Ultrasonic and Spectroscopic Techniques for the Measurement of the Elastic Properties of Nanoscale Materials

1. Introduction

Solid materials exist when the atoms find a configuration in which their potential energy has at least a local minimum, around which a stability region exists. Irrespective of the high number of degrees of freedom, and with rare exceptions, in the neighborhood of any minimum a more or less narrow interval exists, in which the potential energy is well represented by quadratic terms, higher order terms becoming negligible. A quadratic potential energy means an elastic restoring force, meaning that, when considering free motion around a stable equilibrium configuration, vibrational excitations are expected, which have a time periodicity. Periodicity can be represented by a frequency, or an angular frequency ω, which is obviously determined by the properties of the physical system, namely in terms of stiffness and inertia. This general consideration applies from the atomic scale up to the full macroscopic scale, at which continuum models are appropriate.

When looking at the atomic motions, stiffness and inertia are given by the interatomic forces and the atomic masses. A distinction can be proposed among vibrational excitations, based on the phase difference between the motions of different atoms. Excitations exist in which when considering atoms at progressively decreasing distances, down to first neighbors, the phase difference between their displacements does not tend to zero. Examples are the vibrations of molecules, or the optical phonons in a crystalline structure. In this case the average displacement evaluated over a group of neighboring atoms does not have a relevant meaning, since the displacement of single atoms can be significantly different from the average one, and since the average can be null (as in the case of isolated molecules) or close to zero even in presence of atomic motions of significant amplitude. Excitations instead exist in which the phase difference between displacements of atoms located at smaller and smaller distances, down to nearest neighbors spacing, are smaller and smaller. Examples are the acoustic phonons in a crystalline structure. In this case the average displacement evaluated over a group of neighboring atoms becomes fully representative of the displacement of single atoms, and is the natural bridge towards a description of the continuum type, with a displacement vector field urt=u(x1,x2,x3,t)which is a continuous function of the position vector r=(x1,x2,x3)and of time t. Excitations of this type are called acoustic excitations.

The description by the continuous vector field of displacement, and consequently by the tensor fields of strain and stress, is appropriate at scales which go from the supramolecular one, of the order of the nanometer or slightly more, at which the above-mentioned average begins to be meaningful, up to the fully macroscopic one. In the elastic continuum model [1, 2, 3] the potential energy is a quadratic function of strains, the coefficients of the expansion being the elements of the tensor of the elastic constants Cijmn, conveniently represented in compact Voigt notation as Ckl. The stiffness being represented by these tensor elements, or by some functions of them which are specific elasticity moduli, and the inertia being represented by the mass density ρ, the equations of motion of the continuous medium are the elastodynamic equations. In the limit of an infinite homogeneous medium, which is symmetric to any translation in position and in time, the fundamental excitations are conveniently taken as waves, which, besides being periodic in time, are periodic in space, as described by a wavelength λ or a wave vector k, with k=2π/λ. These fundamental solutions are characterized by dispersion relations ωk, which are determined by the stiffness and the inertia properties [1, 2, 3].

It has been recognized, since long ago, that the measurement of the vibrational excitations gives access to these properties [4]. In particular, if the inertial properties (atomic masses or mass densities) are known, the stiffness properties can be measured. In both the atomic and the continuum case, they represent the curvature of the potential energy in the neighborhood of its minimum, and therefore they contain information, respectively, about the interatomic bonding and about the stiffness of solids, in its usual meaning. A whole wealth of experimental techniques has therefore been developed, which exploit vibrational excitations to measure material properties [4]. A general advantage of all the measurement techniques based on vibrational excitations is that they can exploit displacements of small amplitude, confined in the neighborhood of the equilibrium position, in which the representation of the potential energy by only the quadratic terms is an excellent approximation. In other words, higher order terms of the potential do not interfere, and, in the case of continua, non-elastic deformation mechanisms, other than the simple, reversible, stretching of interatomic bonds, are not activated.

Vibrational excitations of non-acoustic type, which cannot be described by a continuous field of displacements, are not treated by mechanics, but rather by solid state physics, which measures them by techniques like Raman spectroscopy, to obtain various information at atomic level, including that concerning the interatomic forces. They are not considered here.

Vibrational excitations of the acoustic type, instead, have the same properties from the supramolecular, or nanometric, scale, up to the kilometric scale and above. Accordingly, techniques based on acoustic excitations are exploited to measure the stiffness properties of objects of various sizes, up to dams, and are exploited for geological investigations. These methods measure the dynamic, or adiabatic, elastic moduli; these moduli do not coincide with the isothermal moduli which are measured in monotonic tests (if strain rate is not too high), but in elastic solids the difference between adiabatic and isothermal moduli seldom exceeds 1% [2].

