- Email: [email protected]
The computer simulation of ISS
Vacuum/volume 39/numbers Printed in Great Britain
24/pages
309 to 314/I
0042-207X/89$3.00+.00 Pergamon Press plc
989
The computer simulation MHou. Universite Libre de Bruxelles CP234, Campus
of ISS de la Plaine, Bd du Triomphe,
B- 7050 Brussels, Belgium
A review is presented of the use of ion surface spectroscopy (ES) in combination with computer simulation in the last half decade. The basic principles of the ISS method and the computer simulation are presented, using either the binary collision approximation or molecular dynamics. Recent progress in modelling that was made possible by a systematic comparison with experiment is then described. Potential parameters are given for some systems. Conversely, examples are outlined that show how experimental data can be interpreted with the support of computer simulation. They concern the study of neutralization, the analysis of ISS energy and angular distributions and surface structure characterization. The references quoted should be sufficient to start a more extensive study of the topics presented in this rather short overview.
1. Introduction According to the suggestion of the American Society for Testing and Materials, the following definition of ‘ion scattering spectroscopy’ (US) will be adopted here’ : ‘a technique to elucidate composition and structure of the outermost atomic layers of a solid material, in which principally monoenergetic, singly charged, low energy (less than 10 keV) probe ions are scattered from the surface and are subsequently detected and recorded as a function of energy or scattering angle or both’. This definition provides no low energy limit to the method. In what follows, however, only high enough energies will be considered so that quantum effects can reasonably be neglected in the elastic scattering. Many other techniques are available for surface analysis and comparison is provided in several reviewszm5. General aspects of atomic surface scattering have been considered in refs 6 and 7 and reviews are also available concerning the general principles of ISS”~“. More specific aspects of ISS are the subject of many recent reviews. Discussions of methods for surface structure studies are provided in refs 12-17. The use of light ion scattering at higher energies for structure analysis is described, among other, in refs 5, 18-20. Charge exchange processes related to ISS between the probe ions and solid surfaces are reviewed in refs 15524. Some reviews are available concerning the use of ISS for the study of atomic adsorption and desorptior?” and, to our knowledge, none is available about the computer simulation of ISS, nor about the influence of ion-induced surface damage on ISS data. It is the purpose of the present paper to review progress performed in the computer simulation of ISS, with special emphasis on the last half decade. Some short comments about the production of surface damage induced by the ions used as a probe will be given, suggesting the plausible relevance of the phenomenon. 2. Basic principles 2.1. The experimental ISS method. The method is extensively described in the reviews quoted above and will only be briefly outlined here. A parallel ion beam with a well defined energy is directed onto a solid surface. As a consequence of the ion-solid interaction, a fraction of the ions may emerge from the irradiated
surface. The ISS technique aims to deduce information about the surface from the angular and energy distributions of the emerging scattered ions. In general, most attention is paid to the structures observed in the energy distributions of ions scattered within a limited solid angle. As the detection of neutral atoms is difficult (but possible) at the low energies involved in ISS, the energy distributions are often measured with an electrostatic prism and only the charged particles are then analysed. These electrostatic analysers may be mobile, then providing, in addition, angular distributions. The collected scattered angular and energy distributions depend on the lattice structure in the vicinity of the surface, the atomic masses, electronic configurations, but also on the instrumental transmission function, which is unfortunately not well described in the literature. Computer simulation significantly contributed to the characterization of surface scattering mechanisms27~34, surface structure?(in refs 39 and 40, higher energies than conventionally used for ISS are used), charge exchange processes37,3y-47 and thermal vibrations 35.46.48 In addition, the comparison between simulated and experimental ion distributions are sometimes used to discuss the validity of computer models’5.37.49m55, or experimental uncertainties4’.*6.57. Comparison between different experimental methods and different simulation codes is given in ref 58. 2.2. Computer simulation in the binary collision approximation. In such an approximation, the ion-solid interaction is described by a sequence of binary collisions between the incident ion and the target atoms. In general, only their asymptotical motion is considered and the scattering and recoiling momentum are determined from the scattering integrals :
s 73
O=n-2h
r2g(r) RO
z = (R; -by”2
_ s
’ dr
m R, g(r)-
’ - (1 -~‘/r’)~
‘!’ dr,
(2)
with
g(r) = [l -b2/r2
- (1 +A)V(r)/(AE,)]“2
and where 0 is the centre of mass scattering
angle, b the impact 309
