estimate labeled as the LGM, whole set of independentlongitudinal elastic constants,guided mode, because and the LB as is summarized labeled as LA, longitudinal in Figure 3acoustic, measuring wave). Their the frequencydetection of the Brillouin peaksprovides associated a sufficiently to large the amount above of mentioned information modesto estimate provides the whole access set of independent to the values elastic of the five constants, because as summarized in Figure 3, measuring the frequency of the Brillouin peaks independentassociated constants, to the namely above mentionedC11, C13, modesC33, C 44providesand C access66. In to particular, the values oneof the has five the independent following selective q C q C constants, namely C11, C13, C33, C44 and C66. In particular, one has the following selective relationship11 66 relationship between the measured velocities and the elastic constants: VLM = ρ , VSHM = ρ C C q V = 11 V = 66 = C44 LM ρ SHM ρ and VRM betweenβ ρ thewith measuredβ, a constant velocities that and depends the elastic on theconstants: other elastic constants, C11, C13 and, C33 . It must be evaluated numerically,C using the model of [25] and usually takes values in the range between about V = β 44 RW ρ 0.9 and 0.95. The exact with value β, a ofconstantβ can that be founddepends according on the other to elastic the following constants C implicit11, C13, C33.expression It must be derived by Dobrzynskievaluated and numerically, Maradudin using [26 the]: model of [25] and usually takes values in the range between about 0.9 and 0.95. The exact value of β can be found according to the following implicit expression derived by Dobrzynski and Maradudin [26]: 2 2 C44 2 C11 C13 4 2 C11 C33(v − )(v − + ) = C44v (v − ) (7) R R R R ρ − ρ− +ρC33= − ρ (7)

Figure 3. Schematic view of the four kind of acoustic modes that can be detected in Brillouin spectra Figure 3. Schematictaken from transparent view of thefilms four of micrometric kind of acousticthickness deposited modes thaton a reflecting can be detectedsubstrate. The in BrillouinRayleigh spectra taken fromwave transparent (RW), the filmslongitudinal of micrometric mode (LM) thicknessand the longitudinal deposited bulk on (LB) a reflecting waves are substrate.polarized in Thethe Rayleigh wave (RW),sagittal the longitudinalplane, while the modeshear horizontal (LM) and mode the (SHM longitudinal) is polarized perpendicular bulk (LB) waves to the sagittal are polarized plane, in the sagittal plane,so that while the latter the gives shear origin horizontal to depolarized mode light (SHM) scattering. is polarized perpendicular to the sagittal plane,

so that theTo latter illustrate gives the origin above to considerations, depolarized lightin Figure scattering. 4, one can see typical spectra relative to an AlN film of 1.7 μm thickness measured on a wide frequency range, up to 100 GHz, where several peaks are present. Some of them, corresponding to the Rayleigh wave (RW) and the longitudinal mode (LM), change their frequency position with θ, according to Equation (5). The peak corresponding to the longitudinal bulk wave (LB) instead, stays almost fixed with changing θ, because it corresponds to the interaction mechanism of bulk phonons, according to Equation (6). A deeper insight into the Appl. Sci. 2018, 8, 124 7 of 19

To illustrate the above considerations, in Figure4, one can see typical spectra relative to an AlN film of 1.7 µm thickness measured on a wide frequency range, up to 100 GHz, where several peaks are present. Some of them, corresponding to the Rayleigh wave (RW) and the longitudinal mode (LM), change their frequency position with θ, according to Equation (5). The peak corresponding to θ Appl.the longitudinalSci. 2018, 8, 124 bulk wave (LB) instead, stays almost fixed with changing , because it corresponds7 of 19 to the interaction mechanism of bulk phonons, according to Equation (6). A deeper insight into characteristicsthe characteristics of “surface” of “surface” waves waves show show that that the the peak peak at atlower lower frequency frequency is is actually actually a a doublet, doublet, corresponding to to scattering from either the Raylei Rayleighgh wave (polarized scattering, p-p) or the shear horizontal mode mode (SHM) (SHM) that that is depolarized is depolarized (p-s) (p-s) [22]. [ 22Measuring]. Measuring the frequency the frequency shift f of shift each f of these each twoof these peaks, two one peaks, can determine one can determinethe phase velocity the phase of the velocity corresponding of the corresponding acoustic modes acoustic according modes to Equationaccording (5). to EquationFor the LB (5). peak, For instead, the LB peak, the correspondi instead, theng corresponding phase velocity phase is obtained velocity by is Equation obtained (6). by × ρ × 2 MeasuringEquation (6). the Measuring LB wave at the normal LB wave incidence at normal permits incidence to obtain permits C33 = toρ obtain . Finally,C33 = the VvalueLB. Finally, of C13 canthe valuebe estimated, of C13 can even be estimated,if less precisely, even iffrom less precisely,the dependence from the of dependenceVLB on the angle of VLB α,on performing the angle measurementsα, performing measurementsat different incidence at different angles. incidence Note however angles. Note that howeverif one uses that the if one above uses procedure, the above disregardingprocedure, disregarding the contribution the contribution of the piezoelectric of the piezoelectric constants, constants,one obtains one the obtains values the of the values so called of the “piezoelectrically-stiffened”so called “piezoelectrically-stiffened” effective effectiveelastic consta elasticnts. constants. If one instead If one wants instead to wants estimate to estimate the “true” the “true”elastic elasticconstants constants values, values, one should one should also consider also consider the contribution the contribution of the of piezoelectric the piezoelectric terms terms in the in numericalthe numerical model model and andthis thisimplies implies that that the thevalues values of C of11, CC1113, C3313 ,areC33 slightlyare slightly modified modified [24]. [24].

p-t θ=60° LM p-p LM LM SHM LB RW θ=70° RW RW

LM RW + SHM SHM SHM LB p-s

θ=45° units) (arb. Intensity Intensity (arb. units) Intensity

LB θ=10°

-90 -60 -30 0 30 60 90 -40 -20 0 20 40 Frequench Shift (GHz) Frequency Shift (GHz)

