1. Introduction

The interaction of ultrafast laser pulses with solid materials (metals, semiconductors and dielectrics) has been a primary field of research with numerous applications. One of the main fields of interest is the generation of ultra-high frequency acoustical pulses in matter with the use of ultrafast laser pulses [1–3]. Various experimental techniques have been implemented to study the response after ultrafast laser illumination [4–6]. Pump-probe optical techniques have been extensively used, in particular the transient reflectivity configuration. Typically, up to a few hundred femtoseconds after laser illumination there is a fast initial electronic excitation of matter. The subsequent changes, in the picoseconds range, are attributed to the lattice reaction which gains energy mainly though excited electrons [7]. The lattice dynamics are governed by nano- and micro-acoustic strain localization that is generated either thermally or non-thermally [8–10]. The spatial extend of the localization depends on the material’s response to the ultrafast laser excitation and the structure of the solid target and can be of the order of few nanometers [11] up to few tens of nanometers. These features are favorable for a number of applications, such as the non-destructive material characterization in the nano- and micro-regime. This is indeed important in the case of biomaterials where probing with electromagnetic radiation of such short wavelength can be fully destructive. For the above reasons studies have been performed to understand the underlying mechanisms of the generation of these nano-acoustic waves. Some studies focus in the first moments of laser-matter electronic interaction in order to understand the role of the exciting laser wavelength [12], chirp [13], duration [14], and fluence [15] on the material response. Furthermore, the role of non-thermal and thermal excited electrons has been studied along with the mechanisms involved in the transfer of the excited electron energy to the lattice [16–19].

In this article, a detailed experimental study is presented on the role of metal thin-film transducers in the characteristics of secondary-induced nano-acoustical pulses in Si monocrystal substrates, and in particular of the mechanisms of efficient transfer of the laser energy to the semiconductor lattice, thus producing giant nano-mechanical strains. For this purpose a degenerate femtosecond pump-probe transient reflectivity technique is employed to study different metal thin-film transducers deposited on Si (100) monocrystal thick substrates. The thin-film metal transducers have been intentionally selected to have distinctively different electronic density of states, electron-phonon coupling factors and acoustical impedances. Experimental results show that giant strains are transferred in Si substrates when the metal transducer possesses high electron-phonon coupling strength, high compressive yield strength and acoustical impedance matched to that of Si. Ti as a transducer material satisfies these, allowing for strain detection using a laser probe wavelength (λ ~795 nm) which is away from the direct band gap value of Si at 3.4 eV where the Si acousto-optic coupling coefficients are higher. Furthermore, a theoretical thermo-mechanical approach is applied and supports the experimental findings.

2. Experimental details

In this study Ti and Ag polycrystalline thin films on Si (100) substrates are used. In particular, 12 nm, 25 nm and 53 nm thick Ti and 12 nm and 25 nm thick Ag films were deposited on 0.5 mm Si (100) substrates by unbalanced dc magnetron sputtering (P_b < 5 × 10⁻⁶ mbar) using Ar gas (purity 99.999%); the film thicknesses were determined by X-ray reflectivity. The films thicknesses were selected as such in order to be above the percolation threshold and be continuous [20]. The degenerate transient reflectivity experimental arrangement is schematically depicted in Fig. 1.

 

Fig. 1 (a) Schematic diagram of excitation and detection scheme in the metal/Si systems, and (b) experimental setup for the femtosecond degenerate pump-probe differential reflectivity method.

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Laser pulses are derived from a Ti:Sapphire-based amplifier laser system, at a repetition rate of 1 kHz, having a FWHM time duration of 35 fs, maximum energy of 1.5 mJ, and center wavelength of ~795 nm. The amplifier’s output is split into pump and probe beams, which are orthogonally polarized by means of a low-dispersion λ/2 waveplate. The pump beam is incident perpendicularly to the metal film surface, loosely focused by means of a spherical broad bandwidth metallic mirror to achieve a fluence of 13 ± 1 mJ/cm², which is within the thermoelastic regime for these metals as is also experimentally confirmed by the means of optical microscopy. The probe beam was incident at an angle of ~40° relative to the pump beam, focused by means of a parabolic metal mirror, while its fluence is kept at significantly lower levels compared to the pump in order to avoid contribution to the excitation dynamics.

