THERMAL WAVE SUPERPOSITION AND REFLECTION PHENOMENA DURING FEMTOSECOND LASER INTERACTION WITH THIN GOLD FILM

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1 Numerical Heat Transfer, Part A, 65: , 2014 Copyright Taylor & Francis Group, LLC ISSN: print/ online DOI: / THERMAL WAVE SUPERPOSITION AND REFLECTION PHENOMENA DURING FEMTOSECOND LASER INTERACTION WITH THIN GOLD FILM Ling Li 1, Ling Zhou 1, and Yuwen Zhang 2 1 School of Energy and Power Engineering, University of Shanghai for Science and Technology, Shanghai, P. R. China 2 Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, Missour, USA For the cases that the characteristic time is comparable to the relaxation time of the energy carriers, heat conduction can be interpreted as a form of wave propagation. In this article, the wave effects of both electron and lattice temperatures during interaction of two symmetric femtosecond laser beams with thin gold film are investigated using a dual hyperbolic two-step model. The results show an obvious overshoot phenomenon of thermal wave and oscillations of both electron and lattice temperatures. The influence of laser parameters on thermal wave propagation is also studied. 1. INTRODUCTION It is different from the classical conduction phenomena in that there are many particular characteristics in the microscale heat transfer realm. The wave effect of heat propagation is one of the intriguing properties in the microscale conduction field. The thermal wave theory proposed by Cattaneo [1] and Vernotte [2] is derived from the second sound that is commonly termed as the non- Fourier effect. The second sound that was first detected experimentally by Peshkov in 1944 using the superfluid liquid helium at 14 K [3] is a quantum mechanical phenomenon, in which heat transfer occurs by a wave-like motion, rather than by the more usual mechanism of diffusion. It is known as the second sound because thermal wave is fluctuations in the density of phonons which is similar to the propagation of sound [4]. However, most successful experiments showing the wave behavior in heat propagation were carried out at low temperature in liquid helium and dielectric crystals [3, 5]. Thus, it was believed that slow speeds in ordinary materials at room temperature would never be measured and, even if they exist, they may be masked by diffusion arising from an effective thermal conductivity associated with modes of heat that already relaxed [6]. However, if the characteristic time of various heat carriers are comparable to Received 26 May 2013; accepted 7 November Address correspondence to Ling Li, School of Energy and Power Engineering, University of Shanghai for Science and Technology, Shanghai , China; liling@usst.edu.cn 1139

2 1140 L. LI ET AL. NOMENCLATURE A e material constant for electron t time, [s] relaxation time, [1/K 2 s] T temperature, [K] B e coefficient for electronic T F Fermi temperature, [K] heat capacity, [J/m 3 K2 ] T i initial temperature, [K] B l material constant for z dimensionless relaxation time, lattice relaxation time [1/K 2 s] thermal diffusivity, [m 2 /s] C heat capacity, [J/m 3 K] dimensionless time G electron-lattice coupling optical penetration depth, [m] factor, [W/m 3 K] b ballistic range, [m] G RT electron-lattice coupling density, [kg/m 3 ] factor at room temperature, relaxation time, [s] [W/m 3 K] dimensionless excess J heat source fluence of temperature laser, [J/m 2 ] excess temperature k thermal conductivity, [W/m K] k B Boltzmann constant, [J/K] Subscripts L thickness of gold film, [m] e electron N number density of atom, eq thermal equilibrium state [m 3 ] F Fermi R reflectivity of gold film i initial R g gas constant for gold, [J/kg K] l lattice Sc heat source of unit L left volume, [W/m 3 ] R right the characteristic energy excitation time [7], such as the process of ultrafast later interaction with metals, the essence of thermal wave can be unveiled from the mask of heat diffusion. To validate the universality of heat wave theory in ultrafast heat transfer, several researches on ultrashort laser interaction with metal were carried out. When metals are subject to the irradiation of ultrashort pulsed laser, there exists nonequilibrium heat transport between electrons and phonons. Kaganov et al. [8] pioneered the two-temperature model based on the observations of difference between electrons and lattice. The rapid advance of femtosecond laser since 1987 provided strong impetus for the reactivation of two-step model due to the laser pulse ( 100 fs) is of the same order of magnitude as the electron-to-phonon relaxation time [9]. The experiment by Brorson et al. revealed that the times of transit across thin gold films of heat pulsed are consistent with wave propagation [10]. Anisimov et al. developed a new model involving the two major macroscopic steps: the absorption of radiation energy by free electrons and subsequent heating of lattice through electron-lattice coupling effect [11]. Qiu and Tien [12] derived a hyperbolic two-step model from Boltzmann transport equation and compared the simulation results with the experiment in reference [10]. Tzou proposed a universal constitutive equation between the heat flux vector and the temperature gradient generalized from the dual-phase-lag concept [13]. A new generalized heat conduction laws based on thermal mass theory and phonon hydrodynamics is proposed by Guo to interpret