Figure 4. (Left) Comparison between two Brillouin spectra, relative to a 1.7 µm thick AlN film, Figure 4. (Left) Comparison between two Brillouin spectra, relative to a 1.7 μm thick AlN film, supported by Si, extending to a large frequency range, collected at different angles of incidence. supported by Si, extending to a large frequency range, collected at different angles of incidence. The The labels LM and LB refer to the longitudinal mode and the longitudinal bulk acoustic waves, labels LM and LB refer to the longitudinal mode and the longitudinal bulk acoustic waves, respectively. respectively. The incoming light was to p-polarized and no polarization analysis was made on the The incoming light was to p-polarized and no polarization analysis was made on the scattered light; scattered light; (Right) Comparison between two Brillouin spectra, corresponding to p-polarized (Right) Comparison between two Brillouin spectra, corresponding to p-polarized incoming light and incoming light and p- or s-polarized scattered light (labeled p-p and p-s, respectively). It is seen that in p- or s-polarized scattered light (labeled p-p and p-s, respectively). It is seen that in the former case one the former case one can detect the peaks corresponding to the surface Rayleigh wave (RW) and to the can detect the peaks corresponding to the surface Rayleigh wave (RW) and to the longitudinal mode (LM), longitudinal mode (LM), while in the latter case, only the peak relative to shear horizontal mode (SHM) while in the latter case, only the peak relative to shear horizontal mode (SHM) is detected. The angle of is detected. The angle of incidence of light was ϑ = 60◦. Adapted with permission, from [24] IEEE, 2017. incidence of light was ϑ = 60°. Adapted with permission, from [24] IEEE, 2017.

Please note that if the studied studied film film has only only two two [27,28], [27,28], or or three three [29] [29] independent independent elastic elastic constants, constants, it is is not not necessary necessary to to look look for for the the depolarized depolarized scatte scatteringring from the SHM (that is much less intense intense than the otherother modesmodes [ 30[30]),]), because because the the information information extracted extracted from from the RW the and RW either and LMeither or LBLM is sufficientor LB is sufficientto determine to determine the two independent the two independent constants constantsC11 and C 44C11. and C44. In any case, even if the polarization analysis of the detected light is not necessary, one should notice that the cross section of the RW and LM can be rather strongly dependent on both the value of the incidence angle and the polarization status of the incoming light. The RW whose cross section mainly relies on the rippling mechanism, can be efficiently detected using an incoming p-polarization and a relatively large angle of incidence (θ ≥ 50°) [31]. Instead, the cross section of the LM depends on the values of the photoelastic constants of the material under consideration. For instance, the cross section of the LM in dielectric glasses is much stronger using s-polarization (rather than p-polarization) Appl. Sci. 2018, 8, 124 8 of 19

In any case, even if the polarization analysis of the detected light is not necessary, one should notice that the cross section of the RW and LM can be rather strongly dependent on both the value of the incidence angle and the polarization status of the incoming light. The RW whose cross section mainly relies on the rippling mechanism, can be efficiently detected using an incoming p-polarization and a relatively large angle of incidence (θ ≥ 50◦)[31]. Instead, the cross section of the LM depends on the values of the photoelastic constants of the material under consideration. For instance, the cross Appl. Sci. 2018, 8, 124 8 of 19 section of the LM in dielectric glasses is much stronger using s-polarization (rather than p-polarization) of the incomingof the incoming light at light relatively at relatively small small angles angles of incidenceof incidence (30 (30°◦ << θθ < 50°). 50◦). This This is isclearly clearly visible visible in in the spectra reportedthe spectrain reported Figure in5, Figure taken 5, from taken Ref. from [27 Ref.]. [27].

