- Email: [email protected]
Depth of origin of atoms sputtered from crystalline targets
Nuclear Instruments and Methods in Physics Research B 180 (2001) 58±65
www.elsevier.nl/locate/nimb
Depth of origin of atoms sputtered from crystalline targets q M.H. Shapiro
a,b,*
, E. Trovato a, T.A. Tombrello
b
a
b
Department of Physics, California State University, P.O. Box 6866, Fullerton, CA 92834-6866, USA Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, CA 91125, USA
Abstract Recently, V.I. Shulga and W. Eckstein (Nucl. Instr. and Meth. B 145 (1998) 492) investigated the depth of origin of atoms sputtered from random elemental targets using the Monte Carlo code TRIM.SP and the lattice code OKSANA. They found that the mean depth of origin is proportional to N 0:86 , where N is the atomic density; and that the most probable escape depth is k0 =2, where k0 is the mean atomic distance. Since earlier molecular dynamics simulations with small crystalline elemental targets typically produced a most probable escape depth of zero (i.e., most sputtered atoms came from the topmost layer of the target), we have carried out new molecular dynamics simulations of sputtered atom escape depths with much larger crystalline targets. Our new results, which include the bcc targets Cs, Rb and W, as well as the fcc targets Cu and Au predict that the majority of sputtered atoms come from the ®rst atomic layer for the bcc(1 0 0), bcc(1 1 1), fcc(1 0 0) and fcc(1 1 1) targets studied. For the high-atomic density targets Cu, Au and W, the mean depth of origin of sputtered atoms typically is less than 0:25k0 . For the low-atomic density targets Cs and Rb, the mean depth of origin of sputtered atoms is considerably larger, and depends strongly on the crystal orientation. We show that the discrepancy between the single-crystal and amorphous target depth of origin values can be resolved by applying a simple correction to the single-crystal results. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 79.20)m; 79.20.Ap; 79.20.Rf Keywords: Sputtering; Sputtered atoms; Depth of origin
1. Introduction The mean depth of origin of sputtered atoms has been investigated for more than 30 years. q Supported in part by the National Science Foundation (grants DMR-9712538 at Cal State Fullerton and DMR9730893 at Caltech). Parts of this work were submitted by E. Trovato as senior thesis in physics at San Diego State University. * Corresponding author. Tel.: +1-714-278-3884; fax: +1-714278-5810. E-mail address: [email protected] (M.H. Shapiro).
However, only a relatively small number of experimental measurements of escape depth are available [2±8]. Instead, most of the available information has come from theoretical [9±15] and simulation studies [1,2,16±30]. One of the interesting features of the simulation results is a discrepancy between mean sputtering depths computed with crystalline targets and those computed with amorphous targets for the same element. One of the primary aims of this paper is to show that a simple correction eliminates that discrepancy.
0168-583X/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 0 1 ) 0 0 4 9 2 - X
M.H. Shapiro et al. / Nucl. Instr. and Meth. in Phys. Res. B 180 (2001) 58±65
Early simulation results with amorphous targets generally produced mean escape depths that were substantially smaller than the value predicted from transport theory [2]. Vicaneck et al. [10] suggested that the discrepancy could be traced to the use of too low a value of the power-law scattering parameter m in the transport theory calculations. Using a more realistic value of m they obtained a value of the mean escape depth that was a factor of two smaller than the previous results from transport theory [2]; namely x0 0:207=N ;
1 3
and N is in atoms=A . However, where x0 is in A even with this adjustment, the mean sputtering depths predicted by transport theory were signi®cantly larger than most simulation and experimental results that were available in the literature. For some cases Eq. (1) predicted mean depths of origin for sputtered atoms that were as much as a factor of ®ve larger than the simulation results [11]. Kelly and Oliva [11] used the multi-component sputtering theory of Sigmund et al. [13] to calculate the fractional emission from beyond the ®rst layer of a sputtering target. They found this to be f k0 4E5 A0 k0 , where 1=A0 L0 is a characteristic depth for sputtering, k0 is the mean atomic R1 spacing, and E5 z 1 t n exp zt dt. Then, using the results for f k obtained from binary collision simulations for Al, V, Cu, U, Ni and Ti targets, molecular dynamics simulations for Cu targets, and experimental results for Li±Cu, O±Ti, In±Ga and C±Ni targets they obtained values of L0 =k0 for about 25 dierent cases. Most of these values fell into a narrow range, and Kelly and Oliva concluded that the mean depth of origin for sputtered atoms could be represented adequately by x2 0:80 0:1k0 :
2
Since the atomic densities for solid targets can range from as little as 0.0086 for Cs to as much as 0.176 for the diamond structure of carbon, Eq. (1) predicts mean escape depths that range from 1.2 for diamond to 48.3 A for Cs, while Eq. (2) A for predicts a much narrower range ± from 0.45 A for Cs. Shulga and Eckstein [1] diamond to 3.9 A noted that most of the existing results for mean
59
