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A simple formalism for the prediction of angular distributions in laser ablation deposition
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.:
Applied Surface North-Holland
Science
...
69 (1993) 133-139
i..... .::.:.. . . . ...‘...‘: :/.. .>,. .: ~ :,:.:., ,.,.,, y::; y ,,:, :+:: :j.. . .,.. . . . . . . . . . . ..,...:; :.:.:I.:
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applied surface science
A simple formalism for the prediction of angular distributions in laser ablation deposition J.C.S.
Kools,
E. van
de Riet
and
’
J. Dieleman
Philips Research Laboratories, P.O. Box 80.000, 5600 JA Eindhouen, Netherlands Received
2 June
1992; accepted
for publication
30 October
1992
An analytical model allowing the quantitative description of gas clouds generated by near-threshold laser ablation is described. The model is based on a continuum description of the expansion of an idealized laser-generated gas cloud. This analysis yields the time- and space-dependent gas density and the angular intensity distribution. Dependencies of the angular intensity distribution on experimental variables (laser fluence, atom mass and laser spot size) are determined. It is found that the laser spot dimensions are most important parameters in determining the angular distribution. For a circular spot, a simple formula is derived which allows quantitative prediction of angular distributions. These predictions are in good agreement with the angular distributions of ablation plumes reported in the literature. Finally, recommendations towards deposition practice are given.
1. Introduction Laser ablation deposition (LAD), also known as pulsed laser deposition (PLD), is a thin-film deposition technique of growing importance [l]. The versatility and power of the technique have been demonstrated 121convincingly by the deposition of high-quality thin films of high-ir, superconductor materials. This has resulted in the introduction of the technique in nearly every major materials science laboratory as a tool for development of complex, high-quality thin-film material stuctures on a laboratory scale with a short optimalization procedure. Extending the possibilities of LAD/PLD to industrial applications still requires solution of some problems, a most important one being the scale-up of the technique. Although homogeneous deposition over 2 and 3 inch wafers has been reported [3,4], most present deposition machines are in general able to deposit homogeneously over surfaces of only a few square centimeters. This is related to the angular
’ Present address: DSA Consultants, CD Waalre, Netherlands. 0169-4332/93/$06.00
Geraniumlaan
0 1993 - Elsevier
Science
11, 5582
Publishers
distribution of the atoms and molecules vaporized by the laser ablation process. Due to expansion of the laser-created gas cloud, this angular distribution is found to be peaked along the target normal. An understanding of this expansion effect and, if possible, quantitative description would enable one to tackle the scale-up problem with judgment, enlarging the opportunities to use LAD as an industrial technique. In this paper, the results of a model [5,6] describing gas clouds generated by laser ablation of solids are used to describe the relationship between (a) angular distributions obtained as in LAD and (b) experimental parameters such as laser spot geometry, laser fluence and particle mass. This leads to a number of practical recipes relevant to deposition practice.
2. Hydrodynamical
model: physical
assumptions
The physical picture on which the model is based will be discussed first. It is shown schematically in fig. 1. The gas cloud history is divided in three stages:
B.V. All rights
reserved
134
Evaporation
J.C.S. Kook et al. / Quantitative
1-D
Expansion
3-D
prediction
Expansion
Fig. 1. Physical picture on which the hydrodynamical model is based. A gas cloud is generated by ultrafast evaporation from a solid. The cloud will at first expand one-dimensionally (1D) and gain a stream velocity. The cloud will then expand threedimensionally (3D) in its center-of-mass (COM) system.
(1) A gas cloud is created by ultrafast desorption from a solid or liquid surface. (2) Immediately after desorption, the kinetic degrees of freedom of the gas cloud equilibrate to a full MB-distribution in its center-of-mass (COM) system, followed by a one-dimensional (1D) expansion of the cloud along the target normal. (3) This is followed by adiabatic three-dimensional (3D) expansion of the gas cloud in its COM reference frame. The two expansion processes (1D and 3D) are described in a continuum hydrodynamical formalism (Navier-Stokes equations). This complicated behavior is caused by an anisotropy induced by the presence of the target. For times shortly after desorption, the gas cloud is in contact with the target, and considerable momentum exchange with the target is thus possible. This results in a pressure of the gas cloud on the target surface, causing the gas cloud to gain momentum along the target normal, thus being accelerated away from the target (1D expansion). The gas cloud also expands into the surrounding vacuum (3D expansion) due to the pressure gradient induced by its density profile. The gas cloud does not change its overall momentum in this process. Although both expansion processes overlap, it is reasonable to make the simplification of a 1D expansion stage followed by a 3D expansion stage, since the pressure on the target is a strongly decreasing function of time. This picture is supported by ultrafast photography of laser ablation plumes. It is found there that, under typical LAD conditions, for the shortest time-scales (t G 100 ns> the expansion of
of angular distributions
in LAD
