Ultrafast solid-liquid-vapor phase change of a thin gold film irradiated by femtosecond laser pulses and pulse trains

Abstract

Effects of different parameters on the melting, vaporization and resolidification processes of thin gold film irradiated by femtosecond pulses and pulse train were systematically studied. The classical two-temperature model was adopted to depict the non-equilibrium heat transfer in electrons and lattice. The melting and resolidification processes, which was characterized by the solid-liquid interfacial velocity, as well as elevated melting temperature and depressed solidification temperature, was obtained by considering the interfacial energy balance and nucleation dynamics. Vaporization process which leads to ablation was described by tracking the location of liquid-vapor interface with an iterative procedure based on energy balance and gas kinetics law. The parameters in discussion included film thickness, laser fluence, pulse duration, pulse number, repetition rate, pulse train number, etc. Their effects on the maximum lattice temperature, melting depth and ablation depth were discussed based on the simulation results.

Keywords

melting / evaporation / nucleation dynamics / nanoscale heat transfer

Introduction

Ultrashort pulsed laser is regarded as an ideal tool for high-resolution, high-quality material processing, fast fabrication and diagnosis because of its ability to deposit high density energy at a given place in a very short period of time. The heat flux of a nanosecond laser pulse can reach as high as 1 TW/m², while a femtosecond one can reach 1 ZW/m². Such a rapid and highly energy densified process leads to unique characteristics to be studied. The traditional phenomenological laws, such as Fourier’s law, assume that heat transfer takes place simultaneously with the onset of temperature gradient. During pico- to femtosecond laser-materials processing, the characteristic times of the heat carriers are comparable to the characteristic energy excitation time [1]. The Fourier’s law becomes invalid for such ultrafast process. In the last two decades, many researchers worked on this problem, to uncover the heat transfer mechanism with extremely short time-scale and high energy intensity [2-6]. Various computational models were put forward and developed to describe the non-equilibrium energy transport phenomena during the process. One of the classical methods is the two-temperature model, which was originally proposed by Anisimov [7] and then rigorously derived by Qiu and Tien [8] based on the Boltzmann equation. The nonequilibrium energy transport between electron and lattice can also be described by the dual-phase-lag model [9,10]. Jiang and Tsai [11] extended the existing two-temperature model to high electron temperatures by using full-run quantum treatments. Chen et al. [12] proposed a semiclassical two-step heating model to investigate thermal transport in metals caused by ultrashort laser irradiation.

Most existing two-temperature models dealt with the case that lattice temperature is well below the melting point and only pure conduction is considered. Under higher laser fluence and/or short pulse, the lattice temperature can exceed the melting point and melting takes place. The liquid phase will be resolidified when the lattice is cooled by conducting heat away. The rapid phase change phenomena induced by ultrashort pulse laser are controlled by nucleation dynamics at the interface, not by interfacial energy balance [13]. Zhang and Chen [14] proposed an implicit, fixed grid interfacial tracking method to solve kinetics controlled rapid melting and resolidification during ultrashort pulse laser interaction with a free-standing metal film. The nonlinear electron heat capacity obtained by Jiang and Tsai [11] and a temperature-dependent electron-lattice coupling factor based on a phenomenological model [15] is incorporated into the interfacial tracking method. When the laser fluence is sufficiently high or when the laser pulse width is in the order of femtosecond, the liquid surface temperature may exceed the saturation temperature and vaporization may take place. Chowdhury and Xu [16] proposed a numerical model to simulate the melting-vaporization-resolidification process during femtosecond laser-metal interaction, which will be adopted in this paper to track the liquid-vapor interface.

In this paper, the entire process of laser-metal interaction—including preheating, melting, vaporization, resolidification and thermalization—will be thoroughly investigated under the frame works of two-temperature model and interfacial tracking method. Temperature and velocity of liquid-vapor interface are controlled by energy balance and gas kinetics law, which will be fulfilled simultaneously through an iterative computational procedure. At the mean time, the effects of various parameters on the maximum lattice temperature, melting depth and ablation depth are discussed based on the simulation results. The parameters in discussion include film thickness, laser fluence, pulse duration, pulse number, repetition rate, pulse train number, etc.

