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Effect of ionization on laser-induced plume self-similar expansion
Applied Surface Science 255 (2009) 4595–4599
Contents lists available at ScienceDirect
Applied Surface Science journal homepage: www.elsevier.com/locate/apsusc
Effect of ionization on laser-induced plume self-similar expansion D. Bennaceur-Doumaz *, M. Djebli 1 Centre de De´veloppement des Technologies Avance´es, B.P. 17 Baba Hassen, 16303 Algiers, Algeria
A R T I C L E I N F O
A B S T R A C T
Article history: Received 7 September 2008 Received in revised form 13 November 2008 Accepted 28 November 2008 Available online 7 December 2008
The dynamics of a laser ablation plume during the first stage of its expansion, just after the termination of the laser pulse is modelled. The one-dimensional expansion of the evaporated material, considered as an ideal fluid, is governed by one-fluid Euler equations. For high energetic ions, the charge separation can be neglected and the hydrodynamics equations solved using self-similar formulation. Numerical solution is obtained, first when the laser fluence range is low enough to deal with a neutral vapor, and in a second stage, when ionization effects on the expansion are taken into account, for different material targets. As a main result, we found that the presence of ions in the evaporated gas enhances the self-similar expansion. ß 2008 Elsevier B.V. All rights reserved.
PACS: 52.30.q 52.38.Mf Keywords: Plume expansion Laser ablation Self-similar
1. Introduction Pulsed laser deposition (PLD) has been found to be a successful and a promising technique for the growth of thin films [1]. In order to understand basic physical principle of this technique, laser ablation process can be divided into three stages: (i) interaction of the laser beam with the target materials leading to the evaporation of the surface layers, (ii) interaction of the evaporated cloud with the incident laser beam resulting in isothermal plasma formation and expansion, and (iii) adiabatic expansion of the plasma and subsequent deposition. The first two stages start with the laser pulse and continue through laser pulse duration. The third regime starts after the termination of the laser pulse. For short laser pulses, a reasonable assumption is that the last stage can be considered separately. To optimize the thin film quality in the PLD process, an understanding of the plume expansion dynamics in this stage and its dependence on the initial conditions is very important [2–4]. Thus much work has been done to understand the many complex physical processes in the ablation plume dynamics and to reproduce the temporal and spatial evolution of the profiles expansion. To this end, various approaches were used by authors to
* Corresponding author at: Division Milieux Ionise´s et Lasers, Centre de De´veloppement des Technologies Avance´es, B.P. 17 Baba Hassen, 16303 Algiers, Algeria. E-mail addresses: [email protected], [email protected] (D. Bennaceur-Doumaz), [email protected] (M. Djebli). 1 Permanent address: Theoretical Phys. Lab., Faculty of Physics, USTHB, B.P. 32 Bab Ezzouar, 16079 Algiers, Algeria. 0169-4332/$ – see front matter ß 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.apsusc.2008.11.089
model laser ablation plumes, including numerical ones such as Monte Carlo [5], Particle In Cell [6], gas dynamics [7] or Molecular Dynamics [8] which are useful for real geometry and gas flows with considerable deviations from local thermal equilibrium (LTE). Analytical approaches, in the other hand, are based on the solutions of the gas dynamics equations, using self-similar theory, where partial differential equations are transformed to ordinary ones. This transformation has been largely used to investigate plasma expansion and given new results [9,10]. When evaporation of the target as well as plasma formation cease, there is no additional input energy while most of the thermal energy of the species is converted into kinetic energy. The generated plume can be described by an adiabatic, self-similar expansion. Anisimov et al. studied plasma expansion into vacuum after the end laser pulse, using a three-dimensional model based on a special solution of gas dynamics equations. The model is semi-analytical and plasma profiles were obtained by numerically solving simple differential equations [11,12]. Singh et al. developed a self-similar model (S-N model) of plasma flow which began with an assumed self-similar profile of plasma velocity and considered the plasma velocity near the target surface irradiated by laser as zero. Based on the conservations of mass and momentum, they derived a set of plasma dynamic equations. This model can explain the plasma expansion preferentially normal to the target surface quantitatively [13]. However, the assumption of velocity is not consistent with the experimental results, that is, the velocity of the plasma near the target surface is not zero, but large. So Chen et al. developed a one-dimensional model of an accelerated expansion and assumed a linear profile of plume velocity [2]. Later, Stapleton