the laser-generated gas cloud is essentially onedimensional [7], whereas for longer time-scales (t 2 300 ns) the expansion is essentially three-dimensional [8]. The model should apply to all systems of transient gas desorption from a solid or liquid target in a vacuum where the following conditions are met: (1) The number of post-desorption collisions is sufficient for hydrodynamics to apply. To get some estimate of this condition, consider the following numerical example: five monolayers of copper (- 1 x 1016 atoms cm-*) are evaporated from a 5000 K surface in 20 ns. The atoms will have typical velocities of about 1000 mss’, resulting in a cloud which has a length of about 20 km along the surface normal. The gas density is then in a pressure of 5 X 10” atoms cmp3 resulting 3.5 X lo5 Pa. Taking a typical collision cross section of 5 x lo-l5 cm-*, the mean free path becomes 0.3 pm. This is significantly smaller than the gas cloud dimensions. As long as the expansion process is mainly lD, both mean free path and smallest gas cloud dimension are approximately linear in time and the ratio cloud dimensions/mean free path does not change. This motivates the treatment of the gas as a continuum. (2) The majority of the atoms are in a neutral state. The flow of charged particles is not considered here. This condition is not fulfilled for laser vaporization phenomena in the higher fluence range such as in the case of laser-generated X-ray sources, ion sources or controlled-fusion devices. For LAD circumstances near the ablation threshold, the ionized fraction is sufficiently low. However, deviations may occur in the case of atoms with very low ionization energies. Since the depositing flux in LAD is in general found to consist mainly of single atoms, only the results for atoms are given here. (3) The particles gain energy as long as momentum exchange with the target is possible. Absorption of laser radiation after considerable 1D expansion has occurred, is not considered here. (4) The expansion is happening in a vacuum. More quantitatively: the mean free path of the ablated particles with respect to collisions with background gas atoms or molecules is large compared to the distances in the deposition geome-
135
J.C.S. Kook et al. / Quantitative prediction of angular distributions in LAD
try. Since these distances are typically on the order of a few cm, the model applies for ablation events in pressures typically below 10-l Pa. This implies that an important class of LAD processes, i.e. reactive LAD processes cannot be described properly by the present model. Nevertheless, it can be concluded that the model applies to a wide class of laser vaporization phenomena, including LAD situations.
3. Hydrodynamical
model: results
The primary quantity of the model is the timeand position-dependent particle density distribution n(xi, t) (or equivalently a velocity distribution f(u, 8)) which comes as a solution of the Navier-Stokes equations. From this quantity, the angular intensity distribution I(6) (6 is the polar angle) can be calculated by integration of the total number of particles moving in one direction: I( 8) dR = i
l>,f( ,
xoizoFig. 2. Dependence for a circular
of the ratio T, /T, on the ratio X, /Z,, laser spot as determined numerically.
The elliptical temperatures are parameters describing the direction-dependent expansion velocity of the cloud for the 3D expansion. It must be emphasized that they should not be interpreted as real physical temperatures. It is found that Txi is linear in TO and independent of m. Furthermore,
c’, 6) du
This yields for small angles:
it can be proven that
Z(6) acosp(~),
(3)
where
(1) where m is the atomic mass (in amu) and the parameters u and T,, T, are called stream velocity and elliptical temperatures, respectively. The meaning of these parameters and their dependence on experimental conditions will be discussed now. The stream velocity u is the velocity of the center-of-mass of the gas cloud after 1D expansion. It is related to the temperature after 1D expansion CT,) by the relation: u = M( ykT,/m)
1’2,
(2)
where M is the dimensionless Mach number of this 1D expansion. It is found [5] experimentally, and motivated theoretically, that A4 = 2.33 for atoms in a well-developed expansion.
The elliptical temperatures are found to be strongly dependent on the dimensions of the gas cloud prior to 3D expansion (called Xi: i = 1, 2, 3). Fig. 2 shows T,/T, as a function of X,/Z, for a circular laser spot. For dimensions of the same order of magnitude, this ratio scales approximately as: T,/T,
= ( XO/Z,)‘.2.
(4)
For a non-circular laser spot, the ratio T,,/T, is plotted as a function of the aspect ratio A = Y,/X, for a few values of X0 in fig. 3. The behavior shown in fig. 3 is typical. The parameters X0 and Y, can be evaluated straightforwardly as the halves of the widths of a gaussian laser spot intensity profile on the surface, and can be measured directly. The evaluation of Z, is less simple, Z, is a measure for the dimensions in the z-direction of the gas cloud at the onset of 3D expansion.
J. C. S. Kook et al. / Quantitative prediction of angular distributions in LAD
136
t
1oy
1
5
I-
: -
1.0
1
0.1 0.1
I
I
I
IIIIJJ
,
,,,,I,
1.0
10 A-
Fig. 3. Dependence of the ratio TY/ 7” on the aspect ratio for an asymmetric laser spot as determined numerically.
A
Values of about 400-500 pm are found to give a reasonable prediction of the experimental data presented [51. A gas cloud with a typical expansion velocity of lo3 m/s will have such dimensions after 400 ns. This is typically the time-scale on which 3D expansion has become important [8]. Further work, both experimental and theoretical, is required to investigate the dependence of Z, on experimental parameters such as the surface temperature, the particle mass, or the density of the particles. For a circular spot, it is possible to combine eqs. (l)-(4) to p =
3.9[1 +2(g2jii4( 3*.
(5)
The dependence of angular intensity distributions 1(6) on the experimental conditions in a deposition experiment as predicted by the model will be discussed now. (1) Atomic mass: The formalism described above applies in principle to the expansion of a gas consisting of one type of atom. If Z, has no dependency on the atomic mass, it follows from eq. (4) that TV/T, is independent of the mass. It is then found from eq. (3) that T,/T, must be independent of the mass, and thus from eq. (2) that mu2/Tz is also independent of the mass and finally that 1(6) is independent of m.
