**4. Mechanical excitation**

130 Acoustic Waves – From Microdevices to Helioseismology

carbon (also known as amorphous diamond), by standard techniques for the production of micro-electromechanical systems. They exploited piezo-electric actuation, and an interferometric technique to measure the oscillation. They were able to perform measurements at variable temperatures, determining the elastic moduli of this material as function of temperature. They analyzed the uncertainty sources, finding that the leading contribution to the uncertainty comes, for the flexural oscillator, from the value of the mass density of this material, while for the torsional oscillator it comes from the exact dimensions

In larger structures, waves at frequencies in the tens of MHz range can be excited and detected by piezoelectric elements, possibly operating simultaneously as actuators and sensors. Excitation can also be performed by a laser pulse; if the pulse is short enough, the upper limit of the measurable frequency range can be set by the piezoelectric sensor. Optical detection techniques are also available. Specific devices like interdigitated transducers (IDTs) can be built by lithographic techniques on, or within, an appropriate layer stack, which must include a piezoelectric layer. Such devices emit and receive waves at the wavelength which resonates with the periodicity of the transducer, typically at micrometric scale. This configuration was exploited to measure the material properties (Bi et al. 2002; Kim et al., 2000), but it is seldom adopted, because it requires the production of a dedicated micro device. Micrometric and sub-micrometric wavelengths correspond to frequencies in the GHz to tens of GHz range. Detection of such frequencies requires optical techniques; excitation of such

The variety of vibration based methods to measure the stiffness of solids the can be classified according to various criteria. In this chapter methods are reviewed grouping them by the main vibration excitation techniques: mechanical excitation, either periodic or by percussion, laser pulse excitation, and inelastic light scattering (Brillouin spectroscopy). Similarly to Raman spectroscopy, Brillouin spectroscopy does not excite vibrations at all, but relies on the naturally occurring thermal motion. This gives access to the broadest band,

In all the techniques based on vibrations the elastic constants themselves are not the direct outcome of the measurement, but are derived from direct measurements of a primary quantity like frequency or velocity, and 'auxiliary' quantities like thickness, or mass density. The uncertainty to be associated to the resulting value of each elastic constant must be evaluated considering the uncertainties associated to each of the raw measurements. For a quantity *q* which is derived from directly measured quantities *a*, *b* and *c*, the uncertainty

σ

 *<sup>a</sup>* /*a* , ( ) σ

σσ

=++ (8)

= , where *A* is a numerical constant, the usual error propagation formula can

σ

2 2 22 *q* 2 22 *a bc q a bc*

αβγ

However, the various uncertainties can have different meanings and consequences. The frequency is typically measured either identifying the frequency of a periodic signal which achieves resonance, or by the spectral analysis of the response to a broadband excitation. In

σ*q*

*<sup>c</sup>* . For a functional dependence of the

 *<sup>b</sup>* /*b* , ( ) σ*<sup>c</sup>* /*c* as

but with small vibration amplitudes, which require time consuming measurements.

σ *<sup>a</sup>* , σ*<sup>b</sup>* and

σ

frequencies can be obtained by laser pulses of short enough duration.

of the thin member undergoing torsion.

**3.3 Precision and accuracy** 

type *f Aa b c* α β γ

depends on the 'primary' uncertainties

be written in terms of the relative uncertainties ( )

σ Mechanical excitation can be either impulsive and broadband, as obtained by a percussion, or narrow band, as provided by a periodic excitation. Most methods exploiting mechanical excitation rely on the identification of the natural frequencies, or resonances, of a structure. With a periodic excitation, such frequencies are identified scanning the excitation frequency until resonance conditions (maximum oscillation amplitude for given excitation force) are detected. With a broadband excitation the response (measured amplitude) is frequency analyzed to identify the resonant frequencies.

Among the methods adopting harmonic excitation, acoustic microscopy (Zinin, 2001) exploits a piezoelectric actuator, typically in the form of an acoustic lens, and often operating also as a transducer. The acoustic lens is mechanically coupled to the sample by a liquid drop. Acoustic microscopy can be operated with imaging purposes; in the quantitative acoustic microscopy version (Zinin, 2001) it aims at measuring the acoustic properties of the sample.

Beside acoustic microscopy, two main types of methods have been developed. The first one measures the bulk properties. It adopts macroscopic homogeneous samples, and can exploit either broadband or narrow band excitation. These methods have also been ruled by norms (ASTM, 2008, 2009). A second group of methods, collectively called Resonance Ultrasound Spectroscopy, aims at measuring the properties of thin supported films. It almost invariably exploits a periodic excitation, whose frequency is swept in order to achieve resonance conditions.