An imaging system is used to monitor the relative position and size of the two beams on the interaction area. The probe beam spot is smaller than the pump spot by a factor of ~4 and located at the central part of the latter. A variable optical delay is introduced between the pump and probe pulses, with a minimum temporal delay step of 0.8 fs. For the detection of reflectivity changes a lock-in detection technique is employed. A mechanical chopper modulates the pump beam at a frequency of 270 Hz. A small portion of the probe beam is split and is directed onto a balanced photodiode along with the probe beam reflected from the sample surface. The output of the photodiode is the difference of these two signals. Care was taken to ensure that the probe beams simultaneously illuminating (with ns resolution) the two inputs of the balanced photodiode were of equal power, so that in the absence of the pump beam the photodiode output was zero. The balanced photodiode output signal is directed to a dual channel lock-in amplifier which detects signal differences at the pump beam modulating frequency. This detection scheme allows for reflectivity changes of the order of ~10⁻⁵ to be resolved. Specially developed software is used to simultaneously control the temporal delay and detection instruments, and for data recording.

3. Results and discussion

Transient reflectivity results for the case of Ti and Ag transducer thin films on Si substrates are given in Fig. 2. In all cases, the initial abrupt reflectivity change is attributed to the excitation and subsequent thermalization and cooling of metal electrons: following laser excitation a portion of the initially excited non-thermal electrons thermalize through electron-electron and electron-phonon interactions. The reflectivity signal decays as thermal and non-thermal electrons transfer their excess energy to the lattice through electron-phonon interactions, diffusion, and ballistic transfer. As shown in the inset of Fig. 2(b) this whole process is much faster for Ti (less than 1 ps) than for Ag (few ps). This is largely attributed to the much different values (approximately two orders of magnitude) of the electron-phonon coupling factor, G, of the two metals. For metal electron temperatures in the range from 300 Κ to 5000 K the electron-phonon coupling factor for Ti increases approximately linearly, from 1.4 × 10¹⁸ Wm⁻³K⁻¹ to 4.0 × 10¹⁸ Wm⁻³K⁻¹, while for Ag remains approximately constant, equal to 2.5 × 10¹⁶ Wm⁻³K⁻¹ [21].

 

Fig. 2 (a) Transient reflectivity signals for three Ti thicknesses (Ti/Si samples). The inset shows the first and second acoustic echoes (53 nm Ti), (b) transient reflectivity signals for two Ag thicknesses (Ag/Si samples). The two signals are vertically offset for clarity. The inset shows the fast (less than 1 ps) and slower (a few ps) electronic signal decay for Ti and Ag metal films, respectively. In all cases, the period of the oscillations (~13 ps) corresponds to the Si Brillouin period.

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The electronic contribution to the transient reflectivity signal is followed by sinusoidal-like oscillations. For the Ti/Si samples [Fig. 2(a)] these oscillations are superimposed onto an exponential decay which lasts for many tens of ps. Such an exponential decay is not observed for the Ag/Si samples [Fig. 2(b)]. This is mainly attributed to the different metal lattice temperature values and temporal evolution. Indeed, after electron-phonon relaxation the Ti lattice temperature is considerably higher than that of Ag and decays within tens of ps (~200 K decrease in ~50 ps, see Fig. 5 in Appendix B). In contrast, the Ag lattice temperature remains approximately constant in the same temporal period. The amplitude of these oscillations is larger for the case of the 12 nm thick Ti films and decreases with increasing film thickness. For the case of the 53 nm Ti film only one such oscillation is clearly distinguished. In Ag/Si samples [Fig. 2(b)] the amplitude of the oscillations is considerably smaller compared to Ti/Si ones of the same metal thickness.

The incident laser pump energy is partially absorbed from the metallic film and is finally converted into metal lattice movement, thus generating an acoustic longitudinal strain wave in the metal film. A small portion of the laser pump beam is transmitted into the Si substrate, typically ~15% for 25 nm metal thickness. This transmitted energy for all cases is not sufficient for any measurable Si excitation, as experimentally verified by transient reflectivity measurements in Si wafers. Strain reflections from the metal/Si interface can appear in the transient reflectivity data at time delays multiples of τ = 2d_m / u_m, where d_m is the metal thin film thickness and u_m is the longitudinal sound velocity in the metal. In the inset of Fig. 2(a) such a reflections are shown in the form of echoes which agree very well with the expected for the longitudinal sound wave velocity in Ti (~6100 m/s). Applying Fourier analysis to all the presented experimental data we were able to identify such echoes except for the cases of the 12 nm and 25 nm Ti films, where the strong sinusoidal-like oscillations might hinder the echoes identification.