3 THERMAL WAVE SUPERPOSITION AND REFLECTION PHENOMENA 1141 the mechanization of microscale heat transfer from a different perspective [14]. Torii and Yang used dimensionless CV model to deal with the effect of laser irradiation on the propagation phenomenon of a thermal wave in a very thin film subject to a symmetrical heating on both sides [15]. When dealing with classical symmetric conduction problems, such as flat film, cylinder and sphere with symmetric initial-boundary conditions and source, a half model is usually considered due to its symmetry of geometry and thermal condition. For the hyperbolic conduction model, such treatment could be challenged because superposition of two waves will be ignored if half of the domain is considered. In the present paper, the dual hyperbolic two step model is employed to simulate a thin gold film subject to the symmetrical irradiations of two beams of femtosecond laser on both sides. The effects of thermal wave on both electron and lattice temperatures during interaction between femtosecond laser pulse and thin gold film will be investigated. In addition, the influence of laser parameters on heat wave propagation will also studied. 2. PHYSICAL MODEL To observe the wave phenomenon during the ultrafast heat conduction, a simulation of a thin gold film subjected to a symmetrical irradiation of femtosecond laser is carried out, as illustrated in Figure 1. A gold film with a thickness of L and an initial temperature of T i is irradiated by two temporal Gaussian lasers with a full width at half maximum (FWHM) pulse width of t p and the energy fluence of J J/m 2 from both sides (x = 0 and L). Since the radius of laser beam is much larger than the thickness of the gold film, the problem can be approximated to be onedimensional, and the dual hyperbolic two step (DHTS) model is given as follows: C e T e t + C e e 2 T e t 2 = ( ) ( T k e Te x e G T x e T l + Sc G e T ) l t t Sc + e t (1) Figure 1. Physical model.

4 1142 L. LI ET AL. T C l l t + C 2 T l l l = ( ) ( T k l Te t 2 x l + G T x e T l + G l T ) l t t (2) where the electron heat capacity C e is approximated by [16] the following: where B e T e T e <T F / 2 2B C e = e T e /3 + C e /3 T F / 2 T e < 3T F / 2 Nk B + C e /3 3T F / 2 (3) T e <T F 3Nk B /2 T e T F C e = B et F / 2 + 3Nk B/2 B e T F / 2 T F T F / 2 T e T F / 2 (4) The thermal conductivity of electron, k e, is [17] as follows: k e = 2 e /4 2 e e 2 e /2 2 e + l (5) where e = T e /T F following: and l = T l /T F. When <<1, Eq. (5) reduces to [18] the ( ) Te k e = k eq T l (6) The bulk thermal conductivity of gold at equilibrium k eq consists of two parts: the electron thermal conductivity, k e, and the lattice thermal conductivity, k l. The former is 99% of k eq, while the latter only take up 1% [19]. The electron-lattice coupling factor G accounts for the heat transfer from hot electrons to lattices through collisions. The temperature-dependent coupling factor is as follows [20]. [ ] Ae G = G RT T B e + T l + 1 l The source term in Eq. (1) is made up by two parts: (7) Sc = S L + S R (8) where S L and S R are the internal heat sources generated by lasers at left and right, respectively. They can be described as follows [21, 22]. ( 1 R S L = 0 94 t p + b 1 e L/ + b J exp x ( ) ) t b t p ( 1 R S R = 0 94 t p + b 1 e L/ + b J exp L x ( ) ) t b t p (9) (10)