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et al. proposed a numerical model which includes the gas dynamics and atomic kinetics of a laser ablation plume, expanding adiabatically in vacuum, using isothermal and isentropic selfsimilar analytical solutions and steady-state collisional radiative equations [14]. The existing self-similar models describing the behavior of expanding laser produced plasma plume are strictly applied to the expansion of neutral gases. However, it has been shown that ionization and recombination processes play an important role in film preparation through their influence on plume expansion and characteristics of deposited particles, and they have to be considered for accurate description of laser ablation dynamics. Indeed, experiments [15] showed the ionization degree of plasma affects significantly the plasma velocity, but little attention was paid on modeling the ionization effect to plasma kinetic behavior generated by laser ablation in previous works. This effect was studied recently by Tan et al. [16,17] where they showed, by comparing between the velocity when there is ionization and the case when it is neglected, that ionization increases the plume expansion velocity. In the present work, we deal with a theoretical investigation, limited to very small temporal and spatial domains just after the ablation pulse, valid in vacuum or in the early stages of the plume expansion in presence of gas background. Our objective is to improve the self-similar existing models [11,13,18] in the PLD expansion process by solving numerically a set of time-dependant ordinary differential equations, deduced by a self-similar model taking into account the ionization of the plume. The nature of material target is also investigated to find its influence on the expansion profiles. 2. Description of the model During the initial stage of plasma expansion, the particle density is sufficiently high and the mean free path of the particles is short (mm) allowing the plasma to behave as a continuum fluid. Many collisions take place between the various particles so the plasma can be considered in local thermal equilibrium. This means that in a sufficiently small region, the electrons, ions and neutrals can be characterized with a common temperature [19]. The existence of LTE requires that the electron-atom and electron-ion collisional processes occur sufficiently fast and dominate radiative processes. In such a system, particles will have Maxwellian velocity distributions and population levels will be distributed according to Boltzmann’s statistics. Initially, for a time where the distance of the laser-induced plasma front from the target surface is smaller than the laser spot dimension, the propagation of plasma can be considered onedimensional and strictly directed perpendicularly to the target surface. It is a reasonable assumption for the early stage of expansion, until an expansion distance of about 1 mm and 100 ns duration [20]. We have used a single fluid approach in which we follow the expansion of a fluid of electrons, ions and neutrals just after the termination of the laser pulse. The expansion is mainly conducted by the pressure rather than the electrostatic potential which may rise from the charge separation. Thus, quasi-neutrality assumption during the whole expansion can be justified for such sufficiently dense plasma [21]. The plasma plume considered as a non-dissipative and nonheat-conductive fluid, is described by Euler equations:
@E @E @v þv þP ¼ 0; @t @x @x
(3)
where Eq. (1) stands for density conservation, n and v are respectively, the density and fluid velocity. The momentum and energy conservations are governed by Eqs. (2) and (3), respectively, where M is the mass of the plume particles, E ¼ Mnðe þ v2 =2Þ is the plume energy density, e ¼ ð1 þ hÞðT=MÞ=ðg 1Þ þ hU i =M is the enthalpy, U i is the ionization energy, h is the ionization fraction of the plume, g is the specific heat ratio of the plume and T is the temperature in eV. Eqs. (1–3) are closed by an ideal gas equation of state such that the plume pressure is given by P ¼ nð1 þ hÞT. For sake of simplicity, we assume to have only singly ionized species, so the following Saha equation estimates the ionization fraction [2]:
h2 1h
¼
2 uþ 2pme T 3=2 exp ðU i =TÞ n u0 h
(4)
uþ and u0 are, respectively, the partition functions of ionized atoms and neutral atoms, me is the electron mass, h is Plank’s constant. 3. Self-similar theory and numerical calculation In general, hydrodynamic equations describing the expansion are difficult to solve numerically, but under certain assumptions, these partial differential equations can be reduced to ordinary differential equations which greatly simplifies the problem. This transformation is based on the assumption that we have a selfsimilar solution, i.e., every physical parameter distribution preserves its shape during expansion and there is no scaling parameter [9,22]. Self-similar solutions usually describe the asymptotic behavior of an unbounded problem and the time t and the space coordinate x appear only in the combination of ðx=tÞ. It means that the existence of self-similar variables implies the lack of characteristic lengths and times. Indeed, this is justified when we deal with the early stage of expansion where charge quasineutrality in the dense plasma is imposed at the beginning of the process. The Debye length then, loses its importance as a characteristic length in the plasma. The self-similar solution of Eqs. (1–4) can be constructed for a quasi-neutral plasma by using the ansatz defined as, nvt v T U N˜ ¼ ; V˜ ¼ ; T˜ ¼ ; U˜ i ¼ i2 : n0 cs Mcs2 Mcs pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where cs is the sound velocity given by cs ¼ g T v =M, T v is the vaporptemperature ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of the target and v, the plasma frequency, v2 ¼ 4pn0 e2 =M, n0 is the initial density of the target. U i is the first ionization potential of the target material. For adiabatic expansion and mono-atomic gases, g ¼ 5=3. We have chosen to write the differential equations in terms of a single dimensionless similarity variable j ¼ x=cs t, then partial derivatives become total derivatives with respect to j. We have
@ 1 d ¼ ; @x cs t dj
@ j d ¼ @t t dj
then, we deduce a set of ordinary differential equations given by
@N˜ ˜ @V˜ ˜ þN N¼0 @j @j
@n @ðnvÞ þ ¼ 0; @t @x
(1)
ðV˜ jÞ
@v @v 1 @P þv þ ¼ 0; @t @x Mn @x
(2)
˜ ˜ ˜ ˜ þ hÞ @T þ N˜ T˜ @h ¼ 0 ˜ þ hÞ @N þ ðV˜ jÞ @V þ Nð1 Tð1
@j
@j
(5)
@j
@j
(6)
D. Bennaceur-Doumaz, M. Djebli / Applied Surface Science 255 (2009) 4595–4599
˜ ˜ ˜ ˜ þ hÞ dV þ 3 ðV˜ jÞð1 þ hÞ dT þ ðV˜ jÞT˜ 3 þ Ui Tð1 2 T˜ dj 2 dj
T˜
dN˜ 3 U˜ i þ þ N˜ 2 T˜ dj
!
dT˜ 2 h ˜ ˜ dh N˜ T˜ ¼0 NT dj j dj hð1 hÞ
!
