A New Synthetical Model of High-Power Pulsed Laser Ablation

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1 Commun. Theor. Phys. (Beijing, China) 48 (2007) pp c International Academic Publishers Vol. 48, No. 1, July 15, 2007 A New Synthetical Model of High-Power Pulsed Laser Ablation ZHANG Duan-Ming, FANG Ran-Ran, LI Zhi-Hua, GUAN Li, LI Li, TAN Xin-Yu, LIU Dan, LIU Gao-Bin, and HU De-Zhi Department of Physics, Huazhong University of Science and Technology, Wuhan , China (Received July 12, 2006; Revised September 21, 2006) Abstract We develop a new synthetical model of high-power pulsed laser ablation, which considers the dynamic absorptance, vaporization, and plasma shielding. And the corresponding heat conduction equations with the initial and boundary conditions are given. The numerical solutions are obtained under the reasonable technical parameter conditions by taking YBa 2Cu 3O 7 target for example. The space-dependence and time-dependence of temperature in target at a certain laser fluence are presented, then, the transmitted intensity through plasma plume, space-dependence of temperature and ablation rate for different laser fluences are significantly analyzed. As a result, the satisfactorily good agreement between our numerical results and experimental results indicates that the influences of the dynamic absorptance, vaporization, and plasma shielding cannot be neglected. Taking all the three mechanisms above simultaneously into account for the first time, we cause the present model to be more practical. PACS numbers: Mf, Jm, Ds Key words: pulsed laser ablation, dynamic absorptance, vaporization, laser produced plasma 1 Introduction Recently, pulsed laser deposition (PLD) technique has become a well-known method employed to grow highquality thin film materials. [1] During PLD process, pulsed laser ablation (PLA) is an important stage, which determines the quality and the efficiency in making thin film. Therefore, the investigation of the corresponding thermodynamic process of PLA is very significant and the relevant researches are developing very fast. [2] In the past several years, many theoretical models of high power pulsed laser ablation were reported. For instance, Hassan [3] and Zhang [4,5] studied the dynamic absorptance, which is a variable quantity but has been regarded as a constant in most literatures. References [6] [9] investigated the effect of vaporization, which has been ignored by many papers. It should be specially pointed out that the vapor absorption and the forthcoming plasma shielding are greatly important. Other models include work by Mao and Russo, [10] Lunney and Jordan, [11] Zhang and Liu, [12] in which the shielding of plasma was studied. However, we have not found any literature that has considered the influence of all the three mechanisms in PLA process simultaneously. As an ablation process involving several physical mechanisms, the works presented above have got some achievements, but it should be more perfect. In this work, the new synthetical heat conduction equations which consider the dynamic absorptance, vaporization, and plasma shielding are conducted to study the thermal phenomena in laser target interaction using finite difference method. And the corresponding heat conduction equations with the initial and boundary conditions are given. The numerical solutions are obtained under the reasonable technical parameter conditions by taking YBa 2 Cu 3 O 7 target for example. Our numerical results much more extraordinarily agree with the experimental data. This fact indicates that all kinds of physical mechanisms in PLA cannot be ignored under the laser fluences given in this paper. 2 Theoretical Model It is quite a complicated physical process for the interaction of high-power pulsed laser with target, involving laser-surface interaction, and plasma formation. [13] When a high-power laser beam irradiates on a target surface, the target will absorb the laser energy and the vapor is formed. The vapor absorbs part of incident laser energy, and then the atoms are ionized in which the external electrons get free, namely, a plasma comes into being. For better understanding the mechanism of pulsed laser ablation, we should investigate the dynamic absorptance, vaporization, and plasma shielding in detail. 