The generated longitudinal acoustic strain wave then enters and propagates inside the Si substrate. The coupling efficiency of the strain into Si is determined by the acoustical impedance matching between the metallic material and Si. In most of our cases, the metal films are sufficiently thin for a fraction of the ~795 nm probe pulse to reach the Si substrate and to probe this strain wave as it propagates inside it. The probing process is as follows [Fig. 1(a)]: a fraction of the probe beam is initially reflected upon incidence on the metal/Si sample (R_pr1). The small fraction of the probe beam that is not absorbed from the metal thin film is transmitted into the Si substrate (T_pr1). The Si absorption at the probe wavelength is very small and in our case can be safely neglected. Part of the probe beam that enters Si will be reflected from the longitudinal strain wave, and travels back towards the metal/Si sample surface experiencing absorption and reflection losses at the interfaces. Eventually, a small fraction of the probe beam will be transmitted out of the metal surface (T_pr2). The observed signal is the interference of R_pr1 and T_pr2 beams, manifested as oscillations. The period of the observed oscillations effectively corresponds to a 2π optical phase difference between the R_pr1 and T_pr2 beams. This prerequisites that the longitudinal acoustic strain has to travel a distance that corresponds to an optical delay of the probe beam ~2.7 fs, which is the probe beam E-field period, E_T. Maxima in the oscillations are observed when the optical time delay between the R_pr1 and T_pr2 pulses is an integer multiple of the E-field period (i.e. k E_T, where k = 1, 2, 3…). Since the E-field of the interfering pulses (R_pr1 and T_pr2) has a Gaussian-like temporal envelope (with τ_FWHM ~35 fs), as the relative temporal delay between them increases, their interference oscillations amplitude decreases. This is the main reason that the observed Brillouin oscillations amplitude decreases at longer time pump-probe time delays [22]. It is well known that for pulses of Gaussian-like temporal envelope the autocorrelation function is also Gaussian-like with FWHM ~√2 times broader than the FWHM of each pulse. For our case this means that a 50% decrease in the oscillations amplitude is expected for ~6 periods after the first observed oscillation maximum. Indeed, this is line with the data presented in Fig. 2. Furthermore, the expected acoustical damping up to the presented 100 ps time-delay (corresponding to ~850 nm acoustic strain travel distance inside Si) is estimated to be no more than 1% based on the acoustic attenuation data presented in [23]. Above that, it is worth noting that the optical absorption of the ~795 nm probe beam for the aforementioned acoustic strain travel distance inside Si is negligible and does not contribute to the decrease of the oscillations amplitude.

Taking into account that the refractive index of Si at the probe wavelength, n_Si = 3.7, the optical path of the probe beam in Si is d_opt = E_T c / n_Si ≈ 219 nm, where c is the speed of light in vacuum. The time required for the longitudinal acoustic strain to travel this distance is${\tau }_{osc}=d_{opt}/2u_{Si}\approx 13\text{ps}$, where u_Si is the longitudinal sound velocity in Si. Another way to understand the same phenomenon is to apply Bragg condition, so that the optical path difference between two successive maxima must be equal to the probe wavelength, λ (constructive interference between R_pr1 and T_pr2). Therefore:

For a quantitative description of the observed signals in our metal/Si cases and in order to extract the net acoustic strain relative strength, analytic optical calculations for thin metal films on Si substrates are employed (see Appendix A), based on the transfer matrix method [24]. This method derives the total reflectance, R, absorptance, A, and transmittance, T, of multi-layered materials taking into account the optical admittance, η_i, of every different medium. This is applied in our different sample systems, assuming that the probe light traveling in the air/metal/Si direction is totally reflected by the strain pulse inside Si and subsequently travels back in the Si/metal/air direction. In such a way we aim to exclude the contribution of the different optical properties of the two systems in the observed signals, thus gaining insight in the actual generated acoustical strain values in these two systems. The results are summarized in Table 1 for each particular sample system. It must be noted that this assumption does not count for the different spatial profiles of strains generated in Si by the different transducers. Based on these assumptions, we also present in Table 1 the expected strain probing efficiency, p_eff, of the Brillouin oscillations for each particular sample system, defined as the percentage of T_pr2 to R_pr1.

