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Depth of origin of sputtered atoms for elemental targets
Nuclear Instruments and Methods in Physics Research B 145 (1998) 492±502
Depth of origin of sputtered atoms for elemental targets V.I. Shulga a, W. Eckstein
b,*
a
b
Institute of Nuclear Physics, Moscow State University, 119899 Moscow, Russian Federation Max-Planck-Institut f ur Plasmaphysik, Boltzmannstrasse 2, D-85748 Garching bei Munchen, Germany Received 17 July 1998; received in revised form 3 September 1998
Abstract The mean depth of origin of sputtered atoms is an important characteristic of the sputtering process. There exist several theoretical and experimental determinations of the escape depth with dierent results. To clear up the situation, in the present work a systematic computer simulation study of the mean depth of origin of sputtered atoms is performed. The Monte Carlo program TRIM.SP and the lattice code OKSANA are applied to calculate the distribution of depth of origin and the dependences of the mean depth of origin on the atomic density, N , projectile energy, E, the angle of incidence, a, projectile and target atomic species, Z1 and Z2 , as well as the simulation model. Ó 1998 Elsevier Science B.V. All rights reserved.
1. Introduction The mean depth of origin of ejected atoms is an important integral characteristic of the sputtering process which can be measured and calculated. Sigmund was the ®rst who made an analytical estimate for the mean escape depth based on the linear cascade theory and the Born-Mayer type of potential. According to [1], the mean escape depth is de®ned by a simple equation x0 3= 4C0 N , 2 and N is the target atomic where C0 1.808 A density. This yields x0 0:415=N ;
1
and N in atoms/A 3 . Later, where x0 is given in A using a more accurate extrapolation of the lowenergy cross-section, Vicanek et al. [2] proposed x1 x0 =2:
While Eqs. (1) and (2) refer to extremely low values of m, where m is the power-law scattering parameter, they are often used as a standard estimate for the mean escape depth. All existing experimental and computer simulation data on depth of origin of elemental and two-component materials were compressed by Kelly and Oliva [3] in a simple relation x2 0:80 0:10 k0 ;
*
Corresponding author. Tel.: 49 8932991259; fax: 49 8932991149; e-mail: [email protected]
2
3
where k0 is the mean atomic distance. We should note that the error 0.10 in Eq. (3) strongly underestimates an actual spread of experimental and
0168-583X/98/$ ± see front matter Ó 1998 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 9 8 ) 0 0 6 2 6 - 0
V.I. Shulga, W. Eckstein / Nucl. Instr. and Meth. in Phys. Res. B 145 (1998) 492±502
simulation data quoted in Ref. [3] and must be replaced at least by 0.25, where 0.25 means one standard deviation. Because k0 N ÿ1=3 , it is evident that Eqs. (1)± (3) suggest dierent dependences x N . The numerical values of x extracted from Eqs. (1)±(3) may also be very dierent. Such a strong disagreement between experimental, analytical and simulation data on depth of origin of sputtered atoms was the main motivation for the round robin simulation of ejection probability in sputtering [4]. Recently, the problem of depth of origin came up again for discussion in connection with the experiment of Wittmaack [5] for the case of near-normal bombardment of a Si target with 2 keV Ne ions. The value of x for 28 Si atoms ejected with 40 eV at 48° to the surface normal, was found to be 2.0 0.4 A. This value correlates very closely with Eq. (3) and with the results of recent simulations [6] but it is much lower than the values which follow from Eqs. (1) and (2). It should be noted that even though the problem of depth of origin was studied analytically very carefully (see also Refs. [7±9]) the data of computer simulations are very scanty and cover only 3 which the region near N 0:05 and 0.09 atoms/A is not enough to say something about the density dependence of x. All this is illustrated by Fig. 1 which provides a representative although not quite exhaustive survey of the current status for elemental targets. To illuminate the problem, in the present work a systematic computer simulation study of the depth of origin of ejected atoms is performed. The Monte Carlo program TRIM.SP and the lattice code OKSANA are applied to calculate the depth of origin of sputtered atoms for dierent bombarding ions with energies 100 eV ± 100 keV and for many solid and liquid targets with the atomic density in 3 the range from 0.008604 (Cs) to 0.1309 atoms/A (B) and even for an ``amorphous diamond'', i.e., an amorphous carbon with the atomic density of 3 ). Eqs. (1)±(3) natural diamond (0.1765 atoms/A do not predict any dependence of the depth of origin on the projectile energy, E, the angle of incidence, a, the projectile and target atomic numbers, Z1 and Z2 , therefore, it was interesting to draw information also about such dependences.
493
Fig. 1. Survey of predicted, calculated and measured values of the mean depth of origin of sputtered atoms for elemental targets. Simulation data (open symbols): 0.6 keV Ar±Cu [18]; 8 keV Ar±Cu [19]; 2±9 keV U±U [20]; 90 keV Cu±Cu [21]; Xe±Ni [25]; 5 keV Ar±Cu [22]; Ar,He±Ti [23]; 5 keV Ar±Cu [24]; 2 keV Ne±Si [6]. Experimental data (®lled circle): 2 keV Ne±Si [5].
2. Simulation codes TRIM.SP (version TRV1C) [25,10] is a Monte Carlo program which assumes a random target. It allows to use three dierent interaction potentials: the WHB (Kr-C) potential [11], the Moliere potential [12], and the ZBL potential [13]. Electronic energy losses can be applied according to ®ve different models: Lindhard±Schar [14], Oen±Robinson [15], an equipartition of the two preceding models, and the tables for H [16] and He [17]. The program is based on the binary collision model. The heat of sublimation is taken for the surface binding energy in the planar surface potential. The free path length, k, is an important parameter in TRIM.SP. In the standard version of the program k k0 . In the present work, k has been changed in such a way that the mean distance of a recoiling atom to the next target atom equals k0 . As a result, the eective mean path length is shortened by a factor t k k0 t: This leads to the following equation:
4
494
2pt4 3
V.I. Shulga, W. Eckstein / Nucl. Instr. and Meth. in Phys. Res. B 145 (1998) 492±502
1 1 3 pt
!
