Femtosecond laser interactions with dielectric materials: insights of a detailed modeling of electronic excitation and relaxation processes

Transcription

1 Femtosecond laser interactions with dielectric materials: insights of a detailed modeling of electronic excitation and relaxation processes Nikita S. Shcheblanov & Tatiana E. Itina Applied Physics A Materials Science & Processing ISSN Appl. Phys. A DOI /s

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3 Appl Phys A DOI /s Femtosecond laser interactions with dielectric materials: insights of a detailed modeling of electronic excitation and relaxation processes Nikita S. Shcheblanov Tatiana E. Itina Received: 3 October 2011 / Accepted: 3 August 2012 Springer-Verlag 2012 Abstract Electronic excitation relaxation processes induced by ultra-short laser pulses are studied numerically for dielectric targets. A detailed kinetic approach is used in the calculations accounting for the absence of equilibrium in the electronic subsystem. Such processes as electron photon phonon, electron phonon and electron electron scatterings are considered in the model. In addition, both laser field ionization ranging from multi-photon to tunneling one, and electron impact (avalanche) ionization processes are included in the model. The calculation results provide electron energy distribution. Based on the time-evolution of the energy distribution function, we estimate the electron thermalization time as a function of laser parameters. The effect of the density of conduction band electrons on this time is examined. By using the average electron energy, a new criterion is proposed based on determined damage threshold in agreement with recent experiments (Sanner et al. in Appl. Phys. Lett. 96:071111, 2010). 1 Introduction Interactions of ultra-short laser pulses with wide band-gap materials are used in numerous areas such as photonics, micromachining, and medicine [2]. For the further development of such applications, as precision nano-structuring, a better understanding of the mechanisms of ultra-short interactions with wide band-gap materials is required. In par- N.S. Shcheblanov T.E. Itina ( ) Laboratoire Hubert Curien, UMR CNRS 5516/Université de Lyon, Bat. F, 18 rue de Professeur Benoît Lauras, Saint-Etienne, France tatiana.itina@univ-st-etienne.fr Fax: ticular, the detailed kinetics of the electronic excitation relaxation processes should be investigated by considering the electron energy distribution function, rather than by assuming equilibrium as was done in many previous models [3]. Previously, both theoretical and experimental studies of ultra-short laser interactions were carried out for dielectric targets. Many models, however, were based on a simplified rate equation [4, 5], whereas a detailed quantum-kinetic approach was used only in several studies [3, 6]. The advantage of the latter technique is that it accounts for the absence of equilibrium in the electron subsystem, which is particularly important for the correct investigation of ultra-short laser interactions. Therefore, herein we propose a model based on a system of Boltzmann transport equations (BTEs). Both theoretical and experimental investigations demonstrated that if laser intensity is above a certain value, the optical breakdown (OB) occurs. This happens when the electron density exceeds a so-called critical value for the OB, n e n o.b. cr, n o.b. cr = ε r ε 0 m cb Ω 2 /e 2 in dielectric materials [4, 5]. Commonly, this intensity value was supposed to be equal to the damage threshold. This assumption, however, requires a careful verification [7, 8]. For instance, in several recent studies the OB-based threshold was found to disagree with the measured damage thresholds [1]. As a result, several modifications of the damage criterion were proposed based on a comparison of the electron energy with the energy of the lattice at melting temperature [8, 9]. To elucidate the laser damage criterion by considering non-equilibrium electron energy distribution, one should know the values of the energy absorbed and the corresponding relaxation time (thermalization process). For this, herein a numerical solution of the BTEs for electron and phonon subsystems is carried out. A series of calculations are performed for quartz. Electron energy distribution is calculated