### **4.1 Measurement of bulk properties**

Macroscopic homogeneous samples are self supporting. They can be tested as free standing samples, provided the disturbances to free oscillations are minimized. Such a minimization includes sample suspension by thin threads, or specimen support by adequate material (cork, rubber), the supports having contact of minimum size, and positions at the nodes of the fundamental vibrational mode of interest, either flexural or torsional. Also the sensor contact, if oscillation is detected by a contact device, must be devised aiming at the minimization of the disturbance induced by the contact. Non contact detection techniques

Acoustic Waves: A Probe for the Elastic Properties of Films 133

thin disks, exciting different modes by different impact points of a hollow zirconia bead, and measuring the response by a microphone. They exploit computations of the resonant frequencies of thin disks performed by others, and they estimate the accuracies of these

In all these cases, from the numerical computations the mode frequencies are tabulated or

mode frequencies depend essentially on Poisson's ratio: the ratios of the observed frequencies allow therefore to identify the modes and to evaluate Poisson's ratio. The

Both Nieves et al. and Alfano & Pagnotta perform detailed analyses of the measurement uncertainties. They both find that, also with their experimental set-up, the frequency measurement has a crucial role in the precision of the obtained moduli. Since Alfano & Pagnotta consider thin plates, they find that the precision of the thickness *t* is also crucial, simply because, thickness *t* being much smaller than the plate size *a*, a small value of the

/ is more difficultly achievable than for *a <sup>a</sup>*

For comparison purposes, Nieves et al. (2000), beside considering the axial modes, also excite, by tangential percussion, the torsional modes, whose frequencies can be computed in closed form. Once the torsional modes are discriminated from the bending modes, the results agree to better than 1%, indicating a precision of this order. They also perform measurements by the pulse-echo method, finding instead discrepancies of 2% or more; they suggest that this method, which measures the propagation velocity of travelling waves, might become intrinsically less accurate when performed in a confined geometry of small size.

The Resonance Ultrasound Spectroscopy (RUS) methods have been developed (Migliori et al., 1993; Ohno, 1976; Schwarz et al., 2005; So et al., 2003) aiming in particular at the measurement of the properties of supported thin films. A recent implementation (Nakamura et al., 2004, 2010) exploits, as film support, a thin plate which, to be measured, is located on a tripod. One of the three legs is rigid, and contains a thermocouple which monitors the sample temperature; the second leg is a piezoelectric actuator, feeding a harmonic excitation whose frequency is swept, the third one is a piezoelectric sensor, which detects the oscillation amplitude. The resonance spectrum, i.e. the oscillation amplitude as function of the frequency, is measured sweeping the excitation frequency. Several peaks are found, sometimes with partial overlaps; their amplitudes, but not their frequencies, depend on the position of the piezoelectric sensor. The measurement is precise enough to clearly detect the difference between measurements in air and in vacuum, and the reproducibility of the resonance frequencies is at the 0.1 % level (Nakamura et al., 2010). The elastic constants are found fitting the computed frequencies to the measured ones; to this end, mode identification is crucial. Identification is performed keeping the excitation frequency at resonance, and scanning the specimen surface by a laser-Doppler interferometer. The map of the out-of-plane displacement of the vibrating specimen is thus obtained, which allows an

The elastic constants of the substrate are previously found by performing the same type of measurement on a bare substrate, and the elastic constants of the film are derived from the measured modifications of the resonance spectrum of the substrate. Since the vibration amplitude of the standing waves in a plate is maximum at the surface, the sensitivity of the

frequency values allow then, by the scaling parameter, to derive the shear modulus.

ρ ( ) π

σ/ .

. The ratios of the

interpolated; Nieves et al. exploit the scaling parameter *G L* / /

computations to 1 % or better.

relative uncertainty *t <sup>t</sup>*

σ

**4.2 Resonance ultrasound spectroscopy** 

unambiguous mode identification.

are available, including all optical techniques, and acoustic techniques: in the proper frequency range, oscillations can be detected through the air, by a microphone, and even excited, with harmonic excitation, by a similar technique, exploiting an audio oscillator and an audio amplifier. The optical techniques, intrinsically contact-less and inertia-less, have the broadest band, only limited by the light detection and analysis apparatus.

Since the full characterization of the elasticity of an isotropic medium requires two independent parameters, it typically requires excitation of at least two modes of different nature. For a specific simple geometry, the slender rod of length *L*, test methods have been regulated by norms (American Society for Testing and Materials [ASTM], 2008, 2009, and other ASTM norms cited by these two). Mainly flexural and torsional modes are considered, for a slender rod of mass *m*, and either rectangular section of width *b* and thickness *t*, or circular section of diameter *D*. Mass density being measured as ρ = *m tbL* /( ) , Eq.(7) takes in this case, for the rectangular section, the forms (ASTM, 2008)

$$E = \left[ m \left/ \left( t b \mathcal{L} \right) \right] \right] \left( f\_f L \right)^2 \times 0.9465 \left( L \left/ t \right)^2 \mathcal{T}\_\mathcal{E} \left( \nu\_\prime L \,/ \, t \right) \tag{9a}$$

$$\mathbf{G} = \left[ m \,/\left( t\mathbf{b}L \right) \right] \left( f\_r \mathbf{L} \right)^2 \times \mathbf{4} \mathbf{T}\_\odot \left( \mathbf{b} \,/\, \mathbf{t}, \mathbf{b} \,/\, \mathbf{L} \right) \tag{9b}$$

where *<sup>f</sup> f* and *<sup>t</sup> f* are the frequencies of respectively the fundamental flexural and torsional modes, and *TE* and *TG* are numerical factors, functions of the indicated dimensionless quantities. Similar formulas hold for rods of circular section. The fundamental frequencies can be identified by either sweeping the frequency of a periodic excitation (ASTM, 2008), or by the broadband excitation by a mechanical percussion (ASTM, 2009), followed by the frequency analysis of the response. In both cases the displacement can be sensed by a contact transducer or by a microphone. The estimates for *E* and *G* being coupled by the value of Poisson's ratio (Eqs. 4 and 9a), an iterative procedure is indicated, to obtain consistent estimates. The algorithm of Eq.(8) applied to Eq. (9a) gives

