**1. Introduction**

Films and thin films are exploited by an ever increasing number of technologies. The properties of films can be different from those of the same material in bulk form, and can depend on the preparation process, and on thickness. Specific techniques are needed for their measurement. Whenever films or thin layers have structural functions, as in micro electro-mechanical systems (MEMS), a precise characterization of their stiffness is crucial for the design of devices. The same can be said for the design of devices which exploit thin layers to support surface acoustic waves (surface acoustic wave filters). More generally, knowledge of the elastic properties is interesting because such properties depend on the structural properties.

The most widespread technique for the mechanical characterization of films, instrumented indentation, induces both elastic and inelastic strains. It also characterizes irreversible deformation, but the extraction of the information concerning the elastic behaviour is non trivial. To overcome this difficulty, several measurement methods have been developed, which exploit vibrations as a probe of the material behaviour. These methods intrinsically involve only elastic strains, and are non destructive. This is true at any length scale, and is peculiarly useful at micrometric and sub-micrometric scales, where the exploitation of other types of probes can become critical.

Both propagating waves and standing waves can be exploited, with excitation which can be either monochromatic (e.g. resonance techniques) or impulsive, therefore broadband, requiring an analysis of the response either in the time domain (the so called picoseconds ultrasonics) or in the frequency domain (the so called laser ultrasonics). The propagation velocities of vibrational modes are obtained, from which the stiffness is derived if an independent value of the mass density (the inertia) is available.

Older resonance techniques have been developed to be operated with thin slabs, also exploiting optical measurements of displacement. In the measurement of films and small structures, the advantages of light, a contact-less and inertia-less probe, are substantial, and are increasingly exploited. The laser ultrasonics technique, commercially available since some years ago, measures waves travelling along the film surface. The so called picosecond ultrasonics technique measures waves travelling across the film thickness; it is a relatively sophisticated optical technique, which exploits femtosecond laser pulses in a pump-and-probe measurement scheme. For best performance it needs the deposition of

Acoustic Waves: A Probe for the Elastic Properties of Films 127

modes, having different polarization vectors and frequencies. In general the phase velocity

In the simplest case, the isotropic solid, the matrix of the elastic constants is fully determined by only two independent quantities; the only non null matrix elements are C*11* = C*22* = C*33*, C*44* = C*55* = C*66*, C*12* = C*13* = C*23* = C*11* - 2C*44*. In this case the shear modulus *G* coincides with

ν

44 12 44 44 11 44 ( ) ( ) 12 44 11 44 *CC C CC C* 32 34

2 2

and *<sup>t</sup>* C / <sup>44</sup> *v* =

transforms Eq. (1) into an eigenfunction / eigenvalue equation of the

. Therefore, a measurement of the frequencies of the acoustic modes

2 4 3 3

In the isotropic case the phase velocities are independent from the direction of **k**, only depending on the relative orientation of **e** with respect to **k**; one of the three modes is longitudinal ( **e k** ) and has velocity *<sup>l</sup> v* , the other two are transversal ( **e k** ⊥ ), are independent (the two polarization vectors are orthogonal) and degenerate: they have the

In the non isotropic case more than two independent quantities are needed to determine the matrix of the elastic constants, and the phase velocities, beside depending on the direction

In a finite geometry the search for standing waves having the harmonic time dependence of

Helmholtz type (Auld, 1990); an appropriate set of basis functions allows to transform this equation into a matrix eigenvalue problem (Nakamura et al., 2004). The eigenvalues are proportional to ω2, the square of the frequencies of the acoustic modes of the structure, or natural frequencies of the structure. In other words, the finiteness of the geometry converts the continuum spectrum of frequencies of the modes of the infinite medium, given by Eq. (2), into the discrete spectrum of the natural frequencies. These frequencies depend on the

In a schematic way: also in non isotropic media the acoustic velocities depend on stiffness

elastic constants and, possibly, direction cosines of **k**. In the simplest case, the one dimensional geometry of length *L*, the standing waves are identified by the constructive self

*CC CC*

*C CC E CC CC G*

( ) <sup>12</sup> 11 44 11 12 11 44

11 12

and bulk modulus *B* are respectively given

+ − = = + − , (3)

<sup>−</sup> = = =− + − , (4)

*C C B CC* <sup>+</sup> = =− , (5)

ρ

, indicating generically by *C* the relevant combination of

/2 ( /2) / (*n* is an integer number), such that

. (6)

<sup>2</sup> <sup>1</sup>

11 44

frequency *f* compatible with the mentioned lower limit for wavelength.