3=2 ÿ1
1;
5
t is determined as 0.909731. The maximum impact parameter pm is given by pm
k0 pt
1=2
:
6
This adjustment which is fully justi®ed gives very good agreement with results from the lattice code for the depth of origin and the sputter yield. The sputter yield is only weakly dependent on the free path length, see Fig. 2. Similarly to TRIM.SP, the program OKSANA [26] is based on the binary collision approximation and takes into account interactions of moving particles simultaneously with several atoms. The simultaneous collisions are described in a linear approximation [27]. The program allows to simulate sputtering of crystalline and amorphous materials but in the present study we dealt only with amorphous targets. As in the MARLOWE pro-
Fig. 2. The mean depth of origin of sputtered atoms, x, and the sputter yield, Y, versus k=k0 for the case of 2 keV Ne bombardment of an amorphous Si target. Filled and open circles show the depth of origin and the sputter yield calculated by the program TRIM.SP. The vertical line marks the value k=k0 0:909731 (see the text). The dashed and dot-dashed lines show the results of simulation with the program OKSANA for an amorphous target [6].
gram [28], an amorphous target is simulated by rotation of a crystalline atomic block, the procedure of rotation being repeated from collision to collision. The OKSANA program can operate with seven dierent potentials including the Born± Mayer and Lenz±Jensen potentials. The standard model for the inelastic energy loss is the Firsov model [29]. Both planar and spherical (isotropic) surface potential barriers can be adopted. An essential dierence between the programs is target setup. Although in both programs each atom occupies the same volume, 1/N, the distribution of atoms is dierent: in the TRIM.SP program the next partners for collisions are located in the plane which is the bottom side of the cylinder [25,10], whereas in the OKSANA code it is assumed that the nearest atoms are located on the ®rst coordination sphere. For simplicity and to be more consistent with the ideology of the TRIM.SP program, the atomic block in the OKSANA program was taken in form of a tetrahedron (as for Si) and only the interatomic distance was changed from target to target to get correct values of the atomic density. It should be mentioned also that the binarycollision dynamics in the two programs is not identical: in TRIM.SP the scattering angles are calculated using the integration procedure `Magic' [25,10], whereas in OKSANA the angles are calculated using the tabulated values of angles found beforehand by solving the equations of motion [26]. In our study we considered eight projectiles (from He to Rn) and about 40 solid and liquid targets (from Li to U) with dierent values of k0 , see Fig. 3. Most simulations were carried out for Si bombarded 3 , k0 2.718 A) (N 0.04978 atoms/A with Ar ions. The Kr±C potential was employed as a standard potential. In all cases the planar potential barrier was used. The values of the surface binding energy Eb were taken from Table 6.1 in Ref. [10]. We could not ®nd in the literature the values of Eb for Rb and Ra, and used Eb 0.87 and 1.84 eV respectively. 0.87 eV is the mean value between the surface binding energies for K and Cs, and 1.84 eV is the surface binding energy for Ba which belongs to the same atomic group as Ra.
V.I. Shulga, W. Eckstein / Nucl. Instr. and Meth. in Phys. Res. B 145 (1998) 492±502
Fig. 3. The mean atomic distance, k0 , versus the target atomic number (open circles). Full circles mark the targets studied in the present work. To calculate k0 for dierent targets, the values of N from Table 6.1 in Ref. [10] were used. The corrected value 3 . of N for Ca is 0.02314 atoms/A
3. Results and discussion 3.1. Si target 3.1.1. Normal incidence Fig. 4a±d plots the escape depth distributions of all sputtered atoms and the atoms sputtered with dierent energies for an amorphous Si target bombarded with 100 eV±100 keV Ar ions at normal incidence. These ®gures show clear evidence that at all bombardment energies the major part of atoms are ejected from a depth of several angstroms, and that there is some most probable escape depth which is dierent from zero. The latter means that in our case the topmost layer of atoms thickness) (we divided the target by layers of 0.2 A is not the major source of sputtered atoms. Another interesting feature (Fig. 4b±d) is a change of which is slope of the total distribution at x 2.5 A The depth distributions close to k0 2.718 A. shown in Fig. 4 were calculated using the program TRIM.SP but simulations with the program OKSANA provide similar results (see below). The depth distributions were used to calculate the mean escape depth, x. For the Ar±Si case the energy dependences of x for all sputtered atoms
495
and for the atoms ejected with dierent energies are shown in Fig. 5. The dependencies for all sputtered atoms calculated with the TRIM.SP and OKSANA programs are similar: in a double logarithmic scale they are almost linear, and hence x may be described by a power law x / Ek with k 0:17 (E is the projectile energy). Note that this value of k is close to the exponent m ' 1=6 which approximates the low-energy stopping cross-section for the Moliere potential [20] and the Kr±C potential [25]. At E > 500 eV the depth of origin increases with increasing energy of ejected atoms which is in good agreement with analytical predictions [30] and simulation data (e.g. [26]). The horizontal straight lines (Fig. 5) were calculated using Eqs. (1)±(3). We can see that Eq. (1) overestimates the mean depth of origin of all sputtered atoms even at E 100 keV whereas Eq. (2) predicts more realistic values of x. Eq. (3) yields reasonable results in the range of bombardment energies 1±10 keV, i.e., at energies typical for simulation and experimental data analysed in Ref. [3]. Fig. 6 plots the mean depth of origin for different projectiles. It is seen that projectiles have only a weak in¯uence on the mean escape depth of all sputtered atoms. This in¯uence gets more pronounced for energetic sputtered atoms which are mainly recoils generated in ion-atom collisions (primary recoils). The lattice code OKSANA provides somewhat higher values of x but the difference decreases for light projectiles and will be even smaller at higher bombardment energies as it can be expected from the results for 2 keV Ne (Fig. 2). 3.1.2. Oblique incidence Fig. 7 shows the mean depth of origin as a function of the angle of incidence a. It is seen that up to 60°, the mean depth of origin for all sputtered atoms is almost independent of a. At very oblique incidence, the values of x are small because at such angles the collision cascades generated by incoming particles are located very close to the surface. More information about the escape depth at oblique incidence can be drawn from Fig. 8 which also compares the results of simulations performed
496
V.I. Shulga, W. Eckstein / Nucl. Instr. and Meth. in Phys. Res. B 145 (1998) 492±502
Fig. 4. Depth distributions of all sputtered atoms and the atoms sputtered with dierent energies for a Si target bombarded with Ar ions at normal incidence. Ion energies: 100 eV (a), 1 keV (b), 10 keV (c), and 100 keV (d). Simulations with the program TRIM.SP: results from 1 500 000 (a), 500 000 (b), 200 000 (c), and 500 000 (d) impacts accumulated. Zero depth corresponds to the uppermost target atoms.