dh ¼0 dj
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(7)
(8)
The self-similar solution is valid as long as the local Debye length is much smaller than the scale length of the self-similar solution cs t. The set of Eqs. (5–8) is solved numerically, using Runge–Kutta method with appropriate initial conditions, for two cases. The first one concerns an expansion of a neutral gas while in the second case the energy is sufficiently high to produce singly ionized atoms. 3.1. Expansion of a neutral vapor The model is used to simulate the ablation of a titanium target by a 25 ns pulse of a KrF excimer with a 248 nm wavelength. At low irradiance, ionization is completely negligible. It is interesting to study the plume dynamics in which the plume consists only of a neutral gas. In this case, h ¼ 0, which is a good approximation for potential ionization U i T v . The initial conditions of the plume expansion in our case, correspond to experimental considerations, at the end of the laser pulse. Investigation is conducted for Ti target of initial density of about 1019 cm3, the neutral atoms are assumed to have an average initial velocity close to the sound velocity and their initial temperature is taken to be the vapor temperature T v ¼3562 K. Knudsen layer effects in front of the target are not considered in this work. Our initial conditions correspond to a certain time and distance from the surface where the self-similar expansion may start. Since the source is localized at the surface, we expect that after a certain time (dt, set to be t ¼ 0) and a transition distance dx near the surface, the system may develop to be self-similar as in free expansion. Furthermore, we use a full numerical solution instead of a semi-analytical model as in Ref. [2] where the selfsimilar theory begins with an assumed linear profile of plume velocity, given by v ¼ vm fa þ ð1 aÞjg, where vm is the maximum expansion velocity, a is a constant, and j ¼ x=vm t, where the expansion ends at j ¼ 1. In other words, the objective of our study is to look for the numerical value of j beyond which the self-similar expansion is no longer valid. Fig. 1 shows that both temperature and density of the plume are decreasing monotonically while the velocity increases almost linearly during the expansion process. This behavior is due to the adiabatic expansion where thermal energy is converted into directed kinetic energy. In this stage, the expansion of the vapor plume is similar to that in vacuum where the plasma expands in a way similar to a supersonic expansion with a free linear behavior. All the expansion profiles of density, velocity and temperature, reach their limit at a self-similar variable value (jl ¼ 1:12) beyond which there is no self-similar solution. We note also for j < j1 ¼ 0:6, we have almost a linear profile as suggested by the free expansion analytical model [23]. For j > j1 , numerical solution exhibits a nonlinear behavior, in particular for the velocity profile which may result from the increase of pressure gradient far away from the source and density depletion. For j > jl , the conditions of LTE may not be valid, this is the limit of the model given in this paper.
Fig. 1. Velocity (dashed line) v=vð0Þ, density n=nð0Þ (solid line) and temperature T=Tð0Þ (dashed dotted line) normalized to their initial values as function of the similarity variable.
and characteristics of deposited particles, and they have to be considered for accurate description of laser ablation dynamics. In the present work, we have studied the influence of this effect on the expansion profiles by introducing Saha equation in the dynamical equations. The results of calculations show from Fig. 2 that the maximum expansion velocity of the ionized plasma is increasing with the initial ionization fraction. The velocity reaches approximately 5.5 times the sound velocity (solid line), while it is 4.5 when h0 ¼ 0:50. During the propagation, the thermal energy of the electrons and ions is converted to kinetic energy which is mainly carried by the heavy ions. The electrons transfer their energy to the ions by coulombian diffusion processes. The ions located at the front of the plasma acquire the largest energy during hydrodynamic acceleration. This small group of high-energy ions transports a significant
3.2. Ionization effects on the expansion profiles Ionization and recombination processes play an important role in film preparation through their influence on plume expansion
Fig. 2. Velocities normalized to their initial values, for h0 ¼ 0:50 (dashed curve), h0 ¼ 0:95 (solid curve) and h0 ¼ 0 (dashed dotted curve) as function of the similarity variable.
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Fig. 4. Temperatures normalized to their initial values, for h0 ¼ 0 (dashed dotted curve), h0 ¼ 0:25 (dashed curve) and h0 ¼ 0:95 (solid curve) as function of the similarity variable. Fig. 3. Densities normalized to their initial values, for h0 ¼ 0 (dashed dotted curve), h0 ¼ 0:95 (solid curve) and h0 ¼ 0:50 (dashed curve) as function of the similarity variable.