Figure 5.FigureExperimental 5. Experimental surf-BLS surf-BLS spectra spectra relative relative to to thethe SiOC:H/Si film, film, taken taken for an for angle an angleof incidence of incidence of light ofθ =light 65 ◦θ (=left 65° panel(left panel) and) and 25◦ 25°(right (right panel panel).). TheThe upper upper spectra spectra were were taken taken using using p-polarized p-polarized incident light, while s-polarized light was used for the lower ones. The peaks corresponding to the Rayleigh incident light, while s-polarized light was used for the lower ones. The peaks corresponding to the surface wave (RW), to the longitudinal guided mode (LGM) and to longitudinal bulk wave (LB) are Rayleighindicated surface on wave the anti-Stokes (RW), to side the of longitudinaleach spectrum. guidedIt can be seen mode that (LGM) the RW pe andak has to longitudinala large cross section bulk wave (LB) arefor indicated p-polarized on incoming the anti-Stokes light and large side angle of each of incidence. spectrum. Instead, It can the be LGM seen has that a larger the cross RW section peak has for a large cross sections-polarized for p-polarized light and small incoming angle of lightincidence. and largeReprinted angle with of permission incidence. from Instead, [27], Elsevier, the LGM 2005. has a larger cross section for s-polarized light and small angle of incidence. Reprinted with permission from [27], Elsevier,3.2. Transparent 2005. Films with Deeply Submicrometric Thickness: Dispersion Curves and Interference Effects A relatively different situation occurs when the film thickness is submicrometric, i.e., comparable 3.2. Transparentto, (or lower Films than) with the DeeplyacousticSubmicrometric wavelength. In the Thickness: usual case Dispersionof a film supported Curves by and a substrate, Interference whose Effects acoustic phase velocities are higher than those of the film (slow film/fast substrate), a number of Adiscrete relatively acoustic different modes, situationnamely the Rayleigh-Se occurs whenzawa and the Love film modes thickness are present is submicrometric, [25] and can be i.e., comparablerevealed to, in (or Brillouin lower than)spectra. the The acoustic former modes wavelength. are polarized In thein the usual sagittal case plane of a(i.e., film in supportedthe plane by a substrate,of incidence whose acoustic of light, phasewhile the velocities latter modes are are higher polarized than perpendicularly those of the film to the (slow above film/fast plane). This substrate), a numbertransformation of discrete from acoustic a spectrum modes, typical namely of thick the films Rayleigh-Sezawa to the discretized and spectrum Love typical modes of arethin presentfilms [25] is seen in the left panel of Figure 6, relative to GaSe films of different thickness [32]. Here, it is seen that and can be revealed in Brillouin spectra. The former modes are polarized in the sagittal plane (i.e., the Rayleigh and a few Sezawa discrete modes are seen for film thickness of 77 and 251 nm. Taking in the planemeasurements of incidence at different of light, angles while of incidence, the latter one modescan obtain are the polarized dispersion perpendicularlycurves shown in Figure to the 6, above plane). Thisright transformationpanel. from a spectrum typical of thick films to the discretized spectrum typical of thin filmsA is detailed seen in theory the left about panel the oftheo Figureretical6 cross, relative section to of GaSe these films modes of has different been presented thickness in [33], [ 32 ]. Here, it is seenand that it has the been Rayleigh shown that and strong a few interference Sezawa discrete effects can modes arise when are seen one considers for film all thickness the possible of 77 and 251 nm.scattering Taking measurementsmechanisms (ripple atdifferent scattering anglesfrom bo ofth incidence,the free surface one canand obtainthe interface, the dispersion elastooptic curves scattering from the interior of the film) [34]. However, these modes are dispersive, so that measurements shown inare Figure usually6, performed right panel. on films of different thicknesses and with different angles of incidence, in A detailedorder to span theory a large about interval the of the theoretical ratio between cross the film section thickness of these and the modes phonon haswavelength. been presented In in [33],the and case it of has a hexagonal been shown elastic that symmetry, strong four interference of the five effective effects elastic can ariseconstants, when namely one C considers11, C13, all the possibleC33 and scattering C44, influence mechanisms the Rayleigh-Sezawa (ripple modes, scattering so that th fromey can both be evaluated the free by surface a best fit procedure and the interface,of elastoopticthe experimental scattering from velocities the interiorto the calculated of the film) disper [34sion]. However,curves. This these procedure modes has arebeen dispersive, used in the so that past for a number of different thin film materials, such as cubic Zinc Selenide [35], hexagonal and cubic measurements are usually performed on films of different thicknesses and with different angles of Boron Nitride [36–38], hexagonal Zinc Oxyde [39], Tin Dioxide [40], Gallium- [32] or Indium- [41] Selenide, and isotropic dielectrics [42–44]. In all the above studies it has been put in evidence that the Appl. Sci. 2018, 8, 124 9 of 19 incidence, in order to span a large interval of the ratio between the film thickness and the phonon wavelength. In the case of a hexagonal elastic symmetry, four of the five effective elastic constants, namely C11, C13, C33 and C44, influence the Rayleigh-Sezawa modes, so that they can be evaluated by a best fit procedure of the experimental velocities to the calculated dispersion curves. This procedure has been used in the past for a number of different thin film materials, such as cubic Zinc Selenide [35], hexagonal and cubic Boron Nitride [36–38], hexagonal Zinc Oxyde [39], Tin Dioxide [40], Gallium- [32] or Indium- [41] Selenide, and isotropic dielectrics [42–44]. In all the above studies it has been put in evidence that the different elastic constants can influence the acoustic modes in a similar way, so that Appl. Sci. 2018, 8, 124 9 of 19 one has to face the problem of the correlation among the fitting parameters. It follows that is useful to exploit filmsdifferent of differentelastic constants thickness can influence (in the range the acoustic between modes a few in a tens similar to away, few so hundreds that one has of nanometers),to face but in histhe case problem a further of the complication correlation among may the arise, fitting if theparame structuralters. It follows and elastic that is differencesuseful to exploit among films films of differentof thickness different thickness are not negligible,(in the range due between forinstance a few tens to to the a few occurrence hundreds of nanometers), a modified but transition in his layer case a further complication may arise, if the structural and elastic differences among films of different at the interfacethickness between are not negligible, substrate due and for film instance material. to the occurrence This is illustrated of a modified in Figure transition7, where layer at it the is shown that theinterface experimental between data substrate for the and ZnO/Si film material. films This fit the is illustrated expected in theoretical Figure 7, where dispersion it is shown curves that fairly well forthe film experimental thicknesses data greater for the thanZnO/Si 150 films nm, fitwhile the expected theyappreciably theoretical dispersion depart fromcurves thefairly same well curves for smallerfor film thicknesses thicknesses [39 greater]. Such than a behavior 150 nm, while (that th wasey appreciably present in thedepart case from of Sithe substrate same curves but for not for a fused quartzsmaller one) thicknesses was interpreted [39]. Such ina behavior terms of (that a reduction was present of thein the effective case of elasticSi substrate constants but not of for the a film in fused quartz one) was interpreted in terms of a reduction of the effective elastic constants of the film a layer near the interface, due to the lattice misfit between the film and the Si substrate. in a layer near the interface, due to the lattice misfit between the film and the Si substrate.

Figure 6. (Left) Brillouin spectra relative to the three GaSe films of different thickness h, for an angle Figure 6. (Left) Brillouin spectra relative to the three GaSe films of different thickness h, for an angle of of incidence θ = 70°. The peaks corresponding to the Rayleigh (R) and Sezawa (S) guided acoustic ◦ incidencemodesθ = 70are. clearly The peaks seen. In corresponding the case of the thicker to the film, Rayleigh h = 990 (R) nm, and the Sezawa discrete Sezawa (S) guided modes acoustic are not modes are clearlyresolved seen. anymore In the case and of the the peak thicker due to film, bulk h long = 990itudinal nm, theacoustic discrete waves Sezawa (LA) ismodes observed; are (Right not resolved) anymoreExperimental and the peak values due of to the bulk phase longitudinal velocity of the acoustic Rayleigh wavesand Sezawa (LA) acoustic is observed; modes (points (Right with) Experimental error values ofbars) the as phase a function velocity of the of ratio the between Rayleigh the andfilm th Sezawaickness and acoustic the acoustic modes wavelength. (pointswith The dispersion error bars) as a functioncurves of the (solid ratio lines) between have been the obtained film thickness from a best and fit the procedure acoustic assuming wavelength. the elastic The constants dispersion of the curves film material as free parameters. Adapted with permission from [32], IOP Publishing, 1999. (solid lines) have been obtained from a best fit procedure assuming the elastic constants of the film materialIn as the free above parameters. examples Adapted of Figures with 6 and permission 7, a best fit from procedure [32], IOP of the Publishing, measured 1999.dispersion curves to the calculated ones permitted to achieve a determination of four of the five independent elastic In theconstants. above As examples for thick offilms Figures discussed6 and in7 ,the a best previous fit procedure section, the of fifth the elastic measured constant dispersion C66 can be curves determined from measurement of the phase velocity of shear horizontal modes (Love modes) in a to the calculated ones permitted to achieve a determination of four of the five independent elastic depolarized scattering experiment, as shown in Figure 8 for an InSe film [41]. Use of an opaque constants.substrate As for (typically thick films crystalline discussed silicon) inis recommended the previous in section, order to thetake fifthadvantage elastic of constantthe presenceC 66of can be determineda reflecting from measurementinterface which ofenhances the phase the Brillouin velocity cross of shear section, horizontal even if long modes acquisition (Love modes)times in a depolarized(several scattering hours) areexperiment, usually necessary. as shown in Figure8 for an InSe film [ 41]. Use of an opaque substrate (typicallyTo conclude crystalline this Section, silicon) let us note is recommended that in the case of ina soft order film toon takea hard advantage substrate [25], of instead, the presence of a reflectingone observes interface only the which RW, which enhances evolves the into Brillouin a leaky wave cross as soon section, as the evenfilm thickness if long exceeds acquisition a few times nanometers, radiating energy into the bulk (as for AlN on Si [45], CaF2 on GaAs [46] or TiN over (several hours) are usually necessary. high-speed steel [47]). As a consequence, the RW peak broadens and finally disappears from the spectra. In these cases, only a relatively limited information about the shear elastic constant C44 can be extracted from the measurement. Appl. Sci. 2018, 8, 124 10 of 19