Physical model

Figure 1 shows the physical model of the problem under consideration. Laser pulses impinge on the right side of a free standing gold film, which has a thickness of L. The range of thickness used in this paper is very small in comparison to the radius of the laser beam; therefore the problem under discussion can be approximated to be one-dimensional. The structures of three modes of laser pulses used in this paper is illustrated in Fig. 2, in which f_rep and t_sep are the repetition rate and time interval between single pulses, respectively. Each single pulse is assumed to be temporally Gaussian. The pulse duration (t_p), is defined as the full width at half maximum (FWHM).

The two-step heating model for free electrons and the lattice are given by [7]

(1)C_{\mathrm{e}}{\displaystyle \frac{\partial T_{\mathrm{e}}}{\partial t}}={\displaystyle \frac{\partial }{\partial x}}\left(k_{\mathrm{e}}{\displaystyle \frac{\partial T_{\mathrm{e}}}{dx}}\right)-G(T_{\mathrm{e}}-T_{\mathrm{l}})+S,

(2)C_{\mathrm{l}}{\displaystyle \frac{\partial T_{\mathrm{l}}}{\partial t}}={\displaystyle \frac{\partial }{\partial x}}\left(k_{\mathrm{l}}{\displaystyle \frac{\partial T_{\mathrm{l}}}{dx}}\right)+G(T_{\mathrm{e}}-T_{\mathrm{l}}).

The heat capacity of electron C_e is approximated by [12]

(3)

where

(4)C_{\mathrm{e}}^{\prime }=B_{\mathrm{e}}{T}_{\mathrm{F}}/{\mathrm{\pi }}^{\mathrm{2}}+{\displaystyle \frac{\mathrm{3}N{k}_{\mathrm{B}}/\mathrm{2}-B_{\mathrm{e}}{T}_{\mathrm{F}}/{\mathrm{\pi }}^{\mathrm{2}}}{T_{\mathrm{F}}-T_{\mathrm{F}}/{\mathrm{\pi }}^{\mathrm{2}}}}(T_{\mathrm{e}}-T_{\mathrm{F}}/{\mathrm{\pi }}^{\mathrm{2}}).

The thermal conductivity of electron k_e can be obtained from [17]

(5)k_{\mathrm{e}}=\chi {\displaystyle \frac{{({\vartheta }_{\mathrm{e}}^{\mathrm{2}}+\mathrm{0.16})}^{\mathrm{5}/\mathrm{4}}({\vartheta }_{\mathrm{e}}^{\mathrm{2}}+\mathrm{0.44}){\vartheta }_{\mathrm{e}}}{{({\vartheta }_{\mathrm{e}}^{\mathrm{2}}+\mathrm{0.092})}^{\mathrm{1}/\mathrm{2}}({\vartheta }_{\mathrm{e}}^{\mathrm{2}}+\eta {\vartheta }_{\mathrm{l}})}},

where {\vartheta }_{\mathrm{e}}=T_{\mathrm{e}}/T_{\mathrm{F}} and {\vartheta }_{\mathrm{l}}=T_{\mathrm{l}}/T_{\mathrm{F}}.

In Eqs. (1) and (2), G is the electron-lattice coupling factor. A phenomenological temperature-dependent G suggested by Chen et al. [15] is adopted:

(6)G=G_{\mathrm{R}\mathrm{T}}\left[{\displaystyle \frac{A_{\mathrm{e}}}{B_{\mathrm{1}}}}(T_{\mathrm{e}}+T_{\mathrm{l}})+\mathrm{1}\right],

where G_RT is the coupling factor at room temperature; A_e and B_l are material constants for the electron relaxation time.

Since the electrons are more likely to collide with liquid atoms than the atoms in solid crystals, in the liquid phase, G is taken to be 20% higher than that of the solid [18].

The laser irradiation is considered as a source term S in Eq. (1):

(7)S={\displaystyle \sum _{i=\mathrm{1}}^K{\displaystyle \sum _{j=\mathrm{1}}^N{\displaystyle \frac{\mathrm{0.94}{J}_i(\mathrm{1}-R)}{t_{\mathrm{p}}(\delta +{\delta }_{\mathrm{b}})[\mathrm{1}-{\mathrm{e}}^{-L/(\delta +{\delta }_{\mathrm{b}})}]}}}}·\exp \left[-{\displaystyle \frac{x}{(\delta +{\delta }_{\mathrm{b}})}}-\mathrm{2.77}{\left({\displaystyle \frac{t-(i-\mathrm{1})/f_{\mathrm{r}\mathrm{e}\mathrm{p}}-(j-\mathrm{1})t_{\mathrm{s}\mathrm{e}\mathrm{p}}}{t_{\mathrm{p}}}}\right)}^{\mathrm{2}}\right],