Table 1. Calculated reflectance R_pr1, transmittance T_pr2 and strain probing efficiency p_eff

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Let us now compare the Ti/Si and Ag/Si cases for the same metal film thickness, always assuming the same induced acoustical strain in Si. We observe that for the 12nm metal films one would expect a signal enhancement by a factor of ~1.4 for Ag/Si with respect to the Ti/Si case, while for the 25 nm metal films one would expect a signal decrease of ~0.8 for Ag/Si compared to the Ti/Si case. This is not, of course, the case of our experimental findings, because of the different induced strain in the different sample systems. For the case of 12nm metal films the experimentally observed Brillouin peak around τ = 20 ps, is ~50 ± 10 times larger for Ti/Si compared to the Ag/Si one, while for the case of 25 nm metal films is ~5 ± 0.5 times larger for Ti/Si compared to the Ag/Si one. Considering the previously calculated probing efficiency (considering same strain spatial profile for the two different transducers) for the two cases the corrected signals are estimated to be ~35 ± 7 times larger for Ti/Si compared to the Ag/Si one (12 nm metal films), and ~6 ± 0.6 times larger for Ti/Si compared to the Ag/Si (25 nm metal films).

To explore the underlying physical mechanisms, and to gain insight on the generated acoustic strains, we have simulated the laser beam heat transfer, the electron excitation, electron-phonon interactions and stress-strain generation for the Ti/Si and Ag/Si cases. A thermo-mechanical model (Appendix B) based on the combination of: (i) a revised Two-Temperature Model (TTM) to account for the inclusion of non-thermal electron dynamics due to the ultrafast (~35 fs) pulse duration [18,19] (see Appendix B for the need of inclusion of contribution of nonthermal electrons), and (ii) on elasticity theory, is used to describe the electron excitation, non-thermal electron generation, electron-phonon relaxation and the spatio-temporal distribution of the stress/strain fields [25]. The used theoretical model aims to provide a direct correlation of the laser beam characteristics, the material properties and the induced thermomechanical effects for a wide range of materials. In the case of 25 nm thick metal films simulations are in agreement with experimental observations and indicate that the resulting maximum lattice temperatures are insufficient to induce either plastic deformation or melting of the material. Therefore, a more complex physical mechanism is not required to be incorporated into the model. The theoretical model predicts the generation of a strain field wave due to the heating of the metal which travels inside the metal/Si system. The different mechanical properties of the metal/semiconductor induce a strain gradient on the interface during the early stages (positive strain to the left of the interface and negative strain to the right). The produced front of the strain propagates inside Si and it is preserving its initial form. Figure 3 illustrates the strain propagation at different time instants. By contrast, the strain form inside the metal film is the result of the interference of two strain waves, one that travels towards Si and another that is reflected from the interface.

 

Fig. 3 Calculated strain pulse distribution in 25 nm metal/Si systems as a function of depth, at different time instants. Solid blue and dot-dashed red lines correspond to Ti/Si and Ag/Si cases, respectively. Vertical dashed line corresponds to the metal/Si interface position.

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In the case of the 12 nm thick metal films the lattice temperatures approach the melting point, while ballistic electron transport may lead to efficient electron-interface scattering [26], which is not taken into account in the TTM. Furthermore, the induced stress exceeds the yield stress of the irradiated material and therefore plastic deformations are anticipated. For these reasons in such cases the developed elasticity model is of limited validity. Therefore, we compare the theoretical findings with the experimentally observed for the case of 25 nm thick cases where the maximum lattice temperature (metal surface) is well below the melting point of both metals (Fig. 5 in Appendix B). The theoretically maximum calculated strain at 20ps for Ti/Si case is at least 4 times higher than that predicted for the Ag/Si case (see Fig. 3). Based on the aforementioned results, giant strains are indeed generated in Si for the Ti/Si systems, experimentally identified as very strong Brillouin oscillations. For 25 nm thick Ti films the amplitude of the acoustic strain has peak values ~0.01, corresponding to a compressional stress of ~1.7 GPa, which is below the crystalline Si yield strength [27]. Furthermore, the spatial extend of the induced strain in Si seems to be quite different for different transducers, as indeed shown in Fig. 3. For example, and for 20 ps where the strain is fully developed inside Si, it is estimated that the spatial extend of the positive part of the strain is ~90 nm for the case of 25 nm thick Ag film. For the case of 25 nm thick Ti film the strain inside Si is much more localized, having an extend of ~35 nm. It is worth noting that the model predicts that for the thicker 53 nm Ti film the strain inside Si has an extend of ~80 nm when fully developed.