with the two programs. In contrast to Fig. 4, the distributions shown in Fig. 8 display a maximum at zero depth and the only exception is the lowenergy sputtered atoms (energies of 0.1±10 eV) which still exhibit a characteristic depth x k0 =2. Such a characteristic depth can be explained by a strong gradient of recoils near the surface and by the depth dependence of the ejection probability of recoils which is almost constant up to x k0 =2 due to passage of recoils through the open surface atomic windows and then sharply decreases with depth [33].
The depth distributions of recoils are shown in Fig. 9a which is similar to Fig. 8a but relates to all cascade recoils (only recoils with energies higher than the surface binding energy were registered; for Si Eb 4:7 eV). We can see that even at a 70° the major part of recoils with energies 4.7± 100 eV is generated deeply inside the target. This is true also for a < 70°. For example, at a 0 the depth distribution of all recoils displays a maxi where the number of recoils exmum at 17 A ceeds the number of recoils at zero depth by a factor of 3.7 (TRIM.SP) and 4.4 (OKSANA); cf.
V.I. Shulga, W. Eckstein / Nucl. Instr. and Meth. in Phys. Res. B 145 (1998) 492±502
Fig. 5. The mean depth of origin of all sputtered atoms and the atoms sputtered with dierent energies as a function of bombardment energy. Simulations with the programs TRIM.SP and OKSANA. The horizontal straight lines were calculated using Eqs. (1)±(3).
Fig. 6. The mean depth of origin of all sputtered atoms and the atoms sputtered with dierent energies as a function of the projectile atomic number. Simulations with the programs TRIM.SP and OKSANA. The horizontal straight lines were calculated using Eq. (3).
Fig. 9a. We explain the gradient of recoils mainly by a strong momentum anisotropy of the cascade near the surface. At a 85°, the depth distributions display a sharp peak at zero depth for all energy intervals. In
497
Fig. 7. The mean depth of origin of all sputtered atoms and the atoms sputtered with dierent energies as a function of the angle of incidence (zero angle corresponds to normal incidence). Simulations with the programs TRIM.SP and OKSANA. The horizontal straight lines were calculated using Eq. (3).
addition, the distributions reveal some wave structure (the wave length k0 ) typical rather for a crystalline than for an amorphous target. We attribute this eect to the fact that at very glancing bombardment a recoiling atom of the uppermost layer receives a momentum directed almost perpendicularly to the surface, therefore, the next collision partner for such a recoil will be at the distance k0 from the surface etc. From Figs. 8 and 9 we can also conclude that simulations with the program TRIM.SP and OKSANA are in overall agreement with each other. The only exception is the very glancing incidence where the depth distributions of all recoils turned out to be quite sensitive to the simulation model (Fig. 9b). 3.2. Dierent targets Considerable eort in our study has been devoted to the target eects. Fig. 10 shows the mean escape depth of sputtered atoms as a function of the target atomic number. At 1 keV bombardment, the values of x for most targets are in the and may be essentially higher for range 1±3 A some targets (K, Cs, Ra).
498
V.I. Shulga, W. Eckstein / Nucl. Instr. and Meth. in Phys. Res. B 145 (1998) 492±502
Fig. 8. Depth distributions of sputtered atoms for a Si target bombarded with 1 keV Ar ions at a 70° (a) and 85° (b). Simulations with the programs TRIM.SP and OKSANA. Results from 700 000 impacts accumulated.
The main conclusion which emerges from Fig. 10 is a strong correlation between the mean depth of origin and the mean atomic distance (Fig. 3). For example, in both ®gures we can see a series of peaks corresponding to alkali metals (Z2 3, 11, 19, 37, and 55) which are characterized by low values of N. Unfortunately, we do not know the values of N and Eb for Fr (Z2 87) which belongs to the group of alkali metals, therefore, an interesting point in Fig. 10 is missing. This lack is partly compensated by a point for Ra (Z2 88); this
Fig. 9. Same as Fig. 8 for all recoils in collision cascades.
element, similarly to alkali metals, has relatively low atomic density and, hence, a high calculated escape depth. According to Eqs. (1) and (2), the product of the mean escape depth and the atomic density should be constant. Fig. 11 suggests that this is not absolutely true. Even though the major part of points is squeezed in the interval x N 0.07±0.09, there are several points with much higher values of x N . The latter group includes points for Li, C(graphite), Be, and B, i.e., for the targets with small Z2 and, correspondingly, with the low stopping power which is in favour for high escape
V.I. Shulga, W. Eckstein / Nucl. Instr. and Meth. in Phys. Res. B 145 (1998) 492±502
Fig. 10. The mean depth of origin of all sputtered atoms and the atoms sputtered with dierent energies as a function of the target atomic number. Simulations with the program TRIM.SP.
depth. The two groups of targets are well separated from each other by a group of gaseous targets, from N to Ne (Z2 from 7 to 10), which are not considered in the present paper. According to Eq. (3), the ratio of the mean escape depth and the mean atomic distance should be constant. Fig. 12 shows that this is not the case
Fig. 11. The product of the mean escape depth and the target atomic density, N, as a function of the target atomic number. Simulations with the program TRIM.SP.