$$\left(\frac{\sigma\_E}{E}\right)^2 = \left(\frac{\sigma\_m}{m}\right)^2 + 3^2 \left(\frac{\sigma\_r}{t}\right)^2 + \left(\frac{\sigma\_b}{b}\right)^2 + 3^2 \left(\frac{\sigma\_L}{L}\right)^2 + 2^2 \left(\frac{\sigma\_f}{f}\right) \tag{10}$$

similar expressions being obtained for ( ) σ *<sup>G</sup>* / *G* and for rods of circular section; such equations allow to derive the uncertainties to be associated to the obtained moduli, from the intrinsic uncertainties of the primary quantities. It was estimated (ASTM, 2008) that the major sources of uncertainty come from the fundamental frequency *f* and from the smallest specimen dimension (thickness or diameter). Uncertainties of the moduli in the 1% range are achievable.

Other free standing sample geometries were considered, and analyzed by detailed finite elements computations, to identify the appropriate values of the numerical factor *N* (see Eq. (7)). Nieves et al. (2000) consider a cylinder with length equal to diameter (*L* = *D*). They excite vibrations by a longitudinal percussion, and detect the displacement by an optical technique; several modes are typically observed. Alfano & Pagnotta (2006) consider instead a thin square plate, and, analyzing the displacement distributions of the first modes, they identify the best positions for the plate supports and for perpendicular percussion. They measure the response by a microphone. D'Evelyn & Taniguchi (1999) similarly exploited

are available, including all optical techniques, and acoustic techniques: in the proper frequency range, oscillations can be detected through the air, by a microphone, and even excited, with harmonic excitation, by a similar technique, exploiting an audio oscillator and an audio amplifier. The optical techniques, intrinsically contact-less and inertia-less, have

Since the full characterization of the elasticity of an isotropic medium requires two independent parameters, it typically requires excitation of at least two modes of different nature. For a specific simple geometry, the slender rod of length *L*, test methods have been regulated by norms (American Society for Testing and Materials [ASTM], 2008, 2009, and other ASTM norms cited by these two). Mainly flexural and torsional modes are considered, for a slender rod of mass *m*, and either rectangular section of width *b* and thickness *t*, or

*E m tbL f L L t T L t* ( ) ( ) *f E* ( )( ) <sup>2</sup> <sup>2</sup> <sup>=</sup> / <sup>×</sup>0.9465 / , /

where *<sup>f</sup> f* and *<sup>t</sup> f* are the frequencies of respectively the fundamental flexural and torsional modes, and *TE* and *TG* are numerical factors, functions of the indicated dimensionless quantities. Similar formulas hold for rods of circular section. The fundamental frequencies can be identified by either sweeping the frequency of a periodic excitation (ASTM, 2008), or by the broadband excitation by a mechanical percussion (ASTM, 2009), followed by the frequency analysis of the response. In both cases the displacement can be sensed by a contact transducer or by a microphone. The estimates for *E* and *G* being coupled by the value of Poisson's ratio (Eqs. 4 and 9a), an iterative procedure is indicated, to obtain

> *mE bt L f b L f E m t*

equations allow to derive the uncertainties to be associated to the obtained moduli, from the intrinsic uncertainties of the primary quantities. It was estimated (ASTM, 2008) that the major sources of uncertainty come from the fundamental frequency *f* and from the smallest specimen dimension (thickness or diameter). Uncertainties of the moduli in the 1% range are

Other free standing sample geometries were considered, and analyzed by detailed finite elements computations, to identify the appropriate values of the numerical factor *N* (see Eq. (7)). Nieves et al. (2000) consider a cylinder with length equal to diameter (*L* = *D*). They excite vibrations by a longitudinal percussion, and detect the displacement by an optical technique; several modes are typically observed. Alfano & Pagnotta (2006) consider instead a thin square plate, and, analyzing the displacement distributions of the first modes, they identify the best positions for the plate supports and for perpendicular percussion. They measure the response by a microphone. D'Evelyn & Taniguchi (1999) similarly exploited

2 2

σσ

σ

*G m tbL* ( ) ( ) *ft G*

ρ

ν(9a)

= / 4/ × , / (9b)

*L T b tb L* ( ) <sup>2</sup>

= *m tbL* /( ) , Eq.(7) takes in

 

 <sup>+</sup> <sup>+</sup> <sup>+</sup> <sup>+</sup> = 

*<sup>G</sup>* / *G* and for rods of circular section; such

σ2

σ

2

2

3 3 2 (10)

the broadest band, only limited by the light detection and analysis apparatus.

circular section of diameter *D*. Mass density being measured as

consistent estimates. The algorithm of Eq.(8) applied to Eq. (9a) gives

2

σ

2 2

σ

achievable.

similar expressions being obtained for ( )

this case, for the rectangular section, the forms (ASTM, 2008)

thin disks, exciting different modes by different impact points of a hollow zirconia bead, and measuring the response by a microphone. They exploit computations of the resonant frequencies of thin disks performed by others, and they estimate the accuracies of these computations to 1 % or better.