2.1 Target Absorbs Laser Energy and Dynamic Absorptance When laser irradiates target surface, several physical processes are involved including reflection, dispersion, and absorption. The absorptance is a physical parameter that can indicate how much laser energy has been absorbed by target surface. In the literatures about PLA, the absorptance of target surface was generally regarded as a constant. [6,8,9,11,12] However, the absorptance is not a constant actually and its dynamic character affects PLA remarkably. [4] Recently, there are some experiments about The project supported by National Natural Science Foundation of China under Grant Nos and , the Natural Science Foundation of Hubei Province under Grant No. 2001ABB099, and the Sunshine Foundation of Wuhan City under Grant No ranranfang@eyou.com

2 164 ZHANG Duan-Ming, FANG Ran-Ran, LI Zhi-Hua, et al. Vol. 48 the dynamic absorptance, especially about the dependence of the absorptance on the incidence angle and target surface temperature. [14,15] For the angle of incident laser is usually 45 and the dependence of absorptance on the incidence angle is weak, we just present the dependence of absorptance on target surface temperature in this paper. The dependence of the dynamic absorptance on the temperature to be in the form [16] β(t) = A 0 + A 1 T (0, t), (1) where the target temperature is a function of target depth x and time t, that is to say, T = T (x, t). x refers to the spatial coordinate in the direction perpendicular to the target surface, T (0, t) is the surface temperature. A 0 is the absorptance at room temperature and is assumed to be constant but to vary with the material, A 1 is a constant which varies with the material too. The value of A 1 for YBa 2 Cu 3 O 7 is K 1. [17] According to Ref. [3], αi(t)(a 0 + A 1 T (0, t)) t 0 F (t) = 0 + A 1 T (0, t))dt (λt (0, t)) 2 αi(t)(a 0 + A 1 T (0, t)) t 0 I(t)(A 0 + A 1 T (0, t))dt, (2) kt (0, t)f (t) δ(t) = I(t) [A 0 + A 1 T (0, t)], (3) T (0, t) t = αt (0, t) F (t) [F (t) 1] δ(t) 2. (4) Substituting Eqs. (2) and (3) into Eq. (4), the timedependent surface temperature will be given as T (0, t) = Ct + [(Ct) 2 + Dt] 1/2, (5) where δ(t) and I(t) are the heat diffusion distance and the laser intensity reaching the target surface at the time t. C = 1.07αI 0 2 A 0 A 1 /k 2, D = (4/3)(I 0 A 0 ) 2 /k 2, α = k/(ρc). k is the thermal conductivity, c is specific heat, and ρ is the material density. Substituting Eqs. (5) into Eq. (1), the dynamic absorptance can be written in the following explicit form: β(t) = A 0 + A 1 [Ct + (C 2 t 2 + Dt) 1/2 ]. (6) 2.2 Formation of Vapor and Vaporization Mechanisms The process of vaporization will begin due to the increasing of the temperature of target when it continuously absorbs laser energy. However, when the temperature of the target surface is more than the melting temperature, namely T T m, the phenomenon of vaporization will become noticeable. The vaporization flux is governed by the Hertz Knudsen equation, [7] so the velocity of vaporization (or the velocity of surface recession) is u(t) = 0.82 p ( m ) 1/2, (7) ρ 2πk b T (0, t) where p is the gas pressure, m is the particle average mass, k b is the Boltzmann constant. Under the limit V lip V gas, V lip and V gas being the molar volume of liquid and gas, respectively. The vapor pressure above the vaporized surface can be estimated with the Clausius Clapeyron equation: [ L ( 1 p = p b exp 1 )], (8) k b T b T (0, t) where p b is 1 atm, L is the latent heat of the target material, and T b is the boiling temperature under the reference pressure p b. 2.3 Formation of Plasma and Plasma Shielding Absorbing part of the incident laser beam, the vapor becomes significantly ionized and leads to plasma formation. When the density and temperature of plasma are so high, the plasma will rapidly expands outward. When the plasma absorbs the laser energy, there are two different photo-absorption processes, namely inverse bremsstrahlung (IB) and photoionization (PI). The IB process involves the absorption of photos by free electrons, which gain energy from the laser beam during collisions with neutral and ionized atoms, thus promoting vapor ionization and excitation through electron collisions with excited- and