 / **k** *f* depends on both the direction of **k** and the polarization vector **e**. In an infinite homogeneous medium, travelling waves of the type given by Eq. (2) exist for any

*v* = = ω

the type *i t e*

(C*ij*/ρ − ω

 λ

by (Every, 2001; Kundu, 2004)

C*44*, while Young modulus *E*, Poisson's ratio

*E*

ν

same velocity *<sup>t</sup> v* (Auld, 1990; Kundu, 2004). The two velocities are

*<sup>l</sup>* C / <sup>11</sup> *v* =

ρ

λ

of **k**, have a more complex dependence on the C*ij* values.

*)* values and on the geometry.

interference condition *Ln n vf* = =

and inertia as in Eqs. (6): <sup>2</sup> *v C*= /

ρ

*f* = ( /2) / / *nC L*

ρ

an interaction layer, possibly combined with microlithography techniques to obtain a patterned layer.

Techniques for optical detection of acoustic vibrations include inelastic scattering of light: Brillouin scattering. Brillouin spectroscopy does not excite vibrations, but relies on thermal excitation, which has the broadest band but small amplitude, and measures the spectrum of inelastically scattered laser light. Brillouin Spectroscopy and Surface Brillouin Spectroscopy are relatively simple optical techniques, which do not require a specific specimen preparation. They operate at sub-micrometric acoustic wavelengths, which are selected by the scattering geometry, and have been exploited to characterize the elastic properties of bulk materials and of films.

An overview of this variety of techniques is presented here, underlining analogies and differences. The increasing demand of precise characterization raised the point of precision and accuracy achievable by vibration based techniques, and specifically by the optical techniques. In the overview, attention is drawn to the steps or the parameters which are the limiting factor for the achievable precision, and to the way of characterizing them. The effects of inaccuracies of the mass density are common to all the techniques based on vibrations, while other sources of uncertainty are more specific to each technique.

### **2. Acoustic modes in elastic solids**

In the continuum description the instantaneous configuration of a solid undergoing deformation can be represented by the displacement vector field **u r**(,)*t* , where ( ) <sup>123</sup> **u** = *uuu* , , , ( ) <sup>123</sup> **r** = *xxx* , , and *t* is time. The local state of the solid being represented by the strain and stress tensors, the linear elastic behaviour is characterized by the tensor of the elastic constants *Cijmn* , which can be conveniently represented by the matrix of the elastic constants C*ij*. When other phenomena, like e.g. viscoelasticity or elasticity of higher orders, can be neglected, the tensor of the elastic constants fully characterizes the stiffness. Inertia is characterized by the mass density ρ. Within a homogeneous linear elastic solid, in the absence of body forces, the equations of motion for the displacement vector field are homogeneous, and read (Auld, 1990; Every, 2001; Kundu, 2004)

$$
\rho \frac{\partial^2 u\_i}{\partial t^2} = \sum\_{j,m,n} \mathcal{C}\_{jlmn} \frac{\partial^2 u\_m}{\partial x\_j \partial x\_n}, \quad i = 1, 2, 3 \quad . \tag{1}
$$

These equations describe vibrational elastic excitations, which are typically called acoustic also in the ultrasonic frequency range. The basic solutions of Eqs.(1), and the most important ones when boundary effects are irrelevant, are the plane acoustic waves, or modes (Auld, 1990; Kundu, 2004), of the form

$$\mathbf{u} = \mathbf{e} \Re \left[ \tilde{A} \exp \left[ i \left( \mathbf{k} \cdot \mathbf{r} - at \right) \right] \right] \quad , \tag{2}$$

where **k** is the wavevector, ω = 2π *f* the circular frequency, *f* the frequency, *A* an arbitrary complex amplitude, and **e** the polarization vector, which is normalized. The continuum description, underlying Eq. (1), is appropriate until the wavelength λ π = 2 / **k** is much larger than the interatomic distances. The three translational degrees of freedom of each infinitesimal volume element correspond, for each wavevector **k**, to three independent

an interaction layer, possibly combined with microlithography techniques to obtain a