In all these cases, from the numerical computations the mode frequencies are tabulated or interpolated; Nieves et al. exploit the scaling parameter *G L* / / ρ ( ) π . The ratios of the mode frequencies depend essentially on Poisson's ratio: the ratios of the observed frequencies allow therefore to identify the modes and to evaluate Poisson's ratio. The frequency values allow then, by the scaling parameter, to derive the shear modulus.

Both Nieves et al. and Alfano & Pagnotta perform detailed analyses of the measurement uncertainties. They both find that, also with their experimental set-up, the frequency measurement has a crucial role in the precision of the obtained moduli. Since Alfano & Pagnotta consider thin plates, they find that the precision of the thickness *t* is also crucial, simply because, thickness *t* being much smaller than the plate size *a*, a small value of the relative uncertainty *t <sup>t</sup>* σ / is more difficultly achievable than for *a <sup>a</sup>* σ/ .

For comparison purposes, Nieves et al. (2000), beside considering the axial modes, also excite, by tangential percussion, the torsional modes, whose frequencies can be computed in closed form. Once the torsional modes are discriminated from the bending modes, the results agree to better than 1%, indicating a precision of this order. They also perform measurements by the pulse-echo method, finding instead discrepancies of 2% or more; they suggest that this method, which measures the propagation velocity of travelling waves, might become intrinsically less accurate when performed in a confined geometry of small size.

### **4.2 Resonance ultrasound spectroscopy**

The Resonance Ultrasound Spectroscopy (RUS) methods have been developed (Migliori et al., 1993; Ohno, 1976; Schwarz et al., 2005; So et al., 2003) aiming in particular at the measurement of the properties of supported thin films. A recent implementation (Nakamura et al., 2004, 2010) exploits, as film support, a thin plate which, to be measured, is located on a tripod. One of the three legs is rigid, and contains a thermocouple which monitors the sample temperature; the second leg is a piezoelectric actuator, feeding a harmonic excitation whose frequency is swept, the third one is a piezoelectric sensor, which detects the oscillation amplitude. The resonance spectrum, i.e. the oscillation amplitude as function of the frequency, is measured sweeping the excitation frequency. Several peaks are found, sometimes with partial overlaps; their amplitudes, but not their frequencies, depend on the position of the piezoelectric sensor. The measurement is precise enough to clearly detect the difference between measurements in air and in vacuum, and the reproducibility of the resonance frequencies is at the 0.1 % level (Nakamura et al., 2010). The elastic constants are found fitting the computed frequencies to the measured ones; to this end, mode identification is crucial. Identification is performed keeping the excitation frequency at resonance, and scanning the specimen surface by a laser-Doppler interferometer. The map of the out-of-plane displacement of the vibrating specimen is thus obtained, which allows an unambiguous mode identification.

The elastic constants of the substrate are previously found by performing the same type of measurement on a bare substrate, and the elastic constants of the film are derived from the measured modifications of the resonance spectrum of the substrate. Since the vibration amplitude of the standing waves in a plate is maximum at the surface, the sensitivity of the

Acoustic Waves: A Probe for the Elastic Properties of Films 135

Fitting the computed dispersion relation to the measured one allows to derive the film properties. The width of the measured frequency interval can allow to to derive the Young modulus and also the film thickness (Schneider et al., 1997, 2000). The uncertainties of the results is evaluated numerically by the fitting procedure. Since the observed propagation distance is of the order of millimetres, the obtained properties are representative of an average over the propagation distance. The performance of the method could be pushed to the measurement of the properties of diamond-like carbon films having thickness down to 5 nm (Schneider et al., 2000). By stretching the observed propagation path to 20 mm it was possible to reduce the uncertainty of the measured propagation velocity of the Rayleigh wave to below 0.25 m/s. It was thus possible to detect the small variation of the Rayleigh velocity (5081 m/s for the bare (001) silicon substrate) induced by the presence of the film. In a different configuration, the laser pulse is focused on a point instead of a line, with consequent excitation of waves which expand in all the radial directions, and a common path interferometer is adopted, whose light collection point scans the specimen surface (Sugawara et al., 2002). It is thus possible to visualize the wavefronts, circular for isotropic samples and non circular for anisotropic ones, with time resolution of the order of

The so called picosecond ultrasonics technique owes its name to the picosecond laser pulses which were available at the time it was introduced (Thomsen et al., 1984, 1986). It is nowadays implemented by femtosecond laser pulses, and is intrinsically suited to characterize thin supported films and multilayers. It follows the optical pump-and-probe scheme (Belliard et al., 2009; Bienville et al., 2006; Bryner et al. 2006; Vollmann et al. 2002). The pump beam, a femtosecond laser pulse, is focused, by a spherical lens, at the specimen surface and, at least partially, absorbed. The focusing spot, a few to tens of micrometers wide, is orders of magnitude larger than the characteristic lengths of the involved phenomena: mainly the optical absorption length, and also the thermal diffusion length and the acoustic wave propagation length, both for a femtosecond time scale. Bryner et al. (2006) estimate that with an aluminium surface, a near infrared laser (800 nm), and a pulse width of 70 fs, the absorption depth and therefore the dominant acoustic wavelength are of the order of 10 nm. Quite consistently, for a similar laser pulse but for Pt and Fe ultra thin films, Ogi et al. (2007) estimate at several THz the upper bound of the frequencies excited by the

The thermal and mechanical fields are thus (almost perfectly) laterally uniform, and essentially one dimensional: the absorbed pulse has a depth of the order of nanometres, and propagates like an acoustic wave with a plane wavefront which travels perpendicular to the surface, towards the specimen depth. At each interface this wave is partly transmitted and partly reflected, according to the acoustical impedances of the layers, and gives rise to

The surface is then probed by the probe beam, much weaker than the pump beam, which reaches the surface with a variable delay, controlled by a delay line. The probe beam, similarly to the acoustic wave, is partly transmitted and partly reflected at each interface, leading, in each layer, to a forward and a backward electromagnetic field, whose complex amplitudes can be calculated using a transfer-matrix formalism. The external reflectivity of the surface is the ratio of the backward to the forward complex amplitudes in the outer

echoes, which return to the surface, where they again are reflected back.

picoseconds and spatial resolution of a few micrometres.

