- Email: [email protected]
Simulation of cluster impacts on silicon surface
Nuclear Instruments
and Methods in Physics
ResearchB 127/128 (1997) 269-272
Simulation of cluster impacts on silicon surface
*,
Z. Insepov a, M. Sosnowski b3 I. Yamada a a Ion Beam Engineering Laboratory, Kyoto University, Sakyo, Kyoto 606. Japan b ECE Departmeni. New Jersey Institute of Technology, Newark, NJ 07102, USA Abstract A new hybrid model, combining Molecular Dynamics (MD) with continuum mechanics and thermodynamics, has been developed for studying collisions of energetic particles with a solid surface. MD describes interaction of atoms in the central impact zone characterized by energetic atomic collisions and non-equilibrium states of matter while the continuum model is applied to a much larger volume outside. Appropriate boundary conditions at the interface of the two regions prevent the appearance of unphysical shock wave reflections. The hybrid model is very efficient in computations as it reduces the number of the system’s degrees of freedom by minimizing the size of the central MD zone. The model was applied to collisions of a few keV Ar clusters containing approximately 100 atoms with Si(lO0) surface. The results show that cluster impacts create craters and local melting and that a number of displaced surface atoms have large lateral velocities. The latter may explain the experimentally observed surface smoothing by cluster bombardment.
1. Introduction Interactions of energetic clusters of atoms with a solid surface exhibit unique phenomena which are different from those observed with energetic monomers and hold promise for applications in surface modification. While experimental data are still sparse [l-3], modeling may help to evaluate these prospects and shed light on the mechanisms involved. Of particular interest are clusters of gaseous elements and compounds consisting of hundreds to thousands of atoms, with energies from a few eV to a few hundreds of eV per cluster atom. We have simulated impacts of Ar clusters of about 100 atoms, with energy up to 100 eV per cluster atom, on SXlOO) surface. In MD simulations equations of motion of interacting particles comprising the physical system being modeled are solved with appropriate boundary conditions. In modeling impacts of energetic clusters on a solid surface, all atoms of the cluster are included in the calculations while the solid target is represented by a “reasonable” number of atoms [4-61. The choice of the latter number is a critical decision in setting up a model: too few atoms may show unphysical effects due to the limited size of the systems but increasing their number has a steep price in computation time and required memory as both grow rapidly with increasing number of the system’s degrees of freedom. Usually, the number of the target atoms in the calculations * Corresponding author.
Fax:
201
596
5680;
e-mail:
[email protected]. 0168-583X/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved PII SO168-583X(96)00938-X
is limited by the available computing power, and the influence of rest of the system is simulated by proper boundaries which allow the flow of energy deposited by the impact to the bulk of the solid. One of the often used techniques, based on stochastic Langevin dynamics, employs two or three atomic layers surrounding the central volume treated by MD. These layers of damped atoms represent a “thermal bath” which simulates heat transfer to the bulk of the target. This technique is very useful in simulation of low energy processes, such as film deposition. Energetic cluster impacts create violent collisions between atoms in the central zone where equivalent temperature and pressure may reach lo5 K and lo6 bar, respectively [6]. For modeling of such events the boundary can be refined by allowing its expansion to keep the average pressure constant for a given modulus of elasticity. The technique is not completely satisfactory as it uses the average pressure, which depends on the system size, and requires the knowledge of materials characteristics, such as compressibility. These parameters cannot be reliably extrapolated from the normal equilibrium conditions to the extreme state of matter in the central collision zone. The problem of boundary conditions can also be examined by considering shock waves created by the energetic cluster impact. Unphysical reflections of the shock waves from the system boundary may show up in MD results, distorting the picture of the investigated process. A system as large as 4X 10’ atoms modeled by MD with periodic boundary conditions and the total impact energy of 10 kV suffers from shock wave reflections [7].