sputtering depths were for targets that fell into a much narrower range of atomic densities (0:04 < N < 0:1). To provide additional information they carried out binary collision simulations on more than 40 targets (amorphous solids and liquids) with atomic densities that covered the full The great range from 0.0086 to 0.176 atoms/A. majority of their mean depth of origin results fell between 0:5k0 and 1:0k0 , with only the results for the lowest density targets lying outside this range. Although the experimental depth of origin data are sparse, the available results generally agree better with the predictions of Eq. (2) than those of Eq. (1). Examples include Wittmaack's recent work on silicon [8], which produced a value of 2:0 0:4 A for the mean sputtering depth of 40 eV Si atoms ejected at 48° to the surface; and, the results of Dumke et al. [4] and Hubbard et al. [5] on liquid Ga± In eutectic targets. The surface layer of these targets is >94% In, while the bulk composition is only 16.5 at.% In. Thus, by examining the relative sputtering yields of indium and gallium atoms mean sputtering depths can be determined directly. The results ran depending on the bombarding ged from 1.8 to 3.3 A, energy and target temperature. Again, these values are reasonably consistent with Eq. (2), but considerably smaller than the prediction of Eq. (1). Simulations with crystalline targets produce much smaller mean depths of origin than simulations with amorphous or liquid targets or the predictions of Eqs. (1) and (2). For example, an early molecular dynamics simulation of 5 keV Ar bombardment of Cu(1 0 0) [24] yielded a mean while a similar depth of origin of less than 0.2 A, simulation with a liquid copper target [26] gave a We believe mean depth of origin of about 1.34 A. that the dierence is due largely to the dierent way in which the top layer of the target is de®ned for crystalline and amorphous targets. For crystalline targets the top layer of the target is located while in an amorphous target at a depth of 0 A, atoms in the ®rst layer are spread uniformly from a to a depth of k0 . Thus, k0 =2 should be depth of 0 A added to depths of origin obtained from crystalline target simulations when they are compared with amorphous target simulation results. We test this assumption by applying the correction to new molecular dynamics simulation re-
60
M.H. Shapiro et al. / Nucl. Instr. and Meth. in Phys. Res. B 180 (2001) 58±65
Table 1 Potential parametersa Parameter
A (eV A) B (A) ra (A)
C0 (eV) C1 (eV/A) 2 ) C2 (eV=A 3 ) C3 (eV=A rb (A) De (eV) 1) b (A re (A) rc (A)
A (eV A) B (A) ra (A) C0 (eV) C1 (eV/A) 2 ) C2 (eV=A 3 ) C3 (eV=A rb (A)
Interaction Ar±Cu
Au±Cu
Cu±Cu
7515.7559 0.1035 2.50
32985.8164 0.07959 2.50
12108.7178 0.09603 1.500 370.1096 )350.7199 74.7303 4.1129 1.988 0.48056 1.40465 2.62768 5.00
Ar±Au
Au±Aub
Au±Au
20473.9551 0.08414 2.50
98843.7 0.06876 2.50
98843.7 0.06876 2.30 47.6734 )44.6822 13.9459 )1.4824 2.50 0.560 1.637 2.922 5.000
Ar±W
Au±W
W±W
19178.135 0.085386 2.50
84170.703 0.06950 2.50
78843.4453 0.070271 2.00 506.4165 )654.7341 283.3908 )40.9762 2.50 0.560 1.637 3.032 5.00
Ar±Cs
Au±Cs
Cs±Cs
14254.0195 0.09108 2.50
62559.3086 0.07283 2.50
45152.1289 0.07711 1.50 1260.8964 )1816.2227
De (eV) 1) b (A re (A) rc (A)
A (eV A) B (A) ra (A)
C0 (eV) C1 (eV/A) 2 ) C2 (eV=A 3 ) C3 (eV=A rb (A) De (eV) 1) b (A re (A) rc (A)
A (eV A) B (A) ra (A)
C0 (eV) C1 (eV/A)
M.H. Shapiro et al. / Nucl. Instr. and Meth. in Phys. Res. B 180 (2001) 58±65
61
Table 1 (Continued) Parameter
Interaction Ar±Cu
Au±Cu
2
) C2 (eV=A 3 ) C3 (eV=A rb (A) De (eV) 1) b (A re (A) rc (A)
A (eV A) B (A) ra (A) C0 (eV) C1 (eV/A) 2 ) C2 (eV=A 3 ) C3 (eV=A rb (A)
Cu±Cu 874.9900 )140.5958 2.00 0.04485 0.41569 7.557 10.00
Ar±Rb
Au±Rb
Rb±Rb
9589.0674 0.09877 2.50
42085.3516 0.07710 2.50
19710.8613 0.08854 1.50 810.7094 )1085.1523 481.0954 )70.1970 2.00 0.04644 0.42981 7.207 10.00
De (eV) 1) b (A re (A) rc (A)
a
Moliere: ij A=r0:35e 0:3r=B 0:55e 1:2r=B 0:1e 6r=B ; r < ra . Cubic spline: Vij C0 C1 r C2 r2 C3 r3 ; ra 6 r < rb . Morse: Vij De e 2b r rc 2e b r re ; rb 6 r < rc ; Vij 0; r P rc . b Incident ion.
sults for Cu, Au, W, Cs and Rb, and to a representative sample of existing simulation results for crystalline targets. These corrected results then are compared to amorphous target results from [1] for the same elements. 2. Simulation model The simulations reported in this paper were done with the SPUT2 molecular-dynamics code [31,32]. This code uses two-body potentials to describe the interactions between the incident ion and target atoms and the interactions between target atoms. Moliere potentials were used for the ion±atom interactions, while Moliere repulsive core potentials connected to attractive Morse wells with cubic splines were used for the atom± atom interactions. The potential parameters are given in Table 1. The Cu±Cu and Au±Au pa-
rameters have been used in previous simulations [33]. The Morse parameters for the bcc targets (Cs, Rb and W) were taken from [34]. The procedure outlined in [35] was used to connect the Morse wells to Moliere core potentials for these cases. To check for any projectile mass eects, simulations were run with both 5 keV Ar ions and 5 keV Au ions normally incident on the targets. The targets used in these simulations were signi®cantly larger than those used in the earlier simulations from which depth of origin results were obtained. Nine-layer targets were used for the Cu(1 1 1), Au(1 1 1) and W(1 1 1) simulations, 36-layer targets were used for Cs(1 1 1) and Rb(1 1 1), 12-layer targets were used for Cu(1 0 0) and Au(1 0 0), and 24-layer targets were used for Cs(1 0 0), Rb(1 0 0) and W(1 0 0). The (1 1 1) targets contained more than 12,000 atoms, while the (1 0 0) targets contained more than 11,000 atoms.