ground-state neutrals. The PI process is the direct photo-ionization of excited atoms. In view of the above mentioned two photo-absorption processes, the total absorption coefficient α p of the plasma can be expressed by the Kramers Unsold equation as [18] α p = 4 2πe 6 (Z + 1) 2 ZN 2 0 λ 3 [ ( hc ) ] 3 exp 1, (9) 3hc 4 m 3/2 e (k b T ) 1/2 λk b T where e, Z, N 0, and c are electron charge, the average charge, the plasma particle density, speed of light in vacuum. h and λ are the Planck constant, and wavelength of the pulsed laser. In this paper, λ is 1064 nm. However, when the wavelength of the pulsed laser is not very short, the absorption primarily occurs by the IB process, equation (9) can be simplified as [18] α p = Z(Z + 1) 2 c 2 N 0 2 λ 2 T 3/2. (10) The plasma particle density is in the range of cm 3, the plasma temperatures are of the order of 10 4 K, the average ion charge Z 2. [19] The model has been based on particle density value of cm 3, and the plasma temperature of T = K. The laser intensity reaching the target surface at the time t is I(t) = I 0 (t) exp( α p H), I 0 (t) is the incident laser

3 No. 1 A New Synthetical Model of High-Power Pulsed Laser Ablation 165 intensity, H is the dimension perpendicular to the target of the expanding plasma. The laser energy is highly absorbed if α p H is large. [19] During the duration of the laser pulse, an isothermal temperature within the plasma is assumed. The plasma length H varies with the increasing time as the plasma expands outward. The dynamic equations of plasma expansion are given by [19,20] [ 1 dx X(t) τ dt + d2 X(t) ] [ 1 dy dt 2 = Y (t) τ dt + d2 Y (t) ] dt 2 [ 1 dz = Z(t) τ dt + d2 Z(t) ] dt 2 = kt 0 m, (11) where m corresponds to the particle mass, T 0 is the plasma temperature, τ is the width of laser pulse. dx/dt, dy /dt, and dz/dt refer to the expansion velocities of the plasma edges X, Y, and Z. We have defined X as the dimension perpendicular to the target of the expanding plasma. The reason that the expansion of the plasma results in a similar elongated plasma shape has been analyzed in detail. [13] We only consider the dimension of the expanding plasma perpendicular to the target. The dynamic equation (11) is numerically solved by a finite difference scheme for computing the plasma length. In our calculations, the initial velocity υ is the thermal particle velocity defined by [21] 8kTs υ = πm, (12) where T s is the surface vaporization temperature. 2.4 Ablation Model The target temperature depends on the incident laser parameters, the thermophysical parameters of the target, the boundary, and initial conditions. Usually, the heat conduction equation is used to describe the target temperature fields. In order to describe the temperature fields more accurately, this paper gives the heat conduction equations with the initial and boundary conditions, which base on the dynamic absorptance, vaporization, and plasma shielding. Before the time t th when the target surface begins to vaporize, the target conforms to cρ T t = x k x + α bβ(t)i 0 (t) exp( α b x), (0 < t < t th ). (13) In the time period from t = t th to t = τ, there is ( T ) cρ t u(t) T x = x k T x + α bβ(t)i 0 (t) exp( α p H) exp( α b x), (t th < t < τ), (14) where α b is the absorption coefficient of the target, τ is the width of laser pulse. In order to describe the incident laser intensity more accurately, this paper adopts the laser incident intensity which can be expressed by a Gauss function, namely I 0 (t) = I 0 e (t τ/2)2 /2σ 2, (15) where I 0 (t) is the maximal laser power density. The temporal shape of the laser pulse may be changed with the parameter σ. The initial condition is λ T x λ T T (x, 0) = T 0. (16) And the boundary conditions are = I(x, t) (0 < t < t th ), (17) x=0 x=0 = Lu(t) + I(x, t) (0 < t < t th ), (18) x where T 0 is the initial temperature uniformly across the target. 