(2) Laser jluence: Provided the etch yield is sufficiently high for a well developed expansion to occur, and the laser fluence F is not so high that plasma effects disturb the neutral atom flow, the only parameter that depends on the fluence is the surface temperature T,. From eq. (2) it can be seen that u* is linear in To, just like Txi. Once again, this condition is only fulfilled if Z,, is independent of T,,. Their respective ratios are then independent of To and thus of T,. In conclusion, it is found (because of eq. (1)) that the functional form of 1(6) is independent of F. (3) Spot dimensions: The elliptical temperatures are sensitive functions of the initial lateral dimensions of the gas cloud (X0, Y,>. These dimensions are dictated by the dimensions of the laser spot. For a circular spot with X0 = Y, of the same order as Z,, it is possible to calculate 1(6) from eq. (5). For an asymmetric spot, it is possible to predict the ratio of the exponents I, in the cosine power law from fig. 3.
4. Comparison terns
to experimental
deposition
pat-
In this section, the predictions of the hydrodynamical model formulated above will be compared to angular intensity distributions Z(6) reported in the literature. These angular intensity distributions are measured by evaluation of the thickness profile of the deposited film. The sticking coefficient of the atoms arriving on the substrate is assumed to be constant for all species under all conditions. Since (1) the sticking coefficient is in general a function of velocity and polar angle of impact, and (2) the velocity of the incoming particles is strongly dependent on the polar angle this simplification might a priori be a source of discrepancies. (1) Atom mass: The formalism applies in principle to the expansion of a cloud of a gas consisting of one type of atom. If one assumes Z, to be independent of the mass of the atom, it follows that I(6) is independent of the mass. Gorbunov and Konov [9] found indeed that ablation of gold and graphite yields the same angular distribution. Akhsakhalyan et al. [lo] investigated a large se-
J.C.S. Kook et al. / Quantitative
prediction
ries of elemental solids for a much higher power regime (a lo9 W cm-2). In contradiction to the previous authors, it was claimed there that the angular distribution has a dependence on the mass of the atom, although the scatter between data points is considerable. Elements with a low ionization energy (Bi, Pb) are found to give different angular distributions, probably due to plasma formation effects at these high fluences. The situation is somewhat more complex for the ablation of a multicomponent solid. The stream velocities of several components of such a target are somewhere in between equality (all types of atoms move in a single gas cloud) and scaling with the square root of the mass u a rn-‘/* (all types of atoms are in different gas clouds). A scaling behavior of the stream velocity implies independency of the angular distribution on the mass, as has been described for the ablation of CdS [ill and SmBa,Cu,O, [12]. (2) Laser fluence: The angular distribution is predicted to be independent of the fluence. A small second-order effect is expected, since a higher fluence causes the ablation threshold to be exceeded for a larger fraction of the spot. Such a weak dependency of the angular distribution on the laser fluence is expected, and indeed found experimentally [9,12,13]. (3) Laser spot dimensions: As mentioned above, the elliptical temperatures, and therefore Z(6) are sensitive functions of the initial dimensions of the cloud, i.e. the geometry of the laser spot. For a circular spot, it is possible to predict the angular intensity distribution from the spot radius X0, using eq. (51, provided 2, is known. If the angular distribution is measured for a series of radii, Z, can be fitted by plotting p versus X,/Z,. This is done in fig. 4 for a series of angular distributions measured for W-ablation by Akhsakhalyan et al. [lo]. The experimental data can be fitted fairly well. The value of Z, is 350 pm, comparable to the values obtained earlier [5] in ARTOF measurements (400 and 500 pm). Since the three fit values obtained for Z, are comparable (350, 400 and 500 pm), it is reasonable to suppose Z, to be a constant quantity. This allows the prediction of angular distributions of atoms ablated by circular laser spots. Table 1
of angular distributions
t
137
in LAD
15
P
10
5
0 10-l
lo0
wh Fig. 4. Comparison power law angular with a varying laser (5)
-
between measured [33] powers p in cosine distributions for Nd:YAG ablation of W spot size with the predictions based on eq. and a value Z, = 350 km.
compares this prediction to a few experimental values of p, taking Z, = 400 pm. Considering the (lack of) accuracy of the input data, the agreement is found to be quite good for these experiments. Cheenne et al. [13] determined the dependence of the angular distribution on the laser spot size at fixed fluence for the case of excimer ablation of high-T, superconductors. Increasing the laser spot from 1 to 8 mm2 caused a change from p = 2.5 to p = 5.5. Substituting Z, = 2.2 mm in eq. (5) gives p = 2.6 and p = 6.0. It is thus found that this case requires a somewhat larger value for Z,. It is not clear what is responsible for this effect. For an asymmetric laser spot, the behavior is more complex: the azimuth-dependent angular intensity distribution is determined by the aspect ratio A. As was noticed experimentally 1161,there Table 1 Comparison between literature results Ref. [91 1141
1151
the angular
distribution
Target
Xa (mm)
P&II
Graphite Gold YBaCuO TIN
0.22 0.22 1.2 0.3
6.4 6.4 16 7.0
(5) and
some
Plll~~, 8 8 >ll 6
The exponent in the cosine power law is calculated from the spot dimensions as given in the paper, and the assumptions M = 2.33 and Z, = 0.4 mm.