At the other extreme of the size scale, some techniques can be pushed to measure microscopic objects, down to carbon nanotubes. Nanomechanics precisely addresses the behavior of microscopic objects; in this chapter we consider measurement techniques based on vibrational excitations of the acoustic type, which are pushed to measure small objects, towards the size at which the same concept of ‘acoustic excitations’ begins to lose significance, as well as that of the strain field. Techniques based on vibrational excitations, and in particular optical techniques, which avoid mechanical contact, are particularly prone to be applied to small objects, for which the contact with actuators or sensors becomes critical. In many cases, the mechanical properties of materials at this scale are of interest for the design and production of microsystems which operate dynamically; in these cases, the dynamic, adiabatic, moduli, are precisely those of interest.

The next section summarizes some basic concepts about free acoustic excitations in finite objects. The following two sections give an overview of several measurement techniques, which are grouped in two categories. First, those that exploit the oscillations of purpose-built testing structures, which are mechanically actuated, often by piezoelectric means. Secondly, the techniques which measure the properties of ultrasonic waves, that are typically excited and/or detected by optical means. These techniques can be further subdivided among those which operate in the time domain and those which operate in the frequency domain.

2. Acoustic excitations in confined media

Displacements and strains in the elastic continuum model in the absence of body forces, obey the elastodynamic equations; for homogeneous media they are [1, 2, 3]

The invariance to any translation in time is the root of the harmonic time dependence of the fundamental solutions, characterized by the circular frequency ω, which allows to transform the equations into the Helmholtz equations. The invariance to any translation in position, in practically infinite media, is the root of the harmonic space dependence of the fundamental solutions, which are traveling monochromatic harmonic waves, characterized by the wave vector k. The dispersion relations ωkare determined by the properties of the medium. The displacement vector uhaving three independent components, the dispersion relation has three branches, which can be classified according to the relative orientations of the vectors uand k, i.e. according to their polarization: in an isotropic medium, one longitudinal and two transversal modes [1, 2, 3].

The elastic continuum model has no intrinsic length scales; as mentioned above it loses significance at the nanometric scale, while in an infinite medium, which also from the geometrical point of view has no intrinsic length scale, it does not have an upper limit of size. The wavelengths span in a continuous way this whole infinite interval, and the corresponding angular frequencies go, in continuity, from null frequencies for λ→∞, i.e. k→0, up a not sharply defined upper limit, corresponding to the shortest meaningful wavelengths. The absence of intrinsic length scales implies that all these wavelengths behave in exactly the same way, and the dispersion relations are simply linear: ω=vk, the velocities being independent from k, i.e. these modes are non dispersive. In the general anisotropic medium the velocities depend on the direction of k, while in the isotropic case they do not, and the velocities vland vtof the longitudinal and transversal modes are respectively [1, 2, 3]

thermodynamic stability requiring that vt<vl.

Rupture of the unconditional translational symmetry by some kind of boundary condition, which introduces some kind of confinement, induces the appearance of further acoustic modes, namely standing waves. The consequences of confinement can be appreciated also without abandoning the relative simplicity of the isotropic model. They are already present, in a paradigmatic way, in the case which is probably the simplest rupture of the infinite translational symmetry: the plane external surface of a semi-infinite medium. The invariance to any translation in time is not altered, and correspondingly the acoustic modes remain periodic in time, associated to a circular frequency ω. In the same way, the invariance of the medium to any translation in the plane of the surface is not modified, and correspondingly the acoustic modes remain periodic, and traveling, in this plane. This periodicity is conveniently represented by a wave vector k∥parallel to the surface, which identifies a direction and a repetition period. Considering media which possess in-plane isotropy, the direction of k∥becomes irrelevant and, in order to simplify the notation, we introduce here the symbol λ∥≡2π/k∥, which seems inappropriate because wavelength is not a vectorial quantity, but must be understood as a compact form of: “the period along the direction of k∥of an acoustic excitation whose wave vector component is k∥”.