To conclude this Section, let us note that in the case of a soft film on a hard substrate [25], instead, one observes only the RW, which evolves into a leaky wave as soon as the film thickness exceeds a few nanometers, radiating energy into the bulk (as for AlN on Si [45], CaF2 on GaAs [46] or TiN over high-speed steel [47]). As a consequence, the RW peak broadens and finally disappears from the spectra. In these cases, only a relatively limited information about the shear elastic constant C44 can be extractedAppl. Sci. from 2018 the, 8, 124 measurement. 10 of 19

Appl. Sci. 2018, 8, 124 10 of 19

FigureFigure 7. (Left 7. ()Left Brillouin) Brillouin spectra spectra from from ZnO ZnO films films ofof differentdifferent thickness thickness on on (001)-cut (001)-cut Si substrate Si substrate for for fixed backscattering angle θ = 45°; (Right) Experimental data of phase velocity of first two Rayleigh fixed backscattering angle θ = 45◦;(Right) Experimental data of phase velocity of first two Rayleigh modesFigure for the 7. (ZnO/SiLeft) Brillouin structure spectra and from theoretical ZnO films disper of differentsion curves, thickness plotted on (001)-cut as a function Si substrate of the forproduct modes for the ZnO/Si structure and theoretical dispersion curves, plotted as a function of the product betweenfixed the backscattering acoustic wavevector angle θ = 45°; and (Right the film) Experimental thickness, data evaluated of phase by ve uselocity of ofelastic first two constants Rayleigh of bulk modes for the ZnO/Si structure and theoretical dispersion curves, plotted as a function of the product betweenZnO the (dashed acoustic lines), wavevector of effective andconstants the film of the thickness, film (dash-dotted evaluated lines) by useand of modified elastic constants constantsof of bulk between the acoustic wavevector and the film thickness, evaluated by use of elastic constants of bulk ZnO (dashedthe film (solid lines), line). of effective Adapted constantswith permission of the from film [39], (dash-dotted IEEE, 1991. lines) and of modified constants of ZnO (dashed lines), of effective constants of the film (dash-dotted lines) and of modified constants of the film (solid line). Adapted with permission from [39], IEEE, 1991. the film (solid line). Adapted with permission from [39], IEEE, 1991.

Figure 8. Brillouin spectra taken at an angle of incidence θ = 65°. From◦ an InSe, 79 nm thick, film. The FigureFigure 8. Brillouinpeaks 8. Brillouin corresponding spectra spectra takento takenthe Rayleigh at at anan angle wave of(R) of incidence and incidence to the θ Love =θ 65°.= mode 65From .(L) From an are InSe, clearly an 79 InSe, seennm thick,in 79 the nm film.p-p thick, The film. The peakspeaksspectrum corresponding corresponding and the p-sto thespectrum, the Rayleigh Rayleigh respectively. wave wave (R) (R)Note and and that to tothethe the ghost Love Love of mode the mode Rayleigh (L) (L)are peak areclearly clearlyis also seen presentseen in the in p-p the p-p spectrumspectrum andin the theand p-s p-sspectrum,the spectrum,p-s spectrum, because respectively. of respectively. the finite extinction Note Note thatratiothat theof the ghost ghost polarization of of the the Rayleigh analyzer. Rayleigh peakReprinted peak is also is with also present present in thein p-s the spectrum,permission p-s spectrum, from because because[41], IOP of thePublishing,of the finite finite extinction1999. extinction ratioratio of of the the polarization polarization analyzer. analyzer. Reprinted Reprinted with with permission from [41], IOP Publishing, 1999. permission3.3. Opaque from Films: [41], IOPRippling Publishing, of the Surface, 1999. Guided Modes and Relevant Thresholds 3.3. OpaqueThe Films: surf-BLS Rippling technique of the canSurface, be used Guided also Modesto deter andmine Relevant the elastic Thresholds constants of metallic films, where the penetration depth is only a few nanometers, so that the ripple mechanism is the dominant one The surf-BLS technique can be used also to determine the elastic constants of metallic films, where the penetration depth is only a few nanometers, so that the ripple mechanism is the dominant one Appl. Sci. 2018, 8, 124 11 of 19