In the literature there are quite a few articles that study the generation of acoustic strain in Si using similar multilayered structures with metallic transducers [28–30]. In all these works, it is emphasized that when a degenerate probing scheme is used (i.e. probe wavelength ~800 nm), the induced strains are not detected in the form of Brillouin oscillations. In these works the probe wavelengths are in the vicinity of the 3.4 eV (365 nm) direct band gap in Si, thus taking advantage of the enhancement of the Si acousto-optic coupling as one approaches the direct band gap and the strong absorption of Si in this wavelength range. In this way the Brillouin oscillations amplitude increases with decreasing probe wavelengths. In our case, although the probing wavelength is far away from this resonance, very strong oscillations are observed. We attribute this to a series of factors: Ti is used as the transducer material, which has high compressive yield strength, high melting point and exhibits fast electron-lattice thermalization with very high electron-phonon coupling factor which increases almost linearly for electron temperatures up to ~5000 K where it has a value of ~4.0 × 10¹⁸ Wm⁻³K⁻¹. Also Ti has a good acoustical impedance matching to Si, therefore the generated acoustic strain in Ti is efficiently transferred to Si (~85%). Furthermore, the pump laser pulse duration (35 fs) is fast enough to excite metal electrons to very high temperatures, while the used fluencies are close to the limits of the elastic regime.

4. Summary

In summary, we have studied in detail the role of metal thin film transducers in acoustic strain generation by ultrafast laser pulses in Ti/Si and Ag/Si layered systems. We employed a degenerate transient reflectivity technique and detected strong Brillouin oscillations in the Ti/Si systems, probing at a wavelength far away from the 3.4 eV direct band gap of Si. An optical transfer matrix method was employed for the qualitative comparison of the relative strains between the aforementioned systems. To support the experimental findings, a thermo-elastic theoretical model was developed, based on the combination of elasticity theory and a revised TTM which includes the nonthermal electron dynamics. In this work, the model is extended to account for the spatiotemporal distribution of the lattice field inside the metal and semiconductor and the heat flux through the interface. Thus, the model allows for the computation of the strain wave propagation between the two materials in a rigorous and systematic way, providing insight into the role of the transducer material.

We have observed giant acoustic strains in Si when Ti is used as the transducer material, manifested experimentally in the form of strong Brillouin oscillations. Such oscillations have never been detected with probe wavelengths far away from the direct band gap of Si, where the acousto-optic coupling is very small, providing a simpler alternative to the existing experimental methods without any need for frequency doubling of the probe beam or use of white-light continuum probe. We explained these observations to be due to the high electron-phonon coupling strength and high stress yield of Ti, as well as the very good acoustic impedance matching to the Si substrate. We have also explored experimentally the limits of the transducer thickness that allow for detection of acoustic strains in the semiconductor substrate, which can be more than twice the optical penetration depth. Our calculations show that in the case of Ti the induced acoustic strain localization in Si is of the order of ~35 nm, approximately 3 times smaller than that of the Ag case. Transducers having such characteristics as Ti are favorable for use in the generation and detection of acoustic strains in semiconductors, and have potential for material characterization applications involving such large localized strains.

Appendix A. Transfer matrix method

Transfer Matrix Method is used for the calculations of the total reflectance, R, absorptance, A, and transmittance, T, of multi-layered materials, taking into account the optical admittance, η_i, of every different medium. The optical admittance is defined as the ratio of the amplitudes of the tangential components of magnetic (H) and electric (E) field vectors:

(2)$${\eta }_i=\frac{H}{E}$$

The optical admittance can be expressed in the following forms for s and p polarized waves, respectively:

(3)$${\eta }_s=\sqrt{\frac{{\epsilon }_o}{{\mu }_o}}{N}_i\cos {\theta }_i,{\eta }_p=\sqrt{\frac{{\epsilon }_o}{{\mu }_o}}{N}_i/\cos {\theta }_i$$

where ε_ο is the permittivity of vacuum, μ_ο is the permeability of vacuum, N_i is the complex index of refraction of the i-th medium, and θ_i is the angle of incidence of light in the i-th medium.