5 THERMAL WAVE SUPERPOSITION AND REFLECTION PHENOMENA 1143 The time t = 0 is defined as the time when the peak of a laser pulse reaches the film surface and simulation starts from t = 2t p. Thus, the initial conditions are as follows. T e x 2t p = T l x 2t p = T i (11) T e t = T l t = 0 (12) The adiabatic boundary conditions are adopted with the assumption that the heat loss from the film surface can be neglected [23]. 3. NUMERICAL METHOD T e x = T l = 0 x = 0 and x = L (13) x The laser irradiation to the thin film is simulated by a finite volume method (FVM) with the implicit scheme. The fixed uniform grid with 2,500 control volumes and a step time of 1 fs are used. The thermophysical properties used in the simulation can be found in reference [24] and the relaxation times of electron and lattice are e = 0 04 ps and l = 0 8 ps, respectively [25]. The discretized form of Eq. (1) is as follows. ] T n+2 e P 2T n+1 e P + Te P n + e = 1 t 2 x P ] T n+2 e W T n+2 e P T n+2 e P T n+2 e S k e w k x e s w x s ( T n+2 e p T n+2 l P + e P Te P n e T n+2 l P T )] l P n 2 t 2 t ( Sc n P ) Scn 0 P e 2 t [ T n+2 e P Te P n C e P 2 t [ G P [ T n+2 +Sc n+2 0 P The discretized form of Eq. (2) can be obtained in a similar manner for the lattice temperature. To validate the numerical method, a comparison between the numerical solution employing the algorithm adopted here and the analytical solution on the one-dimensional heat conduction problem [13] is provided. The nondimensional CV model and the initial-boundary condition are given as follows. (14) 2 2 = + z 2 0 < <1 >0 (15) 2 = 0 and = 0at = 0 = 1at = 0

6 1144 L. LI ET AL. Figure 2. Comparison between numerical and analytical results (Z = 0 05, = = 0at = 1 where the dimensionless variables are defined as follows. = T T 0 = x T w T 0 L = t L 2 / and z = L 2 / (16) The results for z = 0 05 at the instant = 0 05 are plotted in Figure 2, which are in excellent agreement with the analytical results obtained by Tzou [13]. Figure 3. Comparison between numerical and experimental results of the front electron temperature (L = 100 nm, J = 10 J/m 2, and t p = 96 fs.