Techniques for optical detection of acoustic vibrations include inelastic scattering of light: Brillouin scattering. Brillouin spectroscopy does not excite vibrations, but relies on thermal excitation, which has the broadest band but small amplitude, and measures the spectrum of inelastically scattered laser light. Brillouin Spectroscopy and Surface Brillouin Spectroscopy are relatively simple optical techniques, which do not require a specific specimen preparation. They operate at sub-micrometric acoustic wavelengths, which are selected by the scattering geometry, and have been exploited to characterize the elastic properties of

An overview of this variety of techniques is presented here, underlining analogies and differences. The increasing demand of precise characterization raised the point of precision and accuracy achievable by vibration based techniques, and specifically by the optical techniques. In the overview, attention is drawn to the steps or the parameters which are the limiting factor for the achievable precision, and to the way of characterizing them. The effects of inaccuracies of the mass density are common to all the techniques based on

In the continuum description the instantaneous configuration of a solid undergoing deformation can be represented by the displacement vector field **u r**(,)*t* , where ( ) <sup>123</sup> **u** = *uuu* , , , ( ) <sup>123</sup> **r** = *xxx* , , and *t* is time. The local state of the solid being represented by the strain and stress tensors, the linear elastic behaviour is characterized by the tensor of the elastic constants *Cijmn* , which can be conveniently represented by the matrix of the elastic constants C*ij*. When other phenomena, like e.g. viscoelasticity or elasticity of higher orders, can be neglected, the tensor of the elastic constants fully characterizes the stiffness. Inertia is

absence of body forces, the equations of motion for the displacement vector field are

, 1,2,3 *i m ijmn jmn j n u u C i t x x*

These equations describe vibrational elastic excitations, which are typically called acoustic also in the ultrasonic frequency range. The basic solutions of Eqs.(1), and the most important ones when boundary effects are irrelevant, are the plane acoustic waves, or modes (Auld,

**u e kr** = ℜ{*Ai t* exp

where **k** is the wavevector, ω = 2π *f* the circular frequency, *f* the frequency, *A* an arbitrary complex amplitude, and **e** the polarization vector, which is normalized. The continuum

larger than the interatomic distances. The three translational degrees of freedom of each infinitesimal volume element correspond, for each wavevector **k**, to three independent

 ( ) ⋅ −ω

. Within a homogeneous linear elastic solid, in the

} , (2)

λ π

= 2 / **k** is much

∂ ∂ = = ∂ ∂∂ . (1)

vibrations, while other sources of uncertainty are more specific to each technique.

ρ

2 2

, ,

homogeneous, and read (Auld, 1990; Every, 2001; Kundu, 2004)

ρ

2

description, underlying Eq. (1), is appropriate until the wavelength

patterned layer.

bulk materials and of films.

**2. Acoustic modes in elastic solids** 

characterized by the mass density

1990; Kundu, 2004), of the form

modes, having different polarization vectors and frequencies. In general the phase velocity *v* = = ω λ / **k** *f* depends on both the direction of **k** and the polarization vector **e**. In an infinite homogeneous medium, travelling waves of the type given by Eq. (2) exist for any frequency *f* compatible with the mentioned lower limit for wavelength.

In the simplest case, the isotropic solid, the matrix of the elastic constants is fully determined by only two independent quantities; the only non null matrix elements are C*11* = C*22* = C*33*, C*44* = C*55* = C*66*, C*12* = C*13* = C*23* = C*11* - 2C*44*. In this case the shear modulus *G* coincides with C*44*, while Young modulus *E*, Poisson's ratio ν and bulk modulus *B* are respectively given by (Every, 2001; Kundu, 2004)

$$E = \frac{\mathbb{C}\_{44} \left( \mathcal{R} \mathbb{C}\_{12} + \mathcal{D} \mathbb{C}\_{44} \right)}{\mathbb{C}\_{12} + \mathbb{C}\_{44}} = \frac{\mathbb{C}\_{44} \left( \mathcal{R} \mathbb{C}\_{11} - \mathbb{4} \mathbb{C}\_{44} \right)}{\mathbb{C}\_{11} - \mathbb{C}\_{44}},\tag{3}$$

$$\nu = \frac{\mathbf{C}\_{12}}{\mathbf{C}\_{11} + \mathbf{C}\_{12}} = \frac{\mathbf{C}\_{11} - 2\mathbf{C}\_{44}}{2\left(\mathbf{C}\_{11} - \mathbf{C}\_{44}\right)} = \frac{E}{2G} - 1 \,, \tag{4}$$

$$B = \frac{\mathbb{C}\_{11} + 2\mathbb{C}\_{12}}{3} = \mathbb{C}\_{11} - \frac{4}{3}\mathbb{C}\_{44} \, \prime \, \tag{5}$$