/VlHou: Computer simulation of ISS parameter, R,, the distance of closest approach, T the time integral, E,, the incident energy, A the ratio of the mass of the recoiling atom to that of the scattered one and V(r) the central pairwise repulsive potential between atoms separated by a distance r. Methods for the numerical integration of equations (I) and (2) can be found in ref 59 and the relation with the asymptotical motion in the laboratory system is described in detail in rcf 60. The role of the time integral in the estimation of the position of the trajectories is most important for the lowest cnergctic collisions and is negligible in the cast of light particles at high energies, when the momentum approximation applies. Since the exact position of the asymptotes of the trajectories arc dependent on the time integral, its role is significant in the calculation of correlated binary collisions, as they take place in single crystals. In order to simulate a single crystal. its translational symmetry is used to determine successive atomic target positions. Collision partners arc chosen on the basis of criteria of which details depend upon the code. A maximum impact parameter is. however, always assumed, which limits the number of collisions to be calculated. The trajectories of recoiling atoms do not need to bc calculated for the purpose of ISS and those of the scattered atoms are cut off either if they escape from the target or if their kinetic energy falls below some threshold value. Several sophistications to this model exist which may, in principle, improve the relation between simulated and real experiments. For instance, in many situations, several candidates as target atoms can meet the criteria to be collision partners at the same step of the trajectory calculation. The quasi-simultaneous interaction between three or more atoms is not handled correctly in the binary collision approximation. Such interactions are however of significant importance in some scattering processes such as the focusing of trajectories and rainbow scattering. An approximate treatment of quasi simultaneous events is given in ref 61. The treatment preserves the total momentum but overestimates somewhat the energy lost by the scattered particle. Provision can be made for inelastic energy loss to electronic excitation, currently based on the Lindhard theory”’ or, as local inelastic energy loss is considered, the Firsov model”’ or the OenRobinson formulah4. Both local and nonlocal inelastic losses may have weighted contributions, the consequences of which are not yet thoroughly studied on the simulation of ISS. A modification of the Firsov model accounting for was found reasonable in several prclarge mass difference? dictionshh but is not very convenient for numerical calculations. The possibility of specific surface modelling”-” allows defect states, reconstruction and relaxation to be taken into account. Finally, the characterization of surface thermal vibrations requires their modelling as well. The description of uncorrelated thermal displacements is conveniently provided by the Debyc Waller modelh7 with the Debye temperature as single parameter. A convenient procedure for sampling displacements with avoiding occasional very large amplitudes is given in ref 65. The treatment of correlated thermal vibrations is discussed in ref 6X. A Fortran algorithm based on a simple correlation model is published in ref 70. An ISS experiment may require more than I nA during 60 s. This represents a dose of 3.75 x IO” incident ions. Experimentalists consider this number necessary in order to get sufficient statistics in energy distributions collected within a limited solid angle. The order of magnitude of the computer time required to fully simulate an experiment on one of the most powerful computers in scalar mode can be evaluated to be about 310
one century, which may reasonably be considered as prohibitive. Therefore, the statistics in experimental data are always bettct than in computer simulations. In this respect, computer simulation does not allow experimental data to bc analyscd in full detail. The comparison between experiment and simulation has thus to be limited to the most salient features. Even though. required computer times may be prohibitive and several procedures to speed up calculations are suggested : significant ones concern the calculation of the scattering integrals which can bc replaced by a table look up procedure as in ref 47. This may represent an improvement of the order of 30% in computing time; in case of axial channelled trajectories, the target search can bc most simplified”, allowing a further reduction of 30% of the required computer time. For the study of some specific scattering mechanisms, two dimensional trajectory simulations may hc sufficient. Two dimensional calculations require about two orders of magnitude less calculation time than three dimensional ones. A special algorithm was recently developed. associating impact arcas to surface scattering mechanisms”.“. which allows cffectivc three dimensional trajectory calculations related to specific scattcring mechanisms. To our knowledge. it requires the smallest computer time and predicts ISS intensities associated to a well defined scattering mechanism with an accuracy better than USLIally reached in experiment. The estimation of one impact arca requires about IO s on an IBM 3090 machine. The validity of the binary collision approximation for the simulation of ISS is assessed by many comparisons between simulation and experiment~7.‘h-‘x,‘” “. The approximation is, however. expected to break down when a characteristic distance in binary collisions, e.g. the distance of closest approach in head on collisions. cannot bc considered as small when compared to the target atoms spacing. In such circumstances, the trajectories cannot be approximated by the asymptotic motion and an altcrnativc method is required. 2.3. Computer simulation with molecular dynamics. With the molecular dynamics method, the motion of a given number of particles is calculated as a function of time. This motion is governed by an interatomic pairwise potential. The method is based on the numerical integration of the Hamilton equations : dr,