499
Fig. 12. The ratio of the mean escape depth and the mean atomic distance, k0 , as a function of the target atomic number. Simulations with the program TRIM.SP.
and that it happens in spite of a strong correlation between k0 and x (Figs. 3 and 10). Nevertheless, Fig. 12 indicates that the numerical values of x=k0 for many targets are quite close to the results of Eq. (3). The dependencies of the mean escape depth on the atomic density are plotted in Fig. 13a±c in a double logarithmic scale for ion energies 0.1, 1 and 10 keV. It is seen that at all energies there are two groups of points which can be approximated by straight lines. This is true both for simulations with the TRIM.SP and OKSANA programs. The origin of these groups was discussed above. The upper group includes Li, C(graphite), Be, B, and C(diamond), i.e., elements with low Z2 ; the lower (more numerous) group is bounded by Cs (the lowest atomic density) and Ni (the highest atomic density). According to Eqs. (1)±(3), the mean depth of origin x is proportional to 1=N n , where n 1 for Eqs. (1) and (2), and n 1/3 for Eq. (3). The results of simulations shown in Fig. 13 con®rm such a proportionality and suggest n 0:86 which is close to Sigmund's value n 1. This means that Sigmund [1] predicted correctly the form of the dependence x N but overestimated the values of x in numerical calculations which were based on a rough approximation of the interatomic potential.
500
V.I. Shulga, W. Eckstein / Nucl. Instr. and Meth. in Phys. Res. B 145 (1998) 492±502
Fig. 13. The mean depth of origin of all sputtered atoms as a function of the target atomic density. Ion energies: 100 eV (a), 1 keV (b), 10 keV (c). Simulations with the programs TRIM.SP and OKSANA.
This statement can well be supported by Fig. 14 which shows the results of simulations with different interatomic potentials including the Born± Mayer potential applied in Ref. [1]. The neglect of inelastic energy losses as done in the analytical estimates increases the escape depth only marginally in the energy range investigated. It was noted above that Eqs. (1) and (2) refer to very low values of the power-law scattering parameter m. At ®xed polar ejection angle and energy, the m-dependent escape depth can be calculated using Eq. 21 from Ref. [7]. For condi-
tions of the experiment [5], this yields the values of (the Moliere±Lindhard x in the range from 5.7 A (the Born±Mayer potential, m 0.21) to 11.6 A potential, m 0.19). These values correlate well and x1 4.2 A but they are much with x0 8.4 A found experimentally. higher than x 2:0 0:4 A Evaluating the origin of this discrepancy requires a detailed analysis of the analytical and simulated dierential depth distributions of sputtered atoms. This work considering the m- and depth-dependent dierential sputter cross sections published recently [9] is in progress.
V.I. Shulga, W. Eckstein / Nucl. Instr. and Meth. in Phys. Res. B 145 (1998) 492±502
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Acknowledgements The authors would like to acknowledge useful discussions of this topic with Peter Sigmund. One of the authors (VIS) acknowledges with thanks the hospitality of the Max-Planck Institut f ur Plasmaphysik, Garching, and the ®nancial support of the Regionalverband Bayern of the German Physical Society. References
Fig. 14. The mean depth of origin of sputtered atoms versus bombardment energy. Simulations with dierent potentials using the program OKSANA (the Born±Mayer and Lenz± Jensen potentials) and TRIM.SP (all other potentials). The Moliere potential was used with the Firsov screening length [31] and taking into account the correction factor proposed by O'Connor and MacDonald [32].
4. Conclusions Computer simulations with the Monte Carlo program TRIM.SP and the lattice code OKSANA have demonstrated that the mean depth of origin of sputtered atoms is a function of many parameters such as atomic density, projectile energy, the angle of incidence, projectile and target atomic numbers. The dependence on atomic density N is proportional to N ÿ0:86 , the values of the depth of origin are generally smaller than given by the analytical theory, and the depth of origin increases with incidence energy. For a Si target bombarded with Ar ions, the energy dependence of the escape depth can be approximated by E0:17 . The depth distribution of sputtered atoms is angle dependent. For normal and not very oblique incidence, the most probable escape depth is about k0 =2. The nonzero value of the most probable escape depth indicates that the topmost atoms are not necessarily the great majority of sputtered atoms. This aspect would be of interest in connection with such processes as surface segregation etc.
[1] P. Sigmund, Phys. Rev. 184 (1969) 383. [2] M. Vicanek, J.J. Jimenez-Rodriguez, P. Sigmund, Nucl. Instr. and Meth. 36 (1989) 124. [3] R. Kelly, A. Oliva, Nucl. Instr. and Meth. 13 (1986) 283. [4] P. Sigmund, M.T. Robinson, M.I. Baskes, M. Hautala, F.Z. Cui, W. Eckstein, Y. Yamamura, S. Hosaka, T. Ishitani, V.I. Shulga, D.E. Harrison, I.R. Chakarov, D.S. Karpuzov, E. Kawatoh, R. Shimizu, S. Valkealahti, R.M. Nieminen, G. Betz, W. Husinsky, M.H. Shapiro, M. Vicanek, H.M. Urbassek, Nucl. Instr. and Meth. 36 (1989) 110. [5] K. Wittmaack, Phys. Rev. B 56 (1997) R5701. [6] L.G. Glazov, V.I. Shulga, P. Sigmund, Surf. Interface Anal. 26 (1998) 512. [7] G. Falcone, P. Sigmund, Appl. Phys. 25 (1981) 307. [8] P. Sigmund, A. Oliva, G. Falcone, Nucl. Instr. and Meth. 194 (1982) 541. [9] M. Vicanek, M.W. Sckerl, P. Sigmund, Nucl. Instr. and Meth. 140 (1998) 61. [10] W. Eckstein, Computer Simulation of Ion-Solid Interaction, Springer, Berlin, 1991. [11] W.D. Wilson, L.G. Haggmark, J.P. Biersack, Phys. Rev. B 15 (1977) 2458. [12] G. Moliere, Z. Naturforsch. A 2 (1947) 133. [13] J.F. Ziegler, J.P. Biersack, U. Littmark, in: J.F. Ziegler (Ed.), The Stopping and Range of Ions in Solids, The Stopping and Range of Ions in Matter, vol. 1, Pergamon Press, New York, 1985, p. 41. [14] J. Lindhard, M. Schar, Phys. Rev. 124 (1961) 128. [15] O.S. Oen, M.T. Robinson, Nucl. Instr. and Meth. 132 (1976) 647. [16] H.H. Andersen, J.F. Ziegler, The Stopping and Ranges of Ions in Matter, vol. 3, Pergamon Press, New York, 1977. [17] J.F. Ziegler, The Stopping and Ranges of Ions in Matter, vol. 4, Pergamon Press, New York, 1977. [18] D.E. Harrison, P.W. Kelly, B.J. Garrison, N. Winograd, Surf. Sci. 76 (1978) 311. [19] D.E. Harrison, in: P. Varga, G. Betz, F.P. Viehb ock, Symp. on Sputtering, Inst. f ur allgemeine Physik Technische Univ., Vienna, 1980, p. 36. [20] M.T. Robinson, J. Appl. Phys. 54 (1983) 2650. [21] M. Rosen, G.P. Mueller, W.A. Fraser, Nucl. Instr. and Meth. 209/210 (1983) 63.