In the direction perpendicular to the planar external surface (the direction of depth), two types of space dependence are instead found. A set of modes takes advantage of the (semi) infiniteness of the medium, and maintains a space periodicity, conveniently represented by a wave vector k⊥perpendicular to the surface. These modes are thus characterized by a full three dimensional wave vector k=k∥+k⊥, and are completely analogous to those of an infinite medium: they are the bulk waves, which are reflected at the surface. There is no upper limit to their wavelengths, and no lower limit to their frequencies. The only novelties are introduced upon reflection: firstly, in the direction perpendicular to the surface the interference between the incident and the reflected waves generates a standing wave pattern. Secondly, the mere superposition of an incident longitudinal wave and its reflected counterpart would not satisfy the stress free boundary conditions. Therefore, upon reflection, an incident longitudinal wave is partially reflected into a longitudinal wave, having the same k∥and a reversed component k⊥, and partially converted into a transversal wave, having the same frequency and the same k∥, but a different value of k⊥, because it has a different velocity [1, 3]. The dual conversion occurs for an incident transversal wave.

However, in the presence of this boundary, the elastodynamic equations admit another set of solutions, which are not periodic in the direction perpendicular to the external surface. They are the surface acoustic waves (SAWs), of which the Rayleigh wave, the only one existing at the free surface of a semi-infinite homogeneous medium, is the prototype. The Rayleigh wave has a displacement field which decays exponentially with depth, with a decay length which is uniquely determined by the elastodynamic equations, and turns out to be very close to λ∥; it is thus confined in the neighborhood of the external surface [1, 3, 5]. The Rayleigh wave also has another character typical of the modes induced by confinement: they do not have a specific polarization. The displacement vector uof the Rayleigh wave has a direction which changes with depth; it always is in the plane identified by k∥and by the normal to the surface, which is called the sagittal plane, but the two components vary with depth in different ways. The Rayleigh wave has its own velocity vR, lower than that of any bulk wave: vR<vt<vl[1, 3, 5]. Acoustic modes having similar properties can also appear at the surface separating two media in perfect adhesion.

In the semi-infinite case the medium still has no intrinsic length scale. Both periods, λ∥and 2π/k⊥, can go in continuity from infinity to the smallest meaningful values, and correspondingly the frequencies go from zero to a very high upper limit. Accordingly, the bulk waves and the Rayleigh wave are non-dispersive, also the velocity vRbeing independent from k∥; the dispersion relation ω=vRk∥is linear. However, for any value of k∥, the value of λ∥, which determines the decay length, somehow sets a length scale. It can approximately be said that the value of vRis mainly determined by the properties of the medium up to a depth λ∥/2; in other words, the Rayleigh wave is a probe which senses the properties of the medium up to that depth [6]. The component k⊥exploring the whole interval from zero to the maximum meaningful value, we have k=k∥+k⊥≥k∥, and λ cannot be below λ∥. For that value of k∥we have bulk modes, whose frequencies go with continuity from vtk∥till very high values, and, below this lower limit, the Rayleigh wave at an isolated frequency vRk∥.

Instead, in media that are finite in at least one dimension, with a size D, the size of the object which supports the excitations sets a reference length scale. The relevance of confinement is determined by the ratio of the wavelength λ to the size D. On the high side, acoustic excitations having a ratio λ/Dsignificantly larger than one are not supported: the size Dsets an upper limit for wavelength. On the lower side, the excitations having a ratio λ/D≪1, down to the lower limit at which the continuum model loses significance, are affected in a negligible way by the finiteness of the medium. For these excitations, the medium is still almost invariant for translations of several wavelengths, and the modes are indistinguishable from traveling periodic waves. However, strictly speaking, they become standing waves, and the difference becomes evident for the excitations whose ratio λ/Dis not far from unity. For these excitations, the medium is definitely not invariant for translations of few wavelengths: the modes cannot have a well-developed periodicity, and are strongly affected by confinement.

Objects having a high aspect ratio, like thin layers or beams, or nanorods, are often of interest: in one or two dimensions their size is D, in the remaining direction(s) it is D′, with D≪D′. The consequences of confinement within Dare evident for excitations having λ∼D, well before the finiteness of D′becomes perceptible. These excitations can be analyzed as if in the other direction(s) the medium was still infinite: the excitations are still periodic in this (these) direction(s), characterized by a wave vector k∥. The effects of the finiteness of D′become evident only when λ∥∼D′.