3.3. Opaque Films: Rippling of the Surface, Guided Modes and Relevant Thresholds

Appl. Sci.The 2018 surf-BLS, 8, 124 technique can be used also to determine the elastic constants of metallic11 films, of 19 where the penetration depth is only a few nanometers, so that the ripple mechanism is the dominant onewhile while the elastooptic the elastooptic coupling coupling is usually is usuallynegligible. negligible. It is, therefore, It is, therefore,recommendable recommendable to use p-polarized to use p-polarizedincoming light incoming and relatively light and large relatively angles large of incidence angles of to incidence maximize to the maximize cross section the cross [31]. section [31]. The firstfirst investigations of opaque filmsfilms were concernedconcerned with Au filmsfilms of thickness from ten to hundreds of nanometers, supported byby glass [[48]48] where the Rayleigh and Sezawa guided modes were detected and the elastic constants estimated from a bestbest fitfit procedure ofof theirtheir dispersiondispersion curves.curves. In a successive study, thethe surf-BLSsurf-BLS cross sectionsection of AuAu filmsfilms onon SiSi [[49]49] waswas alsoalso analyzed,analyzed, showingshowing thatthat the rippleripple mechanismmechanism gives the mainmain contributioncontribution to the scatteringscattering process, even ifif thethe elastoopticelastooptic contribution isis notnot insignificant.insignificant. FurtherFurther studiesstudies ofof metallicmetallic filmsfilms dealtdealt withwith Al/NaClAl/NaCl [50] [50] (where the authors were were unable unable to to fit fit our our experimental experimental results results to obtain to obtain the elastic the elastic constants constants independently, independently, since sincedifferent different combinations combinations of the Cij of lead the to Cij similar lead tospectra) similar and spectra) with Mo and [51] with or Ni Mo [52] [ 51films.] or More Ni [52 recently,] films. Moreother recently,investigation other on investigation the limits and on accuracy the limits of and the accuracyestimation of of the the estimation constants ofhave the been constants proposed, have beenwith proposed,reference also with the reference uncertainty also theconnec uncertaintyted with connectedthe film thickness with the [38,53,54]. film thickness [38,53,54]. Particularly relevant, during during the the late late eighties eighties an andd the the early early nineties, nineties, were were the the investigations investigations of ofmetallic metallic superlattices, superlattices, i.e., i.e.,multilayers multilayers with withmany many repetitions repetitions of elemental of elemental nanometric nanometric bilayers, bilayers, where whereanomalous anomalous values valuesof the elastic of the mo elasticduli moduliwere reported were reported by conventional by conventional methods. methods. Among the Among systems the systemssuccessfully successfully investigated investigated we recall here we recallperiodic here superlattices: periodic superlattices: Mo/Ta [55], Cu/Nb Mo/Ta [56] [55 (quasi-periodic], Cu/Nb [56] (quasi-periodic[57]), Ag/Pd [58], [57 Fe/Pd]), Ag/Pd [59], [Co./Cu58], Fe/Pd [60]. [ 59Co./Au], Co./Cu [61], [ 60Ag/Ni]. Co./Au [62], FeNi/Cu [61], Ag/Ni and [ 62FeNi/Nb], FeNi/Cu [63] andand FeNi/NbTa/Al [64] [(quasi-periodic63] and Ta/Al [65]). [64] (quasi-periodicIn all these systems, [65]). the In overall all these film systems, thickness the was overall typically film hundreds thickness of wasnanometers, typically while hundreds the artificial of nanometers, periodicity while ranged the artificialfrom a few periodicity nanometers ranged to a few from tens a few of nanometers. nanometers toTherefore, a few tens the of superlattice nanometers. was Therefore, modeled theas an superlattice effective medium was modeled with hexagonal as an effective symmetry medium and with five hexagonaleffective elastic symmetry constants, and five whose effective dependence elastic constants, on the artificial whose dependenceperiodicity was on the investigated. artificial periodicity Some of wasthese investigated. constants, such Some as ofthe these shear constants, constant suchC44, was as the found shear to constantbe appreciablyC44, was dependent found to beof the appreciably artificial dependentperiodicity, of because the artificial of strong periodicity, interface because effects. of The strong analysis interface was effects.generally The based analysis on the was analysis generally of basedthe dispersion on the analysis curves of of the Rayleigh dispersion and curves Sezawa of Rayleighmodes, as and shown Sezawa in modes,Figure 9 as for shown the case in Figure of Ag/Ni9 for thesuperlattices. case of Ag/Ni Four superlattices. of the five constants Four of the elastic five constants elastic(C11, C constants13, C33 and( C1144), couldC13, C 33beand obtainedC44) could by a bebest obtained fit procedure by a best of the fit measured procedure freq of theuencies measured on the frequencies calculated curves, on the calculatedwhile C66 was curves, not accessible while C66 wassince not it only accessible affects since Love it modes, only affects not detec Loveted modes, in the not case detected of metallic in the samples. case of metallic samples.

Figure 9. (Left) Brillouin spectrum taken onon anan Ag/NiAg/Ni supe superlattice,rlattice, 500 500 nm nm thick, thick, having a periodicity of 2 nm, supported by a glass substrate. The Rayl Rayleigheigh and four Sezawa modesmodes are seen. The incidence Λ angle is 6767°,◦, which correspondscorresponds toto aa valuevalue ofof h/h/Λ = 1.8; 1.8; ( (RightRight)) Experimental Experimental (circles) and calculated (solid(solid line)line) velocityvelocity dispersiondispersion ofof thethe RayleighRayleigh andand Sezawa modesmodes forfor the same sample. Adapted with permission from [[62],62], AIPAIP Publishing,Publishing, 1992.1992.

In Figure 10 it is shown the effect of the different constants on the different modes, so one can In Figure 10 it is shown the effect of the different constants on the different modes, so one can see for instance that the shear constant C44 gives the main contribution to the velocity of the Rayleigh see for instance that the shear constant C gives the main contribution to the velocity of the Rayleigh mode at large film thickness, while its contribution44 to high-order modes is rather limited. One can also see that the contribution of C11 and C33 to the various modes is similar, so that it is not straightforward to disentangle these two constants from each other in the fitting procedure. Appl. Sci. 2018, 8, 124 12 of 19 mode at large film thickness, while its contribution to high-order modes is rather limited. One can also see that the contribution of C11 and C33 to the various modes is similar, so that it is not straightforward to disentangleAppl. Sci. 2018, 8 these, 124 two constants from each other in the fitting procedure. 12 of 19