62
M.H. Shapiro et al. / Nucl. Instr. and Meth. in Phys. Res. B 180 (2001) 58±65
In contrast to the close-packed fcc targets, which can remain quasi-stable for several picoseconds when two-body potentials are used, the more open bcc targets tend to become spherical rather quickly. We were able to compensate for this eect by keeping the cuto time for the bcc simulations to 1 ps for Cs and Rb and to 750 fs for W, and by using a 2 eV threshold for placing an atom on the ``moving atom'' list in the SPUT2 code. (In the SPUT2 code forces are computed between a pair of atoms only if one or both members of the pair are moving. To be considered moving a minimum threshold energy must be transferred to the atom in a collision.) The output from the SPUT2 code includes information that identi®es the crystal layer in which each sputtered atom was located at the beginning of the simulation for that impact. The depth of the layer from which the sputtered atom originated was taken to be the ``depth of origin'' for that sputtered atom, even though the particular atom may have reached a deeper point in the target at some time before being sputtered. 3. Results Mean depth of origin (x) results for the 20 simulation cases studied are given in Table 2. In addition, we have included in Table 2 simulation results for six additional single-crystal cases reported in the
literature [16,17,24,36]. The depths of origin re for the depth ported in Table 2 are referenced to 0 A of the ®rst layer of each target. The dierence between the mean depths of origin obtained with Ar bombardment and those obtained with Au bombardment for the same target are relatively small. As can be seen in the table our Cu depth of origin results are in good agreement with the values obtained in earlier simulations that were done with much smaller targets. In fact, all the single-crystal Cu depth of origin results are less than 0.2 A. Our single-crystal Au depth-of-origin results are in reasonably good agreement with the only single-crystal result for Au available from the literature (700 eV Xe bombardment of an Au(1 0 0) surface [36]). In most cases x is larger for the (1 0 0) target orientation than for the more densely packed (1 1 1) orientation. This is much more pronounced for the three bcc targets than for the two fcc targets. In Table 3, k0 =2 has been added to the values of x from Table 2, so that our depth-of-origin results and those from the earlier single-crystal simulations can be compared to results from the amorphous target simulations of [1] and to the predictions of Eqs. (1) and (2). The analytical sputtering theory predictions from Eq. (1) signi®cantly overestimate the mean depth of origin for all of the targets in the table. Eq. (2), which was obtained semi-empirically, tends to overestimate the
Table 2 Mean depth of origin of sputtered atoms from single-crystal targets (uncorrected)a Incident ion target Orientation/target
Ar(1 0 0) (A)
Au(1 0 0) (A)
Ar(1 1 1) (A)
Au(1 1 1) (A)
Cu Cub Cuc Cud Au Aue W Rb Cs
0.151 0.000
0.160
0.125
0.199 0.179 1.531 2.484
0.195 0.273
0.307
0.091 0.083 0.187 0.111 0.200
1.498 5.112 5.421
1.319 4.310 7.370
0.187 1.854 1.846
Referenced to a ®rst layer located at 0.0 A. Simulation with 600 eV Ar [16]. c Simulation with 8 keV Ar [17]. d Simulation with 5 kev Ar [24]. e Simulation with 700 eV Xe [36]. a
b
Xe(1 0 0) (A)
0.184
M.H. Shapiro et al. / Nucl. Instr. and Meth. in Phys. Res. B 180 (2001) 58±65
63
Table 3 Mean depth of origin of sputtered atoms from single-crystal targets (corrected)a Incident ion target Orientation/target
Ar(1 0 0) (A)
Au(1 0 0) (A)
Cu Cuc Cud Cue Au Auf W Rb Cs
1.290 1.138
1.299
1.334 1.558
1.592
2.751 7.374 7.872
2.572 6.572 9.821
Xe(1 0 0) (A)
1.468
Eq. (2) (A)
Ref. [1]b (A)
2.450
1.822
0.952
1.484
3.518
2.055
1.30
1.432 3.793 4.935
3.263 19.21 24.44
2.004 3.619 3.922
1.22 5.60 6.88
Ar(1 1 1) (A)
Au(1 1 1) (A)
1.230 1.222 1.326 1.250 1.484
1.264
1.440 4.116 4.297
Eq. (1) (A)
Referenced to a ®rst layer located at k0 /2 A. The values from [1] were obtained from the ®t to the results of amorphous target simulations carried out at 1 keV bombarding energy. c Simulation with 600 eV Ar [16]. d Simulation with 8 keV Ar [17]. e Simulation with 5 kev Ar [24]. f Simulation with 700 eV Xe [36]. a
b
mean depths of origin for the high-density targets (Au, Cu and W); and, it underestimates the mean depths of origin for the two low-density targets (Cs and Rb). Even with this systematic deviation, the predictions of Eq. (2) are closer to the simulation results than the predictions from Eq. (1). In Table 4, we compare averages of the corrected single-crystal results for each target to the corresponding values from Eqs. (1) and (2), and [1]. The amorphous target simulation results from [1] are in good agreement with the averaged singlecrystal results when the k0 =2 correction is included. With the exception of tungsten, where the amorphous target simulation result is about 40% lower than the averaged single-crystal results, the agreement is better than 25%. The theoretical predictions from Eq. (1) overestimate the averaged single-crystal results by more than a factor of two in all cases. The semi-empirical predictions from
Eq. (2) deviate systematically from the averaged single-crystal results, but by not more than about 40% over the range of target densities3 covered 3 (Cs)±0:085 atoms=A ). How(0:0085 atoms=A ever, the N 1 density dependence of Eq. (1) is in better agreement with the simulation results than the N 1=3 dependence of Eq. (2). 4. Discussion 4.1. Most probable escape depths Shulga and Eckstein [1] note that the most probable escape depth in their simulations with amorphous targets for normal incidence is about k0 =2. They argue from this observation that ``the topmost atoms are not necessarily the great majority of sputtered atoms''. However, in our
Table 4 Comparison of averaged corrected single target results with theoretical results and amorphous target simulations Target
Averaged results from Table 3 (A)
Eq. (1) (A)
Eq. (2) (A)
Ref. [1] (A)
Cu Au W Rb Cs
1.26 1.52 2.04 5.46 6.73
2.450 3.518 3.263 19.21 24.44
1.822 2.055 2.004 3.619 3.922
0.952 1.30 1.22 5.60 6.88
64
M.H. Shapiro et al. / Nucl. Instr. and Meth. in Phys. Res. B 180 (2001) 58±65
single-crystal simulations and in those previously published [16,17,24,36] the majority of sputtered atoms are found to arise from the ®rst layer of the target, even for the low-density bcc targets Rb and Cs. Thus, the single-crystal simulations yield a However, the most probable escape depth of 0 A. same correction ± namely, adding k0 =2 to the single-crystal value brings both single-crystal and amorphous target results into agreement. When the diuse nature of amorphous targets is taken into account, there is little dierence in either mean or most probable depths of origin as computed with crystalline and amorphous targets, provided that crystalline target results are corrected by adding k0 =2. Thus, both the single-crystal results and the amorphous target results show that the majority of sputtered atoms come from the ®rst layer of target atoms. 4.2. Density dependence of the depth of origin Shulga and Eckstein ®nd that the dependence of their mean depth of origin data on atomic density is proportional to N 0:86 . They suggest that this is consistent with the predictions of analytical sputtering theory given by Eq. (1). In the lower panel of Fig. 1 we have plotted all the corrected single-crystal mean depth of origin results from Table 3. We ®nd that the density dependence for the (1 0 0) targets is proportional to N 0:80 , which compares quite well with the N 0:86 dependence found by Shulga and Eckstein. However, the density dependence for the (1 1 1) targets is proportional to N 0:57 . This may be related to the relative opacities of the (1 0 0) and (1 1 1) surfaces. The bcc(1 0 0) surfaces are signi®cantly less opaque than the bcc(1 1 1) surfaces, and there are large systematic dierences in the mean depths of origin for the two crystal orientations for all three bcc targets studied. This can be seen clearly in the upper panel of Fig. 1 where the (1 0 0) and (1 1 1) results have been averaged for each of the targets investigated. In contrast, there are only slight differences between the two orientations of the fcc targets. In Fig. 2 we have plotted the averaged and corrected single-crystal results from Table 4 along with the N 0:86 curve that ®ts the amorphous target
Fig. 1. (a) Corrected mean depth-of-origin results for atoms sputtered from all the (1 0 0) and (1 1 1) targets listed in Table 3. The symbols represent the following: (}) Ar projectiles on (1 0 0) targets, (+) Au projectiles on (1 0 0) targets, () Xe projectiles on Au(1 1 1), () Ar projectiles on (1 1 1) targets, (M) Au projectiles on (1 1 1) targets. The straight lines represent power-law ®ts to the results. The (1 0 0) results are proportional to N 0:80 , while the (1 1 1) results are proportional to N 0:57 . (b) Averaged, corrected (1 0 0) and (1 1 1) single-crystal mean depth of origin results from Table 3. In this case (}) represents (1 0 0) targets, and (+) represent (1 1 1) targets. The straight lines have the same meaning as in the lower panel.