Again, the Rayleigh waves paradigmatically indicate the consequences of a finite size. We can consider a slab, whose thickness is the characteristic size D, and which can be considered infinite in the other directions. Bulk waves with λ/D≪1are affected in a negligible way by the finiteness of the medium, as well as the SAWs having λ∥≪D. At such wavelengths, the Rayleigh waves are confined in the neighborhood of the two surfaces, and do not interact with each other, while the bulk waves have a component k⊥which is now discretized (the period 2π/k⊥must be an integer sub-multiple of D) but with a narrow spacing. Instead, when λincreases and approaches D, on one side this discretization becomes relevant, and, on the other side, the tails of the displacement fields of the Rayleigh waves at the two surfaces superpose. The two SAWs then merge into modes which are typical of the slab, with displacement fields which extend throughout the thickness, their depth dependence being not periodic. Among them, the bending modes of the slab [7, 8], reminiscent of the bending modes of a membrane, which are different from the transversal bulk modes, which are shear modes, not bending modes. The dispersion relation ω=ωk∥has discrete branches which correspond to these modes. They are nonlinear, i.e. dispersive, meaning that their velocities depend on the period λ∥, more precisely on the ratio λ∥/D, i.e. on the product Dk∥; they asymptotically tend to the linear dispersion relations of the infinite medium when the product Dk∥becomes large.

In the case of a film supported by a substrate, which can be generalized to stratified media, the layer thickness sets the characteristic size D. Bulk waves in any layer, with λ/D≪1, and the Rayleigh wave at the surface of the outermost layer with λ∥≪D, are indistinguishable from those in a semi-infinite medium of the same material. Instead, waves with λ, or λ∥, comparable to, or larger than D(if the substrate is semi-infinite λcan go to infinity) have a displacement field which extends over various layers, and have properties which depend on the properties of the various layers. In particular, the Rayleigh wave becomes a generalized Rayleigh wave, whose depth dependence is affected by the transition from the film to the substrate. When λ∥≫Dthe decay length is also much larger than the film thickness, and the displacement field of the generalized Rayleigh wave is mostly in the substrate. In this limiting case the generalized Rayleigh wave approaches the Rayleigh wave of the bare substrate, only slightly modified by the presence of the supported layer. Depending on the properties of the layers, namely on their acoustic velocities, other SAWs can be supported. For instance, an acoustically slow layer can act as a waveguide, confining some other modes, like the Sezawa modes. These modes are reminiscent of the modes of a slab, but instead of having stress free boundary conditions they have, on the substrate side, continuity boundary conditions, and tails of the displacement field which extend into the adjacent layers. Obviously also these modes are dispersive, their velocities depending on the product Dk∥, or, in the case of several layers, on the products with the various thicknesses. The dispersion relations for the various branches can be numerically computed, as functions of these products and of the elastic constants and mass densities of all the layers [9, 10].

A wide slab, of thickness D, can be cut into a stripe of width comparable to D: confinement thus occurs in two directions. More generally, it is the case of a slender cylinder, of circular or non-circular cross section, of lateral size D, which can still be treated as infinite in the third direction. The parallel wave vector k∥remains fully meaningful, although, having exclusively the axial direction, its vectorial character becomes redundant. New modes appear, namely the torsional ones, beside the bending or flexural ones, and the dilatational ones. In the case of circular cylinders they obey the Pochammer-Cree Equations [11, 12], in which, as it often happens in cylindrically symmetric cases, Bessel functions play a crucial role. More general cases can be analyzed by the so-called xyz algorithm [13, 14], which has been applied to rectangular [15], circular [16] and hexagonal [17] cases, and also to more complex cases like superlattices [18].

The dispersion relation ω=ωk∥has a significant number of branches, which asymptotically tend to the linear dispersion relations of the infinite medium when the product Dk∥becomes large. For smaller values of Dk∥, instead, these branches remain separate, and nonlinear. In particular, several branches have a non-null frequency for k∥=0; they correspond to modes, like e.g. the radial breathing modes, which can have a non-traveling character [12]. Furthermore, for small values of Dk∥, the dispersion relations of most of these modes have a slope much smaller than the velocities vtor vl, meaning a low group velocity, and in some cases even a negative slope [19].

Finally, for objects whose aspect ratio is not far from unity, and therefore in which confinement is along all the three directions to a size of the same order D, the general considerations apply. Namely, all the excitations have the character of discrete standing waves. For ratios λ/D≪1the discretization has a narrow spacing, the modes are almost indistinguishable from those of an infinite medium, and can still be characterized by a wavevector k, or k∥. Instead, for ratios λ/Dnot far from unity the modes, in general, do not have a regular periodicity, and any wavevector loses its meaning. The modes have clearly separate frequencies, are strongly affected by confinement, and strongly depend on the shape of the object. Note that for objects which become nanometric the regime in which λ/D≪1does not exist, because it would mean wavelengths at which the continuum model breaks down.