7 THERMAL WAVE SUPERPOSITION AND REFLECTION PHENOMENA 1145 Figure 4. Comparison between numerical and experimental results of the rear electron temperature (L = 100 nm, J = 10 J/m 2, and t p = 96 fs. In addition, a 100-nm thin gold film under irradiation of a laser of J = 10 J/m 2, t p = 96 fs is simulated and is compared with the experimental results in reference [10]. Figures 3 and 4 show the time-dependent front and rear electron temperature curves, respectively; they are almost in line with the experiment results. 4. RESULTS AND DISCUSSION Simulation of a 100-nm gold film subject to irradiation of two laser beams from the opposite directions at the same time is carried out. The laser parameters are as follows: R = 0 6 = 20 6nm b = 105 nm t p = 20 fs and J = 1J/m 2. Figure 5 indicates the spatial distribution of excess electron temperature e at different times, where e = T e T e L and T e L is the electron temperature at the left boundary (x = 0). At time 0 fs and 50 fs, since the boundaries are subject to the laser irradiation directly, the internal temperature is lower than the boundary temperature, thus e 0. However, at 100 fs and 150 fs, the internal temperature is higher than the boundary and consequently e 0. Then at the time of 200 fs and 250 fs, the excess temperature e becomes negative again. Finally, at the time of 300 fs, e becomes zero again, which indicate uniform electron temperature in the thin film. Figures 6 and 7 show the time-dependent excess electron and lattice temperatures at the middle of thin film (x = L/2). It is obvious that thermal oscillation is observed. The reason to lead to the oscillation of electron and lattice temperatures attributes to the characteristics of the DHTS model that is composed of two inter-coupling CV conduction equations for electron and lattices. Since the CV conduction model is wave equation, just like the governing equation of damping forced string vibration, the characteristics of heat propagation according to CV equation are similar to waves that have the property of superposition. According to Figure 6, before the time of 73 fs, the excess electron temperature e is negative because the boundary temperature is higher than the center temperature. Then,

8 1146 L. LI ET AL. Figure 5. Spatial distribution of excess electron temperature at different times (J = 1J/m 2, t p = 20 fs, and L = 100 nm). the laser irradiations generate two waves which propagate at opposite directions. When these two waves meet each other (73 fs <t<167 fs), the amplitude increases, which makes the center temperature higher than the surface temperature, i.e., e > 0. After the transient encounter, these waves go to their own destinations, which makes e negative again. When they reach to the other side of the surface (t = 203 fs), the reflection phenomenon take place and the adiabatic boundaries rebound these waves. Then, they encounter with each other at t = 261 fs and the wave Figure 6. Excess electron temperature at the center of the film (J = 1J/m 2, t p = 20 fs, and L = 100 nm).

9 THERMAL WAVE SUPERPOSITION AND REFLECTION PHENOMENA 1147 Figure 7. Excess electron temperature at the center of the film (J = 1J/m 2, t p = 20 fs, and L = 100 nm. superposition takes place once more, so the e turns positive again. The interval time of the first two crests of e is t c 1 2 = 192 fs. The propagation velocity of heat wave is c = e / e, which is a single function of thermal diffusivity. The average propagation velocity between these two crests is c AVE 1 2 = m/s. The wavelength that the thermal wave propagates at the interval of two crests can be obtained as S 1 2 = t c 1 2 c AVE 1 2 = nm, and the interval time between the second crest and third one is t c 1 2 = the average propagation velocity is c AVE 1 2 = m/s and the wave path is S 2 3 = nm. Both the wavelengths narrowly equal to the thickness of the gold film, 100 nm. Simulations for different film thickness L and puls width t p are then carried out and the comparison of S 1 2 and L are summarized in Tables 1 and 2. According to Table 1, is the relative difference between S 1 2 and L, i.e. = S 1 2 L /L which increases with increasing L. That is because increasing L causes the inner temperature distribution more non-uniform, which makes the average velocity c AVE 1 2 deviate the real propagation velocity of thermal wave. However, when t p is increased while L is kept at 100 nm (see Table 2), will change complexly. Figure 8 shows the curve of changing with t p. It can be seen that when t p is less than 60 fs, decreases as t p increases. That is because the source term, Sc, decreases with t p increases, which makes the temperature distribution more uniform. However if 60 < t p 160, will increase with increasing t p because the thermal waves come across at the first time within pulse width and the source of laser influences the time when first Table 1. Comparison of S 1 2 and L t p = 20 fs, J = 1J/m 2 L(nm) S 1 2 (nm) (%)