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[22] M.H. Shapiro, P.K. Ha, T.A. Tombrello, D.E. Harrison, R.P. Webb, Radiat. E. 89 (1985) 243. [23] M.J. Pellin, C.E. Young, D.M. Gruen, Y. Aratono, A.B. Dewald, Surf. Sci. 151 (1985) 477. [24] M.H. Shapiro, D.Y. Lo, P.K. Ha, T.A. Tombrello, Nucl. Instr. and Meth. 3 (1986) 348. [25] J.P. Biersack, W. Eckstein, Appl. Phys. A 34 (1984) 73. [26] V.I. Shulga, Radiat. E. 70 (1983) 65; 82 (1984) 169; 84 (1985) 1. [27] M. Hou, M.T. Robinson, Nucl. Instr. and Meth. B 132 (1976) 641.
[28] M.T. Robinson, I.M. Torrens, Phys. Rev. B 9 (1974) 5008. [29] O.B. Firsov, Zh. Expt. Teor. Fiz. 36 (1959) 1517 [Sov. Fys. JETP 36 (1959) 1076]. [30] P. Sigmund, in: R. Behrisch (Ed.), Sputtering by Particle Bombardment ± I, Springer, Berlin, 1981, p. 9. [31] O.B. Firsov, Zh. Expt. Teor. Fiz. 33 (1957) 696 [Sov. Fys. JETP 6 (1958) 534]. [32] D.J. O'Connor, R.J. MacDonald, Radiat. E. 34 (1977) 247. [33] V.I. Shulga, in preparation.
Depth of origin of sputtered atoms for elemental targets V.I. Shulga a, W. Eckstein
b,*
a
b
Institute of Nuclear Physics, Moscow State University, 119899 Moscow, Russian Federation Max-Planck-Institut f ur Plasmaphysik, Boltzmannstrasse 2, D-85748 Garching bei Munchen, Germany Received 17 July 1998; received in revised form 3 September 1998
Abstract The mean depth of origin of sputtered atoms is an important characteristic of the sputtering process. There exist several theoretical and experimental determinations of the escape depth with dierent results. To clear up the situation, in the present work a systematic computer simulation study of the mean depth of origin of sputtered atoms is performed. The Monte Carlo program TRIM.SP and the lattice code OKSANA are applied to calculate the distribution of depth of origin and the dependences of the mean depth of origin on the atomic density, N , projectile energy, E, the angle of incidence, a, projectile and target atomic species, Z1 and Z2 , as well as the simulation model. Ó 1998 Elsevier Science B.V. All rights reserved.
1. Introduction The mean depth of origin of ejected atoms is an important integral characteristic of the sputtering process which can be measured and calculated. Sigmund was the ®rst who made an analytical estimate for the mean escape depth based on the linear cascade theory and the Born-Mayer type of potential. According to [1], the mean escape depth is de®ned by a simple equation x0 3= 4C0 N , 2 and N is the target atomic where C0 1.808 A density. This yields x0 0:415=N ;
1
and N in atoms/A 3 . Later, where x0 is given in A using a more accurate extrapolation of the lowenergy cross-section, Vicanek et al. [2] proposed x1 x0 =2:
While Eqs. (1) and (2) refer to extremely low values of m, where m is the power-law scattering parameter, they are often used as a standard estimate for the mean escape depth. All existing experimental and computer simulation data on depth of origin of elemental and two-component materials were compressed by Kelly and Oliva [3] in a simple relation x2 0:80 0:10 k0 ;
*
Corresponding author. Tel.: 49 8932991259; fax: 49 8932991149; e-mail: [email protected]
2
3
where k0 is the mean atomic distance. We should note that the error 0.10 in Eq. (3) strongly underestimates an actual spread of experimental and
0168-583X/98/$ ± see front matter Ó 1998 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 9 8 ) 0 0 6 2 6 - 0
V.I. Shulga, W. Eckstein / Nucl. Instr. and Meth. in Phys. Res. B 145 (1998) 492±502
simulation data quoted in Ref. [3] and must be replaced at least by 0.25, where 0.25 means one standard deviation. Because k0 N ÿ1=3 , it is evident that Eqs. (1)± (3) suggest dierent dependences x N . The numerical values of x extracted from Eqs. (1)±(3) may also be very dierent. Such a strong disagreement between experimental, analytical and simulation data on depth of origin of sputtered atoms was the main motivation for the round robin simulation of ejection probability in sputtering [4]. Recently, the problem of depth of origin came up again for discussion in connection with the experiment of Wittmaack [5] for the case of near-normal bombardment of a Si target with 2 keV Ne ions. The value of x for 28 Si atoms ejected with 40 eV at 48° to the surface normal, was found to be 2.0 0.4 A. This value correlates very closely with Eq. (3) and with the results of recent simulations [6] but it is much lower than the values which follow from Eqs. (1) and (2). It should be noted that even though the problem of depth of origin was studied analytically very carefully (see also Refs. [7±9]) the data of computer simulations are very scanty and cover only 3 which the region near N 0:05 and 0.09 atoms/A is not enough to say something about the density dependence of x. All this is illustrated by Fig. 1 which provides a representative although not quite exhaustive survey of the current status for elemental targets. To illuminate the problem, in the present work a systematic computer simulation study of the depth of origin of ejected atoms is performed. The Monte Carlo program TRIM.SP and the lattice code OKSANA are applied to calculate the depth of origin of sputtered atoms for dierent bombarding ions with energies 100 eV ± 100 keV and for many solid and liquid targets with the atomic density in 3 the range from 0.008604 (Cs) to 0.1309 atoms/A (B) and even for an ``amorphous diamond'', i.e., an amorphous carbon with the atomic density of 3 ). Eqs. (1)±(3) natural diamond (0.1765 atoms/A do not predict any dependence of the depth of origin on the projectile energy, E, the angle of incidence, a, the projectile and target atomic numbers, Z1 and Z2 , therefore, it was interesting to draw information also about such dependences.