Two main geometries are of interest, both for fundamental studies and for technological applications. Firstly, the planar geometry of thin films, in which confinement is in only one direction, the critical size Dis the thickness, the wider size D′being the lateral extension of the layer. The typical examples are the supported films and the resonators, either in the form of a clamped membrane or of a cantilever, or a bridge. Secondly, the linear geometry of beams and of nanorods, in which confinement is in two directions, the critical size Dis the diameter, the wider size D′being the length. The two characteristic lengths, Dand D′, identify two length ranges; acoustic excitations can be probed at the two length scales, which obviously correspond to two frequency ranges.

Ultrasonic waves of wavelength comparable to Dare not affected by the finiteness of D′, or by the precise shape of the object supporting them: they see the other dimensions as infinite. The properties of the waves depend on the material properties, and the value of Ddetermines the existence of the discrete modes, which can be observed. In some cases the properties of the waves are also affected by the value of D, as it happens for supported films when the ratio λ/Dis close to unity: in this case the displacement field of the wave appreciably penetrates into the substrate, and is affected by its properties.

Instead, for acoustic excitations at the scale of D′, the effective stiffness for the vibrational modes depends on the material properties, but also, crucially, on the value of D. The properties of the waves thus depend on the material properties and on the value of D, while the value of D′determines the existence of the discrete modes, which can be observed. In fact the fundamental, and higher order, oscillations of a cantilever can be seen as standing flexural waves, of wavelength comparable to of D′, and of effective stiffness dependent on D.

The two ranges of wavelength identify two classes of measurements. A first class exploits purpose-built testing structures, which determine the value of D′; they are probed at effective wavelengths of this order. Typically, their oscillations are mechanically actuated by piezoelectric means. The next section is devoted to them. A second class exploits ultrasonic waves at wavelengths comparable to D. Their excitation is achieved by short laser pulses, or simply by thermal motion. Their detection typically requires optical means, and can be performed either in the time domain or in the frequency domain.

3. Measurement techniques based on testing structures

As repeatedly underlined, the elastic continuum model is meaningful down to almost the molecular scale. Techniques based on acoustic excitations are therefore, in principle, applicable to objects down to the nanometric scale. Their effective application depends on the availability of appropriate transducers to excite and to detect the relevant acoustic excitations. In the case of macroscopic objects, the typical techniques are based on piezoelectric transducers, which can be exploited for both excitation and detection. Specific devices are available, and specific instruments, like acoustic microscopes. An alternative, for what concerns detection, is offered by laser Doppler vibrometry. It has the advantages of the optical techniques: light is a massless probe, contactless, which does not load the measured object, is free from own resonances, has a bandwidth that is essentially determined by electronics (the light sensor and the amplification). Furthermore, it can measure small objects, and can measure surfaces which are difficultly accessible, or on which the application of a detector is not possible, e.g. because of their temperature.

In the case of small objects, down to micrometric or sub-micrometric scale, the exploitation of a separate measurement device in mechanical contact becomes impossible. Mechanical actuation by piezoelectric means is still possible either in the case of resonators, by an external actuator which shakes the whole assembly containing the resonant structure, or by inclusion, by nanofabrication techniques, of a piezoelectric (nano)layer as an integral part of the structure being tested. MEMS/NEMS of this type can also be actuated by electrostatic means. In some cases, the test system can have a structure analogous to that of a complete device, and comprehend, beside the possibility of excitation by piezoelectric or electrostatic means, the ability of measurement, e.g. by capacitive detection or by an interdigitated transducer (IDT). Except for these cases, optical detection is mandatory. Laser Doppler vibrometry is the measurement technique of choice, if the measured system has a flat surface of sufficient size. Otherwise, interferometric techniques have been exploited.

A specific case is that of resonators: a significant effort is under way to produce high quality resonators, mainly because of their great potential for applications in sensing, signal processing, and quantum physics. The properties of a resonator can be measured in a static way, by measuring the deflection of a cantilever, or a membrane, or by measuring its resonant frequency. When a resonator is reduced to a small size, typically in the shape of a cantilever, a bridge, or a clamped membrane, its surface to volume ratio increases. Therefore phenomena, which otherwise are minor or negligible, and are not accounted for by q. (1), become non negligible; namely anelastic effects, possibly connected to internal friction phenomena, and surface phenomena, like surface tension or environmental effects [20, 21]. In particular, interaction with the environment, typically by adsorption of molecules, including water, made available by relative humidity, is the physico-chemical basis for the development of sensors. The development of high performance sensors based on nanoresonators [22, 23] is not considered here. We merely note that research in this direction has led to doubly clamped resonators of thickness down to 22 nm and aspect ratio up to 5000; the measurement of their deflection requires an interferometric technique, and their motion due to Brownian thermomechanical techniques becomes detectable [24].