3 Results and Discussions Under vacuum conditions, YBa 2 Cu 3 O 7 target is irradiated by the fluences range of 1 15 J/cm 2 Gaussian profile laser, of which the wavelength and width are 1064 nm and 13 ns respectively. [22] Under the initial condition (16) and the boundary conditions (17) and (18), equations (14) and (15) are numerically solved by a finite difference scheme. Table 1 is the thermal and optical properties of YBa 2 Cu 3 O 7 from Refs. [22] and [23]. Table 1 Thermal and optical properties of YBa 2Cu 3O 7. Average mass of vaporized species (m) 89 g Density (20C) (ρ) 4.55 g/cm 3 Sunlimation temperature (at 10 5 Pa) Latent heat (L) 2173 K kl/mol Absorptance (A 0 ) 0.85 Specific heat (c) 2.46 J/cm 3 K Thermal conductivity (k) 3.2 W/m K Absorption coefficient (α b ) cm 1 Temperature conductivity (α) cm 2 /s 3.1 The Space- and Time-Dependence of Temperature in Target The space- and time-dependence of temperature in target for the incident laser fluence of 5 J/cm 2 is represented in Fig. 1. We can see from Fig. 1, that at a fixed location, the temperature firstly increases along with the ablation time t, then the temperature decreases when it reaches the maximum. And the maximum temperature appears at about 6.5 ns. This changing tendency is concerned with the tendency of Gaussian profile of pulsed laser. However, the temperature only decreases along with the ablation time t when the pulsed laser of rectangle profile is

4 166 ZHANG Duan-Ming, FANG Ran-Ran, LI Zhi-Hua, et al. Vol. 48 adopted. [8] The evolvement of target temperature in this paper is consistent with the fact. Fig. 1 The space- and time-dependence of temperature in target for the laser fluence of 5 J/cm 2. In Fig. 2, we can find that the less the distance x, the faster the change rate of temperature is. The reason is that when the value of x is smaller, more laser energy will be absorbed by target. Furthermore, we can also see in time period 0 t 8 ns, for the fixed time, the temperature decreases along with the depth. In time period 8 ns t 13 ns (13 ns is the pulse duration), for the certain time, the temperature increases along with the ablation depth, and after it reaches a peak it begins to decrease. This phenomenon is called subsurface heating which will be explained in the following Subsec We can see from Fig. 3 that the solid curve expresses the relative laser energy reaching the target surface without regard for the plasma shielding. The other curves which consider the plasma shielding corresponding to the laser fluences of 2 J/cm 2, 8 J/cm 2, 15 J/cm 2, respectively. In the same period, the higher the incident laser fluence, the little the relative laser energy reaching the target surface is. In other words, the higher the incident laser fluence is, the more noticeable the plasma shielding is. Furthermore, we can find that at the beginning of the ablation, four curves nearly overlap. The reason is that during this period, the relaxation time of vaporization is considered. There is no plasma existing at the surface during this stage, so the transmitted intensity is a constant and the plasma shielding hardly emerges. However, during the time period of t th t τ, the four curves detach. We can analyze the phenomenon as follows: in this time period, there is vapor around the surface, thereafter, the plasma forms. The IB photo-absorption processes shields part of laser energy and forms the plasma shielding. Fig. 3 The relative laser energy reaching the target surface for three different laser fluences. Fig. 2 The space-temperature in target for the laser fluence 5 J/cm Formation of Plasma and Plasma Shielding The higher the incident laser pulse is, the more noticeable the plasma shielding is. We represent the plasma shielding by the relative laser energy that reaches the target surface, namely the ratio of the actual laser energy reaching the target surface to the peak value of the laser energy. The higher the laser fluence and the shorter the relaxation time t th, the more remarkable the plasma shielding is. For the high laser fluence, the plasma shielding happens in the prior time, the slope coefficient of the curve diminishes (the increment rate of relative laser energy that reaches the target surface is decreasing). To sum up, the higher the laser fluence, the more remarkable the plasma shielding, so the less the transmitted relative laser energy is. 3.3 Subsurface Heating We have seen in Fig. 1, in the time period 8 ns t 13 ns, for the certain time, the temperature increases then it decreases along with the ablation depth, namely, the subsurface heating phenomenon which is the most remarkable when the pulsed laser finishes.