138
J.C.S. Kook et al. / Quantitative prediction of angular distributions in LAD
is a “flip-over”: the longest direction of the laser spot results in the shortest direction of the deposition pattern, and vice versa. Afonso et al. [16] have measured the azimuthal dependence of the peaking of a Ge ablation cloud ablated by a spot with A = 2.5. They find a ratio of 7.5 for the power factor in the cosine law for x and y directions. Fig. 3, combined with eq. (l), predicts a ratio of 4.5.
5. Conclusions tion practice
and predictions
towards
deposi-
From the extensive discussion of the deposition pattern given above, it must be clear that the formalism presented above gives a fairly good description of angular intensity distributions I(6). A number of “rules of thumb” relevant to deposition practice will be presented now: Laser spot geometry: The analysis given above is for a Gaussian elliptical gas cloud generated by an elliptical laser spot. In practice significant inhomogeneities in the laser spot may occur. The analysis given above does no longer apply then, but it is clear that the same gas expansion effects play a role. The homogeneity of the deposition geometry is thus directly related to the homogeneity of the laser spot. This is clearly illustrated in fig. 5. This figure shows the thickness profiles (visible as interference rings) of two BN films deposited on Si(100). Film a is deposited using an elliptical Gaussian laser spot, resulting in a smooth elliptical deposition pattern. Film b is deposited using an asymmetric laser spot, resulting in an asymmetric deposition pattern. For a homogeneous laser spot, it is possible to manipulate the distribution pattern in a predictable way using the spot dimensions. For a given set of dimensions (2X,, 2Y,), the deposition pattern can then be estimated using eqs. (1) and (5) and figs. 2 and 3. Other experimental parameters: The fact that the angular intensity distribution is in principle independent of the other experimental parameters (laser fluence and atomic mass), makes the deposition uniformity stable towards small instabilities in laser fluence. Another conclusion is
Fig. 5. Comparison between the deposition uniformity (visible as interference rings) of two BN films deposited on Si(100) using (a) an elliptical laser spot and (b) an irregularly shaped laser spot.
that changing the target does not result in another deposition uniformity, a feature which is important when multilayers have to be deposited. As a last remark, it is important to note that the present analysis does not include the effects of laser-induced target surface morphology, since the target is assumed to be flat. The development
J.C.S. Kook et al. / Quantitative prediction of angular distributions
of laser-induced surface roughness upon prolonged irradiation will cause the depositing flux to be directed along the local target normal, which does not have to coincide with the macroscopic target normal.
References [l] MRS Bull. XVII (2) (February 1992). Special issue on PLD/LAD. [2] T.S. Baller, G.N.A. van Veen and H. van Hal, Appl. Phys. A 46 (1988) 215; D. Dijkkamp, T. Venkatesan, X.D. Wu, S.A. Shaheen, N. Jisrawi, Y.H. Min-Lee, W.L. McLean and M. Croft, Appl. Phys. Lett. 51 (1987) 619. [3] S.R. Foltyn, R.E. Muenchausen, R.C. Dye, X.D. Wu, L. Luo, D.W. Cooke and R.C. Taber, Appl. Phys. Lett. 59 (1991) 1374. [4] J. Greer and H.J. van Hook, MRS Proc. 191 (1990) 171. [5] J.C.S. Kools, PhD Thesis, Eindhoven University of Technology, Eindhoven, 1992.
[‘J J.C.S.
in LAD
139
Kools, T.S. Baller, S.T. De Zwart and J. Dieleman, J. Appl. Phys. 71 (1992) 4547. 171 A. Gupta, B. Braren, K.G. Casey, B.W. Husey and R. Kelly, Appl. Phys. Lett. 59 (1991) 1302. and P.L. Ventzek, Appl. Phys. Lett. 58 [81 R. Gilgenbach (1991) 1597. [91 A.A. Gorbunov and V.I. Konov, Sov. Phys. Tech. Phys. 34 (1989) 1271. S.V. Gaponov, V.I. Luchin and A.P. m A.D. Akhsakhalyan, Chirimanov, Sov. Phys. Tech. Phys. 33 (1988) 1146. [ill A. Namiki, T. Kawai and K. Ichige, Surf. Sci. 166 (1986) 129. C. Liang, S.A. Shaheen, M. [=I R.A. Neifeld, S. Gunapala, Croft, J. Price, D. Simons and W.T. Hill, Appl. Phys. Lett. 54 (1988) 954. [I31 A. Cheenne, J. Perriere, F. Kerherve, G. Hauchecorne, E. Fogarassy and C. Fuchs, MRS Proc. 191 (1990) 229. X.D. Wu, A. Inam and J.B. Wachtman, [141 T. Venkatesan, Appl. Phys. Lett. 52 (1988) 1193. E. van de iI51 J.C.S Kools, C.J.C. Nillesen, S.H. Brongersma, Riet and J. Dieleman, J. Vat. Sci. Technol. A 10 (1992) 1809. [161 C.N. Afonso, R. Serna, F. Catalina and D. Bermejo, Appl. Surf. Sci. 46 (1990) 249.