However, resonant structures can built specifically to the purpose of a precise measurement of the properties of the materials which constitute them. In order to measure the properties of tetrahedral amorphous carbon (also known as diamond-like carbon), Czaplewski et al. built, by standard techniques for the production of MEMS, several resonators, with critical dimension down to 75 nm [25]. Excitation was in some cases electrostatic, and in other cases by shaking by an external piezoelectric actuator. Detection was, depending on the in-plane or out-of-plane deflection, by a laser deflection technique similar to that used by AFMs, or by an interferometric technique. Their results allowed them to analyze dissipation and to discuss various possible mechanisms. Exploiting flexural and torsional oscillators, and the same interferometric measurement technique, they also determined the elastic moduli as function of temperature [26]. In order to measure the properties of an assembly of only carbon nanotubes (called ‘forest of nanotubes’), self-sustained only by their entwining and internal interactions, Hassan et al. built by this material cantilevers, of 1 mm length, which were electrostatically excited, and whose motion was detected by laser Doppler vibrometry [27].

The advantages of miniaturization led to explore also the sizes at which the elastic continuum approach is no longer adequate, and a Molecular Dynamics approach is more appropriate [28]. Interestingly, for membranes which become nanometric but have significantly larger lateral extension, the continuum approach remains useful, both for the theoretical analysis and the measurement technique. Membranes of nanocrystalline diamond and of piezoelectric aluminum nitride, and bilayer membranes, with thicknesses down to 220 nm and diameters up to 1 mm, have been investigated by Knoebber et al. [29] by both a static and a vibrational technique. The static technique, of more macroscopic character, was the bulge test, in which the deflection of the circular clamped membrane under gas pressure is measured; the implementation was optical, the deflection being optically measured by white light interferometry. The vibrational technique involved excitation by an external piezoelectric stack, and detection by laser Doppler vibrometry. The analysis shows that the results from the dynamical technique are less sensitive to the geometrical inaccuracies of the tested membrane. Similarly, the properties of bilayers obtained by growing nanocrystalline diamond on aluminum nitride thin films, of about 200 + 200 nm thickness, were measured producing microresonators, either cantilevers or bridges, of length up to 50 μm, piezoelectrically actuated and measured by laser Doppler vibrometry [30]. Similar nanocrystalline diamond/aluminum nitride membranes, of thicknesses of the order of hundreds of nanometers, were still characterized by the bulge test on circular clamped membranes, of radii up to 1 mm [31].

Also in the analysis of a typical 2-D material, MoS₂, resonators have been built in the form of clamped membranes, of radii of 2 and 3 μm, ranging from a single layer, i.e. a truly atomic thickness, up to over 90 layers. The membranes were obtained over pre-patterned circular holes in a Si substrate, and acted as ‘drums’. Their motion was measured exploiting the vibrating drum membrane and the bottom of circular hole as the two mirrors of an interferometer. The continuum mechanics approach turned out to be useful in the interpretation of the experimental results, which showed the transition from the membrane regime, in which the restoring force in the oscillation is supplied by the membrane tension, to the plate regime, in which the restoring force is supplied by the bending stiffness of the plate [32].

A whole class of devices exploits the intergiditated transducers (IDTs) to launch and resonantly detect surface acoustic waves. A piezoelectric layer is a crucial component of these devices. When the lithographic techniques to produce IDTs is available, devices have been produced specifically for the aim of measuring the properties of the material which constitute them. Measurements have been performed on various forms of artificial diamond (nanocrystalline diamond, nitrogenated diamond-like carbon), of particular interest for devices exploiting surface waves and IDTs because of their high acoustic velocity [33, 34, 35].