4 N.S. Shcheblanov, T.E. Itina as a function of time. The mean electron energy is investigated as a function of laser parameters. The thermalization time is examined and a damage criterion is proposed based on the mean electron energy rather than on the free electron density. 2 Model Our model is based on a system of Boltzmann transport equations (BTEs) for electrons and phonons [3]: f t = f t + f pi t + f e ph pht t + f e e t, imp f(t = 0) = f 0, g β = g β (1) t t, ph e pht g β (t = 0) = g β,0 where f is free electron distribution function, g β is phonon distribution function for β th phonon mode, pi is photoionization collision integral (CI) [3], e ph pht is electron phonon photon CI which describes pure electron phonon collisions and absorbing laser energy by free electrons due to electron phonon collisions [3, 10], e e is electron electron CI [11], imp CI describes the impact ionization process [11], ph e pht is phonon electron photon CI [3, 10]. All CIs are written with the assumptions of parabolic electron bands and for the monochromatic wave E(t) = E 0 cos(ωt). Here, for simplicity, we consider only local effects based on the local value of the laser field. No spatial dependency is introduced in the model. We use a complete Keldysh expression for the PI rate Ẇ pi in [m 3 sec 1 ] as follows: Dawson s integral; K and E are complete elliptic integrals of the first and second kinds, respectively. And PI CI is given by f t = 2π pi M pi (ε, ε v ) 2 f v (ε v ) ( 1 f(ε) ) δ(ε ε v l pi Ω), (4) Mpi (ε, ε v ) 2 = π 4 Ẇ pi 2m 2 r m cb ε δ ε,ε kin, where f v (ε v ) is valence electron distribution function and f v (ε v ) = 1 (since the maximum of free electron density is much smaller than the valence electron density); ε kin = m r /m cb (l pi Ω Δ). In the calculations, the laser pulse with a top-hat temporal pulse shape is assumed. 3 Results and discussion First, we investigate the electron energy distribution (electron distribution function, or DF). The calculated DFs are shown in Fig. 1. Upon the photoionization (Fig. 1, firstpick on the left) and laser-light absorption (oscillations in the figure), the distribution is far from the equilibrium Fermi function. The observed oscillations are caused by the photon absorption by conduction-band electrons due to their scattering on phonons. It takes from around 30 to 50 fs to reach equilibrium at laser pulse duration of 100 fs and laser intensity of W/cm 2. Then, we demonstrate the dependency of the thermalization time on laser pulse duration (Fig. 2). One can see in the figure that the maximum can be observed when the pulse Ẇ pi = 2Ω 9π ( ) mr Ω 3/2 Q(γ, x) γ 2 ( K(γ 2 ) E(γ 2 ) exp πl pi E(γ 1 ) π [ ( exp 2K(γ 1 ) Q(γ, x) = m=0 ), (2) πm K(γ 2) E(γ 2 ) E(γ 1 ) ( φ π 2 l )] pi x + m, (3) 2K(γ 1 )E(γ 1 ) ) where x = Δ/ Ω; 1/m r = 1/m cb + 1/m vb is the reduced mass; m cb and m vb are conduction band and valence band electron masses, respectively; l pi =[ Δ/ Ω + 1] (here, [...] stands for the integer part); Δ = πγ 2Δ 2 E(γ 1 ) is the effective ionization potential; Δ is the energy gap; γ = Ω m r Δ is the Keldysh parameter; γ 1 = 1 ; γ 1+γ 2 2 = γ 1+γ 2 ee 0 ; φ is the Fig. 1 Calculated electron energy distributions at different instants of time. The calculation parameters are laser wavelength λ = 400 nm, energy gap Δ = 9eV(SiO 2 ), and laser intensity I 0 = W/cm 2

