Finite element simulation of pulsed laser ablation of titanium carbide

Applied Surface Science 253 (2007) 7810–7814 www.elsevier.com/locate/apsusc

Finite element simulation of pulsed laser ablation of titanium carbide V. Oliveira *, R. Vilar Departamento de Engenharia de Materiais, Instituto Superior Te´cnico, Av. Rovisco Pais no. 1, 1049-001 Lisboa, Portugal Available online 27 February 2007

Abstract In the present paper, a 2D finite element model based on the heat-conduction equation and on the Hertz-Knudsen equation for vaporization was developed and used to simulate the ablation of TiC by Nd:YAG and KrF pulsed laser radiation. The calculations were performed for fluences of 8 and 10 J/cm2, which according to experimental results obtained previously, correspond to large increases of the ablation rate. The calculated maximum surface temperature of the target for both lasers is higher than the estimated value of TiC critical temperature, corroborating the hypothesis that the increase of the ablation rate is explained by the explosive boiling mechanism. # 2007 Elsevier B.V. All rights reserved. PACS : 61.80.Ba; 81.05.Je; 81.15.Fg Keywords: Laser ablation; Finite element simulation; Titanium carbide

1. Introduction Titanium carbide (TiC) is a transition metal carbide widely used as a coating material because of its unique combination of physical and chemical properties, such as high melting point, high hardness and high wear resistance [1]. Several techniques have been used to deposit TiC thin films, including chemical vapor deposition (CVD) [2], plasma-enhanced CVD [3], laser CVD [4], ion-beam assisted deposition (IBAD) [5], plasma spraying [6], magnetron sputtering [7] and pulsed laser deposition (PLD) [8–14]. Among these methods, the latter process is particularly promising because of its simplicity, flexibility and capability to preserve the stoichiometry of the material [15]. To optimize the quality of TiC films deposited by PLD, it is of fundamental importance to understand the ablation mechanisms involved. For instance, D’Alessio and co-workers [10,11] reported a non-linear variation of the ablation rate of TiC as a function of the radiation fluence, using a frequency doubled NdYAG laser (l = 532 nm). In the fluence range between the ablation threshold (0.5 J/cm2) and 8 J/cm2 the ablation rate initially increases with fluence, then becomes approximately constant, as expected from a plasma mediated vaporization

* Corresponding author. Tel.: +351 218418137; fax: +351 218418120. E-mail address: [email protected] (V. Oliveira). 0169-4332/$ – see front matter # 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.apsusc.2007.02.101

process [16]. Above 8 J/cm2, the ablation rate increases drastically and the deposited films become covered by numerous droplets of resolidified material. A similar behaviour was observed more recently by Oliveira et al. [17] in a study of the pulsed laser ablation of TiC using a KrF excimer laser (l = 248 nm) but the threshold for increasing ablation rate was 10 J/cm2. Similar variations of the ablation rate were also observed in the PLD of other materials, and explained by the occurrence of explosive boiling (or phase explosion) [18–20]. According to Martynyuk, Kelly and Miotello [21–23], explosive boiling is the primary material ablation mechanism for sufficiently high fluences and short pulse durations. In these conditions, the target will reach temperatures near the thermodynamic critical temperature of the material. At this point, the rate of homogeneous bubble nucleation rises catastrophically and the target makes a transition from superheated liquid to an equilibrium mixture of vapor and liquid droplets. In the present paper, a 2D finite element model (FEM) was developed and applied to the simulation of the pulsed laser ablation of TiC using Nd:YAG (l = 532 nm) and KrF (l = 248 nm) laser radiation. The calculations were performed for fluences of 8 and 10 J/cm2, and the possibility of explosive boiling was investigated. The results achieved suggest that explosive boiling is probably the mechanism responsible for the large increase of the ablation rate observed experimentally at those fluences.