10 1148 L. LI ET AL. Table 2. Comparison of S 1 2 and L (L = 100 nm, J = 1J/m 2 t p (fs) S 1 2 (nm) (%) t 1 (fs) t 2 (fs) peak temperature takes place, while the second peak is out of the pulse width. When the t p is increased, although Sc decreases, the time when first superposition takes place will approach the main influence zone of source ( t p <t<t p ). If t>160 fs, the pulse width will include both peaks; thus, will decrease as t p increases. Table 2 also shows the time when the superpositions take place, where t 1 and t 2 represent the first and second superposition times, respectively. Since the excess electron temperature e oscillates, the excess lattice temperature l also fluctuates with the change of electron temperature due to the coupling term, G T e T l, in lattice energy equation (2). Figure 9 shows that the time at which maximum excess electron and lattice temperatures are reached. The difference between the two peak times is almost the same, about 70 fs, for all laser fluencies. In other words, there is a 70 fs delay for the lattice temperature to respond to the oscillation of electron temperature. Figure 10 shows the excess electron temperature at the center of the film for different laser pulse fluence J. It can be seen that the oscillation period of e increases as J increases because thermal diffusivity e decreases as electron temperature increases, which makes the wave velocity c to decrease. However, the maximum excess electron temperature may increase or decrease with increasing J. To see this more clearly, Figure 11 shows the change of maximum temperature with Figure 8. Relative difference versus t p L = 100 nm, J = 1J/m 2.

11 THERMAL WAVE SUPERPOSITION AND REFLECTION PHENOMENA 1149 Figure 9. Times of the maximum excess electron and lattice temperatures versus J t p = 20 fs, L = 100 nm. laser fluence. When J 3J/m 2, the maximum of e increases as J increase, while if J>3J/m 2, the curve reverse the course. The reason to lead to this phenomenon is that when J increases, the Sc will increase, thus the amplitude of e becomes greater. Furthermore, the increasing J will also lead to a non-uniform electron temperature distribution, which will cover the superposition phenomenon of thermal Figure 10. Histories of excess electron temperature at center of film for different laser pulse fluence J L = 100 nm.

12 1150 L. LI ET AL. Figure 11. Maximum excess electron temperature at the center of film versus laser pulse fluence J (t p = 20 fs, L = 100 nm. wave. Figure 12 shows the maximum excess electron temperature at the center of the film at different laser pulse width t p. It can be see that when t p 20 fs e will increase as t p rises. However, when t p > 20 fs e will decrease with increasing t p. Figure 13 shows excess lattice temperature at the center of film for different laser fluencies. It can be seen that there is a minor disturbance of the lattice Figure 12. Maximum excess electron temperature at the center of film versus laser pulse width t p (J = 5 J/m 2, L = 100 nm.

13 THERMAL WAVE SUPERPOSITION AND REFLECTION PHENOMENA 1151 Figure 13. Histories of excess lattice temperature at different laser pulse fluence J t p = 20 fs L= 100 nm. temperature due to the oscillation of electron temperature, then the excess lattice temperature remain unchanged; when J increases, the balanced excess lattice temperature decreases. That is because at the beginning of laser irradiation the electron temperature soars especially the surface temperature; this makes the coupling factor Gincrease and the surging electron energy will soon be transferred to the lattice. Thus, before 400 fs, the lattice temperature will respond to the change of electron temperature quickly. But after that since the laser irradiation ceases, the electron temperature drops and electron and lattice becomes thermal equilibrium, which makes coupling effect very insignificant, i.e., G [ ( T e T l + Te )] l T l t t 0. Furthermore, since the lattice temperature ( becomes ) more uniform after 400 fs, the T influence of diffusion term is limited, i.e., kl x x 0. Therefore, the transient term approximately equals to zero and after 400 fs, the lattice temperature almost remains unchanged. When J increases, the energy absorbed by film increases which makes the balanced excess temperature lower. 5. CONCLUSION In this article, a dual hyperbolic two step model is employed to simulate heat conduction in a thin gold film subject to the symmetrical irradiations of two femtosecond laser beams on both sides. When a 100-nm gold film is subject to irradiation of two laser beams of t p = 20 fs J = 1J/m 2, a clear wave overshoot and oscillation phenomenon of electron temperature can be observed; the lattice temperature also oscillates slightly. The electron temperature overshoot results from the superposition of two thermal waves and the reason of electron temperature oscillation is due to the thermal wave reflection: the thermal waves are rebounded by the adiabatic boundaries; the rebounded waves meet with each other, and the superposition takes place again. The oscillation of the electron temperature causes the lattice temperature to fluctuate. The time delay from oscillations of electron