II. MODELING/SIMULATION/THEORY
210
2. Insepov et al./Nucl. Instr. and Meth. in Phys. Res. B 127/ 128 (1997) 269-272
2. The model
To solve these problems related to boundary conditions we have developed a hybrid model utilizing MD in the central collision zone, and continuum mechanics and thermodynamics outside. The shock wave theory is used in the model to establish the minimum size of the central zone, i.e. the location of the boundary between the volume where atomic collisions are modeled by MD and the outside, treated as continuum. The hybrid model reduces the number of the required degrees of freedom, significantly reducing computation time. According to the shock wave theory [8], the distance L from the center of the cluster impact to where the shock wave becomes thermally equilibrated can be estimated as L = 10’i3 d, where d is the cluster diameter. For argon clusters with 100 atoms d = 30 w and L is of the order of 100 A. The shock wave travels distance L in time T= L/v, where v is the cluster velocity. For argon clusters with kinetic energy of 20 eV per atom u = lo4 m/s and time T= 1 ps. The hybrid model establishes the boundary of MD volume at L and follows the system evolution with MD calculations for the time at least several times 7. The volume beyond radius L from the point of impact is treated as a continuum, using a finite elements method to solve continuum mechanics equations:
(la) where uj is the displacement vector of the jth cell, a,, is the stress tensor and p is the solid density [9]. Heat transfer is described by:
dT(r, t)
-
dt
= ,y AT(r,
t).
(lb)
where x is the thermal diffusivity and T the temperature at position r and time t. Eqs. (la) and (lb) are coupled by the dependence of stress tensor ojk on temperature:
(‘cl (‘4 where (Yis the thermal expansion coefficient, K and 5 the bulk elastic modulus and viscosity, p and n are the sheer modulus and viscosity while uik is the strain tensor. For silicon at room temperature x = 1 cm2/s [lo], so that the characteristic heat transfer time t = L2/,y = 1 ps is similar to the characteristic deformation time. This is convenient for numerical computation as it allows the use of common time steps for calculating thermal and mechanical variables.
Eqs. (1) have been used recently to solve analytically a one-dimensional problem of a heavy ion track in an amorphous target [I 11. Here they are used to solve numerically a two-dimensional problem in cylindrical geometry defined by the symmetry of the system. In cylindrical coordinates Eqs. ( 1) become: 1 dT(r,
X
z; t) dt
a2T(r, =
z; t)
ar2 +-
1 aT(r, r
z; t) ar
+
@T(r,
z; t) az2
’ @a>
aT piir=
-K%
ur
-I_2+;7+
I aur
a2u, -+Jr2
a2uz a22 )
G’b)
w> The boundary conditions, i.e. the values of displacements and temperature at the interface with the central volume were obtained from MD calculations. This is an important detail of the method which, as far as we know, is here applied for the first time to the study of impacts on a solid surface. Similar ideas were used recently in the studies of complex fluid flows [12]. In the MD calculations we used the Buckingham potential to represent two-body forces between cluster atoms and cluster atoms and atoms of the target while interactions between Si atoms were represented by the Stillinger-Weber potential. The cluster was modeled by cutting a spherical volume from an fee argon lattice with initial temperature set to zero. The clusters used in simulations contained about 100 atoms and had kinetic energy of a few keV. The cylindrical target model contained 25000 atoms in the central MD zone while the continuum mechanics calculations extended to ten times larger volume.
3. Results MD calculations give position and momenta of all cluster atoms and the target atoms in the central zone, which provides a wealth of information on the collision process and allow one to obtain a number of parameters of interest. Fig. 1 represents the time dependence of the target temperature, at its center CT,) and at the boundary of the
2. lnsepov et al./Nucl.
Instr. and Meth. in Phys. Res. B 127/
271
128 (1997) 269-272
Y ti
2 fd f w c
600
300
01 0
2
I
3 40
TIME, es Fig. 1. Time dependence of the mean temperature T, and the temperature T,,,, at the boundary of the MD volume. Circles show the temperature of the adjacent mesh points in the continuum.
central zone (T,). The latter values were taken as the boundary conditions for the continuum region calculation for different time steps, as shown in the figure by circles representing the temperature of the first layer of mesh cells in the continuum. Temperature values are averaged over z. Fig. 2 shows the angular distributions of the displaced target atoms at two different times after the beginning of a cluster impact. The cluster contained 79 Ar atoms with initial kinetic energy of 50 eV per atom. The initial stage of the impact (at 308 fs) is characterized by a remarkable lateral motion of the atoms. After several picoseconds the distribution changes as the velocity component normal to the surface increases. Fig. 2 shows also that the result is not very sensitive to the size of the central zone: the dotted line represents MD calculations with half the number of the target atoms. These results are different from those obtained with standard MD calculations, using the velocity mscaling technique or Langevin dynamics boundary conditions, which do not show significant emission from the surface at later stages of the impact. One of the remarkable and unexpected effects of energetic clusters is surface smoothing, which has been observed on a number of targets irradiated with gas cluster ions [2]. The effect is unlike that observed with monomer
0
15
30 N&n
45
60
tJ
Fig. 2. The number of displaced target atoms as a function of the angle between their velocity vector and the surface normal. The lower curve corresponds to 0.3 ns and the apper curves to 8.1 ns after the impact. The dotted line was obtained with half of the atoms in MD calculations.