In our case there are three layers, namely air (i = 1), a thin metallic layer (i = 2), and a Si substrate (i = 3), therefore two interfaces (air/metal and metal/Si). The requirement that the tangential components of E and H fields at each interface must be continuous, coming from the solution of Maxwell equations, results in analytic equations about electromagnetic field components at each interface. If E_a/m, H_a/m and E_m/Si, H_m/Si are the electric and magnetic field tangential components at the air/metal and metal/Si interfaces, respectively, the equations can be written in matrix notation as:

(4)$$\left[\begin{array}{c}B\\ C\end{array}\right]=\left[\begin{array}{c}\frac{E_{a/m}}{E_{m/Si}}\\ \frac{H_{a/m}}{E_{m/Si}}\end{array}\right]=\left[\begin{array}{cc}\cos \delta & \frac{\text{i}\sin \delta }{{\eta }_2}\\ \text{i}{\eta }_2\sin \delta & \cos \delta \end{array}\right]\times \left[\begin{array}{c}1\\ {\eta }_3\end{array}\right]$$

where $\delta =\frac{2\pi N_2{d}_m\cos {\theta }_2}{\lambda }$ is the phase shift that undergoes the light wave traversing the thin metal film [31], and B and C are the normalized E- and H-fields, respectively, at the air/metal interface. For three-layer systems as the ones used here, where an absorbing thin metal film is between two non-absorbing materials, the total reflectance of the system, the transmittance into the third medium and the absorptance from the thin metal film is given by the equations:

(5)$$R=\left(\frac{{\eta }_{1}B-C}{{\eta }_{1}B+C}\right){\left(\frac{{\eta }_{1}B-C}{{\eta }_{1}B+C}\right)}^{\ast }$$

(6)$$T=\frac{4{\eta }_1\mathrm{Re}\left[{\eta }_3\right]}{\left({\eta }_{1}B+C\right){\left({\eta }_{1}B+C\right)}^{\ast }}$$

(7)$$A=\frac{4{\eta }_1\mathrm{Re}\left[B{C}^{\ast }-{\eta }_3\right]}{\left({\eta }_{1}B+C\right){\left({\eta }_{1}B+C\right)}^{\ast }}$$

where asterisk denotes the complex conjugate. The above expressions satisfy the requirement that$R+T+A=1$. A significant point that must be stressed is that in this treatment all the individual electric and magnetic field components, coming from successive reflections and transmissions inside the metal/Si system, are added coherently together to give the resultant fields at a particular point.

Appendix B. Revised two-temperature model and incorporation of mechanical response

To describe the influence of the ultrafast electron dynamics in the relaxation procedure after irradiation of the Ti/Si (or Ag/Si) system, a revised version of the Two-Temperature Model is used to account for the early and transient interaction of the non-thermal electron distribution with the electron and lattice baths [18,19]. Hence, the following set of equations is employed to investigate the spatio-temporal distribution of the produced thermalized electron (T_e) and lattice (T_L) temperatures of the assembly:

(8)$$C_e^{(2)}\frac{\partial T_e^{(2)}}{\partial t}=\overrightarrow{\nabla }·\left(k_e^{(2)}\overrightarrow{\nabla }{T}_e^{(2)}\right)-G_{}^{(2)}\left(T_e^{(2)}-T_L^{(2)}\right)+\frac{\partial U_{ee}}{\partial t}$$

(9)$$C_L^{(2)}\frac{\partial T_L^{(2)}}{\partial t}=G^{(2)}\left(T_e^{(2)}-T_L^{(2)}\right)+\frac{\partial U_{eL}}{\partial t}-(3{\lambda }^{(2)}+2{\mu }^{(2)}){{\alpha }^{\prime }}^{(2)}{T}_L^{(2)}{\displaystyle \sum _{j=1}^3{\dot{\epsilon }}^{(2)}{}_{jj}}$$