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Fig. 1. Survey of predicted, calculated and measured values of the mean depth of origin of sputtered atoms for elemental targets. Simulation data (open symbols): 0.6 keV Ar±Cu [18]; 8 keV Ar±Cu [19]; 2±9 keV U±U [20]; 90 keV Cu±Cu [21]; Xe±Ni [25]; 5 keV Ar±Cu [22]; Ar,He±Ti [23]; 5 keV Ar±Cu [24]; 2 keV Ne±Si [6]. Experimental data (®lled circle): 2 keV Ne±Si [5].
2. Simulation codes TRIM.SP (version TRV1C) [25,10] is a Monte Carlo program which assumes a random target. It allows to use three dierent interaction potentials: the WHB (Kr-C) potential [11], the Moliere potential [12], and the ZBL potential [13]. Electronic energy losses can be applied according to ®ve different models: Lindhard±Schar [14], Oen±Robinson [15], an equipartition of the two preceding models, and the tables for H [16] and He [17]. The program is based on the binary collision model. The heat of sublimation is taken for the surface binding energy in the planar surface potential. The free path length, k, is an important parameter in TRIM.SP. In the standard version of the program k k0 . In the present work, k has been changed in such a way that the mean distance of a recoiling atom to the next target atom equals k0 . As a result, the eective mean path length is shortened by a factor t k k0 t: This leads to the following equation:
4
494
2pt4 3
V.I. Shulga, W. Eckstein / Nucl. Instr. and Meth. in Phys. Res. B 145 (1998) 492±502
1 1 3 pt
!
3=2 ÿ1
1;
5
t is determined as 0.909731. The maximum impact parameter pm is given by pm
k0 pt
1=2
:
6
This adjustment which is fully justi®ed gives very good agreement with results from the lattice code for the depth of origin and the sputter yield. The sputter yield is only weakly dependent on the free path length, see Fig. 2. Similarly to TRIM.SP, the program OKSANA [26] is based on the binary collision approximation and takes into account interactions of moving particles simultaneously with several atoms. The simultaneous collisions are described in a linear approximation [27]. The program allows to simulate sputtering of crystalline and amorphous materials but in the present study we dealt only with amorphous targets. As in the MARLOWE pro-
Fig. 2. The mean depth of origin of sputtered atoms, x, and the sputter yield, Y, versus k=k0 for the case of 2 keV Ne bombardment of an amorphous Si target. Filled and open circles show the depth of origin and the sputter yield calculated by the program TRIM.SP. The vertical line marks the value k=k0 0:909731 (see the text). The dashed and dot-dashed lines show the results of simulation with the program OKSANA for an amorphous target [6].
gram [28], an amorphous target is simulated by rotation of a crystalline atomic block, the procedure of rotation being repeated from collision to collision. The OKSANA program can operate with seven dierent potentials including the Born± Mayer and Lenz±Jensen potentials. The standard model for the inelastic energy loss is the Firsov model [29]. Both planar and spherical (isotropic) surface potential barriers can be adopted. An essential dierence between the programs is target setup. Although in both programs each atom occupies the same volume, 1/N, the distribution of atoms is dierent: in the TRIM.SP program the next partners for collisions are located in the plane which is the bottom side of the cylinder [25,10], whereas in the OKSANA code it is assumed that the nearest atoms are located on the ®rst coordination sphere. For simplicity and to be more consistent with the ideology of the TRIM.SP program, the atomic block in the OKSANA program was taken in form of a tetrahedron (as for Si) and only the interatomic distance was changed from target to target to get correct values of the atomic density. It should be mentioned also that the binarycollision dynamics in the two programs is not identical: in TRIM.SP the scattering angles are calculated using the integration procedure `Magic' [25,10], whereas in OKSANA the angles are calculated using the tabulated values of angles found beforehand by solving the equations of motion [26]. In our study we considered eight projectiles (from He to Rn) and about 40 solid and liquid targets (from Li to U) with dierent values of k0 , see Fig. 3. Most simulations were carried out for Si bombarded 3 , k0 2.718 A) (N 0.04978 atoms/A with Ar ions. The Kr±C potential was employed as a standard potential. In all cases the planar potential barrier was used. The values of the surface binding energy Eb were taken from Table 6.1 in Ref. [10]. We could not ®nd in the literature the values of Eb for Rb and Ra, and used Eb 0.87 and 1.84 eV respectively. 0.87 eV is the mean value between the surface binding energies for K and Cs, and 1.84 eV is the surface binding energy for Ba which belongs to the same atomic group as Ra.