fraction of absorbed energy. For a plasma containing a certain amount of neutrals, energy is transferred from ions to neutrals by collisions. The ions are then decelerated and as a consequence the maximum velocity is reduced in comparison with the case where the gas is fully ionized. This effect is shown in Fig. 2(dashed curve) where the ionization fraction is 0.50. We have noticed also that the self-similar expansion increases with ionization. This may result from the acceleration of the lighter particles which leave first the bulk plume, the other species are then attracted to keep the quasi-neutrality assumption [24]. It is worth to say that the expanding gas goes beyond the limit corresponding to a neutral gas, (jl ¼ 1:12), we have jl ¼ 2:45 for h0 ¼ 0:50 and jl ¼ 3:7 for h0 ¼ 0:95. In Fig. 3 we have plotted the density versus the similarity variable for different initial ionization degrees. We recall the density of Fig. 1 which is also seen in Fig. 3(dotted dashed line) for h0 ¼ 0 but with another scaling. In this case, we see that the expansion exhibits two behaviors. The energy emitted from the target by radiation is less than the ionization potential then it is absorbed by the expanding gas. At the first stage, where the gas is sufficiently close to the heated target, the density increases as a result of thermal pressure. This can also be seen from Fig. 1(solid line) for j < 0:6. Far away from the source (j > 0:6), we found the well known free expansion of decreasing density. This is not true when the ionization is taken into consideration. The density decreases when j increases and the excess energy is used to ionize the neutrals. For a large initial ionization fraction (h0 ¼ 0:95) we deal with the expansion of a fully ionized gas. The expansion of an ionized gas can result from the combination of two effects, the selfconsistent electric field created by lighter particles which first leave the bulk of the plasma, and the thermal pressure [25]. When we compare the two curves for h0 ¼ 0:5 and h0 ¼ 0:95, the decreasing of the density is more important when initial ionization is smaller. This may be result from the recombination effects. Two types of recombination can occur in the plasma: radiative recombination where an electron and an ion are associated to form an atom by emitting a photon, and a three-body recombination which involves two electrons and one ion, one of the electrons recombines with the ion, forming an atom. The excess energy
released in the second process is transferred to the other electron [21]. Each of the two processes depends on the ion charge state Z, the ion and electron densities, and the electron temperature. If the following inequality holds ne ð3 1013 T 3:75 Þ=Z cm3, ne is the electron density, three-body recombination will be more important than radiative recombination [26]. Since the density of the plume is still high in early stages of expansion, and the plasma confined to small spatial dimensions, three-body recombination will eventually dominate radiative recombination, due to the rapid drop in electron temperature as the plasma expands. The ions located at the front of the plasma acquire the largest energy during hydrodynamic acceleration and the interaction time for recombination is very much reduced. This small group of high-energy ions transports a significant fraction of absorbed energy. The ions located in the inner plume layers are accelerated much less due to hydrodynamic expansion. They remain much longer in the denser state, which is being subjected to strong recombination, which in turn enhances the emission from these regions [27,28]. In Fig. 4, we have plotted the normalized temperature to its initial value deduced from Saha equation versus the similarity variable. The same effect seen in the density is observed in the temperature evolution. The temperature drop is slow with
Fig. 5. Ionization fraction as function of the similarity variable.
D. Bennaceur-Doumaz, M. Djebli / Applied Surface Science 255 (2009) 4595–4599
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expansion jl decreases. The target receiving the same energy from the laser source, if this energy is higher than the ionization potential, any excess is used to sustain the gas and provide a larger expansion.
4. Summary To conclude, laser ablated one-dimensional plume expansion is investigated using one-fluid model. Based on self-similar theory, numerical investigation is conducted for neutral gas and a singly ionized one. It is found as a consequence of the ionization that the expansion limit is extended. The solution shows that the profiles exhibit two behaviors, a common linear one given by previous works, and a nonlinear profile away from the source target. This nonlinear behavior starts to appear when the density depletion becomes visible. It is also found that the chemical nature of the target material plays a role in the plume expansion. The limit of self-similar expansion is decreasing, for higher normalized ionization potential. Fig. 6. Velocities as function of the similarity variable for different material targets. (Solid curve) stands for Ti, (dashed curve) for Al and (dashed dotted curve) for Cu.
ionization because the cooling due to expansion is balanced by energy regained from three-body recombination processes of the ions and the electrons. In this situation a very small three-body recombination rate can lead to sufficient transfer of energy to the electrons to make a large change in the electron temperature decay rate. This will effectively reduce the cooling rate caused by adiabatic expansion [28]. Fig. 5 shows the degree of ionization as a function of the similarity variable. Assuming an initially fully ionized plasma (h0 1), like the temperature, we have noticed the ionization fraction decreasing very fast with the expansion by a factor of ten of magnitude as a consequence of recombination processes. Furthermore, we have found that the self-similar behavior of all the expansion profiles when ionization is taken into account, is more extended than for a neutral vapor. 3.3. Influence of the material target on the expansion To study the influence of material target on the expansion, we assumed the plasma initially formed in the interaction of the laser with the target completely ionized. We performed the calculations with three target materials, namely, titanium, aluminum and copper. The results depend mainly on the normalized ionization potential U˜ i . Fig. 6 gives the evolution of the velocity expansion versus the similarity variable, for the three targets. As the normalized ionization potential increases (13.3 for Ti, 14.94 for Al and 18.94 for Cu), the limit of the self-similar