(10)$$C_L^{(3)}\frac{\partial T_L^{(3)}}{\partial t}=\overrightarrow{\nabla }·\left(k_L^{(3)}\overrightarrow{\nabla }{T}_L^{(3)}\right)-(3{\lambda }^{(3)}+2{\mu }^{(3)}){{\alpha }^{\prime }}^{(3)}{T}_L^{(3)}{\displaystyle \sum _{j=1}^3{\dot{\epsilon }}^{(3)}{}_{jj}}$$

where the subscripts e and L are associated with electrons and lattice, respectively, k_e is the thermal conductivity of the electrons, C_e and C_L are the heat capacity of electrons and lattice, respectively, G is the electron-phonon coupling factor while superscripts i correspond to the Ag/Ti (i = 2) and Si materials (i = 3), respectively, and T₀ = 300 K. On the other hand, ε_jj correspond to components of thermally induced strains while λ, μ, and α' are the Lamé constant, shear modulus and thermal expansion coefficients of the materials. Due to the large radius of the laser beam compared to the thickness of the irradiated metal films, the solution of the equations can be simplified by assuming a one dimensional approach. Therefore, the following equation can be used for two materials with density ρ⁽ⁱ⁾ (i = 2, 3) to determine the spatio-temporal distribution of thermally induced lattice displacement v:

(11)$${\rho }^{(i)}\frac{{\partial }^2{v}^{(i)}}{\partial t^2}=({\lambda }^{(i)}+2{\mu }^{(i)})\frac{{\partial }^2{v}^{(i)}}{\partial z^2}-(3{\lambda }^{(i)}+2{\mu }^{(i)}){{\alpha }^{\prime }}^{(2)}\frac{\partial T_L^{(i)}}{\partial t}$$

where the strain and stress along the z-axis are given by the expressions:

(12)$${\epsilon }^{(i)}=\frac{\partial v^{(i)}}{\partial z},{\sigma }_z{}^{(i)}=({\lambda }^{(i)}+2{\mu }^{(i)}){\epsilon }^{(i)}-(3{\lambda }^{(i)}+2{\mu }^{(i)}){{\alpha }^{\prime }}^{(2)}\left(T_L^{(i)}-T_0\right)$$

respectively. The energy densities per unit time transferred from the non-thermal electrons to thermal electrons (∂U_ee/∂t) and lattice (∂U_eL/∂t) require modification with respect to the initial model to account for the dynamic character of the absorption coefficient during irradiation that alters the absorption:

(13)$$\begin{array}{l}\frac{\partial }{\partial t}\{\begin{array}{c}{U}_{ee}\\ U_{eL}\end{array}\}=\frac{2A\sqrt{\ln 2}J}{\sqrt{\pi }{\left(h\nu \right)}^2{t}_p}{\displaystyle \underset{0}{\overset{t}{\int }}[\frac{1}{1-\exp \left(-\frac{dm}{{\alpha }^{-1}+\Lambda }\right)}}\frac{1}{{\alpha }^{-1}+\Lambda }\\ \begin{array}{ccc}& & \times \end{array}\exp \left(-4\ln 2{\left(\frac{t^{\prime }-t_0}{t_p}\right)}^2\right)\exp \left(-{\displaystyle \underset{0}{\overset{z}{\int }}\frac{1}{{\alpha }^{-1}+\Lambda }d{z}^{\prime }}\right)\{\begin{array}{c}{H}_{ee}(t-t^{\prime })\\ H_{eL}(t-t^{\prime })\end{array}\}]d{t}^{\prime }\end{array}$$

where J is the fluence of the laser beam, hν is the one-photon energy, t_p is the pulse duration, A is the absorbance of the laser energy, d_m is the metal film thickness, Λ is the ballistic depth that is characteristic for each metal, α is the absorption coefficient and t₀ = –3t_p while H_ee and H_ep are functions that contain details and parameters related to the transient creation of non-thermal electron distribution (for a more analytical description see [18,19]).

To emphasize on the significance of the nonthermal interaction with electrons and lattice, the source term that originally is placed in the part of the TTM that describes the electron bath has been split into two terms: (i) one that describes the rate of energy density transferred via electron-phonon scattering from the nonthermal electron distribution induced by the laser pulse in to the lattice ($\partial U_{ep}/\partial t$), (ii) one that describes the rate of energy density transferred via electron-electron scattering from the nonthermal electron distribution induced by the laser pulse in to the thermal($\partial U_{ee}/\partial t$). It is obvious that both terms are important but it is becoming more important for materials with higher electron-phonon coupling (Ti).