-20
0
20
40
Y, A” Fig. 3. A typical crater in Si target 10 ps after an impact of Ar,, cluster with kinetic energy of 50 eV per atom.
ions irradiation and is clearly related to different kinetics of collisions of the two projectiles with a solid. Cluster impact results in transfer of kinetic energy to a large number of surface atoms, which leads to their displacement and local melting. Fig. 3 shows a crater, 10 ps after an impact of Ar,, cluster with kinetic energy of 50 eV per atom. Craters left after argon cluster impacts on a number of targets were observed with STM and AFM scanning techniques. The depth of craters and the height of the rims around them may define the residual surface roughness after cluster irradiation, which is on the order of 10 A. To explain smoothing of a surface, which is much rougher initially, we are considering two mechanisms: (1) “lateral sputtering” which would preferentially sputter the ridges and peaks of the surface topography and transport the material to the valleys, and (2) enhanced surface diffusion due to the presence of energetic surface atoms. MD simulations show that atoms with high lateral momenta emerge from the cluster impact zone and they may contribute to both of these mechanisms.
4. Summary A new hybrid model combining Molecular Dynamics (MD) with continuum mechanics and thermodynamics has been applied to studies of energetic cluster impacts on a solid surface. The method significantly reduces computation time and avoids problems of unphysical shock wave reflections from the edge of the system treated by MD with conventional boundary conditions. Impacts of Ar clusters with about 100 atoms and energies up to 100 eV/atom on Si(100) surface were modeled. The results show that the impacts impart high lateral momenta to the target atoms resulting in the characteristic shape of the angular distribution of the displaced atoms. The lateral emission of target atoms (“lateral sputtering”) from a Cu surface bombarded with argon clusters which has been observed experimentally has qualitatively similar distribution [13]. The phenomena may be also responsible for the
II. MODELING/SIMULATlON/THEORY
272
2. Insepov et al./Nucl.
Instr. andMeth.
experimentally observed surface smoothing with cluster ion beams. The details of the smoothing mechanisms remain yet to be worked out.
References
(111. Yamada, W.L. Brown, J.A. Northby and M. Sosnowski, Nucl. Instr. and Meth. B 82 (1993) 223.
I211. Yamada, J. Matsuo, Z. Insepov, D. Takeuchi, M. Akizuki and N. Toyoda, J. Vat. Sci. Technol. A 14 (1996) 1. 131I. Yamada and J. Matsuo, Proc. MRS Fall Meeting, November 1995, Boston and Proc. MRS Spring Meeting, April 1996, San Francisco, in print. [41H. Hsieh, R.S. Averback, H. Sellers and C.P. Flynn, Phys. Rev. B 4.5 (1992) 4417.
in Phys. Res. B 127/128
(1997) 269-272
[S] G.H. Gilmer and C. Rolland, Radiation Effects and Defects in Solid 130-131 (1994) 321. [6] Z. Insepov and 1. Yamada, Nucl. Instr. and Meth. B 99 (1995) 248. [7] M. Moseler, private communication. [8] Ya.B. Zeldovich and Yu.P. Raiser, Physics of shock waves and high-temperature (Academic Press, New York, 1967) p. 653. [9] L.D. Landau and E.M. Lifshitz, Theory of elastisity (Pergamon, Oxford, 1986). [lo] D.R. Lide, ed. Handbook of chemistry and physics (CRC, London, 1993) p. 4-l. [ 111A.I. Ryazanov, A.E. Volkov and S. Klaumnzer, Phys. Rev. B 51 (1995) 12107. 1121 S.T. O’Connel and P.A. Thompson, Phys. Rev. E 52 (1995) R5792. [13] I. Yamada, J. Matsuo, Z. Insepov, D. Takeuchi and N. Toyoda, J. Vat. Sci. Technol. A 14 (1996) 781.