In the isotropic case the phase velocities are independent from the direction of **k**, only depending on the relative orientation of **e** with respect to **k**; one of the three modes is longitudinal ( **e k** ) and has velocity *<sup>l</sup> v* , the other two are transversal ( **e k** ⊥ ), are independent (the two polarization vectors are orthogonal) and degenerate: they have the same velocity *<sup>t</sup> v* (Auld, 1990; Kundu, 2004). The two velocities are

$$
v\_{l} = \sqrt{\mathbb{C}\_{11}/\rho} \quad \text{and} \quad v\_{t} = \sqrt{\mathbb{C}\_{44}/\rho} \; . \tag{6}$$

In the non isotropic case more than two independent quantities are needed to determine the matrix of the elastic constants, and the phase velocities, beside depending on the direction of **k**, have a more complex dependence on the C*ij* values.

In a finite geometry the search for standing waves having the harmonic time dependence of the type *i t e* − ω transforms Eq. (1) into an eigenfunction / eigenvalue equation of the Helmholtz type (Auld, 1990); an appropriate set of basis functions allows to transform this equation into a matrix eigenvalue problem (Nakamura et al., 2004). The eigenvalues are proportional to ω2, the square of the frequencies of the acoustic modes of the structure, or natural frequencies of the structure. In other words, the finiteness of the geometry converts the continuum spectrum of frequencies of the modes of the infinite medium, given by Eq. (2), into the discrete spectrum of the natural frequencies. These frequencies depend on the (C*ij*/ρ*)* values and on the geometry.

In a schematic way: also in non isotropic media the acoustic velocities depend on stiffness and inertia as in Eqs. (6): <sup>2</sup> *v C*= /ρ , indicating generically by *C* the relevant combination of elastic constants and, possibly, direction cosines of **k**. In the simplest case, the one dimensional geometry of length *L*, the standing waves are identified by the constructive self interference condition *Ln n vf* = = λ /2 ( /2) / (*n* is an integer number), such that *f* = ( /2) / / *nC L* ρ. Therefore, a measurement of the frequencies of the acoustic modes

Acoustic Waves: A Probe for the Elastic Properties of Films 129

It has always been recognized that since the phase velocities of acoustic waves and the natural frequencies of the acoustic modes depend on stiffness and inertia, their measurement gives access, by Eq.(6) or Eq. (7), to the elastic constants C*ij*, if the mass density

, and possibly the geometry, are known. Many experimental methods have been devised, which exploit vibrations to measure the elastic properties of solids. 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% (Every, 2001). Furthermore, when the elastic constants are needed to design a device which operates dynamically, like most microdevices, the dynamic moduli are exactly the ones which are

Some methods measure the wave propagation velocity by measuring the transit time over a finite, macroscopic distance, other methods measure the frequency of standing modes defined by the sample geometry, or the frequency of propagating waves of well defined wavelength. The excitation can be either monochromatic, at a frequency which typically should be adjustable until resonance conditions are achieved, or broadband. The latter is typically obtained by an impulsive excitation, which can be provided by a mechanical percussion or by a laser pulse. Generally, the response to a broadband excitation is spectrally analyzed. The availability of ultrafast lasers (femtosecond laser pulses) also allows

In homogeneous specimens each propagation velocity, or the frequency of each standing wave, has a single value, from which the corresponding elastic modulus can be derived. In non homogeneous specimens, typically in supported films, each propagation velocity can

a finite interval of frequency or wavelength, and the film properties can be obtained fitting

The various experimental methods operate in different frequency ranges. The range is determined by both the excitation and the detection techniques, and is strictly correlated to the spatial resolution. It is worth remembering that the acoustic velocities in typical elastic solids like metals and ceramics are of the order of a few km/s = mm×MHz, and that a phase velocity *v* links the frequency *f* to a characteristic length *L*, which can be a characteristic dimension of a structure supporting standing waves, or the wavelength of a travelling wave. Characteristic lengths of centimetres imply frequencies in the tens of kHz range, which are easily excited by a mechanical percussion and measured by a microphone; Nieves et al. (2000) estimate at around 0.1 MHz the upper limit of the frequencies excited by the mechanical percussion, with a steel ball of a very few millimetres. Piezoelectric actuators,

Characteristic lengths of several micrometers correspond to frequencies in the tens of MHz range. Structures of this size can be built by micromachining techniques, and their vibration can be excited and detected by capacitive actuators and sensors. Measurement techniques of this type are essentially a miniaturization of the vibrating reed technique (Kubisztal, 2008). Czaplewski et al. (2005) built flexural and torsional resonators of tetrahedral amorphous

ω

(*k*) or *v*( ) λ

can be measured over

an analysis in the time domain, by pump-and-probe techniques.

depend on wavelength or frequency. Dispersion relations

the computed dispersion relations to the measured ones.