Although the laser beam wavelength (i.e. ~795 nm) is sufficient to excite carriers inside the silicon, the amount of the excitation is not enough to modify substantially the results and therefore, any contribution was not taken into account.

To provide an accurate description of the underlying mechanism after irradiation with ultrafast pulses, it is important to treat the thermophysical properties and the electron-phonon coupling coefficient, that appear in the model as temperature dependent parameters (see Fig. 4). To include explicit functions of the thermal parameters on the electron temperature, theoretical data for the metals (Ag and Ti) calculated by Lin et. al [21] were fitted with polynomials [19] while the relevant expressions of the substrate (Si) are provided in the literature [25].

 

Fig. 4 Calculated temporal evolution for the heat sources for (a) Ti, and (b) Ag.

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The computation of the heat electron conductivity for Ag was performed by means of a general expression [32]:

(14)$$k_e=\frac{{\left[{({\theta }_e)}^2+0.16\right]}^{5/4}\left[{({\theta }_e)}^2+0.44\right]{\theta }_e}{{\left[{({\theta }_e)}^2+0.092\right]}^{1/2}{\left[{({\theta }_e)}^2+\eta {\theta }_L\right]}^{5/4}}\chi ,{\theta }_e=\frac{T_e}{T_F},{\theta }_L=\frac{T_L}{T_F}$$

where, in the case of silver, the parameters that appear in the expression are the Fermi temperature T_F = 6.38 × 10⁴ K, χ = 500 WK⁻¹s⁻¹, and η = 0.1715.

By contrast, a different approach is followed for Ti as the χ and η parameters are not known. More specifically, the procedure that is followed to overcome the difficulty is by using alternative expressions [33] bearing in mind that a polynomial of first order could adequately describe the temperature dependence of the electron heat capacity and electron-phonon coupling coefficient:

(15)$$k_e=k_{e0}\frac{T_e}{\frac{A_e}{B_L}{(T_e)}^2+T_L},G=G_0\left[\frac{A_e}{B_L}(T_e+T_L)+1\right]$$

where the ratio A_e/B_L (that appears in the expression that is pertinent to the electron relaxation time ${\tau }_e=1/B_L{T}_L+A_e{(T_e)}^2$), k_e0, G₀ that are not known can be specified by an appropriate minimization procedure that ensures that the resulting values for the electron heat capacity and electron-phonon coupling coefficient coincide with the values provided by Lin et al [21]. The above procedure constitutes a more general approach as in practice the values of the maximum electron temperatures in Ti (i.e. on the interface) do not exceed 3000 K and a polynomial of first order could adequately describe the evolution of the electron heat capacity and electron-phonon coupling coefficient.

Furthermore, we note that due to the ultrafast character of the irradiation, the optical characteristics of the irradiated material vary within the pulse duration. Therefore, an appropriate procedure that was developed in previous studies will be used to compute the transient absorption coefficient of the metal films [19,34].

The numerical solution of Eq. (8) – Eq. (13) was performed using a finite difference method scheme where it is assumed that there exists perfect thermal contact between the two metals, and electron temperature and heat flux is continuous on the interface, thus interface resistance is neglected. The thickness of the second layer is assumed to be large enough that laser beam will be attenuated before it reaches the back side of the film; this allows assuming von Neumann boundary conditions and to ignore heat losses at the front and back surfaces of the assembly. With respect to the mechanical response of the Ag/Si or Ti/Si, we assume stress free conditions on the surface of the metal film, zero displacement and speed initially while it is also assumed that stress field is continuous across the interface.

Typical results for metal electron and lattice temperature are shown in Figs. 5 and 6.

 

Fig. 5 Calculated temporal evolution of electron (solid red lines) and lattice (dashed blue lines) temperatures at the surface of (a) 25nm Ti thin film, and (b) 25nm Ag thin film, on Si substrates.

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Fig. 6 Calculated spatio-temporal evolution of lattice temperature of (a) 25nm Ti thin film, and (b) 25nm Ag thin film, on Si substrates.

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Acknowledgments

The authors acknowledge financial support through the Operational Program “Education and Lifelong Learning,” Action Archimedes III (sub-action 19: “Innovative optoacoustic device for 3d spatiotemporal micro-characterization of composite materials based on ultrafast laser pulses”) co-financed by the European Union (European Social Fund) and Greek national funds (National Strategic Reference Framework 2007–2013).

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