References [1] D.B. Chrisey, C.K. Hubler, Pulsed Laser Deposition of Thin Films, Wiley, New York, 1994. [2] K.R. Chen, J.N. Leboeuf, R.F. Wood, D.B. Geohegan, J.M. Donato, C.L. Liu, A.A. Puretzky, Phys. Rev. Lett. 75 (1995) 4706. [3] V. Nossov, I. Smurov, G. Flamant, J. Phys. D 37 (2004) 185f. [4] F. Vidal, S. Laville, T.W. Johnston, O. Barthelemy, M. Chaker, B.L. Drogoff, J. Margot, Spectrochem. Acta B 56 (2001) 973. [5] F. Garrelie, J. Aubreton, A. Catherinot, J. Appl. Phys. 83 (1998) 5075. [6] T. Nedelea, J. Urbassek, Phys. Plasmas 9 (2002) 3209. [7] A.V. Gusanov, A.G. Gnedovets, I. Smurnov, J. Phys. D, Appl. Phys. 88 (2000) 4532. [8] P.A. Atanasov, N.N. Nedialkov, S.E. Imamova, A. Ruf, H. Hugel, F. Dausinger, P. Bergeret, Appl. Surf. Sci. 186 (2002) 369. [9] Ch. Sack, H. Schamel, Phys. Reports 156 (1987) 311. [10] M. Djebli, S. Bahamida, R. Annou, J. Plasmas 9 (2002) 4107. [11] S.I. Anisimov, D. Bauerle, B.S. Luk’yanchuk, Phys. Rev. B 48 (1993) 12076. [12] S.I. Anisimov, B.S. Luk’yanchuk, A. Luchesw, Appl. Surf. Sci. 96–98 (1996) 24. [13] R.K. Singh, J. Narayan, Phys. Rev. B 41 (1990) 8843. [14] M.W. Stapleton, A.P. McKiernan, J.-P. Mosniera, J. Appl. Phys. 97 (2005) 064904. [15] X. Tan, D. Zhang, S. Feng, Z. Li, G. Li, L. Li, L. Dan, Physica A 363 (2006) 307. [16] X. Tan, D. Zhang, Z. Li, G. Liu, R. Lili, Fan, Phys. Lett. A 370 (2007) 64. [17] X. Tan, D. Zhang, S. Feng, Z. Li, G. Liu, R. Fang, Chin. Phys. Letters 25 (2008) 198. [18] N. Arnold, J. Gruber, J. Heitz, Appl. Phys. A 69 (1999) 87. [19] M. Capitelli, F. Capitelli, A. Eletskii, Spectrochim. Acta B 55 (2000) 559. [20] A.V. Gusanov, A.G. Gnedovets, I. Smurnov, J. Phys. D, Appl. Phys. 32 (1999) 2162. [21] M. Capitelli, A. Casalova, G. Colona, A. de Giacomo, Spectrochim. Acta B 59 (2004) 271. [22] Ya.B. Zel’dovich, Yu.P. Raizer, Physics of Shock Waves and High-Temperature Phenomena, Academic Press, New York, 1966. [23] L.D. Landau, E.M. Lifshitz, Fluid Mechanics, Addison-Wesley, New York, 1959. [24] A.V. Gusarov, I. Smurov, J. Phys. D 36 (2003) 2962. [25] B. Kechouri, M. Djebli, J. Plasmas 13 (2006) 112106. [26] J.L. Weaver, U. Feldman, A.N. Mostovych, J.F. Seely, D. Colombant, G. Holland, Rev. Sci. Instrum. 74 (2003) 5084. [27] S.S. Harilal, C.V. Bindhu, M.S. Tillack, F. Najmabadi, A.C. Gaeris, J. Appl. Phys. 93 (2003) 2380. [28] S.S. Harilal, B.O. Shay, M.S. Tillack, J. Appl. Phys. 98 (2005) 013306.
Contents lists available at ScienceDirect
Applied Surface Science journal homepage: www.elsevier.com/locate/apsusc
Effect of ionization on laser-induced plume self-similar expansion D. Bennaceur-Doumaz *, M. Djebli 1 Centre de De´veloppement des Technologies Avance´es, B.P. 17 Baba Hassen, 16303 Algiers, Algeria
A R T I C L E I N F O
A B S T R A C T
Article history: Received 7 September 2008 Received in revised form 13 November 2008 Accepted 28 November 2008 Available online 7 December 2008
The dynamics of a laser ablation plume during the first stage of its expansion, just after the termination of the laser pulse is modelled. The one-dimensional expansion of the evaporated material, considered as an ideal fluid, is governed by one-fluid Euler equations. For high energetic ions, the charge separation can be neglected and the hydrodynamics equations solved using self-similar formulation. Numerical solution is obtained, first when the laser fluence range is low enough to deal with a neutral vapor, and in a second stage, when ionization effects on the expansion are taken into account, for different material targets. As a main result, we found that the presence of ions in the evaporated gas enhances the self-similar expansion. ß 2008 Elsevier B.V. All rights reserved.
PACS: 52.30.q 52.38.Mf Keywords: Plume expansion Laser ablation Self-similar
1. Introduction Pulsed laser deposition (PLD) has been found to be a successful and a promising technique for the growth of thin films [1]. In order to understand basic physical principle of this technique, laser ablation process can be divided into three stages: (i) interaction of the laser beam with the target materials leading to the evaporation of the surface layers, (ii) interaction of the evaporated cloud with the incident laser beam resulting in isothermal plasma formation and expansion, and (iii) adiabatic expansion of the plasma and subsequent deposition. The first two stages start with the laser pulse and continue through laser pulse duration. The third regime starts after the termination of the laser pulse. For short laser pulses, a reasonable assumption is that the last stage can be considered separately. To optimize the thin film quality in the PLD process, an understanding of the plume expansion dynamics in this stage and its dependence on the initial conditions is very important [2–4]. Thus much work has been done to understand the many complex physical processes in the ablation plume dynamics and to reproduce the temporal and spatial evolution of the profiles expansion. To this end, various approaches were used by authors to
* Corresponding author at: Division Milieux Ionise´s et Lasers, Centre de De´veloppement des Technologies Avance´es, B.P. 17 Baba Hassen, 16303 Algiers, Algeria. E-mail addresses: [email protected], [email protected] (D. Bennaceur-Doumaz), [email protected] (M. Djebli). 1 Permanent address: Theoretical Phys. Lab., Faculty of Physics, USTHB, B.P. 32 Bab Ezzouar, 16079 Algiers, Algeria. 0169-4332/$ – see front matter ß 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.apsusc.2008.11.089
model laser ablation plumes, including numerical ones such as Monte Carlo [5], Particle In Cell [6], gas dynamics [7] or Molecular Dynamics [8] which are useful for real geometry and gas flows with considerable deviations from local thermal equilibrium (LTE). Analytical approaches, in the other hand, are based on the solutions of the gas dynamics equations, using self-similar theory, where partial differential equations are transformed to ordinary ones. This transformation has been largely used to investigate plasma expansion and given new results [9,10]. When evaporation of the target as well as plasma formation cease, there is no additional input energy while most of the thermal energy of the species is converted into kinetic energy. The generated plume can be described by an adiabatic, self-similar expansion. Anisimov et al. studied plasma expansion into vacuum after the end laser pulse, using a three-dimensional model based on a special solution of gas dynamics equations. The model is semi-analytical and plasma profiles were obtained by numerically solving simple differential equations [11,12]. Singh et al. developed a self-similar model (S-N model) of plasma flow which began with an assumed self-similar profile of plasma velocity and considered the plasma velocity near the target surface irradiated by laser as zero. Based on the conservations of mass and momentum, they derived a set of plasma dynamic equations. This model can explain the plasma expansion preferentially normal to the target surface quantitatively [13]. However, the assumption of velocity is not consistent with the experimental results, that is, the velocity of the plasma near the target surface is not zero, but large. So Chen et al. developed a one-dimensional model of an accelerated expansion and assumed a linear profile of plume velocity [2]. Later, Stapleton