5 No. 1 A New Synthetical Model of High-Power Pulsed Laser Ablation 167 Figure 4 shows the temperature profiles within the target at the time when the laser ends. The values of the subsurface heating, defined as T = T max T s (T max being the maximum temperature in spatial distribution). We can see from Fig. 4 that the values are K, 4000 K, and 1000 K for different laser fluences of 15 J/cm 2, 8 J/cm 2, and 2 J/cm 2, respectively. In other words, the higher the laser fluence, the higher the value of the subsurface heating is. Fig. 4 The calculated temperature profiles within the target at the time when the laser ends. The temperature within the target is determined by three factors: the heating by absorbing the laser energy, the surface cooling due to the vaporization, and the inner cooling by the heat diffusion. The absorption of the laser energy can be governed by the Beer Lambert law I(x, t) = β(t)i(t) e α bx. The smaller the absorption coefficient α b, the greater depth the laser energy penetrates. The absorption coefficient α b of YBa 2 Cu 3 O 7 is very small (Table 1), so the depth by the laser energy penetrated is greater. Commonly, the rate of temperature equalization along the distance with time controlled by the temperature conductivity α = k/ρc. The α value for YBa 2 Cu 3 O 7 is 0.011, so the heat is conducted to the bulk depth slowly. As for the vaporization process, the rate of the vaporization is so high, so much more energy is took away. Therefore, the three factors result in a common tendency for YBa 2 Cu 3 O 7 : although the target surface absorbing the energy of the laser, the ascend trendy of the surface temperature is slowly due to the forceful vaporization; no vaporization phenomena happen in the subsurface, and the rate of the temperature equalization along the distance with time is very slowly, so the temperature ascends quickly and the subsurface temperature is higher than the surface. The higher the laser fluence, the more remarkable the vaporization and the more energy accumulated in the subsurface, the higher the value of the subsurface heating is. Certainly, the higher the laser fluence, the more energy is absorbed by the target, the temperature within the target ascends quickly. We can see from Fig. 4, the higher the laser fluence, the higher the curve locates. 3.4 Mass Removal Per Pulse as a Function of Laser Fluence In Fig. 5, the black dot and the black triangle express the experimental data [22] have been obtained with different ablation spot diameters: 0.5 mm and 0.6 mm. Curve 1 is the numerical result, which is given by the new synthetical model in this paper. Curve 2 does not consider the dynamic absorptance. Curve 3 does not consider the plasma shielding and the dynamic absorptance. We can see from Fig. 5 that the three curves all ascend with the laser fluence, however, the trends of ascending are different. The trends of curve 1 and curve 2 are identical, the slope coefficient of the curves is becoming small and small, while the slope coefficient of curve 3 remains a constant. We can analyze the phenomenon as follows. For the same laser fluence, curve 3 locates the highest position because it does not consider the roles of plasma shielding and the dynamic absorptance. Curve 2 locates above curve 1 because curve 2 does not consider the dynamic absorptance. There is evidence that for the certain laser fluence and the oxide superconductor, the dynamic absorptance results in the decreasing of absorptance. Fig. 5 fluence. Mass removal per pulse as a function of laser Among the three curves, curve 1 of our numerical results is in better agreement with the experimental data. Curve 2 does not consider the dynamic absorptance, the difference between the experimental data and curve 2 is small. This phenomenon indicates that the influence of the dynamic absorptance is small. The difference between the experimental data and curve 3 is apparent because the dynamic absorptance or the plasma shielding is not considered. This phenomenon implies that the influence of the plasma shielding is large. The influence of the plasma shielding is much more notable than the influence of dynamic absorptance.