.:
Applied Surface North-Holland
Science
...
69 (1993) 133-139
i..... .::.:.. . . . ...‘...‘: :/.. .>,. .: ~ :,:.:., ,.,.,, y::; y ,,:, :+:: :j.. . .,.. . . . . . . . . . . ..,...:; :.:.:I.:
.,.,,,
,,
,.,
>:.:
.:.
y::::.::_j:
applied surface science
A simple formalism for the prediction of angular distributions in laser ablation deposition J.C.S.
Kools,
E. van
de Riet
and
’
J. Dieleman
Philips Research Laboratories, P.O. Box 80.000, 5600 JA Eindhouen, Netherlands Received
2 June
1992; accepted
for publication
30 October
1992
An analytical model allowing the quantitative description of gas clouds generated by near-threshold laser ablation is described. The model is based on a continuum description of the expansion of an idealized laser-generated gas cloud. This analysis yields the time- and space-dependent gas density and the angular intensity distribution. Dependencies of the angular intensity distribution on experimental variables (laser fluence, atom mass and laser spot size) are determined. It is found that the laser spot dimensions are most important parameters in determining the angular distribution. For a circular spot, a simple formula is derived which allows quantitative prediction of angular distributions. These predictions are in good agreement with the angular distributions of ablation plumes reported in the literature. Finally, recommendations towards deposition practice are given.
1. Introduction Laser ablation deposition (LAD), also known as pulsed laser deposition (PLD), is a thin-film deposition technique of growing importance [l]. The versatility and power of the technique have been demonstrated 121convincingly by the deposition of high-quality thin films of high-ir, superconductor materials. This has resulted in the introduction of the technique in nearly every major materials science laboratory as a tool for development of complex, high-quality thin-film material stuctures on a laboratory scale with a short optimalization procedure. Extending the possibilities of LAD/PLD to industrial applications still requires solution of some problems, a most important one being the scale-up of the technique. Although homogeneous deposition over 2 and 3 inch wafers has been reported [3,4], most present deposition machines are in general able to deposit homogeneously over surfaces of only a few square centimeters. This is related to the angular
’ Present address: DSA Consultants, CD Waalre, Netherlands. 0169-4332/93/$06.00
Geraniumlaan
0 1993 - Elsevier
Science
11, 5582
Publishers
distribution of the atoms and molecules vaporized by the laser ablation process. Due to expansion of the laser-created gas cloud, this angular distribution is found to be peaked along the target normal. An understanding of this expansion effect and, if possible, quantitative description would enable one to tackle the scale-up problem with judgment, enlarging the opportunities to use LAD as an industrial technique. In this paper, the results of a model [5,6] describing gas clouds generated by laser ablation of solids are used to describe the relationship between (a) angular distributions obtained as in LAD and (b) experimental parameters such as laser spot geometry, laser fluence and particle mass. This leads to a number of practical recipes relevant to deposition practice.
2. Hydrodynamical
model: physical
assumptions
The physical picture on which the model is based will be discussed first. It is shown schematically in fig. 1. The gas cloud history is divided in three stages:
B.V. All rights
reserved
134
Evaporation
J.C.S. Kook et al. / Quantitative
1-D
Expansion
3-D
prediction
Expansion
Fig. 1. Physical picture on which the hydrodynamical model is based. A gas cloud is generated by ultrafast evaporation from a solid. The cloud will at first expand one-dimensionally (1D) and gain a stream velocity. The cloud will then expand threedimensionally (3D) in its center-of-mass (COM) system.
(1) A gas cloud is created by ultrafast desorption from a solid or liquid surface. (2) Immediately after desorption, the kinetic degrees of freedom of the gas cloud equilibrate to a full MB-distribution in its center-of-mass (COM) system, followed by a one-dimensional (1D) expansion of the cloud along the target normal. (3) This is followed by adiabatic three-dimensional (3D) expansion of the gas cloud in its COM reference frame. The two expansion processes (1D and 3D) are described in a continuum hydrodynamical formalism (Navier-Stokes equations). This complicated behavior is caused by an anisotropy induced by the presence of the target. For times shortly after desorption, the gas cloud is in contact with the target, and considerable momentum exchange with the target is thus possible. This results in a pressure of the gas cloud on the target surface, causing the gas cloud to gain momentum along the target normal, thus being accelerated away from the target (1D expansion). The gas cloud also expands into the surrounding vacuum (3D expansion) due to the pressure gradient induced by its density profile. The gas cloud does not change its overall momentum in this process. Although both expansion processes overlap, it is reasonable to make the simplification of a 1D expansion stage followed by a 3D expansion stage, since the pressure on the target is a strongly decreasing function of time. This picture is supported by ultrafast photography of laser ablation plumes. It is found there that, under typical LAD conditions, for the shortest time-scales (t G 100 ns> the expansion of
of angular distributions
in LAD