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et al. proposed a numerical model which includes the gas dynamics and atomic kinetics of a laser ablation plume, expanding adiabatically in vacuum, using isothermal and isentropic selfsimilar analytical solutions and steady-state collisional radiative equations [14]. The existing self-similar models describing the behavior of expanding laser produced plasma plume are strictly applied to the expansion of neutral gases. However, it has been shown that ionization and recombination processes play an important role in film preparation through their influence on plume expansion and characteristics of deposited particles, and they have to be considered for accurate description of laser ablation dynamics. Indeed, experiments [15] showed the ionization degree of plasma affects significantly the plasma velocity, but little attention was paid on modeling the ionization effect to plasma kinetic behavior generated by laser ablation in previous works. This effect was studied recently by Tan et al. [16,17] where they showed, by comparing between the velocity when there is ionization and the case when it is neglected, that ionization increases the plume expansion velocity. In the present work, we deal with a theoretical investigation, limited to very small temporal and spatial domains just after the ablation pulse, valid in vacuum or in the early stages of the plume expansion in presence of gas background. Our objective is to improve the self-similar existing models [11,13,18] in the PLD expansion process by solving numerically a set of time-dependant ordinary differential equations, deduced by a self-similar model taking into account the ionization of the plume. The nature of material target is also investigated to find its influence on the expansion profiles. 2. Description of the model During the initial stage of plasma expansion, the particle density is sufficiently high and the mean free path of the particles is short (mm) allowing the plasma to behave as a continuum fluid. Many collisions take place between the various particles so the plasma can be considered in local thermal equilibrium. This means that in a sufficiently small region, the electrons, ions and neutrals can be characterized with a common temperature [19]. The existence of LTE requires that the electron-atom and electron-ion collisional processes occur sufficiently fast and dominate radiative processes. In such a system, particles will have Maxwellian velocity distributions and population levels will be distributed according to Boltzmann’s statistics. Initially, for a time where the distance of the laser-induced plasma front from the target surface is smaller than the laser spot dimension, the propagation of plasma can be considered onedimensional and strictly directed perpendicularly to the target surface. It is a reasonable assumption for the early stage of expansion, until an expansion distance of about 1 mm and 100 ns duration [20]. We have used a single fluid approach in which we follow the expansion of a fluid of electrons, ions and neutrals just after the termination of the laser pulse. The expansion is mainly conducted by the pressure rather than the electrostatic potential which may rise from the charge separation. Thus, quasi-neutrality assumption during the whole expansion can be justified for such sufficiently dense plasma [21]. The plasma plume considered as a non-dissipative and nonheat-conductive fluid, is described by Euler equations:
@E @E @v þv þP ¼ 0; @t @x @x
(3)
where Eq. (1) stands for density conservation, n and v are respectively, the density and fluid velocity. The momentum and energy conservations are governed by Eqs. (2) and (3), respectively, where M is the mass of the plume particles, E ¼ Mnðe þ v2 =2Þ is the plume energy density, e ¼ ð1 þ hÞðT=MÞ=ðg 1Þ þ hU i =M is the enthalpy, U i is the ionization energy, h is the ionization fraction of the plume, g is the specific heat ratio of the plume and T is the temperature in eV. Eqs. (1–3) are closed by an ideal gas equation of state such that the plume pressure is given by P ¼ nð1 þ hÞT. For sake of simplicity, we assume to have only singly ionized species, so the following Saha equation estimates the ionization fraction [2]:
h2 1h
¼
2 uþ 2pme T 3=2 exp ðU i =TÞ n u0 h
(4)
uþ and u0 are, respectively, the partition functions of ionized atoms and neutral atoms, me is the electron mass, h is Plank’s constant. 3. Self-similar theory and numerical calculation In general, hydrodynamic equations describing the expansion are difficult to solve numerically, but under certain assumptions, these partial differential equations can be reduced to ordinary differential equations which greatly simplifies the problem. This transformation is based on the assumption that we have a selfsimilar solution, i.e., every physical parameter distribution preserves its shape during expansion and there is no scaling parameter [9,22]. Self-similar solutions usually describe the asymptotic behavior of an unbounded problem and the time t and the space coordinate x appear only in the combination of ðx=tÞ. It means that the existence of self-similar variables implies the lack of characteristic lengths and times. Indeed, this is justified when we deal with the early stage of expansion where charge quasineutrality in the dense plasma is imposed at the beginning of the process. The Debye length then, loses its importance as a characteristic length in the plasma. The self-similar solution of Eqs. (1–4) can be constructed for a quasi-neutral plasma by using the ansatz defined as, nvt v T U N˜ ¼ ; V˜ ¼ ; T˜ ¼ ; U˜ i ¼ i2 : n0 cs Mcs2 Mcs pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where cs is the sound velocity given by cs ¼ g T v =M, T v is the vaporptemperature ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of the target and v, the plasma frequency, v2 ¼ 4pn0 e2 =M, n0 is the initial density of the target. U i is the first ionization potential of the target material. For adiabatic expansion and mono-atomic gases, g ¼ 5=3. We have chosen to write the differential equations in terms of a single dimensionless similarity variable j ¼ x=cs t, then partial derivatives become total derivatives with respect to j. We have
@ 1 d ¼ ; @x cs t dj
@ j d ¼ @t t dj
then, we deduce a set of ordinary differential equations given by
@N˜ ˜ @V˜ ˜ þN N¼0 @j @j
@n @ðnvÞ þ ¼ 0; @t @x
(1)
ðV˜ jÞ
@v @v 1 @P þv þ ¼ 0; @t @x Mn @x
(2)
˜ ˜ ˜ ˜ þ hÞ @T þ N˜ T˜ @h ¼ 0 ˜ þ hÞ @N þ ðV˜ jÞ @V þ Nð1 Tð1
@j
@j
(5)
@j
@j
(6)
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˜ ˜ ˜ ˜ þ hÞ dV þ 3 ðV˜ jÞð1 þ hÞ dT þ ðV˜ jÞT˜ 3 þ Ui Tð1 2 T˜ dj 2 dj
T˜
dN˜ 3 U˜ i þ þ N˜ 2 T˜ dj
!