results from [1]. As we noted in Section 3, the agreement between our corrected single-crystal results and the amorphous target results of Shulga and Eckstein [1] is quite good. Neither the depth of origin relationship proposed by Vicanek et al. [10], nor that proposed by Kelly and Oliva [11] adequately ®t the simulation results (cf. Table 4). However, the N 1 density dependence obtained by Vicanek et al. [10] is a much closer approximation to both our averaged singlecrystal simulation results and the amorphous target simulation results than the N 1=3 density dependence suggested by Kelly and Oliva [11].
M.H. Shapiro et al. / Nucl. Instr. and Meth. in Phys. Res. B 180 (2001) 58±65
Fig. 2. Averaged, corrected single-crystal mean depth of origin results from Table 4. The straight line represents the N 0:86 density dependence obtained in [1].
5. Summary There are three principal conclusions from our results. First, the discrepancy between depth of origin results from simulations with crystalline and amorphous targets can be traced to dierence in the de®nition of the location of atoms in the top layer of the targets. When appropriate corrections are made for this dierence, both crystalline and amorphous target simulation results are in good agreement. Second, depth of origin results from bcc crystalline targets appear to be dependent on crystal orientation. The more opaque bcc(1 1 1) targets exhibit signi®cantly smaller mean depths of origin than the corresponding bcc(1 0 0) targets. Third, the absolute values of x predicted by Eq. (1) clearly are too large; however, the N 1 density dependence from the analytic theory is in better agreement with the simulation results than the N 1=3 dependence of Eq. (2). References [1] V.I. Shulga, W. Eckstein, Nucl. Instr. and Meth. B 145 (1998) 492. [2] A.R. Krauss, D.M. Gruen, A.B. DeWald, J. Nucl. Mater. 121 (1984) 398. [3] M.J. Pellin, C.E. Young, D.M. Gruen, Y. Aratono, A.B. Dewald, Surf. Sci. 151 (1985) 477. [4] M.F. Dumke, T.A. Tombrello, R.A. Weller, R.M. Housley, E.H. Cirlin, Surf. Sci. 124 (1983) 407.
65
[5] K.M. Hubbard, R.A. Weller, D.L. Weathers, T.A. Tombrello, Nucl. Instr. and Meth. B 36 (1989) 395. [6] T.B. Lill, W.F. Callaway, M.J. Pellin, D.M. Gruen, Phys. Rev. Lett. 73 (1994) 1719. [7] T.B. Lill, W.F. Callaway, Z. Ma, M.J. Pellin, Surf. Sci. 322 (1995) 361. [8] K. Wittmaack, Phys. Rev. B 56 (1997) R5701. [9] P. Sigmund, Phys. Rev. 184 (1969) 383. [10] M. Vicanek, J.J. Jimenez-Rodriguez, P. Sigmund, Nucl. Instr. and Meth. B 36 (1989) 124. [11] R. Kelly, A. Oliva, Nucl. Instr. and Meth. B 13 (1989) 283. [12] G. Falcone, P. Sigmund, Appl. Phys. 25 (1981) 307. [13] P. Sigmund, A. Oliva, G. Falcone, Nucl. Instr. and Meth. B 194 (1982) 541. [14] M. Vicanek, M.W. Sckerl, P. Sigmund, Nucl. Instr. and Meth. B 140 (1998) 61. [15] Z.L. Zhang, Nucl. Instr. and Meth. B 149 (1999) 272. [16] D.E. Harrison Jr., P.W. Kelly, B.J. Garrison, N. Winograd, Surf. Sci. 76 (1978) 311. [17] D.E. Harrison Jr., in: P. Varga, G. Betz, F.P. Viehb ock (Eds.), Inst. f ur allgemeine Physik, Technische Univ., Vienna, 1980, p. 36. [18] D.M. Gruen, A.R. Krauss, M.H. Mendelsohn, S. Susman, J. Nucl. Mater. 111±112 (1982) 831. [19] A.R. Kraus, D.M. Gruen, N.Q. Lam, A.B. DeWald, J. Nucl. Mater. 128±129 (1984) 570. [20] M.T. Robinson, J. Appl. Phys. 54 (1983) 2650. [21] M. Rosen, G.P. Mueller, W.A. Fraser, Nucl. Instr. and Meth. B 209±210 (1983) 63. [22] M. Rosen, R.H. Bassel, Nucl. Instr. and Meth. B 2 (1984) 592. [23] J.P. Biersack, W. Eckstein, Appl. Phys. A 34 (1984) 73. [24] M.H. Shapiro, P.K. Ha, T.A. Tombrello, D.E. Harrison, R.P. Webb, Radiat. E. 89 (1985) 243. [25] M.J. Pellin, C.E. Young, D.M. Gruen, Y. Aratono, A.B. Dewald, Surf. Sci. 151 (1985) 477. [26] M.H. Shapiro, D.Y. Lo, P.K. Ha, T.A. Tombrello, Nucl. Instr. and Meth. B 13 (1986) 348. [27] L.G. Glazov, V.I. Shulga, P. Sigmund, Surf. Interface Anal. 26 (1998) 512. [28] V.I. Shulga, Nucl. Instr. and Meth. B 152 (1999) 49. [29] G. Betz, R. Kirchner, W. Husinsky, F. Rudenauer, H.M. Urbassek, Radiat. E. Def. Sol. 130 (1994) 251. [30] M.H. Shapiro, K.R. Bengtson, T.A. Tombrello, Nucl. Instr. and Meth B 103 (1995) 123. [31] M.H. Shapiro, T.A. Tombrello, D.E. Harrison Jr., Nucl. Instr. and Meth. B 30 (1988) 152. [32] M.H. Shapiro, T.A. Tombrello, Nucl. Instr. and Meth. B 84 (1994) 453. [33] M.H. Shapiro, T.A. Tombrello, Nucl. Instr. and Meth. B 58 (1991) 161 (Section 4.4). [34] I.M. Torrens, Interatomic Potentials, Academic Press, New York, 1972 (Chapter 4). [35] W. Eckstein, Computer Simulation of Ion±Solid Interactions, Springer, Berlin, 1991. [36] M.J. Pellin, J.W. Burnett, Pure Appl. Chem. 65 (1993) 2361 (Table 3).