c?H
dt =
$p,= 111,
+4
-2H
(jt
=
&.,
p, 6 V(r) =
(jr,
3
where
H =
:,T:+V(r)
is the Hamiltonian. The method was applied to solids for the first time in radiation damage studies by Gibson C/ ~1” and the integration of the equations of motion were integrated stepwise in time. Problems inherent to the method and characteristic of a statistical system evolving toward thermodynamical equilibrium arc very nicely reviewed in ref 74. Since. owing to computer limitations, the Hamilton equations can only be integrated for a limited number of particles. geometrical continuation may be required. Rcnormalization of velocities and/or atomic volumes may be necessary, according to the choice of boundary conditions. The situ-
M Hou; Computer simulation
of ISS
ation is simplified as far as ISS is concerneds4 since the trajectory segments relevant to surface scattering are short and the elapsed time between the first impact and the emergence of the scattered particle is small when compared to the time required to reach equilibrium. As a result, the technique is easier and appreciably less time consuming for ISS than for radiation damage studies. Nevertheless, the same evaluation as performed for the binary collision approximation would suggest that to simulate an experimental spectrum with molecular dynamics may require a couple of millennia. Therefore, the impact area method mentioned above, which is already used in conjunction with the binary collision approximation could be of great help when molecular dynamics is necessary. In the Hamilton equations, the potential is usually constructed as a sum of pairwise central potentials. This was suggested to be a limitation to the molecular dynamics method on the basis of a comparison between simulation and alkali-ion hyperthermal scattering experiments53. An additional repulsion of the alkali ion from a tungsten cluster was estimated to have a significant influence on the angular reflection distribution. As can be seen from the present section, no existing simulation model may presently be assessed as entirely workable for the discussion of ISS experiments although it is found helpful in the understanding of several experimental facts. Therefore, the next section is subdivided into two parts. In the first, it will be shown how ISS experiments help in improving the computer models. Some examples will show how computer simulation helped to understand experimental data in the second. The overview is quite brief, but relevant references are provided. 3. Selected progress in computer modelling and ISS 3.1. ISS for modelling. Several authors use available experimental ISS results in order to assess their computer model. This is the case in refs 45,49 and 52, which represent a sample of codes using the binary collision approximation49*52 and molecular dynamics45. All three codes are found to provide reasonable agreement with the experimental data selected by the authors. In ref 52, the same experiment 75 is simulated as in ref 45 and, in addition, both binary collision codes49,52 are compared with the experimental data presented in ref 76. These latter codes are based on quite similar approximations. Both neglect the time integral (see equation 2) in the trajectory calculation and only non-local inelastic energy losses are considered on the basis of Lindhard’s theoryh3. No special care is taken in the distinction between distance of closest approach and impact parameter. The former is underestimated in ref 49, the latter overestimated in ref 52. Despite these approximations, reasonable qualitative agreement is found with the results in ref 76, which reports energy distributions obtained using a time of flight method, thus allowing to detect and analyse both neutrals and ions. The number of trials seems sufficient in ref 49 to detect a discrepancy between the experimental and simulated peak to background ratios. Although the most salient experimental features are reproduced by the simulations, the nature of the background is less investigated”. A systematic study of its origin began recently’“. Neither the binary collision approximation, nor molecular dynamics are found sufficient to reproduce the experimental results75. The authors of both papers suggest this major discrepancy to be a consequence of neutralization, which is not accounted for in the simulations nor controlled in the experiment. The point is not definitively settled, however. Detailed trajectory
analysis showed that the main features observed in molecular dynamics calculations are reproduced in the binary collision approximation, at least qualitatively. This may not be true, however, when very low (hyperthermal) energies are involved33,53.54. One of the most critical parameters in the simulation of ISS is the potential. The Moliere approximation to the Thomas-Fermi potential79 is quite often used in relation with the binary collision approximation. It is a screened Coulomb potential and can be written as