**3.2 Vibration excitation and detection techniques** 

**3. Stiffness measurements 3.1 Vibration based methods** 

needed in the design process.

and sensors are also available.

ρ

allows to derive ( ) ( ) <sup>2</sup> <sup>2</sup> *C fL n* = ρ / /2 . Also in more complex geometries, the dependence is of the same type

$$\mathbf{C} = \boldsymbol{\rho} \left( \boldsymbol{f} \mathbf{L} \right)^{2} \text{N} \tag{7}$$

where *L* is now a characteristic length of the structure (for a slender rod, essentially one dimensional, the length), and *N* is a dimensionless numerical factor which, beside the mode order *n*, can depend on dimensionless quantities like geometrical aspect ratios or Poisson's ratio. The factor *N* also depends on the character of the mode whose frequency *f* is being measured, and therefore on the specific modulus *C* which is involved.

Structures can be finite in one or two dimensions and practically infinite in others, as it happens e.g. in a slab or a long cylinder. The free surface of an otherwise homogeneous solid is a case of semi-infiniteness along a single dimension. The translational symmetry is broken in the direction perpendicular to the surface, and new phenomena, absent in the infinite medium, are found: the reflection of bulk waves, and the existence of surface acoustic waves. Namely, at a stress free surface Eqs. (1) admit a further solution: the Rayleigh wave, the paradigm of the surface acoustic waves (SAWs). Such waves have peculiar characters (Farnell & Adler, 1972): a displacement field confined in the neighborhood of the surface, with the amplitude which declines with depth, a wavevector parallel to the surface, and a velocity lower than that of any bulk wave, such that the surface wave cannot couple to bulk waves, and does not lose its energy irradiating it towards the bulk. Pseudo surface acoustic waves can also exist, which violate this last condition. The velocity *<sup>R</sup> v* of the Rayleigh wave cannot be given in closed form; in the isotropic case a good approximation is (Farnell & Adler, 1972)

$$
v\_{\mathbb{R}} \equiv v\_t \frac{0.862 + 1.14\nu}{1 + \nu} \; . \tag{8}$$

The continuum model of a homogeneous solid does not contain any intrinsic length scale. Accordingly, all the solutions for this model are non dispersive, meaning that the velocities (Eqs. (6) and (8)) are independent from wavelength (or frequency).

More complex modes occur in non homogeneous media. Layered media are a particularly relevant case, in which new types of acoustic modes can occur; namely, modes confined around the interfaces and modes which are essentially guided by one layer or another, like the Sezawa waves. In this case, also in the continuum model the physical system has an intrinsic length scale, identified by the layer thicknesses. For wavelengths much smaller than the thicknesses wave propagation occurs within each layer as if it was infinite, with reflections and refractions at the surfaces. Instead, for wavelengths comparable to, or larger than, the thicknesses, the acoustic modes extend over several layers, and are modes of the whole structure. Such modes are dispersive: their velocities depend on wavelength, or, more precisely, on the wavelength to thickness ratio(s). Also the simplest surface wave, the Rayleigh wave of a bare homogeneous substrate, is modified by a layer deposited on it, and becomes dispersive: the propagation velocity depends on wavelength, therefore on frequency. The velocities of the acoustic modes in layered structures can be numerically computed, as non trivial functions of the properties of the substrate and the layer(s), and of the wavelength to thickness ratio. The dispersion relations ω(*k*) or *v f* ( ) are thus obtained.