dT˜ 2 h ˜ ˜ dh N˜ T˜ ¼0 NT dj j dj hð1 hÞ
!
dh ¼0 dj
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(7)
(8)
The self-similar solution is valid as long as the local Debye length is much smaller than the scale length of the self-similar solution cs t. The set of Eqs. (5–8) is solved numerically, using Runge–Kutta method with appropriate initial conditions, for two cases. The first one concerns an expansion of a neutral gas while in the second case the energy is sufficiently high to produce singly ionized atoms. 3.1. Expansion of a neutral vapor The model is used to simulate the ablation of a titanium target by a 25 ns pulse of a KrF excimer with a 248 nm wavelength. At low irradiance, ionization is completely negligible. It is interesting to study the plume dynamics in which the plume consists only of a neutral gas. In this case, h ¼ 0, which is a good approximation for potential ionization U i T v . The initial conditions of the plume expansion in our case, correspond to experimental considerations, at the end of the laser pulse. Investigation is conducted for Ti target of initial density of about 1019 cm3, the neutral atoms are assumed to have an average initial velocity close to the sound velocity and their initial temperature is taken to be the vapor temperature T v ¼3562 K. Knudsen layer effects in front of the target are not considered in this work. Our initial conditions correspond to a certain time and distance from the surface where the self-similar expansion may start. Since the source is localized at the surface, we expect that after a certain time (dt, set to be t ¼ 0) and a transition distance dx near the surface, the system may develop to be self-similar as in free expansion. Furthermore, we use a full numerical solution instead of a semi-analytical model as in Ref. [2] where the selfsimilar theory begins with an assumed linear profile of plume velocity, given by v ¼ vm fa þ ð1 aÞjg, where vm is the maximum expansion velocity, a is a constant, and j ¼ x=vm t, where the expansion ends at j ¼ 1. In other words, the objective of our study is to look for the numerical value of j beyond which the self-similar expansion is no longer valid. Fig. 1 shows that both temperature and density of the plume are decreasing monotonically while the velocity increases almost linearly during the expansion process. This behavior is due to the adiabatic expansion where thermal energy is converted into directed kinetic energy. In this stage, the expansion of the vapor plume is similar to that in vacuum where the plasma expands in a way similar to a supersonic expansion with a free linear behavior. All the expansion profiles of density, velocity and temperature, reach their limit at a self-similar variable value (jl ¼ 1:12) beyond which there is no self-similar solution. We note also for j < j1 ¼ 0:6, we have almost a linear profile as suggested by the free expansion analytical model [23]. For j > j1 , numerical solution exhibits a nonlinear behavior, in particular for the velocity profile which may result from the increase of pressure gradient far away from the source and density depletion. For j > jl , the conditions of LTE may not be valid, this is the limit of the model given in this paper.
Fig. 1. Velocity (dashed line) v=vð0Þ, density n=nð0Þ (solid line) and temperature T=Tð0Þ (dashed dotted line) normalized to their initial values as function of the similarity variable.
and characteristics of deposited particles, and they have to be considered for accurate description of laser ablation dynamics. In the present work, we have studied the influence of this effect on the expansion profiles by introducing Saha equation in the dynamical equations. The results of calculations show from Fig. 2 that the maximum expansion velocity of the ionized plasma is increasing with the initial ionization fraction. The velocity reaches approximately 5.5 times the sound velocity (solid line), while it is 4.5 when h0 ¼ 0:50. During the propagation, the thermal energy of the electrons and ions is converted to kinetic energy which is mainly carried by the heavy ions. The electrons transfer their energy to the ions by coulombian diffusion processes. The ions located at the front of the plasma acquire the largest energy during hydrodynamic acceleration. This small group of high-energy ions transports a significant
3.2. Ionization effects on the expansion profiles Ionization and recombination processes play an important role in film preparation through their influence on plume expansion
Fig. 2. Velocities normalized to their initial values, for h0 ¼ 0:50 (dashed curve), h0 ¼ 0:95 (solid curve) and h0 ¼ 0 (dashed dotted curve) as function of the similarity variable.