6 168 ZHANG Duan-Ming, FANG Ran-Ran, LI Zhi-Hua, et al. Vol. 48 Figure 5 indicates that the ablation rate given by the theoretical model in this paper is in agreement with the experimental data. The new synthetical model was a much more satisfactory theoretical framework in the nanosecond high-power laser ablation. 4 Conclusions (i) A new synthetical model of high-power pulsed laser ablation has been developed in this paper, which considers the dynamic absorptance, vaporization, and plasma shielding. And the heat-flow equations with the initial and boundary conditions are all given. (ii) Along with the increasing of the laser fluence, especially after the relaxation time, the plasma shielding happens, so the relative laser energy which reaches the target surface becomes little and little. (iii) The distinct subsurface heating phenomenon happens because the phenomenon of vaporization at the target surface is obvious, the absorption coefficient α b and the temperature conductivity α of YBa 2 Cu 3 O 7 are small. (iv) The satisfactorily good agreement between our numerical results and experimental results confirms that the influences of the dynamic absorptance, vaporization, and plasma shielding cannot be neglected. References [1] D.B. Chrisey and G.K. Hubler, Pulsed Laser Deposition of Thin Films, Wiley, New York (1994). [2] J.C. Miller and R.F. Haglund, Laser Ablation and Deposition, Academic Press, San Diego (1998). [3] A.F. Hassan, M.M. El-nicklawy, et al., Optics and Laser Technology 25 (1993) 155. [4] Duan-Ming Zhang, Li Li, et al., Acta Phys. Sin. 54 (2005) 1283 (in Chinese). [5] Li Li, Duan-Ming Zhang, et al., Phys. Stat. Sol(a) 203 (2006) 906. [6] P. Andrea and M. Antonio, Phys. Rev. E 50 (1994) [7] Quan-Ming Lu, S. Mao, Xiang-Lei Mao, Appl. Phys. Lett. 80 (2002) 3072 [8] Zhang Duan-Ming, Tan Xin-Yu, et al., Physica B 357 (2005) 348. [9] Xin-Yu Tan and Duan-Ming Zhang, Physica B 358 (2005) 86. [10] X. Mao and R.E. Russo, Appl. Phys. A 64 (1997) 1. [11] J.G. Lunney and R. Jordan, Appl. Surf. Sci. 941 (1998) 127. [12] Duan-Ming Zhang, Dan Liu, et al., Physica B 362 (2005) 82. [13] D.M. Zhang, L. Guan, and Z.H. Li, Chin. Phys. Soc. 52 (2003) 242. [14] Y.L. Huang, F.H. Yang, et al., Chin. J. of Laser 30 (2003) 449. [15] L.B. Yang, X.F. Liu, et al., High Power Laser and Particle Beams 6 (1994) 99. [16] M. Sparks and E. Loh, J. Opt. Soc. Am. 69 (1979) 847. [17] CRC Handbook of Chemistry and Physics, ed. David R. Lide, CRC Press, Boca Raton, FL (1992). [18] B. Zel dovich Ya and P. Raizer Yu, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, Academic Press, New York (1966) Chap. 5. [19] R.K. Singh and J. Narayan, Phys. Rev. B 41 (1990) [20] Duan-Ming Zhang, Zhi-Hua Li, et al., Sci. in China (Series A) 44 (2001) [21] F. Garrelie, J. Aubreton, and A. Catherinot, J. Appl. Phys. 83 (1998) [22] N.M. Bulgakova and A.V. Bulgakov, Appl. Phys. A. 73 (2001) 199. [23] Deepika Bhattacharya, R.K. Singh, and P.H. Holloway, J. Appl. Phys. 70 (1991) 5433.