**5.2 Picosecond ultrasonics** 

laser pulse.

method is better than the ratio of film thickness to the support thickness. In particular, the sensitivity of each resonance frequency to each elastic constant is assessed evaluating numerically the derivatives *ij* ∂ ∂ *f* / *C* , and from these values the uncertainties Δ*Cij* are derived from the estimated uncertainties Δ*f* as

$$
\Delta \mathbf{C}\_{\vec{\eta}} = \frac{1}{\partial f \;/\partial \mathbf{C}\_{\vec{\eta}}} \Delta \mathbf{f} \; \; . \tag{11}
$$

Deposited thin films often have a significant texture, with one crystalline direction preferentially oriented perpendicularly to the substrate surface, and random in-plane orientations, resulting, at a scale larger than that of the single crystallite, in in-plane isotropy, with different out-of-plane properties. This type of symmetry corresponds to the hexagonal symmetry, in which the tensor of the elastic constants is determined by five independent quantities. Among these, the resonance frequencies turn out to be almost insensitive to C*44*, which therefore remains not determined, while the highest sensitivity is to C*11* (Nakamura et al., 2010).

### **5. Laser pulse excitation**

A laser pulse which is absorbed induces a sudden local heating which, by thermal expansion, produces an impulsive mechanical loading. Such a mechanical impulse excites waves in a broad frequency range, which are then detected. The accessible frequency band can be limited by either the excitation bandwidth or the detection device. According to the nature of the material being investigated, the deposition of an interaction layer can be needed. In particular, a short absorption depth is required, in order to excite a pulse which has small spatial and temporal duration, and therefore a broad band. The power density threshold for ablation must also be considered: measurements are typically conducted with high repetition rate pulses, and if ablation occurs the specimen undergoes a continuous modification during the measurement.

Two main configurations have been developed up to maturity; they differ for the propagation geometry and for the technique to detect vibrations. They are respectively called laser ultrasonics and picoseconds ultrasonics.

### **5.1 Laser ultrasonics**

The so called laser ultrasonics technique is mainly exploited to characterize thin supported films. Oscillations are excited by a focused laser pulse, and propagation along the surface is detected, measuring the surface displacement at a distance from excitation pulse ranging from millimetres to centimetres. It is mainly the Rayleigh wave, modified by the presence of the film, which is excited and detected. The laser pulse, typically of nanosecond duration, is focused by a cylindrical lens on a line. The sudden expansion of this line source launches surface waves of limited divergence, propagating along the surface, perpendicularly to the focusing line. The component of the surface displacement normal to the surface itself can be measured, at various distances from the line source, by optical interferometry (Neubrand & Hess, 1992; Withfield et al., 2000), or, in a simpler and more robust way, by a piezoelectric sensor (Lehmann et al., 2002; Schneider et al., 1997, 1998, 2000).

The recorded displacement is frequency analyzed, yielding the dispersion relation *fv* )( for a frequency interval that can extend over a full frequency decade (e.g. 20 to 200 MHz).

method is better than the ratio of film thickness to the support thickness. In particular, the sensitivity of each resonance frequency to each elastic constant is assessed evaluating numerically the derivatives *ij* ∂ ∂ *f* / *C* , and from these values the uncertainties Δ*Cij* are

> *C f f C* 1 / Δ= Δ

Deposited thin films often have a significant texture, with one crystalline direction preferentially oriented perpendicularly to the substrate surface, and random in-plane orientations, resulting, at a scale larger than that of the single crystallite, in in-plane isotropy, with different out-of-plane properties. This type of symmetry corresponds to the hexagonal symmetry, in which the tensor of the elastic constants is determined by five independent quantities. Among these, the resonance frequencies turn out to be almost insensitive to C*44*, which therefore remains not determined, while the highest sensitivity is to

A laser pulse which is absorbed induces a sudden local heating which, by thermal expansion, produces an impulsive mechanical loading. Such a mechanical impulse excites waves in a broad frequency range, which are then detected. The accessible frequency band can be limited by either the excitation bandwidth or the detection device. According to the nature of the material being investigated, the deposition of an interaction layer can be needed. In particular, a short absorption depth is required, in order to excite a pulse which has small spatial and temporal duration, and therefore a broad band. The power density threshold for ablation must also be considered: measurements are typically conducted with high repetition rate pulses, and if ablation occurs the specimen undergoes a continuous

Two main configurations have been developed up to maturity; they differ for the propagation geometry and for the technique to detect vibrations. They are respectively

The so called laser ultrasonics technique is mainly exploited to characterize thin supported films. Oscillations are excited by a focused laser pulse, and propagation along the surface is detected, measuring the surface displacement at a distance from excitation pulse ranging from millimetres to centimetres. It is mainly the Rayleigh wave, modified by the presence of the film, which is excited and detected. The laser pulse, typically of nanosecond duration, is focused by a cylindrical lens on a line. The sudden expansion of this line source launches surface waves of limited divergence, propagating along the surface, perpendicularly to the focusing line. The component of the surface displacement normal to the surface itself can be measured, at various distances from the line source, by optical interferometry (Neubrand & Hess, 1992; Withfield et al., 2000), or, in a simpler and more robust way, by a piezoelectric

The recorded displacement is frequency analyzed, yielding the dispersion relation *fv* )( for a frequency interval that can extend over a full frequency decade (e.g. 20 to 200 MHz).