the laser-generated gas cloud is essentially onedimensional [7], whereas for longer time-scales (t 2 300 ns) the expansion is essentially three-dimensional [8]. The model should apply to all systems of transient gas desorption from a solid or liquid target in a vacuum where the following conditions are met: (1) The number of post-desorption collisions is sufficient for hydrodynamics to apply. To get some estimate of this condition, consider the following numerical example: five monolayers of copper (- 1 x 1016 atoms cm-*) are evaporated from a 5000 K surface in 20 ns. The atoms will have typical velocities of about 1000 mss’, resulting in a cloud which has a length of about 20 km along the surface normal. The gas density is then in a pressure of 5 X 10” atoms cmp3 resulting 3.5 X lo5 Pa. Taking a typical collision cross section of 5 x lo-l5 cm-*, the mean free path becomes 0.3 pm. This is significantly smaller than the gas cloud dimensions. As long as the expansion process is mainly lD, both mean free path and smallest gas cloud dimension are approximately linear in time and the ratio cloud dimensions/mean free path does not change. This motivates the treatment of the gas as a continuum. (2) The majority of the atoms are in a neutral state. The flow of charged particles is not considered here. This condition is not fulfilled for laser vaporization phenomena in the higher fluence range such as in the case of laser-generated X-ray sources, ion sources or controlled-fusion devices. For LAD circumstances near the ablation threshold, the ionized fraction is sufficiently low. However, deviations may occur in the case of atoms with very low ionization energies. Since the depositing flux in LAD is in general found to consist mainly of single atoms, only the results for atoms are given here. (3) The particles gain energy as long as momentum exchange with the target is possible. Absorption of laser radiation after considerable 1D expansion has occurred, is not considered here. (4) The expansion is happening in a vacuum. More quantitatively: the mean free path of the ablated particles with respect to collisions with background gas atoms or molecules is large compared to the distances in the deposition geome-
135
J.C.S. Kook et al. / Quantitative prediction of angular distributions in LAD
try. Since these distances are typically on the order of a few cm, the model applies for ablation events in pressures typically below 10-l Pa. This implies that an important class of LAD processes, i.e. reactive LAD processes cannot be described properly by the present model. Nevertheless, it can be concluded that the model applies to a wide class of laser vaporization phenomena, including LAD situations.
3. Hydrodynamical
model: results
The primary quantity of the model is the timeand position-dependent particle density distribution n(xi, t) (or equivalently a velocity distribution f(u, 8)) which comes as a solution of the Navier-Stokes equations. From this quantity, the angular intensity distribution I(6) (6 is the polar angle) can be calculated by integration of the total number of particles moving in one direction: I( 8) dR = i
l>,f( ,
xoizoFig. 2. Dependence for a circular
of the ratio T, /T, on the ratio X, /Z,, laser spot as determined numerically.
The elliptical temperatures are parameters describing the direction-dependent expansion velocity of the cloud for the 3D expansion. It must be emphasized that they should not be interpreted as real physical temperatures. It is found that Txi is linear in TO and independent of m. Furthermore,
c’, 6) du
This yields for small angles:
it can be proven that
Z(6) acosp(~),
(3)
where
(1) where m is the atomic mass (in amu) and the parameters u and T,, T, are called stream velocity and elliptical temperatures, respectively. The meaning of these parameters and their dependence on experimental conditions will be discussed now. The stream velocity u is the velocity of the center-of-mass of the gas cloud after 1D expansion. It is related to the temperature after 1D expansion CT,) by the relation: u = M( ykT,/m)
1’2,
(2)
where M is the dimensionless Mach number of this 1D expansion. It is found [5] experimentally, and motivated theoretically, that A4 = 2.33 for atoms in a well-developed expansion.
The elliptical temperatures are found to be strongly dependent on the dimensions of the gas cloud prior to 3D expansion (called Xi: i = 1, 2, 3). Fig. 2 shows T,/T, as a function of X,/Z, for a circular laser spot. For dimensions of the same order of magnitude, this ratio scales approximately as: T,/T,
= ( XO/Z,)‘.2.
(4)
For a non-circular laser spot, the ratio T,,/T, is plotted as a function of the aspect ratio A = Y,/X, for a few values of X0 in fig. 3. The behavior shown in fig. 3 is typical. The parameters X0 and Y, can be evaluated straightforwardly as the halves of the widths of a gaussian laser spot intensity profile on the surface, and can be measured directly. The evaluation of Z, is less simple, Z, is a measure for the dimensions in the z-direction of the gas cloud at the onset of 3D expansion.
J. C. S. Kook et al. / Quantitative prediction of angular distributions in LAD
136
t
1oy
1
5
I-
: -
1.0
1
0.1 0.1
I
I
I
IIIIJJ
,
,,,,I,
1.0
10 A-
Fig. 3. Dependence of the ratio TY/ 7” on the aspect ratio for an asymmetric laser spot as determined numerically.
A
Values of about 400-500 pm are found to give a reasonable prediction of the experimental data presented [51. A gas cloud with a typical expansion velocity of lo3 m/s will have such dimensions after 400 ns. This is typically the time-scale on which 3D expansion has become important [8]. Further work, both experimental and theoretical, is required to investigate the dependence of Z, on experimental parameters such as the surface temperature, the particle mass, or the density of the particles. For a circular spot, it is possible to combine eqs. (l)-(4) to p =
3.9[1 +2(g2jii4( 3*.
(5)
The dependence of angular intensity distributions 1(6) on the experimental conditions in a deposition experiment as predicted by the model will be discussed now. (1) Atomic mass: The formalism described above applies in principle to the expansion of a gas consisting of one type of atom. If Z, has no dependency on the atomic mass, it follows from eq. (4) that TV/T, is independent of the mass. It is then found from eq. (3) that T,/T, must be independent of the mass, and thus from eq. (2) that mu2/Tz is also independent of the mass and finally that 1(6) is independent of m.