5 Femtosecond laser interactions with dielectric materials: insights of a detailed modeling Fig. 2 Thermalization time as a function of pulse duration. The calculation parameters are the same as in Fig. 1 duration τ 55 fs. This behavior can be explained as follows. When the density of free electrons is small, the collision rate responsible for absorption and photoionization, which are responsible for the disordering, is larger than the rate of electron electron (e e) collisions which are responsible for the thermalization. With increasing laser pulse duration (keeping the same laser field, or intensity), the electron density increases, and the e e collisions become more frequent. As a result, electrons reach more rapidly the equilibrium state when laser pulses are longer. The impact ionization has a minor role because the calculated mean energy of a free electron ε is not above 3 ev by the end of the pulse (see Fig. 3), which is not sufficiently high for the impact ionization. In fact, the required energy should be higher than ε th = Δ( 1+2μ 1+μ ) 13.5 ev for SiO 2, where μ = m cb /m vb. We note that there was an experimental indication of the presence of high-energy population in photoelectron spectra [12]. In our model, we consider only transitions to the conduction band, not to the continuum. In addition, our model is based on a simple parabolic model of the conduction band, which can limit the gain in the electron energy. The model modifications that can allow us to explain these results require more detailed investigations and will be presented elsewhere. Interestingly, laser energy absorption also depends on laser parameters. One can see in Fig. 3 that the optimum value of laser intensity can be observed in our calculations. The decay in the absorption is observed for laser intensities because of the screening effect. In fact, when the electron density exceeds a critical value n e n o.b. cr [4, 5, 13], the laser-irradiated dielectric is transformed in an absorbing and reflecting material. Finally, it remains to discuss the damage criterion. Our calculations demonstrate that if the criterion is based on the Fig. 3 Mean kinetic energy of free electron at fixed fields (laser fluences). The calculation parameters are the same as in Fig. 1 OB concept, or n e n o.b. cr, the critical electron density typically does not exceed 1 2 % of valence electrons. This criterion is based only on electron density, whereas electron energy is disregarded, as well as material structure. We emphasize that even if the critical density is reached at the end of the laser pulse, the density of high-energetic conductionband electrons can be insufficient for the effective electronimpact ionization. In this case, no further growth of electron density can occur [8]. It should be noted that only extensive bond-breaking guarantees the laser damage, rather than OB itself. Therefore, we propose another criterion based on the following energy-based considerations. To obtain such a criterion, we propose to compare the total energy of free electrons per atom, or electron energy density Q e, with the energy density of lattice at melting temperature, Q ph (T m ). For simplicity, the latter is represented as the energy of atoms at melting temperature: V at Q e > 3k B T m, (5) where Q e = 1 (2π) (Δ + ε(k))f (k) d 3 k = n 3 e ε tot is the total energy density of free electrons in ev/m 3, Q ph (T ) = β 1 (2π) 3 ωβ (q)g β (q, T ) d 3 q is the energy density of latticeinev/m 3 ; ε tot = Δ + ε, and V at is the mean volume of atom. In fact, the lattice heating is calculated as a result of the excitation of phonon subsystem. Then, in order to calculate the energy density of lattice at melting temperature, we just suppose that the phonon system is at equilibrium and use the Bose Einstein distribution for the phonon subsystem. In addition, the molar heat of melting is 10 % of the amount of heat required to melt the matter and comparable with the energy required for bond-breaking due to ionization, which is taken into account. That is why we do not add additional value to the energy density of lattice at melting temperature. Thus, we obtain the following condition for