4. Measurement techniques based on ultrasonic waves

Various measurement techniques based on vibrations and acoustic waves rely on impulsive mechanical excitation of vibrational modes or of waves. Mechanical excitation by an impact has been analyzed in detail [36, 37], and standards have been issued concerning it [38]. When extending these techniques towards nanomechanics, and in particular to thin films, wave excitation by laser pulses is their natural evolution. Absorption of a laser pulse, of duration τ, induces, by thermal expansion, a sudden expansion: a strain pulse, which propagates away at the speed vlof longitudinal sound. The mechanism of phonon generation has been analyzed in detail [39]. The geometry of this pulse depends on the absorption length ζ of the optical pulse, on the distance vlτtraveled by the strain pulse during the laser pulse, and on the lateral size of the region on which the laser beam is focused. Metals have the shortest absorption lengths, of some nanometers, and have longitudinal sound velocities of few km/s, i.e. few μm/ns. With optical pulses of the order of the ns, heating and thermal expansion occur until the pulse has traveled a distance of few μm.

In the case of a thin supported film, of thickness Dof some micrometers, or even less, a ns laser pulse launches a strain pulse which is originated in the whole thickness of the film. Such a strain pulse travels parallel to the surface, and, if the film has a sufficient lateral extension, can be detected at a distance of millimeters, measuring the transit time. This is the regime of the so-called laser ultrasonics, in which the laser is focused by a cylindrical lens. A line source is therefore present in the film, which launches two SAWs traveling in opposite directions, on a surface whose lateral size D′is large. The ratio λ∥/Dcan be smaller than one, implying that the displacement field is essentially confined within the film, or larger, meaning that the displacement fields of the SAWs penetrate significantly into the substrate. In the latter case a typical objective of the measurement is the detection of the (small) modification of the Rayleigh wave of the bare substrate.

With optical pulses of less than the picosecond, instead, heating and thermal expansion occur only until the strain pulse has traveled a distance of the order of the nm, i.e. less than the absorption length. The pulse displacement during excitation is thus negligible, and heating, and thermal expansion, occur only in a thin surface skin, of depth ζ, i.e.few nm. This is the regime of the so-called picosecond ultrasonics, in which the laser is focused by a spherical lens to a spot whose width is of several micrometers, i.e. orders of magnitude wider that the depth within which thermal expansion occurs. In the case of a film of thickness D, a plane wave is therefore launched: a strain pulse, of lateral extension of several μm, which extends over a depth of ζ, has leading wavelength of this same order, and travels in the direction perpendicular to the surface. Except for truly nanometric films, in this case λ/D≪1, and the finiteness of Dbecomes relevant thanks to the reflection of the pulse when it reaches the opposite surface. If the film is supported the reflection is only partial, part of the pulse being transmitted; the echo however returns to the surface, where it can be detected, and where it is again reflected back. The transit time, of the order of the ns, or less, is measured by a pump-and-probe technique, in which a short, weaker, ‘probe’ pulse follows the ‘pump’ pulse with a variable delay, and detects the time evolution of the transient induced by the pump pulse, scanning the delay until the exhaustion of the transient itself.

The technique has also been exploited for nanostructures which do not have a planar surface extending over the several micrometers of the focused spot; nanorods are an example. In this case the geometry of the strain pulse is not the simple planar one outlined above, and is rather dictated by the specific geometry of the illuminated object(s). The detected signal contains however relevant information, whose interpretation typically requires the modeling of the specific dynamic structure of the investigated objects.

In both the above techniques a transient is induced by a pulse which is short, i.e. strongly localized in time, and also strongly localized in space, resulting in a broadband and localized pulse, whose transit is easily detected. In a different technique a similar short pulse is exploited, which produces a transient, but illuminates a wider segment of the surface with a periodic pattern, and has therefore a narrow band. Periodicity is obtained splitting the pump laser pulse in two beams, and recombining them at the surface of the sample, forming an interference pattern. The same impulsive thermal expansion thus occurs with a periodic space modulation. The periodicity, which here is called λ∥and which in the literature dealing with this technique is more often called Λ, is selected by the interference geometry, as well as its direction: the wavevector k∥is selected by the illumination geometry. The technique is applicable to samples having a planar surface whose lateral size D′is much larger than the periodicity λ∥.