V. Oliveira, R. Vilar / Applied Surface Science 253 (2007) 7810–7814

leading to the ablation rate v given by:

2. Numerical simulation 2.1. Background

vðTÞ ¼ ð1  bÞ

In the absence of convective and radiative energy transport, the temperature distribution T induced by absorption of laser radiation in a material is given by the heat conduction equation, which, in 2D, can be written as [24]: rðTÞCp ðTÞ

@Tðx; y; tÞ ¼ r½kðTÞrTðx; y; tÞ þ Qðy; tÞ @t

(1)

where x and y are the space coordinates and r, Cp, k the mass density, specific heat at constant pressure and thermal conductivity of the target material. The source term Q(y, t) represents the laser energy absorbed by the sample and is expressed as: Qðy; tÞ ¼ I s ð1  RÞa expðayÞ

(2)

where R and a are the reflectivity and the absorption coefficient of the target material, y the spatial coordinate in the direction normal to the sample surface and Is is the temporal laser irradiance at the sample surface. According to Bulgakov and Bulgakova [25], Is is given by: I s ðtÞ ¼ IðtÞ exp½LðtÞ

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(3)

where L is the optical thickness of the ablation plume, is given by L(t) = ah(t) + bEa(t). In this expression, a and b are timeindependent coefficients, h the ablation depth and Ea is the density of the radiation absorbed by the plasma. The laser beam temporal profile I(t) is supposed to be Gaussian with full width at half maximum t. For materials with a metallic electronic structure, such as TiC [26], ablation occurs essentially by thermal ablation processes [24]. According to Kelly and Miotello [22,23], there are three regimes of thermal ablation: vaporization, heterogeneous boiling and explosive boiling (or phase explosion). Only vaporization and explosive boiling are compatible with the time scale of nanosecond pulse duration laser. Since explosive boiling only occurs when the target reaches temperatures near the thermodynamic critical values of the material [21–23], the ablation mechanism assumed in the model is vaporization. The flow of material vaporized from the surface follows the Hertz-Knudsen equation,

   rffiffiffiffiffiffiffiffiffiffiffiffiffiffi m p0 LV 1 1 exp  2pkB T r kB T B T

(4)

where TB is the boiling temperature at the pressure p0, kB the Boltzmann constant, b the back flux coefficient and LV is the latent heat of vaporization of the material. In Eq. (4), it is assumed that no vapor is present in the atmosphere and that the vapor pressure above the vaporized material is given by the Clausius-Clapeyron equation. 2.2. Finite element model (FEM) To simulate the ablation of TiC, a finite element numerical model was developed using the commercial package ANSYS [27]. The target is represented by a mesh of finite elements that changes over time so as to simulate materials removal. This time-dependent problem was solved sequentially, as a series of 1 ns time steps, linked together by using the output of step n as the initial conditions for problem n + 1. The target initial temperature is 298 K. If the temperature of an element is higher than the melting temperature (Tm) at the end of a particular step, melting is assumed to have occurred and the latent heat of melting (Lm) is taken into account in the calculation. Similarly, ablation is assumed to occur when the temperature of the surface elements is higher than the boiling temperature. If this happens an ablation depth h is calculated from Eq. (4) and compared with the element thickness, Dy. If h > Dy the element is supposed to be vaporized and the surface temperature of the remaining material corrected to take into account the latent heat of vaporization. The geometry used for the simulations is illustrated in Fig. 1. To minimize the computer processing time the target is supposed rectangular with 10 mm  3 mm and only half of the target is simulated because of the axial symmetry of the problem. Moreover, only half of the simulated region is irradiated in order to account for the lateral heat losses to the non-irradiated part of the sample (Fig. 1). The size of the elements is 2 nm  10 nm in the upper half of the substrate and 10 nm  10 nm in its lower half. The use of a thinner mesh in the upper part of the

Table 1 TiC parameters [1,28,29] Tm (K)

Tb (K)

r (kg/m3)

Lm (J/kg)

LV (J/kg)

b

3340

5080

4910

10 6

10 7

0.18

Table 2 Temperature-dependent parameters [1]

Fig. 1. Geometry for laser ablation of TiC.

T (K)

300

400

800

1200

1600

2000

2400

>2400

K [W/(m K)] Cp [J/(kg K)]

23 550

24 650

31 830

36 890

40 900

43 915

45 930

45 930

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Table 3 Lasers characteristics, optical properties, ablation rates and plasma parameters Laser

l (nm)

t (ns)

R

a (m1)

v ðmm=pulseÞ

a (m1)

b (m2)

Nd:YAG KrF

532 248

10 30

0.46 0.43

5.9  10 7 1.1  10 8

0.16 0.14

2  10 6 4  10 5

7  104 5  104

substrate allows a more precise estimation of the material removal. 2.3. TiC properties The values of the TiC properties used in the calculation are presented in Tables 1–3. The reflectivity (R) and absorption coefficient (a) were calculated from the real and imaginary part of the refraction index n and k taken from ref. [30], using the equations: a¼

4pk ; l

R¼

ðn  1Þ2 þ k2 ðn þ 1Þ2 þ k2

(6)

where l is the radiation wavelength. Since the latent heat of vaporization (LV) of TiC is not known by the authors, it was estimated from the values reported for metals with similar boiling temperature [31]. Finally, the plasma parameters a and b were adjusted to have coincidence between the experimental ablation depth reported in refs. [10,17] and the calculated ablation depth.