6 N.S. Shcheblanov, T.E. Itina 4 Conclusions Fig. 4 Calculated and measured damage thresholds are shown. The black solid and red dash curves correspond to optical breakdown criterion and thermal criterion, respectively. Here, laser wavelength λ = 800 nm, energy gap Δ = 9eV(SiO 2 ). The blue dash-dot curve shows the results of experiments by Sanner et al. [1] the required number of free electrons: n e >n th cr, nth cr = 3k BT m V at ε tot. (6) Note that material properties are taken into account in this criterion. For wide-gap dielectrics and for short pulses when photo-ionized electrons have no time to absorb addition energy, i.e. ε tot l pi Ω, we can rewrite our condition as the following: n e >n th cr, nth cr 3k BT m V at l pi Ω. (7) To check the validity of this criterion, we perform a comparison with the previous experimental findings (Fig. 4) [1]. In our calculations, we set laser pulse duration as a constant, so that laser intensity rises with fluence. The results show good agreement with experimental data for the laser pulses longer than 50 fs. Below this value, pulse duration may become comparable with the optical cycle s one and photoionization process can become different from the one described in the paper. In our calculations, the density of conduction band electrons rises from to /m 3 and can overcome the critical density. However, as one may notice in the figure, when we introduce our damage criterion, the damage takes place when free electron density is smaller than the critical density. After the damage, the band structure changes and additional effects play a role in the absorption, which are disregarded here. In summary, we have investigated ultra-short laser interactions with quartz by accounting for the absence of equilibrium in the electronic subsystem. The thermalization time has been shown to depend on laser parameters, such as pulse duration and laser fluence. This result has been attributed to the difference in free electron densities that are reached. In addition, local screening effect has been revealed in our investigation of laser absorption efficiency as a function of laser parameters. Only this effect can be connected with the OB conditions, rather than laser damage. If one uses the optical breakdown (OB) criterion for the free electron density, one obtains about one percent from the total amount of valence-band electrons. This criterion contains no information about the energy, so it is difficult to say something about bond-breaking. This fact is confirmed by our calculations. Finally, we have proposed a new criterion for laser damage, which is based on the effect of lattice melting rather than on the OB. The calculated damage threshold agrees much better with experiments than that based on the OB. The proposed damage criterion is simple enough and can be used in many models. Acknowledgements The authors acknowledge the help of Grant ANR Ultra-sonde 0010 BLAN They are also grateful to CINES for computer support (under Project C ). NSS acknowledges the Ministry of National Education (France) for the support of his Ph.D. research. References 1. N. Sanner, O. Utéza, B. Chimier, M. Sentis, P. Lassonde, F. Légaré, J.C. Kieffer, Toward determinism in surface damaging of dielectrics using few-cycle laser pulses. Appl. Phys. Lett. 96, (2010) 2. K. Sugioka, M. Meunier, A. Piqué, Laser Precision Microfabrication (Springer, Berlin, 2010) 3. A. Kaiser, B. Rethfeld, M. Vicanek, G. Simon, Microscopic processes in dielectrics under irradiation by subpicosecond laser pulses. Phys. Rev. B 61, (2000) 4. N.M. Bulgakova, R. Stoian, A. Rosenfeld, Laser-induced modification of transparent crystals and glasses. Quantum Electron. 40(11), (2010) 5. A.A. Manenkov, A.M. Prokhorov, Laser-induced damage in solids. Usp. Fiz. Nauk 148, (1986). [Sov. Phys. Usp., 29, , 1986] 6. J. Zeller, A.J. Sabbah, M. Mero, P.M. Alsing, J. McIver, W.G. Rudolph, Femtosecond dynamics of highly excited dielectric thin films. Proc. SPIE 5273, 515 (2004) 7. E.N. Glezer, Y. Siegal, L. Huang, E. Mazur, Laser-induced bandgap collapse in GaAs. Phys. Rev. B 51, 6959 (1995) 8. T. Apostolova, Theoretical study of sub-to-picosecond laser pulse interaction with dielectrics, semiconductors and semiconductor heterostructures. J. Phys. Conf. Ser. 113, (2008) 9. B. Chimier, O. Utéza, N. Sanner, M. Sentis, T. Itina, P. Lassonde, F. Legare, F. Vidal, J.C. Kieffer, Damage and ablation thresholds of fused silica in femtosecond regime: relevant physical criteria and mechanisms. Phys. Rev. B 84, (2011)

7 Femtosecond laser interactions with dielectric materials: insights of a detailed modeling 10. E.M. Epshtein, Scattering electrons by phonons in a strong radiationfield.fiz.tverd. Tela (Leningr.)11, (1970). [Sov. Phys. Solid State, 11, , 1970] 11. B.K. Ridley, Quantum Processes in Semiconductors (Springer, Berlin, 1999) 12. H. Bachau, A.N. Belsky, I.B. Bogatyrev, J. Gaudin, G. Geoffroy, S. Guizard, P. Martin, Yu.V. Popov, A.N. Vasil ev, B.N. Yatsenko, Electron heating through a set of random levels in the conduction band of insulators induced by femtosecond laser pulses. Appl. Phys. A, Mater. Sci. Process. 98(3), (2010) 13. S.C. Jones, P. Braunlich, R.T. Casper, X.-A. Shen, P. Kelly, Recent progress on laser-induced modifications and intrinsic bulk damage of wide-gap optical materials. Opt. Eng. 28, (1989)