where K is the number of pulse trains, N is the number of pulses in each train, t_sep is separation time between each single pulse, f_rep is the repetition rate, R is the reflectivity of the thin film, \delta is the optical penetration depth, J is the laser pulse fluence, {\delta }_{\mathrm{b}} is the ballistic depth of the electrons, and \mathrm{1}-{\mathrm{e}}^{-L/(\delta +{\delta }_{\mathrm{b}})} is to correct the finite film thickness effect. For single pulse irradiation, Both K and N are set to be 1. For the multiple pulse irradiations, K is set to be 1.

For a metal at its thermal equilibrium state, the thermal conductivity, k_eq, is the sum of the electron thermal conductivity, k_e, and the lattice thermal conductivity, k_l. In most cases k_e dominates k_eq because free electrons contribute to the majority of heat conduction, For gold, k_l is usually taken to be 1% of k_eq [19], i.e.,

(8)k_{\mathrm{l}}=\mathrm{0.01}{k}_{\mathrm{e}\mathrm{q}}.

A uniform temperature distribution is set to be the initial condition:

(9)T_{\mathrm{e}}(x,-\mathrm{2}{t}_{\mathrm{p}})=T_{\mathrm{l}}(x,-\mathrm{2}{t}_{\mathrm{p}})=T_{\mathrm{0}}.

On the right side of the film which receives laser energy, the heat loss caused by radiation will be considered while on the other side adiabatic boundary condition is applied:

(10){{\displaystyle \frac{\partial T_{\mathrm{e}}}{\partial x}}|}_{x=\mathrm{0}}={{\displaystyle \frac{\partial T_{\mathrm{e}}}{\partial x}}|}_{x=L}={{\displaystyle \frac{\partial T_{\mathrm{l}}}{\partial x}}|}_{x=\mathrm{0}}=\mathrm{0},

(11){q_{\mathrm{R}}^{″}|}_{x=L}=\sigma ϵ(T_{\mathrm{s}\mathrm{u}\mathrm{r}}^{\mathrm{4}}-T_{\mathrm{\infty }}^{\mathrm{4}}).

Before evaporation takes place, T_sur is the surface lattice temperature at x = L. After evaporation begins, T_sur is the liquid-vapor interface temperature, which varies with the heating condition.

The energy balance at the solid-liquid interface is [20]

(12)k_{\mathrm{l},\mathrm{s}}{\displaystyle \frac{\partial T_{\mathrm{l},\mathrm{s}}}{\partial x}}-k_{\mathrm{l},\mathrm{l}}{\displaystyle \frac{\partial T_{\mathrm{l},\mathrm{l}}}{\partial x}}={\rho }_{\mathrm{l}}{h}_{\mathrm{m}}{u}_{\mathrm{s}\mathrm{l}},x=s(t).

where T_l,s and T_{\mathrm{l},\mathrm{l}} are solid and liquid lattice temperature respectively, \rho is mass density, h_{\mathrm{m}} is latent heat of fusion, and u_s is solid-liquid interfacial velocity. The additional interfacial velocity due to the density change during melting and resolidification has been considered.

For rapid melting and solidification processes, the velocity of the interface is dominated by nucleation dynamics, instead of being dominated by the energy balance, Eq. (12). For ultrashort-pulsed laser melting of gold, the velocity of the solid-liquid interface is [18]

(13)u_{\mathrm{s}\mathrm{l}}=V_{\mathrm{0}}\left[\mathrm{1}-\exp \left(-{\displaystyle \frac{h_{\mathrm{m}}}{R_{\mathrm{g}}{T}_{\mathrm{m}}}}{\displaystyle \frac{T_{\mathrm{l},\mathrm{I}}-T_{\mathrm{m}}}{T_{\mathrm{l},\mathrm{I}}}}\right)\right],

where V₀ is the maximum interface velocity, R_g is the gas constant for the metal, and T_l,I is the interfacial temperature which is higher than the normal melting point, T_m, during melting and lower than T_m during solidification.