The time evolution of the reflectivity (or, possibly, of the transmittance) is measured in one point, either by a continuous measurement by a fast detector, or, more frequently, by the pump-and-probe technique. The measured signal typically has a slowly declining component, corresponding to the total energy deposited by the pulse, which eventually diffuses towards the adjacent parts of the sample, and a periodic component. The periodic component is due to the SAWs which are launched in the directions of k∥and −k∥, and which form a standing wave, eventually damped, both because the two SAWs travel away, and because of intrinsic damping mechanisms, like the thermoelastic one. By the spectral analysis of the observed signal one point of the dispersion relation ω=ωk∥is obtained, hence the name of ‘transient grating spectroscopy’. The full dispersion relation is obtained scanning the wavevector k∥, by adjusting the geometry of the interfering beams; at least in principle, a wide interval of k∥is accessible. In the case of a film of thickness D, the interval from λ∥/D≪1to λ∥/D≫1can sometimes be scanned. It must however be noted that this is a case in which the characteristic length Dis dictated, more than by the geometry of the sample, by the geometry of the excitation, i.e. by the periodicity λ∥determined by the experimental set-up. In fact, as noted above, SAWs of wavevector k∥are sensitive to the properties of the medium only up to depth of approximately λ∥/2.

In all the above techniques a ‘pump’ laser pulse induces a transient, whose time evolution is measured, with a time resolution down to the picosecond scale. The transient grating technique deserves the ‘spectroscopy’ name because it relies on the successive spectral analysis of the measured time signal. Instead, the vibrational spectroscopies investigate the steady state excitation of vibrational modes, due to thermal motion. They exploit continuous, not pulsed, lasers, and detect the component of the scattered light which has undergone a frequency shift, because it was inelastically scattered by the excitations present in the sample. Obviously, inelastic scattering involves some energy exchange between the optical and the acoustic fields, but, due to the smallness of the scattering cross section, this is minor, in comparison to the thermal energy. Only in special cases, namely highly confined nanostructures, the exchange can become significant. The most widespread vibrational spectroscopies, namely Raman spectroscopy and infrared spectroscopy, are not considered here because they measure vibrational excitations of non acoustic type.

Instead, Brillouin scattering is precisely the inelastic light scattering by the vibrational excitations of the acoustic type, and Brillouin spectroscopy measures it. In a manner analogous to that of Raman spectroscopy, the sample is illuminated by a laser beam, and the scattered light is collected; the spectral analysis singles out the minor fraction which has undergone inelastic scattering by the excitations in the sample. The scattering geometry selects an exchanged wavevector k, or k∥in the case of SAWs. Therefore, also in this technique, in the case of SAWs the experimental technique selects a periodicity λ∥which essentially dictates the length scale D, and the depth over which the properties of the medium are interrogated. A Brillouin spectrum supplies a point of the dispersion relation ω=ωkfor each of the branches which give a measurable peak. With visible light, and typical properties of solid materials, the observed frequencies, i.e. the frequency shifts of light, range from few GHz to tens of GHz, i.e. a wavenumber in the range of the cm⁻¹, to be compared with the typical frequency shifts observed in Raman spectroscopy, in the range of hundreds of cm⁻¹. Accordingly, the diffraction gratings, which are the common spectrum analysers in Raman spectroscopy, do not have a sufficient resolution, and other techniques must be employed. The tandem multipass Fabry-Perot interferometer has become the standard apparatus in Brillouin spectroscopy [40]. The small observed frequencies (1 cm⁻¹ means 30 GHz or 1.4°K) make the Stokes and anti-Stokes parts of the spectra symmetric, to a difference from what happens in Raman spectroscopy.

The dispersion relations ω=ωkof the traveling waves, whenever a wavevector kcan meaningfully be identified, and the frequencies of the standing waves, can be theoretically predicted, from the simplest ones of Eqs.(2) to the more complex ones which can only be numerically computed. They are all functions of the properties of the medium (or the media involved). Therefore, whenever and however they are measured, they provide access to the properties, which can be found fitting the computed dispersion relations, or the frequencies, to the measured ones. Obviously the amount of obtainable information depends on the amount, and the quality, of the experimental evidence, and on the simpler or more sophisticated way of treating it. The obtainable information ranges from a semi-quantitative comparison for a single parameter, to a complete elastic characterization of a layer.

5. Conclusions

Nanotechnology, and nanodevices, identify a rapidly growing technological field. For the development of nanodevices a precise knowledge of the elastic properties of materials is of utmost importance, also because materials obtained at the nanoscale have properties which do not coincide with those of their bulk counterpart. Accordingly, a variety of techniques have been developed, which have proven able to investigate the material properties at the nanoscale. Most of these techniques rely on the interaction between optical electromagnetic fields, and mechanical acoustic fields. An overview has been given of these various techniques, offering elements for the evaluation of their appropriateness for different characterization needs.

Acknowledgments

This work was supported by the project ‘SpaceSolarShield’, funded by Cariplo Foundation, Milano, Italy, project 2018-1780.