Figure 10. Relative changes of the phase velocity of the Rayleigh and first three Sezawa modes produced Figure 10. Relative changes of the phase velocity of the Rayleigh and first three Sezawa modes produced by a 1% increase of each elastic constant vs. the ratio between film thickness h and phonon wavelength. by a 1% increase of each elastic constant vs. the ratio between film thickness h and phonon wavelength. Calculations refer to an Ag/Ni superlattice film supported by a glass substrate. Reprinted with permission Calculationsfrom [62], AIP refer Publishing, to an Ag/Ni 1992. superlattice film supported by a glass substrate. Reprinted with permission from [62], AIP Publishing, 1992. A relatively easier situation may occur if the total thickness of the superlattice film becomes much largerA relatively than the light easier wavelength, situation so may that the occur number if the oftotal Sezawa thickness modes is of very the large superlattice and the spectrum film becomes muchbecomes larger almost than continuum, the light wavelength, as it would happen so that at the surface number of ofa semi-infinite Sezawa modes medium. is very In this large case, and the as illustrated in Figure 11, the projection of the vertical acoustic displacement at the free surface spectrum becomes almost continuum, as it would happen at the surface of a semi-infinite medium. contains the discrete peak of the Rayleigh surface wave, followed by a continuum of modes. In thisInterestingly, case, as illustratedin this continuum in Figure spectrum 11, the one projection can identify of the a verticaldip, labeled acoustic Longitudinal displacement Threshold at the free surface(LT), containsoccurring thein correspondence discrete peak ofof thethe Rayleighfrequency surfaceof the longitudinal wave, followed acoustic by wave a continuum propagating of modes. Interestingly,parallel to the in free this surface continuum (i.e., where spectrum one would one ha canve identifyobserved athe dip, LM labeled peak in Longitudinala transparent film). Threshold (LT),The occurring experimental in correspondence counterpart of this of calculation the frequency is in ofFigure the longitudinal12, where the acousticspectrum waverelative propagating to a parallel5 μm tothick the AgNi free superlattice surface (i.e., film where is shown one would[66]. It is have clear observedthat, in addition the LM to peakthe RW in peak, a transparent one can film). Thesee experimental the dip corresponding counterpart to the of LT this and calculation directly de isrive in Figureunambiguously 12, where the the value spectrum of the longitudinal relative to a 5 µm thickelastic AgNi constant superlattice C11 = film. is shown [66]. It is clear that, in addition to the RW peak, one can see the The same methodology, based on the detection of the discrete RW peak and the dip associated dip corresponding to the LT and directly derive unambiguously the value of the longitudinal elastic to the LT has been also used to selectively obtain the elastic constants C11 and C44 in polycrystalline constant C = ρV2 . Ta/Al superlattices11 LT with hexagonal elastic symmetry [67]. TheFinally, same also methodology, for opaque semiconductors based on the detectionwith a cubic of elastic the discrete symmetry, RW such peak as andepitaxial the dipInGaAs associated to thefilms LT grown has been on GaAs also analyzed used to selectivelyin Ref. [68], the obtain elastic the constants elastic constants have beenC obtained11 and C looking44 in polycrystalline at both Ta/Althe RW superlattices and the LT. with In addition, hexagonal also elastic the edge symmetry of the continuum [67]. shoulder (transverse threshold, TT) thatFinally, is put alsoin evidence for opaque in Figure semiconductors 13, was measur withed. a More cubic recently, elastic symmetry,it has been suchshown as that, epitaxial taking InGaAs filmsadvantage grown onof the GaAs RW analyzed peak, the inTT Ref. edge [68 and], the the elasticdip of the constants LT, a simple have and been robust obtained fitting looking procedure at both the RWhas and been the presented LT. In addition, for determining also the the edge three of indepen the continuumdent elastic shoulder constants (transverse of a cubic threshold,thick film from TT) that is putsurf-BLS, in evidence using in InAs0.91Sb0.09 Figure 13, was as measured.the case study More [69]. recently, it has been shown that, taking advantage To summarize the discussion of this Section about the potentiality of the technique, we provide in of the RW peak, the TT edge and the dip of the LT, a simple and robust fitting procedure has been Table 1 a schematic presentation of the experimental methodology that is suitable to gain information presented for determining the three independent elastic constants of a cubic thick film from surf-BLS, about the elastic constants of thin films, considering their thickness and their transparency. However, usingbefore InAs0.91Sb0.09 concluding this as section, the case we study want [also69]. to put in evidence a few weak aspects of surf-BLS. The firstTo one summarize is the limited the accuracy discussion in the of thisdetermination Section about of the the phase potentiality velocities and of the of the technique, phonon lifetime we provide in Table(attenuation),1 a schematic which presentation are derived offrom the the experimental frequency po methodologysition and the line that width is suitable of the Brillouin to gain informationpeaks aboutin the the spectra, elastic constantsrespectively. of The thin accuracy films, considering in the determination their thickness of these and quantities their transparency. is limited to However,be Appl. Sci. 2018, 8, 124 13 of 19

before concluding this section, we want also to put in evidence a few weak aspects of surf-BLS. The first

Appl. Sci.one 2018 is, the8, 124 limited accuracy in the determination of the phase velocities and of the phonon13 of lifetime 19 (attenuation), which are derived from the frequency position and the line width of the Brillouin peaks aboveinAppl. about the Sci. spectra, 20180.5%, 8, 124in respectively. the best cases, The due accuracy to the in intrinsic the determinationlimitations in the of thesefrequency quantities and wavevector is13 limitedof 19 to be resolution.above It about follows 0.5% that in the bestderived cases, elastic due toconstants the intrinsic are affected limitations by uncertainty in the frequency larger andthatwavevector about 1%. Anotherresolution.above about relatively It0.5% follows in weak the that best point the cases, derivedis duethe torather elasticthe in longtrinsic constants acquisition limitations are affected intimes the frequencyof by each uncertainty spectrum, and wavevector larger that thatmay about range1%. resolution.from Another a few It tens follows relatively of minutes that weakthe toderived several point elastic is hours the constants rather (in the long worstare affected acquisition case ofby shear uncertainty times horizontal of eachlarger (Love) spectrum, that about modes). that may Finally,range1%. the Another fromunavoidable a relatively few tens heating ofweak minutes pointof the tois sample the several rather region hours long by (inacquisition the the focused worst times case laser of of each shearlight spectrum, should horizontal bethat taken (Love) may into modes). range from a few tens of minutes to several hours (in the worst case of shear horizontal (Love) modes). account.Finally, In fact, the typical unavoidable incident heating powers of are the betw sampleeen region100 and by 300 the mW focused for conventional laser light should measurements be taken into Finally, the unavoidable heating of the sample region by the focused laser light should be taken into account. In fact, typical incident powers are between 100 and 300 mW for conventional measurements and betweenaccount. 1In and fact, 10 typical mW incidentfor micro-focused powers are betwones.een Th 100is causes and 300 a mWtemperature for conventional increase measurements ranging from a and between 1 and 10 mW for micro-focused ones. This causes a temperature increase ranging from a few degreesand between to a few 1 and tens 10 ofmW degrees, for micro-focused depending ones. on theThis specific causes a incident temperature power increase and onranging the dissipation from a characteristicsfewfew degreesdegrees of tothe to a a fewspecimen. few tens tens of ofdegrees, degrees, depending depending on the on specific the specific incident incident power and power on the and dissipation on the dissipation characteristicscharacteristics of of the the specimen. specimen.

Figure 11. Calculated vertical component of the power spectrum of surface phonons, for a semi-infinite Figure 11. Calculated vertical component of the power spectrum of surface phonons, for a semi-infinite Figure 11. Calculated vertical component of the power spectrum of− 1surface phonons, for a semi-infinite hexagonal medium (the phonon wavevector is fixed at 0.0226 nm ). The− vertical line represents the hexagonalhexagonal medium medium (the phonon (the phonon wavevector wavevector is fixed is fixedat 0.0226 at 0.0226 nm−1). nm The1 ).vertical The vertical line represents line represents the the position of the Rayleigh wave (RW); the arrow indicates the dip corresponding to the longitudinal position of the Rayleigh wave (RW); the arrow indicates the dip corresponding to the longitudinal positionthreshold of the (LT).Rayleigh Reprinted wave with (RW); perm theission arrow from in [66],dicates Elsevier, the dip1992. corresponding to the longitudinal thresholdthreshold (LT). Reprinted (LT). Reprinted with perm withission permission from from[66], Elsevier, [66], Elsevier, 1992. 1992.