3. Results The temperature distributions in the target generated by a pulse of Nd:YAG laser radiation 10, 20, 30 and 60 ns after the beginning of the laser pulse are depicted in Fig. 2. After 10 ns the target surface temperature (1500 K) is lower than the boiling temperature of TiC (5080 K), so no significant removal of material occurs up to this moment. On the contrary, after 20 ns (Fig. 2b) ablation has begun and a crater about 0.13 mm deep is predicted. The surface temperature reaches its maximum value (about 14,000 K, Fig. 2b) a temperature much higher than the TiC boiling temperature. The maximum ablation depth (about 0.16 mm) is reached after 30 ns (Fig. 2c), but the surface temperature decreased to 5500 K. Finally, after 60 ns (Fig. 2d) temperature continues to decrease, mainly because of heat conduction to the bulk. The simulation results of TiC ablation using a KrF laser for 40, 55, 70 and 120 ns after the beginning of the laser pulse are presented in Fig. 3. For this type of laser, ablation starts about 40 ns after the beginning of the laser pulse (Fig. 3a), later than the Nd:YAG laser. The maximum surface temperature (about

Fig. 2. Temperature distribution as a function of time for a TiC target processed with Nd:YAG laser radiation (l = 532 nm) at 8 J/cm2: (a) 10 ns, (b) 20 ns, (c) 30 ns and (d) 60 ns. t = 10 ns.

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Fig. 3. Temperature distribution as a function of time for a TiC target processed with KrF laser radiation (l = 248 nm) at 10 J/cm2: (a) 40 ns, (b) 55 ns, (c) 70 ns and (d) 100 ns. t = 30 ns.

12,000 K) is reached after 55 ns (Fig. 3b) and ablation ends after 70 ns (Fig. 3c). The ablation depth is about 0.14 mm (Fig. 3d). 4. Discussion Since KrF laser pulses are longer than Nd:YAG laser pulses (t = 30 and 10 ns, respectively), for similar pulse energy the average power density of the second laser is about three times higher as compared to the first one. However, the ablation rate and the maximum surface temperature obtained with both lasers are similar. This is explained by the higher fluence and radiation absorption coefficient of the excimer laser, which balances its longer pulses. On the other hand, since the model takes into consideration two-dimensional heat conduction to the bulk, the isotherms that are closer to the surface are not straight lines but appear curved near the limit of the irradiated area, and the crater border is not step-like but rather a ramp. To discuss the hypothesis of explosive boiling being responsible for the large increase of the ablation rate observed experimentally at some threshold fluence, we must first estimate the critical temperature of TiC. Since, its value could not be found in the literature, an empirical equation based on the latent-heat of vaporization proposed by Martynyuk for metallic materials [32] was used. The critical temperature of TiC estimated from the equation is about 10,000–10,500 K. Since this value is of the same order of magnitude comparing to the maximum surface temperature estimated for both type of lasers, explosive boiling can be an explanation of the variation of the ablation rates observed experimentally at these fluences.

5. Conclusions A 2D finite element model was developed to simulate the PLA of TiC targets by Nd:YAG (l = 532 nm, t = 10 ns) and KrF (l = 248 nm, t = 30 ns) laser radiation at 8 and 10 J/cm2, respectively. The calculations were performed for fluences, which correspond to large increases of the ablation rate observed experimentally. The maximum surface temperature of the target is found to be of the order of magnitude of the critical temperature of TiC for both types of lasers, corroborating the hypothesis that explosive boiling is a plausible explanation for the variation of the ablation rate observed experimentally at these fluences. Since explosive boiling is normally accompanied by the intensive ejection of numerous droplets, this mechanism should be avoided in order to obtained thin films of good quality. Acknowledgements V. Oliveira acknowledges financial support from the Portuguese Fundac¸a˜o para a Cieˆncia e Tecnologia (FCT). The authors also would like to acknowledge R. Teghil for providing valuable information on the laser ablation of TiC by Nd:YAG laser. References [1] H.O. Pierson, Handbook of Refractory Carbides and Nitrides, Noyes Publications, New Jersey, 1996.

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