### **3. Stiffness measurements**

128 Acoustic Waves – From Microdevices to Helioseismology

( )<sup>2</sup>

where *L* is now a characteristic length of the structure (for a slender rod, essentially one dimensional, the length), and *N* is a dimensionless numerical factor which, beside the mode order *n*, can depend on dimensionless quantities like geometrical aspect ratios or Poisson's ratio. The factor *N* also depends on the character of the mode whose frequency *f* is being

Structures can be finite in one or two dimensions and practically infinite in others, as it happens e.g. in a slab or a long cylinder. The free surface of an otherwise homogeneous solid is a case of semi-infiniteness along a single dimension. The translational symmetry is broken in the direction perpendicular to the surface, and new phenomena, absent in the infinite medium, are found: the reflection of bulk waves, and the existence of surface acoustic waves. Namely, at a stress free surface Eqs. (1) admit a further solution: the Rayleigh wave, the paradigm of the surface acoustic waves (SAWs). Such waves have peculiar characters (Farnell & Adler, 1972): a displacement field confined in the neighborhood of the surface, with the amplitude which declines with depth, a wavevector parallel to the surface, and a velocity lower than that of any bulk wave, such that the surface wave cannot couple to bulk waves, and does not lose its energy irradiating it towards the bulk. Pseudo surface acoustic waves can also exist, which violate this last condition. The velocity *<sup>R</sup> v* of the Rayleigh wave cannot be given in closed form; in the isotropic case a

0.862 1.14

The continuum model of a homogeneous solid does not contain any intrinsic length scale. Accordingly, all the solutions for this model are non dispersive, meaning that the velocities

More complex modes occur in non homogeneous media. Layered media are a particularly relevant case, in which new types of acoustic modes can occur; namely, modes confined around the interfaces and modes which are essentially guided by one layer or another, like the Sezawa waves. In this case, also in the continuum model the physical system has an intrinsic length scale, identified by the layer thicknesses. For wavelengths much smaller than the thicknesses wave propagation occurs within each layer as if it was infinite, with reflections and refractions at the surfaces. Instead, for wavelengths comparable to, or larger than, the thicknesses, the acoustic modes extend over several layers, and are modes of the whole structure. Such modes are dispersive: their velocities depend on wavelength, or, more precisely, on the wavelength to thickness ratio(s). Also the simplest surface wave, the Rayleigh wave of a bare homogeneous substrate, is modified by a layer deposited on it, and becomes dispersive: the propagation velocity depends on wavelength, therefore on frequency. The velocities of the acoustic modes in layered structures can be numerically computed, as non trivial functions of the properties of the substrate and the layer(s), and of

+

ν

ν

ω

(*k*) or *v f* ( ) are thus obtained.

<sup>1</sup> *R t v v*

≅

(Eqs. (6) and (8)) are independent from wavelength (or frequency).

the wavelength to thickness ratio. The dispersion relations

*C* = ρ

measured, and therefore on the specific modulus *C* which is involved.

/ /2 . Also in more complex geometries, the dependence is

*fL N* (7)

+ . (8)

allows to derive ( ) ( ) <sup>2</sup> <sup>2</sup> *C fL n* = ρ

good approximation is (Farnell & Adler, 1972)

of the same type

### **3.1 Vibration based methods**

It has always been recognized that since the phase velocities of acoustic waves and the natural frequencies of the acoustic modes depend on stiffness and inertia, their measurement gives access, by Eq.(6) or Eq. (7), to the elastic constants C*ij*, if the mass density ρ, and possibly the geometry, are known. Many experimental methods have been devised, which exploit vibrations to measure the elastic properties of solids. 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% (Every, 2001). Furthermore, when the elastic constants are needed to design a device which operates dynamically, like most microdevices, the dynamic moduli are exactly the ones which are needed in the design process.

Some methods measure the wave propagation velocity by measuring the transit time over a finite, macroscopic distance, other methods measure the frequency of standing modes defined by the sample geometry, or the frequency of propagating waves of well defined wavelength. The excitation can be either monochromatic, at a frequency which typically should be adjustable until resonance conditions are achieved, or broadband. The latter is typically obtained by an impulsive excitation, which can be provided by a mechanical percussion or by a laser pulse. Generally, the response to a broadband excitation is spectrally analyzed. The availability of ultrafast lasers (femtosecond laser pulses) also allows an analysis in the time domain, by pump-and-probe techniques.

In homogeneous specimens each propagation velocity, or the frequency of each standing wave, has a single value, from which the corresponding elastic modulus can be derived. In non homogeneous specimens, typically in supported films, each propagation velocity can depend on wavelength or frequency. Dispersion relations ω(*k*) or *v*( ) λ can be measured over a finite interval of frequency or wavelength, and the film properties can be obtained fitting the computed dispersion relations to the measured ones.