Figure 12. Brillouin spectrum from an unsupported Ag/Ni superlattice, 5 μm thick. Both the Rayleigh wave peak (R) and the dip corresponding to the longitudinal threshold (LT) are observable; the incidence angle is 67.5”, which corresponds to a value of h/Λ = 18. Reprinted with permission from [62], AIP Figure 12. Brillouin spectrum from an unsupported Ag/Ni superlattice, 5 µm thick. Both the FigurePublishing, 12. Brillouin 1992. spectrum from an unsupported Ag/Ni superlattice, 5 μm thick. Both the Rayleigh wave peakRayleigh (R) and wave the peak dip corresponding (R) and the dip to correspondingthe longitudinal to threshold the longitudinal (LT) are observable; threshold (LT) the incidence are observable; angle theis 67.5”, incidence which angle corresponds is 67.5”, to which a value corresponds of h/Λ = to18. a Reprinted value of h/withΛ = permission 18. Reprinted from with [62], permission AIP Publishing,from [ 621992.], AIP Publishing, 1992.

Appl. Sci. 2018, 8, 124 14 of 19 Appl. Sci. 2018, 8, 124 14 of 19

TableTable 1. Methods 1. Methods of determination of determination of the of elastic the elastic consta constantsnts for films for filmsof different of different symmetry, symmetry, thickness thickness and transparency.and transparency. “Thin” and “Thin” “thick” and refer “thick” to films refer whose to films thickness whose is below thickness about is 500–600 below aboutnm and 500–600 above nm aboutand 1 micron. above aboutRW: Rayleigh 1 micron. wave; RW: LM: Rayleigh longitudinal wave; mode; LM: longitudinal LB: longitudinal mode; bulk; LB: LT: longitudinal Longitudinal bulk; LT: Threshold.Longitudinal Threshold. Symmetry Isotropic Cubic Hexagonal Symmetry Isotropic Cubic Hexagonal Independent constants C11 C44 C11 C12 C44 C11 C13 C22 C44 C66 ThinIndependent transparent constants C11 and C44 Cfrom11 C44 the C11 C12 andC11 C44C12 fromC44 the C11 C13 CC3311 andC13 CC4422 fromC44 C the66 C and C from the dispersion C C and C from the C C C and C from the Thinfilm transparent * filmdispersion * 11 of the44 RW and SW dispersion11 12 of RW44 and SW dispersion11 13 of33 RW and44 SW of the RW and SW dispersion of RW and SW dispersion of RW and SW Thick transparent C11 from the LM and/or LB C11 from the LM, C44 from C11 from the LM, C44 from the C from the LM and/or LB C C from the LM, C from C from the LM, C from the RW, Thickfilm transparent ** film ** 11C44 from the RW 44 the RW,11 C12 from the44 LB RW,11 C13 and C33 from44 the LB from the RW the RW, C12 from the LB C13 and C33 from the LB C11 and C44 from the C11 C12 and C44 from the C11 C13 C33 and C44 from the Thin opaque film * C and C from the dispersion C C and C from the C C C and C from the Thin opaque film * dispersion11 of the44 RW and SW dispersion11 12 of RW44 and SW dispersion11 13 of33 RW and44 SW of the RW and SW dispersion of RW and SW dispersion of RW and SW C44 from the RW and C11 from C11 from the LT, C12 and C44 C44 from the RW and C11 Thick opaque film ** C from the RW and C from C from the LT, C and C Thick opaque film ** 44 the LT 11 from11 the RW and12 TT 44 C fromfrom the RW the and LT C from the LT the LT from the RW and TT 44 11 * It is necessary to know the thickness and the mass density of the film; ** It is necessary to know the * It is necessary to know the thickness and the mass density of the film; ** It is necessary to know the mass density mass ofdensity the film. of Moreover,the film. inMoreover, order to exploit in order the informationto exploit the coming information from the LBcoming mode from in transparent the LB mode films, itin is also transparentnecessary films, to know it is thealso refractive necessa index.ry to know the refractive index.

FigureFigure 13. Experimental 13. Experimental Brillouin Brillouin spectra spectra (dots) (dots) taken taken from froma InGaAs/InP a InGaAs/InP superlattice superlattice with periodicity with periodicity 3.1 nm3.1 and nm total and thickness total thickness 201.5 201.5nm with nm Qs with along Qs along [100] [100](a) and (a )along and along [110] [110](b), for (b an), for angle an angle of incidence of incidence ◦ θ = 67.5°.θ = 67.5The solid. The lines solid correspond lines correspond to the to calculat the calculateded perpendicular perpendicular components components of the of surface the surface phonon phonon powerpower spectrum, spectrum, convoluted convoluted with a with Gaussian a Gaussian function function of width of width 0.6 GHz 0.6 (the GHz instrumental (the instrumental resolution). resolution). The RayleighThe Rayleigh wave peak wave (RW) peak the (RW) dip corresponding the dip corresponding to the longitudinal to the longitudinal threshold (LT) threshold and the shoulder (LT) and the startingshoulder at the startingtransverse at the thresh transverseold (TT) threshold are indicated. (TT) are Reprinted indicated. with Reprinted permission with permissionfrom [68], fromIOP [68], Publishing,IOP Publishing, 1996. 1996.

3.4. Multilayered3.4. Multilayered Structures Structures ComingComing back backto the to complex the complex layered layered structure structure of current of current devices devices suchsuch as acoustic as acoustic resonators resonators (Figure(Figure 1), one1), oneway way to achieve to achieve a complete a complete knowledge knowledge of the of elastic the elastic properties properties of the of stack the stack is to isprepare to prepare and analyzeand analyze a series a series of single of single films filmswith withthe differ the differentent materials, materials, studying studying each eachof them of them independently independently accordingaccording to the to procedure the procedure described described in the in previous the previous paragraphs. paragraphs. Alternatively, Alternatively, one can one even can evendirectly directly measuremeasure the entire the entirestack in stack a surf-BLS in a surf-BLS experiment experiment and compare and comparethe frequencies the frequencies of the detected of the modes detected to thosemodes calculated to those solving calculated acoustic solving propagation acoustic in propagation the multilayer in the by multilayera proper numerical by a proper approach. numerical The approach.case of a Thebilayer case on of aa bilayer substrate on a was substrate successf wasully successfully analyzed, analyzed, for instance, for instance, in Refs. in Refs.[39,70]. [39 , 70]. The difficultyThe difficulty that thatmay mayarise ariseis that is thatin this in case this casethe number the number of fitting of fitting parameters parameters is so islarge so large (a few (a few elasticelastic constants constants for each for eachconstitutive constitutive layer) layer) that, that,likely, likely, it will it not will be not possible be possible to achieve to achieve a meaningful a meaningful set ofset best of fit best parameters. fit parameters.