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Fig. 4. Temperatures normalized to their initial values, for h0 ¼ 0 (dashed dotted curve), h0 ¼ 0:25 (dashed curve) and h0 ¼ 0:95 (solid curve) as function of the similarity variable. Fig. 3. Densities normalized to their initial values, for h0 ¼ 0 (dashed dotted curve), h0 ¼ 0:95 (solid curve) and h0 ¼ 0:50 (dashed curve) as function of the similarity variable.
fraction of absorbed energy. For a plasma containing a certain amount of neutrals, energy is transferred from ions to neutrals by collisions. The ions are then decelerated and as a consequence the maximum velocity is reduced in comparison with the case where the gas is fully ionized. This effect is shown in Fig. 2(dashed curve) where the ionization fraction is 0.50. We have noticed also that the self-similar expansion increases with ionization. This may result from the acceleration of the lighter particles which leave first the bulk plume, the other species are then attracted to keep the quasi-neutrality assumption [24]. It is worth to say that the expanding gas goes beyond the limit corresponding to a neutral gas, (jl ¼ 1:12), we have jl ¼ 2:45 for h0 ¼ 0:50 and jl ¼ 3:7 for h0 ¼ 0:95. In Fig. 3 we have plotted the density versus the similarity variable for different initial ionization degrees. We recall the density of Fig. 1 which is also seen in Fig. 3(dotted dashed line) for h0 ¼ 0 but with another scaling. In this case, we see that the expansion exhibits two behaviors. The energy emitted from the target by radiation is less than the ionization potential then it is absorbed by the expanding gas. At the first stage, where the gas is sufficiently close to the heated target, the density increases as a result of thermal pressure. This can also be seen from Fig. 1(solid line) for j < 0:6. Far away from the source (j > 0:6), we found the well known free expansion of decreasing density. This is not true when the ionization is taken into consideration. The density decreases when j increases and the excess energy is used to ionize the neutrals. For a large initial ionization fraction (h0 ¼ 0:95) we deal with the expansion of a fully ionized gas. The expansion of an ionized gas can result from the combination of two effects, the selfconsistent electric field created by lighter particles which first leave the bulk of the plasma, and the thermal pressure [25]. When we compare the two curves for h0 ¼ 0:5 and h0 ¼ 0:95, the decreasing of the density is more important when initial ionization is smaller. This may be result from the recombination effects. Two types of recombination can occur in the plasma: radiative recombination where an electron and an ion are associated to form an atom by emitting a photon, and a three-body recombination which involves two electrons and one ion, one of the electrons recombines with the ion, forming an atom. The excess energy
released in the second process is transferred to the other electron [21]. Each of the two processes depends on the ion charge state Z, the ion and electron densities, and the electron temperature. If the following inequality holds ne ð3 1013 T 3:75 Þ=Z cm3, ne is the electron density, three-body recombination will be more important than radiative recombination [26]. Since the density of the plume is still high in early stages of expansion, and the plasma confined to small spatial dimensions, three-body recombination will eventually dominate radiative recombination, due to the rapid drop in electron temperature as the plasma expands. The ions located at the front of the plasma acquire the largest energy during hydrodynamic acceleration and the interaction time for recombination is very much reduced. This small group of high-energy ions transports a significant fraction of absorbed energy. The ions located in the inner plume layers are accelerated much less due to hydrodynamic expansion. They remain much longer in the denser state, which is being subjected to strong recombination, which in turn enhances the emission from these regions [27,28]. In Fig. 4, we have plotted the normalized temperature to its initial value deduced from Saha equation versus the similarity variable. The same effect seen in the density is observed in the temperature evolution. The temperature drop is slow with
Fig. 5. Ionization fraction as function of the similarity variable.
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expansion jl decreases. The target receiving the same energy from the laser source, if this energy is higher than the ionization potential, any excess is used to sustain the gas and provide a larger expansion.
4. Summary To conclude, laser ablated one-dimensional plume expansion is investigated using one-fluid model. Based on self-similar theory, numerical investigation is conducted for neutral gas and a singly ionized one. It is found as a consequence of the ionization that the expansion limit is extended. The solution shows that the profiles exhibit two behaviors, a common linear one given by previous works, and a nonlinear profile away from the source target. This nonlinear behavior starts to appear when the density depletion becomes visible. It is also found that the chemical nature of the target material plays a role in the plume expansion. The limit of self-similar expansion is decreasing, for higher normalized ionization potential. Fig. 6. Velocities as function of the similarity variable for different material targets. (Solid curve) stands for Ti, (dashed curve) for Al and (dashed dotted curve) for Cu.
ionization because the cooling due to expansion is balanced by energy regained from three-body recombination processes of the ions and the electrons. In this situation a very small three-body recombination rate can lead to sufficient transfer of energy to the electrons to make a large change in the electron temperature decay rate. This will effectively reduce the cooling rate caused by adiabatic expansion [28]. Fig. 5 shows the degree of ionization as a function of the similarity variable. Assuming an initially fully ionized plasma (h0 1), like the temperature, we have noticed the ionization fraction decreasing very fast with the expansion by a factor of ten of magnitude as a consequence of recombination processes. Furthermore, we have found that the self-similar behavior of all the expansion profiles when ionization is taken into account, is more extended than for a neutral vapor. 3.3. Influence of the material target on the expansion To study the influence of material target on the expansion, we assumed the plasma initially formed in the interaction of the laser with the target completely ionized. We performed the calculations with three target materials, namely, titanium, aluminum and copper. The results depend mainly on the normalized ionization potential U˜ i . Fig. 6 gives the evolution of the velocity expansion versus the similarity variable, for the three targets. As the normalized ionization potential increases (13.3 for Ti, 14.94 for Al and 18.94 for Cu), the limit of the self-similar
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