www.elsevier.nl/locate/nimb
Depth of origin of atoms sputtered from crystalline targets q M.H. Shapiro
a,b,*
, E. Trovato a, T.A. Tombrello
b
a
b
Department of Physics, California State University, P.O. Box 6866, Fullerton, CA 92834-6866, USA Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, CA 91125, USA
Abstract Recently, V.I. Shulga and W. Eckstein (Nucl. Instr. and Meth. B 145 (1998) 492) investigated the depth of origin of atoms sputtered from random elemental targets using the Monte Carlo code TRIM.SP and the lattice code OKSANA. They found that the mean depth of origin is proportional to N 0:86 , where N is the atomic density; and that the most probable escape depth is k0 =2, where k0 is the mean atomic distance. Since earlier molecular dynamics simulations with small crystalline elemental targets typically produced a most probable escape depth of zero (i.e., most sputtered atoms came from the topmost layer of the target), we have carried out new molecular dynamics simulations of sputtered atom escape depths with much larger crystalline targets. Our new results, which include the bcc targets Cs, Rb and W, as well as the fcc targets Cu and Au predict that the majority of sputtered atoms come from the ®rst atomic layer for the bcc(1 0 0), bcc(1 1 1), fcc(1 0 0) and fcc(1 1 1) targets studied. For the high-atomic density targets Cu, Au and W, the mean depth of origin of sputtered atoms typically is less than 0:25k0 . For the low-atomic density targets Cs and Rb, the mean depth of origin of sputtered atoms is considerably larger, and depends strongly on the crystal orientation. We show that the discrepancy between the single-crystal and amorphous target depth of origin values can be resolved by applying a simple correction to the single-crystal results. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 79.20)m; 79.20.Ap; 79.20.Rf Keywords: Sputtering; Sputtered atoms; Depth of origin
1. Introduction The mean depth of origin of sputtered atoms has been investigated for more than 30 years. q Supported in part by the National Science Foundation (grants DMR-9712538 at Cal State Fullerton and DMR9730893 at Caltech). Parts of this work were submitted by E. Trovato as senior thesis in physics at San Diego State University. * Corresponding author. Tel.: +1-714-278-3884; fax: +1-714278-5810. E-mail address: [email protected] (M.H. Shapiro).
However, only a relatively small number of experimental measurements of escape depth are available [2±8]. Instead, most of the available information has come from theoretical [9±15] and simulation studies [1,2,16±30]. One of the interesting features of the simulation results is a discrepancy between mean sputtering depths computed with crystalline targets and those computed with amorphous targets for the same element. One of the primary aims of this paper is to show that a simple correction eliminates that discrepancy.
0168-583X/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 0 1 ) 0 0 4 9 2 - X
M.H. Shapiro et al. / Nucl. Instr. and Meth. in Phys. Res. B 180 (2001) 58±65
Early simulation results with amorphous targets generally produced mean escape depths that were substantially smaller than the value predicted from transport theory [2]. Vicaneck et al. [10] suggested that the discrepancy could be traced to the use of too low a value of the power-law scattering parameter m in the transport theory calculations. Using a more realistic value of m they obtained a value of the mean escape depth that was a factor of two smaller than the previous results from transport theory [2]; namely x0 0:207=N ;
1 3
and N is in atoms=A . However, where x0 is in A even with this adjustment, the mean sputtering depths predicted by transport theory were signi®cantly larger than most simulation and experimental results that were available in the literature. For some cases Eq. (1) predicted mean depths of origin for sputtered atoms that were as much as a factor of ®ve larger than the simulation results [11]. Kelly and Oliva [11] used the multi-component sputtering theory of Sigmund et al. [13] to calculate the fractional emission from beyond the ®rst layer of a sputtering target. They found this to be f k0 4E5 A0 k0 , where 1=A0 L0 is a characteristic depth for sputtering, k0 is the mean atomic R1 spacing, and E5 z 1 t n exp zt dt. Then, using the results for f k obtained from binary collision simulations for Al, V, Cu, U, Ni and Ti targets, molecular dynamics simulations for Cu targets, and experimental results for Li±Cu, O±Ti, In±Ga and C±Ni targets they obtained values of L0 =k0 for about 25 dierent cases. Most of these values fell into a narrow range, and Kelly and Oliva concluded that the mean depth of origin for sputtered atoms could be represented adequately by x2 0:80 0:1k0 :
2
Since the atomic densities for solid targets can range from as little as 0.0086 for Cs to as much as 0.176 for the diamond structure of carbon, Eq. (1) predicts mean escape depths that range from 1.2 for diamond to 48.3 A for Cs, while Eq. (2) A for predicts a much narrower range ± from 0.45 A for Cs. Shulga and Eckstein [1] diamond to 3.9 A noted that most of the existing results for mean
59