and Methods in Physics
ResearchB 127/128 (1997) 269-272
Simulation of cluster impacts on silicon surface
*,
Z. Insepov a, M. Sosnowski b3 I. Yamada a a Ion Beam Engineering Laboratory, Kyoto University, Sakyo, Kyoto 606. Japan b ECE Departmeni. New Jersey Institute of Technology, Newark, NJ 07102, USA Abstract A new hybrid model, combining Molecular Dynamics (MD) with continuum mechanics and thermodynamics, has been developed for studying collisions of energetic particles with a solid surface. MD describes interaction of atoms in the central impact zone characterized by energetic atomic collisions and non-equilibrium states of matter while the continuum model is applied to a much larger volume outside. Appropriate boundary conditions at the interface of the two regions prevent the appearance of unphysical shock wave reflections. The hybrid model is very efficient in computations as it reduces the number of the system’s degrees of freedom by minimizing the size of the central MD zone. The model was applied to collisions of a few keV Ar clusters containing approximately 100 atoms with Si(lO0) surface. The results show that cluster impacts create craters and local melting and that a number of displaced surface atoms have large lateral velocities. The latter may explain the experimentally observed surface smoothing by cluster bombardment.
1. Introduction Interactions of energetic clusters of atoms with a solid surface exhibit unique phenomena which are different from those observed with energetic monomers and hold promise for applications in surface modification. While experimental data are still sparse [l-3], modeling may help to evaluate these prospects and shed light on the mechanisms involved. Of particular interest are clusters of gaseous elements and compounds consisting of hundreds to thousands of atoms, with energies from a few eV to a few hundreds of eV per cluster atom. We have simulated impacts of Ar clusters of about 100 atoms, with energy up to 100 eV per cluster atom, on SXlOO) surface. In MD simulations equations of motion of interacting particles comprising the physical system being modeled are solved with appropriate boundary conditions. In modeling impacts of energetic clusters on a solid surface, all atoms of the cluster are included in the calculations while the solid target is represented by a “reasonable” number of atoms [4-61. The choice of the latter number is a critical decision in setting up a model: too few atoms may show unphysical effects due to the limited size of the systems but increasing their number has a steep price in computation time and required memory as both grow rapidly with increasing number of the system’s degrees of freedom. Usually, the number of the target atoms in the calculations * Corresponding author.
Fax:
201
596
5680;
e-mail:
[email protected]. 0168-583X/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved PII SO168-583X(96)00938-X
is limited by the available computing power, and the influence of rest of the system is simulated by proper boundaries which allow the flow of energy deposited by the impact to the bulk of the solid. One of the often used techniques, based on stochastic Langevin dynamics, employs two or three atomic layers surrounding the central volume treated by MD. These layers of damped atoms represent a “thermal bath” which simulates heat transfer to the bulk of the target. This technique is very useful in simulation of low energy processes, such as film deposition. Energetic cluster impacts create violent collisions between atoms in the central zone where equivalent temperature and pressure may reach lo5 K and lo6 bar, respectively [6]. For modeling of such events the boundary can be refined by allowing its expansion to keep the average pressure constant for a given modulus of elasticity. The technique is not completely satisfactory as it uses the average pressure, which depends on the system size, and requires the knowledge of materials characteristics, such as compressibility. These parameters cannot be reliably extrapolated from the normal equilibrium conditions to the extreme state of matter in the central collision zone. The problem of boundary conditions can also be examined by considering shock waves created by the energetic cluster impact. Unphysical reflections of the shock waves from the system boundary may show up in MD results, distorting the picture of the investigated process. A system as large as 4X 10’ atoms modeled by MD with periodic boundary conditions and the total impact energy of 10 kV suffers from shock wave reflections [7].