*ij*

∂ ∂ . (11)

*ij*

derived from the estimated uncertainties Δ*f* as

C*11* (Nakamura et al., 2010).

**5. Laser pulse excitation** 

modification during the measurement.

**5.1 Laser ultrasonics** 

called laser ultrasonics and picoseconds ultrasonics.

sensor (Lehmann et al., 2002; Schneider et al., 1997, 1998, 2000).

Fitting the computed dispersion relation to the measured one allows to derive the film properties. The width of the measured frequency interval can allow to to derive the Young modulus and also the film thickness (Schneider et al., 1997, 2000). The uncertainties of the results is evaluated numerically by the fitting procedure. Since the observed propagation distance is of the order of millimetres, the obtained properties are representative of an average over the propagation distance. The performance of the method could be pushed to the measurement of the properties of diamond-like carbon films having thickness down to 5 nm (Schneider et al., 2000). By stretching the observed propagation path to 20 mm it was possible to reduce the uncertainty of the measured propagation velocity of the Rayleigh wave to below 0.25 m/s. It was thus possible to detect the small variation of the Rayleigh velocity (5081 m/s for the bare (001) silicon substrate) induced by the presence of the film. In a different configuration, the laser pulse is focused on a point instead of a line, with consequent excitation of waves which expand in all the radial directions, and a common path interferometer is adopted, whose light collection point scans the specimen surface (Sugawara et al., 2002). It is thus possible to visualize the wavefronts, circular for isotropic samples and non circular for anisotropic ones, with time resolution of the order of picoseconds and spatial resolution of a few micrometres.

### **5.2 Picosecond ultrasonics**

The so called picosecond ultrasonics technique owes its name to the picosecond laser pulses which were available at the time it was introduced (Thomsen et al., 1984, 1986). It is nowadays implemented by femtosecond laser pulses, and is intrinsically suited to characterize thin supported films and multilayers. It follows the optical pump-and-probe scheme (Belliard et al., 2009; Bienville et al., 2006; Bryner et al. 2006; Vollmann et al. 2002). The pump beam, a femtosecond laser pulse, is focused, by a spherical lens, at the specimen surface and, at least partially, absorbed. The focusing spot, a few to tens of micrometers wide, is orders of magnitude larger than the characteristic lengths of the involved phenomena: mainly the optical absorption length, and also the thermal diffusion length and the acoustic wave propagation length, both for a femtosecond time scale. Bryner et al. (2006) estimate that with an aluminium surface, a near infrared laser (800 nm), and a pulse width of 70 fs, the absorption depth and therefore the dominant acoustic wavelength are of the order of 10 nm. Quite consistently, for a similar laser pulse but for Pt and Fe ultra thin films, Ogi et al. (2007) estimate at several THz the upper bound of the frequencies excited by the laser pulse.

The thermal and mechanical fields are thus (almost perfectly) laterally uniform, and essentially one dimensional: the absorbed pulse has a depth of the order of nanometres, and propagates like an acoustic wave with a plane wavefront which travels perpendicular to the surface, towards the specimen depth. At each interface this wave is partly transmitted and partly reflected, according to the acoustical impedances of the layers, and gives rise to echoes, which return to the surface, where they again are reflected back.

The surface is then probed by the probe beam, much weaker than the pump beam, which reaches the surface with a variable delay, controlled by a delay line. The probe beam, similarly to the acoustic wave, is partly transmitted and partly reflected at each interface, leading, in each layer, to a forward and a backward electromagnetic field, whose complex amplitudes can be calculated using a transfer-matrix formalism. The external reflectivity of the surface is the ratio of the backward to the forward complex amplitudes in the outer