To characterize the vaporization process, the Clausius-Clapeyron equation is employed to describe the slope of saturation pressure-temperature curve, with the assumption of ideal gas and thermal equilibrium:

(14){\displaystyle \frac{\mathrm{d}p}{\mathrm{d}{T}_{\mathrm{l}\mathrm{v}}}}={\displaystyle \frac{p{h}_{\mathrm{l}\mathrm{v}}(T_{\mathrm{l}\mathrm{v}})}{R_{\mathrm{u}}{T}_{\mathrm{l}\mathrm{v}}^{\mathrm{2}}}},

where R_u is the universal gas constant, which is related to boiling temperature T_{\mathrm{l}\mathrm{v}} as

(15)h_{\mathrm{l}\mathrm{v}}(T_{\mathrm{l}\mathrm{v}})=h_{\mathrm{l}\mathrm{v}\mathrm{0}}\sqrt{\mathrm{1}-{\left({\displaystyle \frac{T_{\mathrm{l}\mathrm{v}}}{T_{\mathrm{c}}}}\right)}^{\mathrm{2}}},

where h_{\mathrm{l}\mathrm{v}\mathrm{0}} is the latent heat of vaporization at absolute zero and T_c the critical temperature [21]. By integrating Eq. (14), the relationship between interfacial temperature and pressure can be obtained as

(16)p=p_{\mathrm{0}}\exp \{-{\displaystyle \frac{{\rho }_{\mathrm{l}}{L}_{\mathrm{0}}}{R_{\mathrm{u}}}}\left[{\displaystyle \frac{\mathrm{1}}{T_{\mathrm{l}\mathrm{v}}}}\sqrt{\mathrm{1}-{\left({\displaystyle \frac{T_{\mathrm{l}\mathrm{v}}}{T_{\mathrm{c}}}}\right)}^{\mathrm{2}}}-{\displaystyle \frac{\mathrm{1}}{T_{\mathrm{b}}}}\sqrt{\mathrm{1}-{\left({\displaystyle \frac{T_{\mathrm{b}}}{T_{\mathrm{c}}}}\right)}^{\mathrm{2}}}\right]-{\displaystyle \frac{{\rho }_{\mathrm{l}}{L}_{\mathrm{0}}}{R_{\mathrm{u}}{T}_{\mathrm{c}}}}\left[\arcsin \left({\displaystyle \frac{T_{\mathrm{l}\mathrm{v}}}{T_{\mathrm{c}}}}\right)-\arcsin \left({\displaystyle \frac{T_{\mathrm{b}}}{T_{\mathrm{c}}}}\right)\right]\}.

The molar evaporation flux j_v at the surface can be calculated by the Hertz-Knudsen-Langmuir equation derived from kinetic theory of gases [22],

(17)j_{\mathrm{v}}={\displaystyle \frac{Ap}{\sqrt{\mathrm{2}\mathrm{\pi }M{R}_{\mathrm{u}}{T}_{\mathrm{l}\mathrm{v}}}}},

where A is an accommodation coefficient that shows which portion of vapor molecules striking the liquid-vapor surface is absorbed by this surface [23]. Xu et al. [21] recommended a value of 0.82 for this coefficient. The liquid-vapor interfacial velocity can be obtained from j_v as given below:

(18)u_{\mathrm{l}\mathrm{v}}={\displaystyle \frac{M{j}_{\mathrm{v}}}{{\rho }_{\mathrm{l}}}}={\displaystyle \frac{AMp}{{\rho }_{\mathrm{l}}\sqrt{\mathrm{2}\mathrm{\pi }M{R}_{\mathrm{u}}{T}_{\mathrm{l}\mathrm{v}}}}}.

Detailed information and discussion on the evaporation model can be found in Refs. [24,25]. It should be noted that because the multiple pulses and pulse train irradiation will significantly decrease the maximum temperature, within the parameter range being studied in this paper, only single pulse mode will cause evaporation to take place.

Numerical procedure

The governing Eqs. (1) and (2) are discretized by the Finite Volume Method [26]. A fixed uniform grid with 2050 control volumes is adopted. The time step is a variable in the numerical solution. The smallest time step is 10^-2t_p, which is implemented during (-2t_p, 2t_p) of each single pulse. The largest time step is 10⁴t_p, which is implemented when the difference between electronic and lattice temperature is less than 1 K. Thermophysical properties used in the calculations can be found in Ref. [14].