(2) Laser jluence: Provided the etch yield is sufficiently high for a well developed expansion to occur, and the laser fluence F is not so high that plasma effects disturb the neutral atom flow, the only parameter that depends on the fluence is the surface temperature T,. From eq. (2) it can be seen that u* is linear in To, just like Txi. Once again, this condition is only fulfilled if Z,, is independent of T,,. Their respective ratios are then independent of To and thus of T,. In conclusion, it is found (because of eq. (1)) that the functional form of 1(6) is independent of F. (3) Spot dimensions: The elliptical temperatures are sensitive functions of the initial lateral dimensions of the gas cloud (X0, Y,>. These dimensions are dictated by the dimensions of the laser spot. For a circular spot with X0 = Y, of the same order as Z,, it is possible to calculate 1(6) from eq. (5). For an asymmetric spot, it is possible to predict the ratio of the exponents I, in the cosine power law from fig. 3.
4. Comparison terns
to experimental
deposition
pat-
In this section, the predictions of the hydrodynamical model formulated above will be compared to angular intensity distributions Z(6) reported in the literature. These angular intensity distributions are measured by evaluation of the thickness profile of the deposited film. The sticking coefficient of the atoms arriving on the substrate is assumed to be constant for all species under all conditions. Since (1) the sticking coefficient is in general a function of velocity and polar angle of impact, and (2) the velocity of the incoming particles is strongly dependent on the polar angle this simplification might a priori be a source of discrepancies. (1) Atom mass: The formalism applies in principle to the expansion of a cloud of a gas consisting of one type of atom. If one assumes Z, to be independent of the mass of the atom, it follows that I(6) is independent of the mass. Gorbunov and Konov [9] found indeed that ablation of gold and graphite yields the same angular distribution. Akhsakhalyan et al. [lo] investigated a large se-
J.C.S. Kook et al. / Quantitative
prediction
ries of elemental solids for a much higher power regime (a lo9 W cm-2). In contradiction to the previous authors, it was claimed there that the angular distribution has a dependence on the mass of the atom, although the scatter between data points is considerable. Elements with a low ionization energy (Bi, Pb) are found to give different angular distributions, probably due to plasma formation effects at these high fluences. The situation is somewhat more complex for the ablation of a multicomponent solid. The stream velocities of several components of such a target are somewhere in between equality (all types of atoms move in a single gas cloud) and scaling with the square root of the mass u a rn-‘/* (all types of atoms are in different gas clouds). A scaling behavior of the stream velocity implies independency of the angular distribution on the mass, as has been described for the ablation of CdS [ill and SmBa,Cu,O, [12]. (2) Laser fluence: The angular distribution is predicted to be independent of the fluence. A small second-order effect is expected, since a higher fluence causes the ablation threshold to be exceeded for a larger fraction of the spot. Such a weak dependency of the angular distribution on the laser fluence is expected, and indeed found experimentally [9,12,13]. (3) Laser spot dimensions: As mentioned above, the elliptical temperatures, and therefore Z(6) are sensitive functions of the initial dimensions of the cloud, i.e. the geometry of the laser spot. For a circular spot, it is possible to predict the angular intensity distribution from the spot radius X0, using eq. (51, provided 2, is known. If the angular distribution is measured for a series of radii, Z, can be fitted by plotting p versus X,/Z,. This is done in fig. 4 for a series of angular distributions measured for W-ablation by Akhsakhalyan et al. [lo]. The experimental data can be fitted fairly well. The value of Z, is 350 pm, comparable to the values obtained earlier [5] in ARTOF measurements (400 and 500 pm). Since the three fit values obtained for Z, are comparable (350, 400 and 500 pm), it is reasonable to suppose Z, to be a constant quantity. This allows the prediction of angular distributions of atoms ablated by circular laser spots. Table 1
of angular distributions
t
137
in LAD
15
P
10
5
0 10-l
lo0
wh Fig. 4. Comparison power law angular with a varying laser (5)
-
between measured [33] powers p in cosine distributions for Nd:YAG ablation of W spot size with the predictions based on eq. and a value Z, = 350 km.
compares this prediction to a few experimental values of p, taking Z, = 400 pm. Considering the (lack of) accuracy of the input data, the agreement is found to be quite good for these experiments. Cheenne et al. [13] determined the dependence of the angular distribution on the laser spot size at fixed fluence for the case of excimer ablation of high-T, superconductors. Increasing the laser spot from 1 to 8 mm2 caused a change from p = 2.5 to p = 5.5. Substituting Z, = 2.2 mm in eq. (5) gives p = 2.6 and p = 6.0. It is thus found that this case requires a somewhat larger value for Z,. It is not clear what is responsible for this effect. For an asymmetric laser spot, the behavior is more complex: the azimuth-dependent angular intensity distribution is determined by the aspect ratio A. As was noticed experimentally 1161,there Table 1 Comparison between literature results Ref. [91 1141
1151
the angular
distribution
Target
Xa (mm)
P&II
Graphite Gold YBaCuO TIN
0.22 0.22 1.2 0.3
6.4 6.4 16 7.0
(5) and
some
Plll~~, 8 8 >ll 6
The exponent in the cosine power law is calculated from the spot dimensions as given in the paper, and the assumptions M = 2.33 and Z, = 0.4 mm.