V.I. Shulga, W. Eckstein / Nucl. Instr. and Meth. in Phys. Res. B 145 (1998) 492±502
Fig. 3. The mean atomic distance, k0 , versus the target atomic number (open circles). Full circles mark the targets studied in the present work. To calculate k0 for dierent targets, the values of N from Table 6.1 in Ref. [10] were used. The corrected value 3 . of N for Ca is 0.02314 atoms/A
3. Results and discussion 3.1. Si target 3.1.1. Normal incidence Fig. 4a±d plots the escape depth distributions of all sputtered atoms and the atoms sputtered with dierent energies for an amorphous Si target bombarded with 100 eV±100 keV Ar ions at normal incidence. These ®gures show clear evidence that at all bombardment energies the major part of atoms are ejected from a depth of several angstroms, and that there is some most probable escape depth which is dierent from zero. The latter means that in our case the topmost layer of atoms thickness) (we divided the target by layers of 0.2 A is not the major source of sputtered atoms. Another interesting feature (Fig. 4b±d) is a change of which is slope of the total distribution at x 2.5 A The depth distributions close to k0 2.718 A. shown in Fig. 4 were calculated using the program TRIM.SP but simulations with the program OKSANA provide similar results (see below). The depth distributions were used to calculate the mean escape depth, x. For the Ar±Si case the energy dependences of x for all sputtered atoms
495
and for the atoms ejected with dierent energies are shown in Fig. 5. The dependencies for all sputtered atoms calculated with the TRIM.SP and OKSANA programs are similar: in a double logarithmic scale they are almost linear, and hence x may be described by a power law x / Ek with k 0:17 (E is the projectile energy). Note that this value of k is close to the exponent m ' 1=6 which approximates the low-energy stopping cross-section for the Moliere potential [20] and the Kr±C potential [25]. At E > 500 eV the depth of origin increases with increasing energy of ejected atoms which is in good agreement with analytical predictions [30] and simulation data (e.g. [26]). The horizontal straight lines (Fig. 5) were calculated using Eqs. (1)±(3). We can see that Eq. (1) overestimates the mean depth of origin of all sputtered atoms even at E 100 keV whereas Eq. (2) predicts more realistic values of x. Eq. (3) yields reasonable results in the range of bombardment energies 1±10 keV, i.e., at energies typical for simulation and experimental data analysed in Ref. [3]. Fig. 6 plots the mean depth of origin for different projectiles. It is seen that projectiles have only a weak in¯uence on the mean escape depth of all sputtered atoms. This in¯uence gets more pronounced for energetic sputtered atoms which are mainly recoils generated in ion-atom collisions (primary recoils). The lattice code OKSANA provides somewhat higher values of x but the difference decreases for light projectiles and will be even smaller at higher bombardment energies as it can be expected from the results for 2 keV Ne (Fig. 2). 3.1.2. Oblique incidence Fig. 7 shows the mean depth of origin as a function of the angle of incidence a. It is seen that up to 60°, the mean depth of origin for all sputtered atoms is almost independent of a. At very oblique incidence, the values of x are small because at such angles the collision cascades generated by incoming particles are located very close to the surface. More information about the escape depth at oblique incidence can be drawn from Fig. 8 which also compares the results of simulations performed
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Fig. 4. Depth distributions of all sputtered atoms and the atoms sputtered with dierent energies for a Si target bombarded with Ar ions at normal incidence. Ion energies: 100 eV (a), 1 keV (b), 10 keV (c), and 100 keV (d). Simulations with the program TRIM.SP: results from 1 500 000 (a), 500 000 (b), 200 000 (c), and 500 000 (d) impacts accumulated. Zero depth corresponds to the uppermost target atoms.
with the two programs. In contrast to Fig. 4, the distributions shown in Fig. 8 display a maximum at zero depth and the only exception is the lowenergy sputtered atoms (energies of 0.1±10 eV) which still exhibit a characteristic depth x k0 =2. Such a characteristic depth can be explained by a strong gradient of recoils near the surface and by the depth dependence of the ejection probability of recoils which is almost constant up to x k0 =2 due to passage of recoils through the open surface atomic windows and then sharply decreases with depth [33].
The depth distributions of recoils are shown in Fig. 9a which is similar to Fig. 8a but relates to all cascade recoils (only recoils with energies higher than the surface binding energy were registered; for Si Eb 4:7 eV). We can see that even at a 70° the major part of recoils with energies 4.7± 100 eV is generated deeply inside the target. This is true also for a < 70°. For example, at a 0 the depth distribution of all recoils displays a maxi where the number of recoils exmum at 17 A ceeds the number of recoils at zero depth by a factor of 3.7 (TRIM.SP) and 4.4 (OKSANA); cf.
V.I. Shulga, W. Eckstein / Nucl. Instr. and Meth. in Phys. Res. B 145 (1998) 492±502
Fig. 5. The mean depth of origin of all sputtered atoms and the atoms sputtered with dierent energies as a function of bombardment energy. Simulations with the programs TRIM.SP and OKSANA. The horizontal straight lines were calculated using Eqs. (1)±(3).
Fig. 6. The mean depth of origin of all sputtered atoms and the atoms sputtered with dierent energies as a function of the projectile atomic number. Simulations with the programs TRIM.SP and OKSANA. The horizontal straight lines were calculated using Eq. (3).
Fig. 9a. We explain the gradient of recoils mainly by a strong momentum anisotropy of the cascade near the surface. At a 85°, the depth distributions display a sharp peak at zero depth for all energy intervals. In
497
Fig. 7. The mean depth of origin of all sputtered atoms and the atoms sputtered with dierent energies as a function of the angle of incidence (zero angle corresponds to normal incidence). Simulations with the programs TRIM.SP and OKSANA. The horizontal straight lines were calculated using Eq. (3).
addition, the distributions reveal some wave structure (the wave length k0 ) typical rather for a crystalline than for an amorphous target. We attribute this eect to the fact that at very glancing bombardment a recoiling atom of the uppermost layer receives a momentum directed almost perpendicularly to the surface, therefore, the next collision partner for such a recoil will be at the distance k0 from the surface etc. From Figs. 8 and 9 we can also conclude that simulations with the program TRIM.SP and OKSANA are in overall agreement with each other. The only exception is the very glancing incidence where the depth distributions of all recoils turned out to be quite sensitive to the simulation model (Fig. 9b). 3.2. Dierent targets Considerable eort in our study has been devoted to the target eects. Fig. 10 shows the mean escape depth of sputtered atoms as a function of the target atomic number. At 1 keV bombardment, the values of x for most targets are in the and may be essentially higher for range 1±3 A some targets (K, Cs, Ra).
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Fig. 8. Depth distributions of sputtered atoms for a Si target bombarded with 1 keV Ar ions at a 70° (a) and 85° (b). Simulations with the programs TRIM.SP and OKSANA. Results from 700 000 impacts accumulated.