sputtering depths were for targets that fell into a much narrower range of atomic densities (0:04 < N < 0:1). To provide additional information they carried out binary collision simulations on more than 40 targets (amorphous solids and liquids) with atomic densities that covered the full The great range from 0.0086 to 0.176 atoms/A. majority of their mean depth of origin results fell between 0:5k0 and 1:0k0 , with only the results for the lowest density targets lying outside this range. Although the experimental depth of origin data are sparse, the available results generally agree better with the predictions of Eq. (2) than those of Eq. (1). Examples include Wittmaack's recent work on silicon [8], which produced a value of 2:0 0:4 A for the mean sputtering depth of 40 eV Si atoms ejected at 48° to the surface; and, the results of Dumke et al. [4] and Hubbard et al. [5] on liquid Ga± In eutectic targets. The surface layer of these targets is >94% In, while the bulk composition is only 16.5 at.% In. Thus, by examining the relative sputtering yields of indium and gallium atoms mean sputtering depths can be determined directly. The results ran depending on the bombarding ged from 1.8 to 3.3 A, energy and target temperature. Again, these values are reasonably consistent with Eq. (2), but considerably smaller than the prediction of Eq. (1). Simulations with crystalline targets produce much smaller mean depths of origin than simulations with amorphous or liquid targets or the predictions of Eqs. (1) and (2). For example, an early molecular dynamics simulation of 5 keV Ar bombardment of Cu(1 0 0) [24] yielded a mean while a similar depth of origin of less than 0.2 A, simulation with a liquid copper target [26] gave a We believe mean depth of origin of about 1.34 A. that the dierence is due largely to the dierent way in which the top layer of the target is de®ned for crystalline and amorphous targets. For crystalline targets the top layer of the target is located while in an amorphous target at a depth of 0 A, atoms in the ®rst layer are spread uniformly from a to a depth of k0 . Thus, k0 =2 should be depth of 0 A added to depths of origin obtained from crystalline target simulations when they are compared with amorphous target simulation results. We test this assumption by applying the correction to new molecular dynamics simulation re-
60
M.H. Shapiro et al. / Nucl. Instr. and Meth. in Phys. Res. B 180 (2001) 58±65
Table 1 Potential parametersa Parameter
A (eV A) B (A) ra (A)
C0 (eV) C1 (eV/A) 2 ) C2 (eV=A 3 ) C3 (eV=A rb (A) De (eV) 1) b (A re (A) rc (A)
A (eV A) B (A) ra (A) C0 (eV) C1 (eV/A) 2 ) C2 (eV=A 3 ) C3 (eV=A rb (A)
Interaction Ar±Cu
Au±Cu
Cu±Cu
7515.7559 0.1035 2.50
32985.8164 0.07959 2.50
12108.7178 0.09603 1.500 370.1096 )350.7199 74.7303 4.1129 1.988 0.48056 1.40465 2.62768 5.00
Ar±Au
Au±Aub
Au±Au
20473.9551 0.08414 2.50
98843.7 0.06876 2.50
98843.7 0.06876 2.30 47.6734 )44.6822 13.9459 )1.4824 2.50 0.560 1.637 2.922 5.000
Ar±W
Au±W
W±W
19178.135 0.085386 2.50
84170.703 0.06950 2.50
78843.4453 0.070271 2.00 506.4165 )654.7341 283.3908 )40.9762 2.50 0.560 1.637 3.032 5.00
Ar±Cs
Au±Cs
Cs±Cs
14254.0195 0.09108 2.50
62559.3086 0.07283 2.50
45152.1289 0.07711 1.50 1260.8964 )1816.2227
De (eV) 1) b (A re (A) rc (A)
A (eV A) B (A) ra (A)
C0 (eV) C1 (eV/A) 2 ) C2 (eV=A 3 ) C3 (eV=A rb (A) De (eV) 1) b (A re (A) rc (A)
A (eV A) B (A) ra (A)
C0 (eV) C1 (eV/A)
M.H. Shapiro et al. / Nucl. Instr. and Meth. in Phys. Res. B 180 (2001) 58±65
61
Table 1 (Continued) Parameter
Interaction Ar±Cu
Au±Cu
2
) C2 (eV=A 3 ) C3 (eV=A rb (A) De (eV) 1) b (A re (A) rc (A)
A (eV A) B (A) ra (A) C0 (eV) C1 (eV/A) 2 ) C2 (eV=A 3 ) C3 (eV=A rb (A)
Cu±Cu 874.9900 )140.5958 2.00 0.04485 0.41569 7.557 10.00
Ar±Rb
Au±Rb
Rb±Rb
9589.0674 0.09877 2.50
42085.3516 0.07710 2.50
19710.8613 0.08854 1.50 810.7094 )1085.1523 481.0954 )70.1970 2.00 0.04644 0.42981 7.207 10.00
De (eV) 1) b (A re (A) rc (A)
a
Moliere: ij A=r0:35e 0:3r=B 0:55e 1:2r=B 0:1e 6r=B ; r < ra . Cubic spline: Vij C0 C1 r C2 r2 C3 r3 ; ra 6 r < rb . Morse: Vij De e 2b r rc 2e b r re ; rb 6 r < rc ; Vij 0; r P rc . b Incident ion.
sults for Cu, Au, W, Cs and Rb, and to a representative sample of existing simulation results for crystalline targets. These corrected results then are compared to amorphous target results from [1] for the same elements. 2. Simulation model The simulations reported in this paper were done with the SPUT2 molecular-dynamics code [31,32]. This code uses two-body potentials to describe the interactions between the incident ion and target atoms and the interactions between target atoms. Moliere potentials were used for the ion±atom interactions, while Moliere repulsive core potentials connected to attractive Morse wells with cubic splines were used for the atom± atom interactions. The potential parameters are given in Table 1. The Cu±Cu and Au±Au pa-
rameters have been used in previous simulations [33]. The Morse parameters for the bcc targets (Cs, Rb and W) were taken from [34]. The procedure outlined in [35] was used to connect the Morse wells to Moliere core potentials for these cases. To check for any projectile mass eects, simulations were run with both 5 keV Ar ions and 5 keV Au ions normally incident on the targets. The targets used in these simulations were signi®cantly larger than those used in the earlier simulations from which depth of origin results were obtained. Nine-layer targets were used for the Cu(1 1 1), Au(1 1 1) and W(1 1 1) simulations, 36-layer targets were used for Cs(1 1 1) and Rb(1 1 1), 12-layer targets were used for Cu(1 0 0) and Au(1 0 0), and 24-layer targets were used for Cs(1 0 0), Rb(1 0 0) and W(1 0 0). The (1 1 1) targets contained more than 12,000 atoms, while the (1 0 0) targets contained more than 11,000 atoms.