### **3.2 Vibration excitation and detection techniques**

The various experimental methods operate in different frequency ranges. The range is determined by both the excitation and the detection techniques, and is strictly correlated to the spatial resolution. It is worth remembering that the acoustic velocities in typical elastic solids like metals and ceramics are of the order of a few km/s = mm×MHz, and that a phase velocity *v* links the frequency *f* to a characteristic length *L*, which can be a characteristic dimension of a structure supporting standing waves, or the wavelength of a travelling wave.

Characteristic lengths of centimetres imply frequencies in the tens of kHz range, which are easily excited by a mechanical percussion and measured by a microphone; Nieves et al. (2000) estimate at around 0.1 MHz the upper limit of the frequencies excited by the mechanical percussion, with a steel ball of a very few millimetres. Piezoelectric actuators, and sensors are also available.

Characteristic lengths of several micrometers correspond to frequencies in the tens of MHz range. Structures of this size can be built by micromachining techniques, and their vibration can be excited and detected by capacitive actuators and sensors. Measurement techniques of this type are essentially a miniaturization of the vibrating reed technique (Kubisztal, 2008). Czaplewski et al. (2005) built flexural and torsional resonators of tetrahedral amorphous

Acoustic Waves: A Probe for the Elastic Properties of Films 131

both cases each frequency reading is associated to a finite degree of uncertainty, mainly due to random errors. In a set of repeated measurements such errors are uncorrelated, and tend to be cancelled by an averaging process; the error of each measurement affects the dispersion of results around the average, but not the average itself. In other words, these errors affect precision, but not accuracy. The accuracy of results is at most affected by the finite accuracy in the calibration of the frequency meter, of whichever nature it be. When then deriving the elastic moduli, the frequency reading can be exploited as such (see e.g. Eq.(7)), or via the determination of a propagation velocity. In both cases, the obtained moduli also depend on further 'auxiliary' parameters. In a very simple case, from Eq. (6) we

the average, but it affects the average itself. This means that it affects accuracy, but not

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

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

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

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

*v* , and the resulting value of C*11* is also affected by the finite uncertainty of

, exploited in the derivation. However, in a set of repeated

does not contribute to the dispersion of results around

have <sup>2</sup> C11 *<sup>l</sup>* = ρ

the best available value of

**4. Mechanical excitation** 

properties of the sample.

**4.1 Measurement of bulk properties** 

conditions.

analyzed to identify the resonant frequencies.

measurements the uncertainty of

ρ

ρ

precision. The same can be said for the sample geometry (see Eq. (7)).

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 of the thin member undergoing torsion.

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

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

### **3.3 Precision and accuracy**

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 *q* depends on the 'primary' uncertainties σ *<sup>a</sup>* , σ *<sup>b</sup>* and σ *<sup>c</sup>* . For a functional dependence of the type *f Aa b c* α β γ = , where *A* is a numerical constant, the usual error propagation formula can be written in terms of the relative uncertainties ( ) σ *<sup>a</sup>* /*a* , ( ) σ *<sup>b</sup>* /*b* , ( ) σ*<sup>c</sup>* /*c* as

$$
\left(\frac{\sigma\_q}{q}\right)^2 = \alpha^2 \left(\frac{\sigma\_s}{a}\right)^2 + \beta^2 \left(\frac{\sigma\_b}{b}\right)^2 + \gamma^2 \left(\frac{\sigma\_c}{c}\right)^2\tag{8}
$$

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 both cases each frequency reading is associated to a finite degree of uncertainty, mainly due to random errors. In a set of repeated measurements such errors are uncorrelated, and tend to be cancelled by an averaging process; the error of each measurement affects the dispersion of results around the average, but not the average itself. In other words, these errors affect precision, but not accuracy. The accuracy of results is at most affected by the finite accuracy in the calibration of the frequency meter, of whichever nature it be. When then deriving the elastic moduli, the frequency reading can be exploited as such (see e.g. Eq.(7)), or via the determination of a propagation velocity. In both cases, the obtained moduli also depend on further 'auxiliary' parameters. In a very simple case, from Eq. (6) we have <sup>2</sup> C11 *<sup>l</sup>* = ρ*v* , and the resulting value of C*11* is also affected by the finite uncertainty of the best available value of ρ, exploited in the derivation. However, in a set of repeated measurements the uncertainty of ρ does not contribute to the dispersion of results around the average, but it affects the average itself. This means that it affects accuracy, but not precision. The same can be said for the sample geometry (see Eq. (7)).