II. MODELING/SIMULATION/THEORY
210
2. Insepov et al./Nucl. Instr. and Meth. in Phys. Res. B 127/ 128 (1997) 269-272
2. The model
To solve these problems related to boundary conditions we have developed a hybrid model utilizing MD in the central collision zone, and continuum mechanics and thermodynamics outside. The shock wave theory is used in the model to establish the minimum size of the central zone, i.e. the location of the boundary between the volume where atomic collisions are modeled by MD and the outside, treated as continuum. The hybrid model reduces the number of the required degrees of freedom, significantly reducing computation time. According to the shock wave theory [8], the distance L from the center of the cluster impact to where the shock wave becomes thermally equilibrated can be estimated as L = 10’i3 d, where d is the cluster diameter. For argon clusters with 100 atoms d = 30 w and L is of the order of 100 A. The shock wave travels distance L in time T= L/v, where v is the cluster velocity. For argon clusters with kinetic energy of 20 eV per atom u = lo4 m/s and time T= 1 ps. The hybrid model establishes the boundary of MD volume at L and follows the system evolution with MD calculations for the time at least several times 7. The volume beyond radius L from the point of impact is treated as a continuum, using a finite elements method to solve continuum mechanics equations:
(la) where uj is the displacement vector of the jth cell, a,, is the stress tensor and p is the solid density [9]. Heat transfer is described by:
dT(r, t)
-
dt
= ,y AT(r,
t).
(lb)
where x is the thermal diffusivity and T the temperature at position r and time t. Eqs. (la) and (lb) are coupled by the dependence of stress tensor ojk on temperature:
(‘cl (‘4 where (Yis the thermal expansion coefficient, K and 5 the bulk elastic modulus and viscosity, p and n are the sheer modulus and viscosity while uik is the strain tensor. For silicon at room temperature x = 1 cm2/s [lo], so that the characteristic heat transfer time t = L2/,y = 1 ps is similar to the characteristic deformation time. This is convenient for numerical computation as it allows the use of common time steps for calculating thermal and mechanical variables.
Eqs. (1) have been used recently to solve analytically a one-dimensional problem of a heavy ion track in an amorphous target [I 11. Here they are used to solve numerically a two-dimensional problem in cylindrical geometry defined by the symmetry of the system. In cylindrical coordinates Eqs. ( 1) become: 1 dT(r,
X
z; t) dt
a2T(r, =
z; t)
ar2 +-
1 aT(r, r
z; t) ar
+
@T(r,
z; t) az2
’ @a>
aT piir=
-K%
ur
-I_2+;7+
I aur
a2u, -+Jr2
a2uz a22 )
G’b)
w> The boundary conditions, i.e. the values of displacements and temperature at the interface with the central volume were obtained from MD calculations. This is an important detail of the method which, as far as we know, is here applied for the first time to the study of impacts on a solid surface. Similar ideas were used recently in the studies of complex fluid flows [12]. In the MD calculations we used the Buckingham potential to represent two-body forces between cluster atoms and cluster atoms and atoms of the target while interactions between Si atoms were represented by the Stillinger-Weber potential. The cluster was modeled by cutting a spherical volume from an fee argon lattice with initial temperature set to zero. The clusters used in simulations contained about 100 atoms and had kinetic energy of a few keV. The cylindrical target model contained 25000 atoms in the central MD zone while the continuum mechanics calculations extended to ten times larger volume.
3. Results MD calculations give position and momenta of all cluster atoms and the target atoms in the central zone, which provides a wealth of information on the collision process and allow one to obtain a number of parameters of interest. Fig. 1 represents the time dependence of the target temperature, at its center CT,) and at the boundary of the
2. lnsepov et al./Nucl.
Instr. and Meth. in Phys. Res. B 127/
271
128 (1997) 269-272
Y ti
2 fd f w c
600
300
01 0
2
I
3 40
TIME, es Fig. 1. Time dependence of the mean temperature T, and the temperature T,,,, at the boundary of the MD volume. Circles show the temperature of the adjacent mesh points in the continuum.
central zone (T,). The latter values were taken as the boundary conditions for the continuum region calculation for different time steps, as shown in the figure by circles representing the temperature of the first layer of mesh cells in the continuum. Temperature values are averaged over z. Fig. 2 shows the angular distributions of the displaced target atoms at two different times after the beginning of a cluster impact. The cluster contained 79 Ar atoms with initial kinetic energy of 50 eV per atom. The initial stage of the impact (at 308 fs) is characterized by a remarkable lateral motion of the atoms. After several picoseconds the distribution changes as the velocity component normal to the surface increases. Fig. 2 shows also that the result is not very sensitive to the size of the central zone: the dotted line represents MD calculations with half the number of the target atoms. These results are different from those obtained with standard MD calculations, using the velocity mscaling technique or Langevin dynamics boundary conditions, which do not show significant emission from the surface at later stages of the impact. One of the remarkable and unexpected effects of energetic clusters is surface smoothing, which has been observed on a number of targets irradiated with gas cluster ions [2]. The effect is unlike that observed with monomer
0
15
30 N&n
45
60
tJ
Fig. 2. The number of displaced target atoms as a function of the angle between their velocity vector and the surface normal. The lower curve corresponds to 0.3 ns and the apper curves to 8.1 ns after the impact. The dotted line was obtained with half of the atoms in MD calculations.