The main conclusion which emerges from Fig. 10 is a strong correlation between the mean depth of origin and the mean atomic distance (Fig. 3). For example, in both ®gures we can see a series of peaks corresponding to alkali metals (Z2 3, 11, 19, 37, and 55) which are characterized by low values of N. Unfortunately, we do not know the values of N and Eb for Fr (Z2 87) which belongs to the group of alkali metals, therefore, an interesting point in Fig. 10 is missing. This lack is partly compensated by a point for Ra (Z2 88); this
Fig. 9. Same as Fig. 8 for all recoils in collision cascades.
element, similarly to alkali metals, has relatively low atomic density and, hence, a high calculated escape depth. According to Eqs. (1) and (2), the product of the mean escape depth and the atomic density should be constant. Fig. 11 suggests that this is not absolutely true. Even though the major part of points is squeezed in the interval x N 0.07±0.09, there are several points with much higher values of x N . The latter group includes points for Li, C(graphite), Be, and B, i.e., for the targets with small Z2 and, correspondingly, with the low stopping power which is in favour for high escape
V.I. Shulga, W. Eckstein / Nucl. Instr. and Meth. in Phys. Res. B 145 (1998) 492±502
Fig. 10. The mean depth of origin of all sputtered atoms and the atoms sputtered with dierent energies as a function of the target atomic number. Simulations with the program TRIM.SP.
depth. The two groups of targets are well separated from each other by a group of gaseous targets, from N to Ne (Z2 from 7 to 10), which are not considered in the present paper. According to Eq. (3), the ratio of the mean escape depth and the mean atomic distance should be constant. Fig. 12 shows that this is not the case
Fig. 11. The product of the mean escape depth and the target atomic density, N, as a function of the target atomic number. Simulations with the program TRIM.SP.
499
Fig. 12. The ratio of the mean escape depth and the mean atomic distance, k0 , as a function of the target atomic number. Simulations with the program TRIM.SP.
and that it happens in spite of a strong correlation between k0 and x (Figs. 3 and 10). Nevertheless, Fig. 12 indicates that the numerical values of x=k0 for many targets are quite close to the results of Eq. (3). The dependencies of the mean escape depth on the atomic density are plotted in Fig. 13a±c in a double logarithmic scale for ion energies 0.1, 1 and 10 keV. It is seen that at all energies there are two groups of points which can be approximated by straight lines. This is true both for simulations with the TRIM.SP and OKSANA programs. The origin of these groups was discussed above. The upper group includes Li, C(graphite), Be, B, and C(diamond), i.e., elements with low Z2 ; the lower (more numerous) group is bounded by Cs (the lowest atomic density) and Ni (the highest atomic density). According to Eqs. (1)±(3), the mean depth of origin x is proportional to 1=N n , where n 1 for Eqs. (1) and (2), and n 1/3 for Eq. (3). The results of simulations shown in Fig. 13 con®rm such a proportionality and suggest n 0:86 which is close to Sigmund's value n 1. This means that Sigmund [1] predicted correctly the form of the dependence x N but overestimated the values of x in numerical calculations which were based on a rough approximation of the interatomic potential.
500
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Fig. 13. The mean depth of origin of all sputtered atoms as a function of the target atomic density. Ion energies: 100 eV (a), 1 keV (b), 10 keV (c). Simulations with the programs TRIM.SP and OKSANA.
This statement can well be supported by Fig. 14 which shows the results of simulations with different interatomic potentials including the Born± Mayer potential applied in Ref. [1]. The neglect of inelastic energy losses as done in the analytical estimates increases the escape depth only marginally in the energy range investigated. It was noted above that Eqs. (1) and (2) refer to very low values of the power-law scattering parameter m. At ®xed polar ejection angle and energy, the m-dependent escape depth can be calculated using Eq. 21 from Ref. [7]. For condi-
tions of the experiment [5], this yields the values of (the Moliere±Lindhard x in the range from 5.7 A (the Born±Mayer potential, m 0.21) to 11.6 A potential, m 0.19). These values correlate well and x1 4.2 A but they are much with x0 8.4 A found experimentally. higher than x 2:0 0:4 A Evaluating the origin of this discrepancy requires a detailed analysis of the analytical and simulated dierential depth distributions of sputtered atoms. This work considering the m- and depth-dependent dierential sputter cross sections published recently [9] is in progress.
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Acknowledgements The authors would like to acknowledge useful discussions of this topic with Peter Sigmund. One of the authors (VIS) acknowledges with thanks the hospitality of the Max-Planck Institut f ur Plasmaphysik, Garching, and the ®nancial support of the Regionalverband Bayern of the German Physical Society. References
Fig. 14. The mean depth of origin of sputtered atoms versus bombardment energy. Simulations with dierent potentials using the program OKSANA (the Born±Mayer and Lenz± Jensen potentials) and TRIM.SP (all other potentials). The Moliere potential was used with the Firsov screening length [31] and taking into account the correction factor proposed by O'Connor and MacDonald [32].
4. Conclusions Computer simulations with the Monte Carlo program TRIM.SP and the lattice code OKSANA have demonstrated that the mean depth of origin of sputtered atoms is a function of many parameters such as atomic density, projectile energy, the angle of incidence, projectile and target atomic numbers. The dependence on atomic density N is proportional to N ÿ0:86 , the values of the depth of origin are generally smaller than given by the analytical theory, and the depth of origin increases with incidence energy. For a Si target bombarded with Ar ions, the energy dependence of the escape depth can be approximated by E0:17 . The depth distribution of sputtered atoms is angle dependent. For normal and not very oblique incidence, the most probable escape depth is about k0 =2. The nonzero value of the most probable escape depth indicates that the topmost atoms are not necessarily the great majority of sputtered atoms. This aspect would be of interest in connection with such processes as surface segregation etc.
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[28] M.T. Robinson, I.M. Torrens, Phys. Rev. B 9 (1974) 5008. [29] O.B. Firsov, Zh. Expt. Teor. Fiz. 36 (1959) 1517 [Sov. Fys. JETP 36 (1959) 1076]. [30] P. Sigmund, in: R. Behrisch (Ed.), Sputtering by Particle Bombardment ± I, Springer, Berlin, 1981, p. 9. [31] O.B. Firsov, Zh. Expt. Teor. Fiz. 33 (1957) 696 [Sov. Fys. JETP 6 (1958) 534]. [32] D.J. O'Connor, R.J. MacDonald, Radiat. E. 34 (1977) 247. [33] V.I. Shulga, in preparation.