In each time step, an iterative procedure will be employed to deal with the nonlinear relationship between electron energy equation, lattice energy equation, solid-liquid and liquid-vapor interfaces. Electron energy Eq. (1) is solved first, using tri-diagonal matrix method (TDMA), and then the lattice energy Eq. (2) is solved. After obtaining an estimated electron and lattice temperature field, the velocity and temperature of solid-liquid interface is obtained by using the method provided in Ref. [14] and is briefly described as

(1) The solid-liquid interfacial temperature T_sl is assumed and the solid-liquid phase interfacial velocity is determined according to the interfacial energy balance;

(2) The interfacial velocity from the nucleation dynamics is obtained from Eq. (13);

(3) The interfacial velocities obtained from Steps (1) and (2) are compared. If the interfacial velocity obtained from the energy balance is higher than that from the nucleation dynamics, the interfacial temperature will be increased; otherwise, the interfacial temperature is decreased.

Steps 1-3 are repeated until the difference between the interfacial velocities obtained from the two methods is less than 10^-5 m/s.

The following iterative procedure will be employed to track the liquid-vapor interface:

(1) Assume an interfacial velocity V_{\mathrm{l}\mathrm{v}}^{*}, then the new interface location s_{\mathrm{l}\mathrm{v}}^{*} is determined;

(2) Solve the energy balance equation at the liquid-vapor interface to obtain the interface temperature T_{\mathrm{l}\mathrm{v}};

(3) According to Eqs. (16) and (18), obtain the new interface velocity V_{\mathrm{l}\mathrm{v}}^{**};

(4) Go to step 2 and use V_{\mathrm{l}\mathrm{v}}^{***} as the new interface velocity. Steps 2-4 are repeated until the difference between the interfacial velocities obtained from two consecutive iterations is less than 10^-5 m/s.

Results and discussion

Vaporization, melting and resolidification of gold film with single pulse irradiation

The melting, evaporation and resolidification processes caused by the single pulse irradiation were studied first. For a single pulse of 0.5 J/cm², the evolution of surface electron temperature, lattice temperature, solid-liquid interface velocity and location, liquid-vapor interface velocity and location are demonstrated in Fig. 3. It can be seen that the electron temperature increases rapidly due to the depositing of laser energy on electrons. Tens of picoseconds later, the lattice temperature increases to a peak caused by the collision of hot electrons with lattices. During this process, melting and evaporation occurs due to the high lattice temperature. After the temperature peak, heat is conducted into the deeper part of the metal film, and surface temperature decreases gradually. The material ablation caused by evaporation stops and a thin layer of metal is lost due to evaporation. After a maximum melting depth is reached, melting ceases and material will be resolidified.

With pulse fluence ranging from 0.2 J/cm² to 0.7 J/cm², the relationship between maximum lattice temperature and maximum melting depth is displayed in Fig. 4. Obviously, a higher pulse fluence leads to a deeper melting. Figure 4 will serve as the base of the comparison between multiple pulses and the single pulse. If the resultant point of a multiple pulses is located in the upper part of Fig. 4, it means with the same maximum temperature, the multiple pulses will achieve deeper melting, and vice-versa.

Single pulse irradiations on film with thicknesses ranging from 500 nm to 1100 nm were simulated. Three different pulse fluence values were adopted: 0.4 J/cm², 0.5 J/cm², and 0.6 J/cm². The relationship between maximum melting depth and film thickness is exhibited in Fig. 5. When the thickness decreases under 600 nm, the melting depth increases dramatically. However, the film thickness shows very little influence on melting depth when it is greater than 800 nm. This means that within the current range of laser energy, thicker film will act almost the same as bulk gold. The evaporation effects are evaluated by ablation depth. As shown in Fig. 6, the ablation depth is much smaller than the melting depth, but its dependence on film thickness is still conclusive. Smaller film thickness will lead to higher ablation depth, which is caused by the higher temperature achieved. When the thickness is greater than 1000 nm, its influence on ablation depth is almost negligible.

Multiple pulses irradiation

By dividing one single pulse into several consecutive smaller laser pulses, more control can be provided for the laser operation, along with some other advantages as described below.