138
J.C.S. Kook et al. / Quantitative prediction of angular distributions in LAD
is a “flip-over”: the longest direction of the laser spot results in the shortest direction of the deposition pattern, and vice versa. Afonso et al. [16] have measured the azimuthal dependence of the peaking of a Ge ablation cloud ablated by a spot with A = 2.5. They find a ratio of 7.5 for the power factor in the cosine law for x and y directions. Fig. 3, combined with eq. (l), predicts a ratio of 4.5.
5. Conclusions tion practice
and predictions
towards
deposi-
From the extensive discussion of the deposition pattern given above, it must be clear that the formalism presented above gives a fairly good description of angular intensity distributions I(6). A number of “rules of thumb” relevant to deposition practice will be presented now: Laser spot geometry: The analysis given above is for a Gaussian elliptical gas cloud generated by an elliptical laser spot. In practice significant inhomogeneities in the laser spot may occur. The analysis given above does no longer apply then, but it is clear that the same gas expansion effects play a role. The homogeneity of the deposition geometry is thus directly related to the homogeneity of the laser spot. This is clearly illustrated in fig. 5. This figure shows the thickness profiles (visible as interference rings) of two BN films deposited on Si(100). Film a is deposited using an elliptical Gaussian laser spot, resulting in a smooth elliptical deposition pattern. Film b is deposited using an asymmetric laser spot, resulting in an asymmetric deposition pattern. For a homogeneous laser spot, it is possible to manipulate the distribution pattern in a predictable way using the spot dimensions. For a given set of dimensions (2X,, 2Y,), the deposition pattern can then be estimated using eqs. (1) and (5) and figs. 2 and 3. Other experimental parameters: The fact that the angular intensity distribution is in principle independent of the other experimental parameters (laser fluence and atomic mass), makes the deposition uniformity stable towards small instabilities in laser fluence. Another conclusion is
Fig. 5. Comparison between the deposition uniformity (visible as interference rings) of two BN films deposited on Si(100) using (a) an elliptical laser spot and (b) an irregularly shaped laser spot.
that changing the target does not result in another deposition uniformity, a feature which is important when multilayers have to be deposited. As a last remark, it is important to note that the present analysis does not include the effects of laser-induced target surface morphology, since the target is assumed to be flat. The development
J.C.S. Kook et al. / Quantitative prediction of angular distributions
of laser-induced surface roughness upon prolonged irradiation will cause the depositing flux to be directed along the local target normal, which does not have to coincide with the macroscopic target normal.
References [l] MRS Bull. XVII (2) (February 1992). Special issue on PLD/LAD. [2] T.S. Baller, G.N.A. van Veen and H. van Hal, Appl. Phys. A 46 (1988) 215; D. Dijkkamp, T. Venkatesan, X.D. Wu, S.A. Shaheen, N. Jisrawi, Y.H. Min-Lee, W.L. McLean and M. Croft, Appl. Phys. Lett. 51 (1987) 619. [3] S.R. Foltyn, R.E. Muenchausen, R.C. Dye, X.D. Wu, L. Luo, D.W. Cooke and R.C. Taber, Appl. Phys. Lett. 59 (1991) 1374. [4] J. Greer and H.J. van Hook, MRS Proc. 191 (1990) 171. [5] J.C.S. Kools, PhD Thesis, Eindhoven University of Technology, Eindhoven, 1992.
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in LAD
139
Kools, T.S. Baller, S.T. De Zwart and J. Dieleman, J. Appl. Phys. 71 (1992) 4547. 171 A. Gupta, B. Braren, K.G. Casey, B.W. Husey and R. Kelly, Appl. Phys. Lett. 59 (1991) 1302. and P.L. Ventzek, Appl. Phys. Lett. 58 [81 R. Gilgenbach (1991) 1597. [91 A.A. Gorbunov and V.I. Konov, Sov. Phys. Tech. Phys. 34 (1989) 1271. S.V. Gaponov, V.I. Luchin and A.P. m A.D. Akhsakhalyan, Chirimanov, Sov. Phys. Tech. Phys. 33 (1988) 1146. [ill A. Namiki, T. Kawai and K. Ichige, Surf. Sci. 166 (1986) 129. C. Liang, S.A. Shaheen, M. [=I R.A. Neifeld, S. Gunapala, Croft, J. Price, D. Simons and W.T. Hill, Appl. Phys. Lett. 54 (1988) 954. [I31 A. Cheenne, J. Perriere, F. Kerherve, G. Hauchecorne, E. Fogarassy and C. Fuchs, MRS Proc. 191 (1990) 229. X.D. Wu, A. Inam and J.B. Wachtman, [141 T. Venkatesan, Appl. Phys. Lett. 52 (1988) 1193. E. van de iI51 J.C.S Kools, C.J.C. Nillesen, S.H. Brongersma, Riet and J. Dieleman, J. Vat. Sci. Technol. A 10 (1992) 1809. [161 C.N. Afonso, R. Serna, F. Catalina and D. Bermejo, Appl. Surf. Sci. 46 (1990) 249.