62
M.H. Shapiro et al. / Nucl. Instr. and Meth. in Phys. Res. B 180 (2001) 58±65
In contrast to the close-packed fcc targets, which can remain quasi-stable for several picoseconds when two-body potentials are used, the more open bcc targets tend to become spherical rather quickly. We were able to compensate for this eect by keeping the cuto time for the bcc simulations to 1 ps for Cs and Rb and to 750 fs for W, and by using a 2 eV threshold for placing an atom on the ``moving atom'' list in the SPUT2 code. (In the SPUT2 code forces are computed between a pair of atoms only if one or both members of the pair are moving. To be considered moving a minimum threshold energy must be transferred to the atom in a collision.) The output from the SPUT2 code includes information that identi®es the crystal layer in which each sputtered atom was located at the beginning of the simulation for that impact. The depth of the layer from which the sputtered atom originated was taken to be the ``depth of origin'' for that sputtered atom, even though the particular atom may have reached a deeper point in the target at some time before being sputtered. 3. Results Mean depth of origin (x) results for the 20 simulation cases studied are given in Table 2. In addition, we have included in Table 2 simulation results for six additional single-crystal cases reported in the
literature [16,17,24,36]. The depths of origin re for the depth ported in Table 2 are referenced to 0 A of the ®rst layer of each target. The dierence between the mean depths of origin obtained with Ar bombardment and those obtained with Au bombardment for the same target are relatively small. As can be seen in the table our Cu depth of origin results are in good agreement with the values obtained in earlier simulations that were done with much smaller targets. In fact, all the single-crystal Cu depth of origin results are less than 0.2 A. Our single-crystal Au depth-of-origin results are in reasonably good agreement with the only single-crystal result for Au available from the literature (700 eV Xe bombardment of an Au(1 0 0) surface [36]). In most cases x is larger for the (1 0 0) target orientation than for the more densely packed (1 1 1) orientation. This is much more pronounced for the three bcc targets than for the two fcc targets. In Table 3, k0 =2 has been added to the values of x from Table 2, so that our depth-of-origin results and those from the earlier single-crystal simulations can be compared to results from the amorphous target simulations of [1] and to the predictions of Eqs. (1) and (2). The analytical sputtering theory predictions from Eq. (1) signi®cantly overestimate the mean depth of origin for all of the targets in the table. Eq. (2), which was obtained semi-empirically, tends to overestimate the
Table 2 Mean depth of origin of sputtered atoms from single-crystal targets (uncorrected)a Incident ion target Orientation/target
Ar(1 0 0) (A)
Au(1 0 0) (A)
Ar(1 1 1) (A)
Au(1 1 1) (A)
Cu Cub Cuc Cud Au Aue W Rb Cs
0.151 0.000
0.160
0.125
0.199 0.179 1.531 2.484
0.195 0.273
0.307
0.091 0.083 0.187 0.111 0.200
1.498 5.112 5.421
1.319 4.310 7.370
0.187 1.854 1.846
Referenced to a ®rst layer located at 0.0 A. Simulation with 600 eV Ar [16]. c Simulation with 8 keV Ar [17]. d Simulation with 5 kev Ar [24]. e Simulation with 700 eV Xe [36]. a
b
Xe(1 0 0) (A)
0.184
M.H. Shapiro et al. / Nucl. Instr. and Meth. in Phys. Res. B 180 (2001) 58±65
63
Table 3 Mean depth of origin of sputtered atoms from single-crystal targets (corrected)a Incident ion target Orientation/target
Ar(1 0 0) (A)
Au(1 0 0) (A)
Cu Cuc Cud Cue Au Auf W Rb Cs
1.290 1.138
1.299
1.334 1.558
1.592
2.751 7.374 7.872
2.572 6.572 9.821
Xe(1 0 0) (A)
1.468
Eq. (2) (A)
Ref. [1]b (A)
2.450
1.822
0.952
1.484
3.518
2.055
1.30
1.432 3.793 4.935
3.263 19.21 24.44
2.004 3.619 3.922
1.22 5.60 6.88
Ar(1 1 1) (A)
Au(1 1 1) (A)
1.230 1.222 1.326 1.250 1.484
1.264
1.440 4.116 4.297
Eq. (1) (A)
Referenced to a ®rst layer located at k0 /2 A. The values from [1] were obtained from the ®t to the results of amorphous target simulations carried out at 1 keV bombarding energy. c Simulation with 600 eV Ar [16]. d Simulation with 8 keV Ar [17]. e Simulation with 5 kev Ar [24]. f Simulation with 700 eV Xe [36]. a
b
mean depths of origin for the high-density targets (Au, Cu and W); and, it underestimates the mean depths of origin for the two low-density targets (Cs and Rb). Even with this systematic deviation, the predictions of Eq. (2) are closer to the simulation results than the predictions from Eq. (1). In Table 4, we compare averages of the corrected single-crystal results for each target to the corresponding values from Eqs. (1) and (2), and [1]. The amorphous target simulation results from [1] are in good agreement with the averaged singlecrystal results when the k0 =2 correction is included. With the exception of tungsten, where the amorphous target simulation result is about 40% lower than the averaged single-crystal results, the agreement is better than 25%. The theoretical predictions from Eq. (1) overestimate the averaged single-crystal results by more than a factor of two in all cases. The semi-empirical predictions from
Eq. (2) deviate systematically from the averaged single-crystal results, but by not more than about 40% over the range of target densities3 covered 3 (Cs)±0:085 atoms=A ). How(0:0085 atoms=A ever, the N 1 density dependence of Eq. (1) is in better agreement with the simulation results than the N 1=3 dependence of Eq. (2). 4. Discussion 4.1. Most probable escape depths Shulga and Eckstein [1] note that the most probable escape depth in their simulations with amorphous targets for normal incidence is about k0 =2. They argue from this observation that ``the topmost atoms are not necessarily the great majority of sputtered atoms''. However, in our
Table 4 Comparison of averaged corrected single target results with theoretical results and amorphous target simulations Target
Averaged results from Table 3 (A)
Eq. (1) (A)
Eq. (2) (A)
Ref. [1] (A)
Cu Au W Rb Cs
1.26 1.52 2.04 5.46 6.73
2.450 3.518 3.263 19.21 24.44
1.822 2.055 2.004 3.619 3.922
0.952 1.30 1.22 5.60 6.88
64
M.H. Shapiro et al. / Nucl. Instr. and Meth. in Phys. Res. B 180 (2001) 58±65
single-crystal simulations and in those previously published [16,17,24,36] the majority of sputtered atoms are found to arise from the ®rst layer of the target, even for the low-density bcc targets Rb and Cs. Thus, the single-crystal simulations yield a However, the most probable escape depth of 0 A. same correction ± namely, adding k0 =2 to the single-crystal value brings both single-crystal and amorphous target results into agreement. When the diuse nature of amorphous targets is taken into account, there is little dierence in either mean or most probable depths of origin as computed with crystalline and amorphous targets, provided that crystalline target results are corrected by adding k0 =2. Thus, both the single-crystal results and the amorphous target results show that the majority of sputtered atoms come from the ®rst layer of target atoms. 4.2. Density dependence of the depth of origin Shulga and Eckstein ®nd that the dependence of their mean depth of origin data on atomic density is proportional to N 0:86 . They suggest that this is consistent with the predictions of analytical sputtering theory given by Eq. (1). In the lower panel of Fig. 1 we have plotted all the corrected single-crystal mean depth of origin results from Table 3. We ®nd that the density dependence for the (1 0 0) targets is proportional to N 0:80 , which compares quite well with the N 0:86 dependence found by Shulga and Eckstein. However, the density dependence for the (1 1 1) targets is proportional to N 0:57 . This may be related to the relative opacities of the (1 0 0) and (1 1 1) surfaces. The bcc(1 0 0) surfaces are signi®cantly less opaque than the bcc(1 1 1) surfaces, and there are large systematic dierences in the mean depths of origin for the two crystal orientations for all three bcc targets studied. This can be seen clearly in the upper panel of Fig. 1 where the (1 0 0) and (1 1 1) results have been averaged for each of the targets investigated. In contrast, there are only slight differences between the two orientations of the fcc targets. In Fig. 2 we have plotted the averaged and corrected single-crystal results from Table 4 along with the N 0:86 curve that ®ts the amorphous target
Fig. 1. (a) Corrected mean depth-of-origin results for atoms sputtered from all the (1 0 0) and (1 1 1) targets listed in Table 3. The symbols represent the following: (}) Ar projectiles on (1 0 0) targets, (+) Au projectiles on (1 0 0) targets, () Xe projectiles on Au(1 1 1), () Ar projectiles on (1 1 1) targets, (M) Au projectiles on (1 1 1) targets. The straight lines represent power-law ®ts to the results. The (1 0 0) results are proportional to N 0:80 , while the (1 1 1) results are proportional to N 0:57 . (b) Averaged, corrected (1 0 0) and (1 1 1) single-crystal mean depth of origin results from Table 3. In this case (}) represents (1 0 0) targets, and (+) represent (1 1 1) targets. The straight lines have the same meaning as in the lower panel.