First of all, two consecutive pulses with identical fluence and a separation time of Δt were studied. The irradiation processes caused by two 0.3 J/cm², 100fs laser pulses with a separation time ranging from 10 ps to 6000 ps were then simulated. The dependence of maximum melting depth on maximum lattice temperature with different separation time is presented in Fig. 7. The arrows indicate the directions toward which the separation time Δt increases. The solid line in Fig. 7 is the result of the single pulse, as shown in Fig. 4. All the results of two pulses are located in the upper side of the solid line, which means that with the same peak temperature, the melting depth of two pulses will always be deeper than that of a single pulse. Furthermore, it can be seen that when the separation time increases from 10 ps, the maximum temperature will decrease without an obvious drop of melting depth. The result points of two pulses deviate from the single pulse line gradually. When the aim of laser irradiation is to obtain deeper melting depth, this is a good trend because the same melting depth can be achieved with a lower temperature, i.e., the residual thermal stress will be lower. However, when the separation time is longer than 280 ps, the result points of two pulses approach to the single pulse line again, the melting depth drops soon with the same temperature. When the separation time equals 800 ps, the difference between two pulses and the single pulse is very small. After that, with the increase of the separation time, the results caused by two pulses will deviate from the solid line again. After comparing this result with the variation of solid-liquid interface velocity, it was found that 280 ps is just the time when the solid-liquid interfacial velocity turn from positive to negative, i.e., when the resolidification process starts; 800 ps is the time when the interfacial velocity achieves maximum negative value. This proves that if a deeper melting depth is the aim of laser metal interaction, the second pulse should be launched when the melting process caused by the first pulse reaches the maximum value. On the other hand, if the second pulse is launched when the resolidification process is about to stop, the two pulses irradiation shows no advantage over the single pulse.

The situations of more than two pulses in one burst were also studied. To compare with a single pulse of 0.6 J/cm², six cases were considered: (a) 2 pulses at 0.3 J/cm² per pulse; (b) 3 pulses at 0.2 J/cm² per pulse; (c) 4 pulses at 0.15 J/cm² per pulse; (d) 5 pulses at 0.12 J/cm² per pulse; (e) 6 pulses at 0.1 J/cm² per pulse; and (f) 10 pulses at 0.06 J/cm² per pulse. The separation time ranged from 1 ps to 15 ps.

Figures 8 and 9 depict the dependence of maximum temperature and melting depth on the number of pulses and separation time. It can be seen that with more pulses per burst, the temperature and melting depth will decrease. The increase of separation time also leads to a lower temperature and melting depth.

If the relationship between the maximum temperature and melting depth caused by the multiple pulses is plotted and compared with that of a single pulse, Fig. 10 can be obtained. It is evident that all the points are located on the upper side of the single pulse line, which means a deeper melting depth will be achieved with the same lattice temperature. And this difference will be larger with higher number of pulses and longer separation time. In comparison to a single 0.6 J/cm² pulse irradiation, the maximum lattice temperature caused by 10 consecutive pulses with a separation time of 15 ps can reduce 22.01%, while the melting depth only decreases 4.93%. This means in practical application, if a deeper melting is wanted, the multiple pulses is preferable than the single pulse, because it will lead to a lower temperature, then a smaller residual stress and distortion.

Pulse train irradiation

As shown in Fig. 2, a laser pulse train consists of several pulse bursts with a repetition rate of 0.5-1 MHz. Each pulse burst contains 3-10 pulses with an interval of 50 ps-10 ns. To compare the result of the pulse train with the single pulse, five different single pulse fluences were used: 0.022, 0.026, 0.030, 0.034, and 0.036 J/cm². The other parameters were kept the same: repetition frequency 1000 Hz, separation time 100 ps, and 10 pulse bursts.

The dependence of maximum lattice temperature and melting depth on the fluence is given in Fig. 11. It is obvious that with a higher laser fluence, a deeper melting will be achieved and a higher temperature will be caused. In the ultrafast laser materials processing, it is desirable that the melting depth can be accurately controlled and the temperature rise should be kept as low as possible to reduce the thermal stress.

To compare the results of the pulse train and the single pulse irradiation, the relationship between the maximum lattice temperature and melting depth is shown in Fig. 12. The lower line is for the single pulse while the upper one is for the pulse train. With the increase of the pulse train power, the melting depth increases rapidly while the maximum lattice temperature increases relatively slowly. It is clear that with the same lattice temperature, the melting depth caused by the pulse trains is much deeper than that caused by the single pulse. For example, the highest lattice temperature caused by a single 0.3 J/cm² pulse irradiation is almost the same as that of the pulse trains which consists of 10 trains with 3 single 0.036 J/cm²-pulses in a train, but its melting depth is only 50 nm, much less than the melting caused by the pulse train, almost 900 nm. On the other hand, to achieve the same melting depth, a pulse train will cause much lower lattice temperature, which is an advantage for laser-materials processing. However, the energy needed for a pulse train is much higher than that of a single pulse. According to Fig. 12, the melting depth caused by a single 0.3 J/cm² pulse is almost the same as that of the pulse train which consists of 10 bursts with 3 single 0.026 J/cm² pulses in a burst. But the total energy for the pulse train is 0.78 J/cm². This is because more energy is used to heat up the deeper part of the film for the case of the pulse train.