-20
0
20
40
Y, A” Fig. 3. A typical crater in Si target 10 ps after an impact of Ar,, cluster with kinetic energy of 50 eV per atom.
ions irradiation and is clearly related to different kinetics of collisions of the two projectiles with a solid. Cluster impact results in transfer of kinetic energy to a large number of surface atoms, which leads to their displacement and local melting. Fig. 3 shows a crater, 10 ps after an impact of Ar,, cluster with kinetic energy of 50 eV per atom. Craters left after argon cluster impacts on a number of targets were observed with STM and AFM scanning techniques. The depth of craters and the height of the rims around them may define the residual surface roughness after cluster irradiation, which is on the order of 10 A. To explain smoothing of a surface, which is much rougher initially, we are considering two mechanisms: (1) “lateral sputtering” which would preferentially sputter the ridges and peaks of the surface topography and transport the material to the valleys, and (2) enhanced surface diffusion due to the presence of energetic surface atoms. MD simulations show that atoms with high lateral momenta emerge from the cluster impact zone and they may contribute to both of these mechanisms.
4. Summary A new hybrid model combining Molecular Dynamics (MD) with continuum mechanics and thermodynamics has been applied to studies of energetic cluster impacts on a solid surface. The method significantly reduces computation time and avoids problems of unphysical shock wave reflections from the edge of the system treated by MD with conventional boundary conditions. Impacts of Ar clusters with about 100 atoms and energies up to 100 eV/atom on Si(100) surface were modeled. The results show that the impacts impart high lateral momenta to the target atoms resulting in the characteristic shape of the angular distribution of the displaced atoms. The lateral emission of target atoms (“lateral sputtering”) from a Cu surface bombarded with argon clusters which has been observed experimentally has qualitatively similar distribution [13]. The phenomena may be also responsible for the
II. MODELING/SIMULATlON/THEORY
272
2. Insepov et al./Nucl.
Instr. andMeth.
experimentally observed surface smoothing with cluster ion beams. The details of the smoothing mechanisms remain yet to be worked out.
References
(111. Yamada, W.L. Brown, J.A. Northby and M. Sosnowski, Nucl. Instr. and Meth. B 82 (1993) 223.
I211. Yamada, J. Matsuo, Z. Insepov, D. Takeuchi, M. Akizuki and N. Toyoda, J. Vat. Sci. Technol. A 14 (1996) 1. 131I. Yamada and J. Matsuo, Proc. MRS Fall Meeting, November 1995, Boston and Proc. MRS Spring Meeting, April 1996, San Francisco, in print. [41H. Hsieh, R.S. Averback, H. Sellers and C.P. Flynn, Phys. Rev. B 4.5 (1992) 4417.
in Phys. Res. B 127/128
(1997) 269-272
[S] G.H. Gilmer and C. Rolland, Radiation Effects and Defects in Solid 130-131 (1994) 321. [6] Z. Insepov and 1. Yamada, Nucl. Instr. and Meth. B 99 (1995) 248. [7] M. Moseler, private communication. [8] Ya.B. Zeldovich and Yu.P. Raiser, Physics of shock waves and high-temperature (Academic Press, New York, 1967) p. 653. [9] L.D. Landau and E.M. Lifshitz, Theory of elastisity (Pergamon, Oxford, 1986). [lo] D.R. Lide, ed. Handbook of chemistry and physics (CRC, London, 1993) p. 4-l. [ 111A.I. Ryazanov, A.E. Volkov and S. Klaumnzer, Phys. Rev. B 51 (1995) 12107. 1121 S.T. O’Connel and P.A. Thompson, Phys. Rev. E 52 (1995) R5792. [13] I. Yamada, J. Matsuo, Z. Insepov, D. Takeuchi and N. Toyoda, J. Vat. Sci. Technol. A 14 (1996) 781.