results from [1]. As we noted in Section 3, the agreement between our corrected single-crystal results and the amorphous target results of Shulga and Eckstein [1] is quite good. Neither the depth of origin relationship proposed by Vicanek et al. [10], nor that proposed by Kelly and Oliva [11] adequately ®t the simulation results (cf. Table 4). However, the N 1 density dependence obtained by Vicanek et al. [10] is a much closer approximation to both our averaged singlecrystal simulation results and the amorphous target simulation results than the N 1=3 density dependence suggested by Kelly and Oliva [11].
M.H. Shapiro et al. / Nucl. Instr. and Meth. in Phys. Res. B 180 (2001) 58±65
Fig. 2. Averaged, corrected single-crystal mean depth of origin results from Table 4. The straight line represents the N 0:86 density dependence obtained in [1].
5. Summary There are three principal conclusions from our results. First, the discrepancy between depth of origin results from simulations with crystalline and amorphous targets can be traced to dierence in the de®nition of the location of atoms in the top layer of the targets. When appropriate corrections are made for this dierence, both crystalline and amorphous target simulation results are in good agreement. Second, depth of origin results from bcc crystalline targets appear to be dependent on crystal orientation. The more opaque bcc(1 1 1) targets exhibit signi®cantly smaller mean depths of origin than the corresponding bcc(1 0 0) targets. Third, the absolute values of x predicted by Eq. (1) clearly are too large; however, the N 1 density dependence from the analytic theory is in better agreement with the simulation results than the N 1=3 dependence of Eq. (2). References [1] V.I. Shulga, W. Eckstein, Nucl. Instr. and Meth. B 145 (1998) 492. [2] A.R. Krauss, D.M. Gruen, A.B. DeWald, J. Nucl. Mater. 121 (1984) 398. [3] M.J. Pellin, C.E. Young, D.M. Gruen, Y. Aratono, A.B. Dewald, Surf. Sci. 151 (1985) 477. [4] M.F. Dumke, T.A. Tombrello, R.A. Weller, R.M. Housley, E.H. Cirlin, Surf. Sci. 124 (1983) 407.
65
[5] K.M. Hubbard, R.A. Weller, D.L. Weathers, T.A. Tombrello, Nucl. Instr. and Meth. B 36 (1989) 395. [6] T.B. Lill, W.F. Callaway, M.J. Pellin, D.M. Gruen, Phys. Rev. Lett. 73 (1994) 1719. [7] T.B. Lill, W.F. Callaway, Z. Ma, M.J. Pellin, Surf. Sci. 322 (1995) 361. [8] K. Wittmaack, Phys. Rev. B 56 (1997) R5701. [9] P. Sigmund, Phys. Rev. 184 (1969) 383. [10] M. Vicanek, J.J. Jimenez-Rodriguez, P. Sigmund, Nucl. Instr. and Meth. B 36 (1989) 124. [11] R. Kelly, A. Oliva, Nucl. Instr. and Meth. B 13 (1989) 283. [12] G. Falcone, P. Sigmund, Appl. Phys. 25 (1981) 307. [13] P. Sigmund, A. Oliva, G. Falcone, Nucl. Instr. and Meth. B 194 (1982) 541. [14] M. Vicanek, M.W. Sckerl, P. Sigmund, Nucl. Instr. and Meth. B 140 (1998) 61. [15] Z.L. Zhang, Nucl. Instr. and Meth. B 149 (1999) 272. [16] D.E. Harrison Jr., P.W. Kelly, B.J. Garrison, N. Winograd, Surf. Sci. 76 (1978) 311. [17] D.E. Harrison Jr., in: P. Varga, G. Betz, F.P. Viehb ock (Eds.), Inst. f ur allgemeine Physik, Technische Univ., Vienna, 1980, p. 36. [18] D.M. Gruen, A.R. Krauss, M.H. Mendelsohn, S. Susman, J. Nucl. Mater. 111±112 (1982) 831. [19] A.R. Kraus, D.M. Gruen, N.Q. Lam, A.B. DeWald, J. Nucl. Mater. 128±129 (1984) 570. [20] M.T. Robinson, J. Appl. Phys. 54 (1983) 2650. [21] M. Rosen, G.P. Mueller, W.A. Fraser, Nucl. Instr. and Meth. B 209±210 (1983) 63. [22] M. Rosen, R.H. Bassel, Nucl. Instr. and Meth. B 2 (1984) 592. [23] J.P. Biersack, W. Eckstein, Appl. Phys. A 34 (1984) 73. [24] M.H. Shapiro, P.K. Ha, T.A. Tombrello, D.E. Harrison, R.P. Webb, Radiat. E. 89 (1985) 243. [25] M.J. Pellin, C.E. Young, D.M. Gruen, Y. Aratono, A.B. Dewald, Surf. Sci. 151 (1985) 477. [26] M.H. Shapiro, D.Y. Lo, P.K. Ha, T.A. Tombrello, Nucl. Instr. and Meth. B 13 (1986) 348. [27] L.G. Glazov, V.I. Shulga, P. Sigmund, Surf. Interface Anal. 26 (1998) 512. [28] V.I. Shulga, Nucl. Instr. and Meth. B 152 (1999) 49. [29] G. Betz, R. Kirchner, W. Husinsky, F. Rudenauer, H.M. Urbassek, Radiat. E. Def. Sol. 130 (1994) 251. [30] M.H. Shapiro, K.R. Bengtson, T.A. Tombrello, Nucl. Instr. and Meth B 103 (1995) 123. [31] M.H. Shapiro, T.A. Tombrello, D.E. Harrison Jr., Nucl. Instr. and Meth. B 30 (1988) 152. [32] M.H. Shapiro, T.A. Tombrello, Nucl. Instr. and Meth. B 84 (1994) 453. [33] M.H. Shapiro, T.A. Tombrello, Nucl. Instr. and Meth. B 58 (1991) 161 (Section 4.4). [34] I.M. Torrens, Interatomic Potentials, Academic Press, New York, 1972 (Chapter 4). [35] W. Eckstein, Computer Simulation of Ion±Solid Interactions, Springer, Berlin, 1991. [36] M.J. Pellin, J.W. Burnett, Pure Appl. Chem. 65 (1993) 2361 (Table 3).