Appl. Sci. 2018, 8, 124 15 of 19

4. New Perspectives from Micro-Focused Surf-BLS: Mapping the Spatial Profile of Acoustic Modes While the conventional surf-BLS technique, presented above, offers the possibility to measure Appl. Sci. 2018, 8, 124 15 of 19 the frequency of acoustic excitations with a well-defined wavenumber in a large frequency range (1–500 GHz),4. New itPerspectives has a poor from lateral Micro-Focused resolution, Surf-BLS: determined Mapping by the the Spatial diameter Profile of of the laser spot, that is usuallyAcoustic about 30–40 Modesµ m. Moreover, the signal is proportional to the intensity of the acoustic field, but it has noWhile phase the conventional sensitivity. surf-BLS This means technique, that pres inented the case above, of offers inhomogeneous the possibility samples,to measure such as artificiallythe frequency defined phononicof acoustic crystalsexcitations (PC) with [ 71a well] or-defined acoustic wavenumber resonators in [ 1a– large4], one frequency cannot range achieve the imaging(1–500 of the GHz), spatial it has profile a poor of lateral the acoustic resolution, field. determ Thisined limitation by the diameter can be of overcome the laser spot, using that the is recently μ developedusually micro-focused about 30–40 m. surf-BLS Moreover, technique, the signal is whoseproportional schematic to the intensity apparatus of the isacoustic drawn field, in but Figure 14, it has no phase sensitivity. This means that in the case of inhomogeneous samples, such as artificially that hasdefined been extensively phononic crystals used (PC) in the [71] last or decadeacoustic toresonators investigate [1–4], the one lateral cannot localization achieve the imaging of spin of waves in ferromagneticthe spatial films profile and of artificialthe acoustic magnonic field. This limitation crystals [can72]. be In overcome fact, using using a the microscope recently developed objective with large numericalmicro-focused aperture, surf-BLS the technique, laser light whose can schematic be focused apparatus down is drawn to a diffraction-limitedin Figure 14, that has spotbeen with a diameterextensively below 300 used nm in the and last a decade submicrometric to investigate depth the lateral of focus.localization In addition,of spin waves phase in ferromagnetic resolution can be achievedfilms in theand caseartificial of externallymagnonic crystals driven [72]. excitations, In fact, using paving a microscope the way objective to the with tridimensional large numerical mapping aperture, the laser light can be focused down to a diffraction-limited spot with a diameter below of magnetic or acoustic excitations by using surf-BLS as a scanning probe technique, in conjunction 300 nm and a submicrometric depth of focus. In addition, phase resolution can be achieved in the with ancase automated of externally scanning driven excitations, stage for thepaving sample. the way A to coaxial the tridimensional viewing system mapping based of magnetic on a collimated or LED lightacoustic source excitations (455 nm by wavelength), using surf-BLS aas beam a scanning expander, probe technique, and a CCD in conjunction camera is with used an toautomated obtain a direct visualizationscanning of stage thelaser for the spot sample. and A of coaxial the sample viewing regionsystem based under on investigation. a collimated LED Let’s light noticesource (455 that, nm although micro-focusingwavelength), permits a beam to expander, achieve and spatial a CCD and camera phase is used resolution, to obtain a thedirect wavevector visualization sensitivity of the laser is lost, becausespot of both and of the the small sample dimension region under of investigation. the focused Let’s spot notice and the that, very although large micro-focusing collection angle. permits Therefore, to achieve spatial and phase resolution, the wavevector sensitivity is lost, because of both the small micro-focused surf-BLS is well suited to analyze the acoustic field driven by an external signal (as in dimension of the focused spot and the very large collection angle. Therefore, micro-focused surf-BLS acousticisresonators), well suited to rather analyze than the acoustic to detect field thermally driven by activated an external phonons. signal (as The in acoustic application resonators), to these aim is initiatedrather and than preliminary to detect thermally results haveactivated been phonons. just presented The application [73]. Further to these progressaim is initiated will be and achieved in the nearpreliminary future, results also because have been the just operating presented [73]. frequencies Further progress of devices will be are achieved expected in the to near increase future, above a few GHz,also so because the availability the operating of afrequencies technique of operating devices are in expected the frequency to increase domain above a (rather few GHz, than so inthe the time availability of a technique operating in the frequency domain (rather than in the time domain as done domain as done up to now [74,75]) can be advantageous. up to now [74,75]) can be advantageous.

Figure 14. (Left) Scattering geometry of the conventional surf-BLS experiment. The wave vectors of Figure 14.the (incident,Left) Scattering reflected geometryand scattered of thelight conventional are indicated, surf-BLStogether with experiment. that of the The acoustic wave surface vectors of the incident,phonon reflected (in red). and The scattered illuminated light area are is indicated, a circle with together a diameter with of tens that of of microns the acoustic (i.e., much surface larger phonon (in red).than The the illuminated wavelength areaof the is dete a circlected phonons) with a diameterso the wavevector of tens conservation of microns is (i.e., guaranteed; much larger (Right than) the wavelengthSchematic of the drawing detected of the phonons) micro-focused so the apparatus. wavevector The conservationlight beams coming is guaranteed; from either (theRight laser) Schematicor drawingthe of auxiliary the micro-focused LED sourcesapparatus. are represented The by light green beams and blue coming beams, from respectively. either the The laser illuminated or the auxiliary area is a spot of about 300 nm diameter and the collection angle is very large, so the conservation of LED sources are represented by green and blue beams, respectively. The illuminated area is a spot of momentum does not apply. about 300 nm diameter and the collection angle is very large, so the conservation of momentum does not apply. Appl. Sci. 2018, 8, 124 16 of 19

5. Conclusions and Future Perspectives An overview on surf-BLS investigation of the elastic properties of single- and multi-layered materials has been presented, with emphasis given to the influence of the thickness and the transparency of the investigated films dependence on the characteristics of the spectra. Depending on these characteristics, different experimental strategies have been outlined, indicating how to get maximum information with minimum effort. In specific cases, it is possible to achieve a complete elastic characterization of the investigated samples: in other cases only a few elastic constants can be determined while others may remain out of the reach of this technique. Moreover, there has been a detailed mapping of the tridimensional spatial profile of the acoustic field in surface- or bulk-acoustic-wave resonators by the recently developed micro-focused scanning apparatus. The above capabilities could significantly contribute to the desired elastic characterization of multilayered acoustic structures operating at microwave frequencies in ubiquitous ICT devices, such as mobile phones. The reason why the exploitation of surf-BLS is still underestimated by industrial manufacturers in this field is, in my opinion, because it is not a standard characterization technique and requires a rather specific expertise. Therefore, I hope that this review paper may be of help especially for new researchers who want to take advantage of the potentialities of surf-BLS and establish a tight connection with industrial partners to contribute to the design and optimization of the next generation of acoustic resonators for the current and forthcoming generation of mobile devices.

Conflicts of Interest: The author declares no conflict of interest.

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