Acoustic Waves: A Probe for the Elastic Properties of Films 137

back and forth the multilayer; such waves show pulses of the order of 10 ps, which correspond to propagation lengths much larger than the superlattice period. These waves therefore see the whole multilayer as an effective medium. The theoretical prediction for the properties of the effective medium and the reflection coefficient at the superlattice / Si substrate interface are confirmed by the experimental findings. They also detect higher frequency oscillations, which correspond to localized waves. The periodic multilayer acts as a Bragg reflector, and opens forbidden gaps in the spectrum. It can confine a mode in the neighbourhood of the outer surface (acoustic-phonon surface modes), but only if the outer layer is the lower acoustic impedance layer (in this case Si), which therefore acts as a perfect reflector. The properties of such modes could be correctly predicted only taking into account the nanometric top silicon oxide layer, which spontaneously forms at he silicon surface. The presence of this additional layer is also consistent with the X-ray reflectivity measurements. The behaviour of a Mo cavity sandwiched between Mo/Si mirrors was also analyzed. The picoseconds ultrasonics technique was also exploited to investigate non laterally homogeneous specimens (Bienville et al., 2006; Mante et al., 2008). One limitation of this technique in the measurement of the elastic constants is that it involves only plane waves travelling perpendicular to the surface, thus allowing only the out-of-plane elastic characterization of the film. To overcome this limitation, Mante et al. and Robillard et al. (2008) proposed a technique by which the film to be characterized, and the aluminium interaction layer deposited on it, are cut by lithographic techniques to obtain a periodic square lattice of square (200 nm × 200 nm) pillars. As confirmed by the measurements performed on square lattices of different lattice constants, the pump pulse also excites, in this nanostructured film, acoustic collective modes of the pillars, which propagate along the surface in various directions. Various branches are measured, from which also the in–plane

properties of the film can be measured, achieving a complete elastic characterization.

speaking, inelastic scattering by these excitations is called Raman scattering.

Aggregates of atoms, from molecules to clusters, to nanoparticles and nanocrystals, up to mesoscopic and macroscopic aggregates, can interact with electromagnetic waves either elastically or inelastically. Inelastic interactions include emission/absorption phenomena, and inelastic scattering. We consider here inelastic scattering by vibrational excitations. At the molecular scale the atomic structure of matter has a crucial role, and quantum phenomena are relevant. At this level, vibrational excitations are the vibrations of molecules, or, in a crystal, the vibrations of the internal degrees of freedom of each unit cell, which form the so called optical branches of the dispersion relation, or optical phonons. Broadly

Aggregates above the nanometric scale also support collective vibrational excitations which begin to resemble to acoustic waves, and can be described by a continuum model. In a crystal, the vibrations of the degrees of freedom of the centre of mass of each unit cell form the so called acoustic branches of the dispersion relation, or acoustic phonons. In the long wavelength limit they are the are acoustic waves, accurately described by the continuum model (Eq. (1)). Broadly speaking, inelastic scattering by these excitations is called Brillouin

Visible light has sub micrometric wavelength. In media, either crystalline or amorphous, which are homogeneous, and therefore translationally invariant, over at least a few micrometres, vibrational excitations of sub-micrometric wavelength have a well defined

**6. Brillouin spectroscopy** 

scattering.

space. This reflectivity is modified by the acoustic strain, by two mechanisms. Firstly, each interface is displaced by the acoustic wave, and, secondly, the refractive index in each layer is modified by the elastic strain, by the acousto-optic (also called photoelastic) coupling. In particular, it must be remembered that the traveling acoustic pulse extends over a depth of a few nanometers, inducing a localized perturbation of the refractive index, which partially scatters the optical beam. Interference can occur between the beam reflected at the outer surface and that reflected by the traveling acoustic pulse.

Interferometric techniques allow to measure the variation of both amplitude and phase of the reflected beam, thus measuring the variation of the complex reflection coefficient. With probe pulses in the femtosecond range, and varying the probe pulse delay, the time evolution of the surface reflectivity can be monitored with high temporal resolution. This time evolution typically shows several features. The diffusion, towards the sample depth, of the heat deposited by the laser pulse gives a slowly varying reflectivity background. The echoes of the acoustic pulse which, after partial reflection at the film/substrate interface, are again reflected at the outer surface are generally visible. The so called Brillouin oscillations, due to the interference between the beam reflected at the outer surface and that reflected by the traveling acoustic pulse, can then be found.

The analysis of the various features allows to characterize the waves which cross the layers travelling perpendicularly to the surface. In the derivation of the film properties, the knowledge of film thickness, typically obtained by X-ray reflectivity, has a crucial role; the uncertainty about thickness is one of the leading terms in the uncertainty of the final results. The achievable resolution depends on the excited wavelength. In copper the absorption depth is larger than the value cited above for aluminium. Since the smallness of the absorption depth determines the localization of the acoustic pulse and the achievable resolution, the deposition of an aluminium interaction layer, which guarantees a very small absorption depth, is a common practice. The interaction layer, typically a few tens of nanometres thick, then participates to the vibrational behaviour of the structure being investigated. Accurate measurements of stiffness therefore require consideration of the effects of the interaction layer, e.g. by measurements with layers of different thicknesses, followed by an extrapolation to null thickness (Mante et al., 2008). Obviously this deconvolution of the effects of the interaction layer contributes to the uncertainty of the final results.

 Near infrared lasers are a common choice, because at shorter wavelength more complex phenomena can occur, which were attributed to electronic interband transitions (Devos & Cote, 2004); obviously, if one is interested in elastic properties, electronic transitions are a spurious effect to be avoided.

By picoseconds ultrasonics it was possible to characterize a layer stack, including a buried layer of about 20 nm thickness (Bryner et al., 2006). The uncertainty for the elastic constants of this layer is estimated at 20-25%, which is however remarkable for a layer of this type. The lowest limit for layer detection is also estimated at about 10 nm thickness. In a different configuration, namely a single Pt or Fe layer on a silicon substrate or a borosilicate glass substrate, Ogi et al. (2007) were able to characterize metallic films of thickness down to 5 nm. They found, at nanometric thicknesses, a dependence of the elastic moduli on thickness. This was explained by the impossibility of plastic flow at such low thicknesses: the elastic strains can thus reach levels which are non reachable in thicker samples, such that higher order elastic constants are no longer negligible.