It should be noted that in Fig. 12 the lattice temperatures for the laser train mode irradiation never reach 2000 K, far blow the normal boiling point of gold, 3127 K. This is true for all the calculations in this paper. Before the lattice temperature reaches 3127 K, the whole film will be melted, at which point the calculation will stop.

Another issue to be addressed is the heat loss at surface. In most computational study on the laser metal interaction, the heat loss at boundaries was neglected, which proved to be reasonable earlier [25]. But for the pulse train irradiation, the time scale is much larger in orders. The pulse train with a repetition rate of 1000 Hz has a time scale of 10¹⁰ times larger than a typical 100-femtosecond laser pulse. Under this condition, the radiation which causes heat loss may play a more important role in the process. Two computations were conducted: one with adiabatic boundary condition while the other includes the radiation heat loss at the boundary. The comparison between maximum lattice temperatures shows no obvious difference. But because the radiation at the surface will cause laser energy to escape from the film, the melting depth will be smaller than the results estimated by models ignoring this factor, as shown in Fig. 13. This means neglecting heat loss at surface will lead to an overestimation on melting depth when the repetition rate is low enough.

From the above discussion, the merits of the pulse train can be observed clearly. But to utilize the laser pulse trains, more parameters need to be controlled than does a single pulse irradiation, such as the repetition rate. The typical repetition rate for femtosecond lasers ranges from 100 Hz to tens of MHz. Numerical simulations are carried out for eight frequencies between 500 Hz and 1 MHz and different pulse fluence, and the results are shown in Fig. 14. It is clear that the repetition rate has little effects on the maximum melting depth. This is because even for a high repetition rate as 1 MHz, there is still enough time between each train for the heat to transfer in the thin film and the melting depth will mainly be decided by the amount of energy deposited on the film.

Conclusion

The entire process of laser-metal interaction—including preheating, melting, vaporization, resolidification and thermalization—was thoroughly investigated under the frame works of two-temperature model and interfacial tracking method. Three modes of laser pulses—including single pulse, multiple pulses and pulse train—were used in the simulation to study the effects of various parameters on the maximum lattice temperature, melting depth and ablation depth.

1) Simulations on single pulse irradiation indicate that a higher laser fluence J leads to a higher liquid-vapor interface temperature, velocity and deeper ablation depth. The ablation depth is very small in comparison with the film thickness and melting depth. The effects of laser fluence on the solid-liquid interface are also studied and the results show the same tendency as for liquid-vapor interface. However, the solid-liquid interfacial velocity is much higher.

2) The calculations of films with different thickness show that when the thickness is less than 600 nm, the melting depth increases significantly. With the laser fluence of 0.5 J/cm², the melting depth doubles when the film thickness increases from 600 nm to 500 nm. However, when the film thickness is larger than 800 nm, the film thickness exerts little influence on melting depth.

3) For the two pulses irradiation, if a deeper melting depth is wanted, the second pulse should be launched when the melting depth caused by the first pulse reaches the peak. In comparison to the single pulse irradiation with the same lattice temperature, this will lead to a greater melting depth. If the second pulse is launched at the time when the resolidification process is about to end, two pulses irradiation shows no difference to a single pulse process. For ablation caused by evaporation, if the second pulse which is launched during the surface temperature reaches the peak, the ablation depth will be smaller than a single pulse.

4) For the laser irradiation with more than two consecutive pulses, if the total laser fluence remains a constant, higher pulse number and longer separation times between pulses will lead to a smaller melting depth, but in comparison with a single pulse that causes the same lattice temperature, the melting depth is much deeper. At the same time, the multiple pulses irradiation shows no advantage in ablation over single pulse with the same peak temperature.

5) Compared to the single-pulse irradiation, the laser pulse train showed good performance in achieving deeper melting depth, especially in higher laser power. The temperature rise caused by laser irradiation increases relatively slowly with the increase of total laser energy deposited on the film. The repetition rate has little influence on the process. With the repetition rate ranges of 500 Hz to 1 MHz, the maximum lattice temperature and melting depth showed little change with other parameters kept unchanged.

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