14 1152 L. LI ET AL. temperature to lattice temperature is about 70 fs. When the laser fluence increases, there is a peak of maximum excess electron temperature at J = 4 J/m 2. In addition, there is also a peak of e when t p is increased. It should be pointed out that this paper does not consider the radiation loss during the interaction of gold film and laser which may influence, at a certain extent, the electron temperature oscillation. However, the loss of energy or the form of boundary condition can only influence the amplitude of oscillation, but could not change the form of thermal wave propagation. FUNDING The authors gratefully acknowledge the financial support from the Chinese National Natural Science Foundation under grant nos , and the US National Science Foundation under grant no. CBET REFERENCES 1. C. Cattaneo, A Form of Heat Conduction Equation which Eliminates the Parabolic of Instantaneous Propagation, Compte Rendus, vol. 247, pp , P. Vemotte, Some Possible Complications in the Phenomena of Thermal Conduction, Compte Rendus, vol. 252, pp , V. Peshkov, Second Sound in Helium II, USSR J. Phys., vol. 8, pp. 381, H. Smith and H. H. Jensen, Transport Phenomena, pp , Clarendon Press, Oxford, B. Bertman and D. J. Sandiford, Second Sound in Solid Helium, Sci. Ameri., vol. 22, pp , D. D. Joseph and L. Preziosi, Heat Waves, Rev. Mod. Phys., vol. 61, pp , P. H. Barret and M. Palmerm, High-Power and Femtosecond Lasers: Properties, Materials and Applications, pp , Nova Science Publishers, New York, M. I. Kaganov, I. M. Lifshitz, and L. V. Tanatarov, Relaxiation Between Electrons and the Crystalline Lattice, Sov. Phys. JETP, vol. 4, no. 2, pp , L. Wang, Advances in Transport Phenomena, pp , Springer, Berlin, S. D. Brorson, J. G. Fujimoto, and E. P. Ippen, Femtosecond Electronic Heat Transfer Dynamics in Thin Gold Films, Phys. Lett., vol. 59, pp , S. I. Anisimov, B. L. Kapeliovich, T. L. Perelman, and Z. Eksp, Electron Emission from Metal Surfaces Exposed to Ultra Short Laser Pulses, Sov. Phys. JETP, vol. 39, pp , T. Q. Qiu and C. L. Tien, Heat Transfer Mechanisms during Short-Pulse Laser Heating of Metals, ASME J. Heat Transfer, vol. 115, pp , D. Y. Tzou, A Unified Field Approach for Heat Conduction From Macro- to Micro- Scales, J. Heat Transfer, vol. 117, pp. 8 16, Y. Dong, B. Cao and Z. Guo, Generalized Heat Conduction Laws based on Thermomass Theory and Phonon Hydrodynamics, J. Appl. Phys., vol. 110, , S. Torii and W. Yang, Heat Transfer Mechanisms in Thin Film with Laser Heat Source, Int. J. Heat Mass Transfer, vol. 48, pp , J. K. Chen, D. Y. Tzou, and J. E. Beraun, A Semiclassical Two-Temperature Model for Ultrafast Laser Heating, Int. J. Heat Mass Transfer, vol. 49, nos. 1 2, pp , S. I. Anisimov and B. Rethfeld, Theory of Ultrashort Laser Pulse Interaction with a Metal, Proc. SPIE, vol. 3093, pp , 1997.

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