Periodic Mo/Si multilayers (superlattices) were investigated by Belliard et al. (2009), exploring various periodicities in the nanometric range. They detect bulk waves crossing

space. This reflectivity is modified by the acoustic strain, by two mechanisms. Firstly, each interface is displaced by the acoustic wave, and, secondly, the refractive index in each layer is modified by the elastic strain, by the acousto-optic (also called photoelastic) coupling. In particular, it must be remembered that the traveling acoustic pulse extends over a depth of a few nanometers, inducing a localized perturbation of the refractive index, which partially scatters the optical beam. Interference can occur between the beam reflected at the outer

Interferometric techniques allow to measure the variation of both amplitude and phase of the reflected beam, thus measuring the variation of the complex reflection coefficient. With probe pulses in the femtosecond range, and varying the probe pulse delay, the time evolution of the surface reflectivity can be monitored with high temporal resolution. This time evolution typically shows several features. The diffusion, towards the sample depth, of the heat deposited by the laser pulse gives a slowly varying reflectivity background. The echoes of the acoustic pulse which, after partial reflection at the film/substrate interface, are again reflected at the outer surface are generally visible. The so called Brillouin oscillations, due to the interference between the beam reflected at the outer surface and that reflected by

The analysis of the various features allows to characterize the waves which cross the layers travelling perpendicularly to the surface. In the derivation of the film properties, the knowledge of film thickness, typically obtained by X-ray reflectivity, has a crucial role; the uncertainty about thickness is one of the leading terms in the uncertainty of the final results. The achievable resolution depends on the excited wavelength. In copper the absorption depth is larger than the value cited above for aluminium. Since the smallness of the absorption depth determines the localization of the acoustic pulse and the achievable resolution, the deposition of an aluminium interaction layer, which guarantees a very small absorption depth, is a common practice. The interaction layer, typically a few tens of nanometres thick, then participates to the vibrational behaviour of the structure being investigated. Accurate measurements of stiffness therefore require consideration of the effects of the interaction layer, e.g. by measurements with layers of different thicknesses, followed by an extrapolation to null thickness (Mante et al., 2008). Obviously this deconvolution of the

effects of the interaction layer contributes to the uncertainty of the final results.

 Near infrared lasers are a common choice, because at shorter wavelength more complex phenomena can occur, which were attributed to electronic interband transitions (Devos & Cote, 2004); obviously, if one is interested in elastic properties, electronic transitions are a

By picoseconds ultrasonics it was possible to characterize a layer stack, including a buried layer of about 20 nm thickness (Bryner et al., 2006). The uncertainty for the elastic constants of this layer is estimated at 20-25%, which is however remarkable for a layer of this type. The lowest limit for layer detection is also estimated at about 10 nm thickness. In a different configuration, namely a single Pt or Fe layer on a silicon substrate or a borosilicate glass substrate, Ogi et al. (2007) were able to characterize metallic films of thickness down to 5 nm. They found, at nanometric thicknesses, a dependence of the elastic moduli on thickness. This was explained by the impossibility of plastic flow at such low thicknesses: the elastic strains can thus reach levels which are non reachable in thicker samples, such that higher

Periodic Mo/Si multilayers (superlattices) were investigated by Belliard et al. (2009), exploring various periodicities in the nanometric range. They detect bulk waves crossing

surface and that reflected by the traveling acoustic pulse.

the traveling acoustic pulse, can then be found.

spurious effect to be avoided.

order elastic constants are no longer negligible.

back and forth the multilayer; such waves show pulses of the order of 10 ps, which correspond to propagation lengths much larger than the superlattice period. These waves therefore see the whole multilayer as an effective medium. The theoretical prediction for the properties of the effective medium and the reflection coefficient at the superlattice / Si substrate interface are confirmed by the experimental findings. They also detect higher frequency oscillations, which correspond to localized waves. The periodic multilayer acts as a Bragg reflector, and opens forbidden gaps in the spectrum. It can confine a mode in the neighbourhood of the outer surface (acoustic-phonon surface modes), but only if the outer layer is the lower acoustic impedance layer (in this case Si), which therefore acts as a perfect reflector. The properties of such modes could be correctly predicted only taking into account the nanometric top silicon oxide layer, which spontaneously forms at he silicon surface. The presence of this additional layer is also consistent with the X-ray reflectivity measurements. The behaviour of a Mo cavity sandwiched between Mo/Si mirrors was also analyzed.

The picoseconds ultrasonics technique was also exploited to investigate non laterally homogeneous specimens (Bienville et al., 2006; Mante et al., 2008). One limitation of this technique in the measurement of the elastic constants is that it involves only plane waves travelling perpendicular to the surface, thus allowing only the out-of-plane elastic characterization of the film. To overcome this limitation, Mante et al. and Robillard et al. (2008) proposed a technique by which the film to be characterized, and the aluminium interaction layer deposited on it, are cut by lithographic techniques to obtain a periodic square lattice of square (200 nm × 200 nm) pillars. As confirmed by the measurements performed on square lattices of different lattice constants, the pump pulse also excites, in this nanostructured film, acoustic collective modes of the pillars, which propagate along the surface in various directions. Various branches are measured, from which also the in–plane properties of the film can be measured, achieving a complete elastic characterization.