V(r) =
(z,z,P’/r)qr/a)
with CD(x) = 0.35 exp (-0.3x)+0.55
exp (-1.2x) f0.1
exp (-6.0x),
(8)
where Yis the interatomic separation and a is the screening length, This parameter was calculated by Firsov” a = 0.8853a,(Z;i2
+z;:‘)-
213,
where a, = 0.529 A is the Bohr radius. Equation (9) is known only to provide an approximate value of the screening length and ISS is most helpful to adjust its magnitude for specific systems. Several methods were used for its quantitative estimate. One is based on the matching of the so-called ‘loop diagrams’ representing the relation between scattering energies and angles for single and multiple scattering peaks observed in energy distributionsasE2. A second makes use of the estimation of shadow cones in the so-called ‘impact collision mode’50, a third adjusts simulated and experimental relative peak intensities3” and a fourth” adjusts the screening length by matching experimental and simulated peak intensity ratios in ISS energy distributions for systems where neutralization effects are negligibly small. Cross checks exist between estimates with these different methods. The results are reported in Table 1. It should be noticed that the values in Table 1 do not deviate systematically from the Firsov equation (9). Also, all these estimates were performed for incident energies between 200 eV and 1 keV. The values in Table 1 are not necessarily valid for significantly different interaction energies since the Moliere screening function (8) is only approximate. Further recent discussions of potential functions are given in refs 50 and 55 and for lower energies in refs 53, 54 and 83. No model potential suggested in the literature is found to be sufficiently accurate for ISS without any parameter adjustment. The Moliere potential has the advantage with regard to others to depend on only one parameter and to allow fast numerical estimate since it requires the calculation of only one exponential. The discussion of experimental data on the basis of simulation is often hampered by charge exchange processes that are not always well known enough to be modelled. Several are reviewed in ref 24. Extensive experimental data are however available concerning the charge fraction of scattered ion beam?‘, Auger Table 1. Screening lengths, a, for the Moliere potential, comparison between computer simulation and ISS
estimated
by
System
a (A)
ref
System
a (A)
ref
Ne-Ag Ne-Ni Ne-W LikMo NapMo
0.105 0.0945 0.094 0.1035 0.1023
82 80 82 50 50
K-Ag K-Au KG K-MO K-W
0.0795 0.0823 0.0902 0.0955 0.0984
55 38, 55, 80 55 50, 55 55
311
M Hou: Computer simulation of ISS ncutrdlization”‘~x’ “. resonant electron cxchange”Y.xx and reionization processes X9” The available data however spread over quite different interaction systems and no universal empirical rule embodies the various possible charge exchange mechanisms which could be incoporated into a general computer model. 3.2. Modelling for ES. Since no accurate general model is available for the description of charge exchange. its contribution to ISS can only be somewhat qualitatively modclled in computer simulationi7~“” “. On the other hand, insofar as a good potential model is used, to compare experimental data with computer simulation not accounting for charge exchange may bring some bcttcr quantitative information about neutralization, as suggested by the work in ref 34. One way to get rid of difficulties inherent to charge exchange is to choose systems where they don’t play a role. To use alkalis as probe ions on metallic surfaces is therefore very useful. For several systems. the rate of charge exchange is negligibly small. Experiment suggests that the charge state of the scattered alkali-ions is trajectory indcpendent’4. As a consequence, a proportionality is expected between simulation and experimental results. When only a small fraction of the incident ions arc neutralized, depending on the scattering geometry, the structure of ISS energy distributions is rather complex. The computer simulation is helpful to resolve this complexity. An extensive scattering mechanism analysis was pcrformed in ref 95. This work, which is essentially experimental was supported by computer simulation all along its progress, providing the necessary confidence in the interpretation of energy distributions”‘. Similar mechanism identification procedures were used by other authors as ~ell”~“.‘~~” “. Such kind of analysis by computer simulation requires three dimensional calculations with a target at least two atomic layers thick and rather large computer times. In many cases, the statistics are not sufficient to extract information about the background. Nevcrtheless, the method proved to be successful when only a few scattering processes contribute to each peak in the energy distributions Therefore alkali-ion surface scattering could be used, in combination with computer simulation for structure analysis” 3X as well as adsorbate distributions”. It is worth noticing that when an adequate geometry is used, the structure of ISS energy distributions is simple enough to be understood without the help of computer simulation. This simplicity is at the basis of the ‘impact collision mode’ of ISS. as described in ref 16. In this context, simulation may however become necessary when the fluxes close to the edge of the shadow cones need to be known and that analytical estimates are not sufficient. ISS is a destructive technique. A short qualitative discussion of the influence of surface defects on ISS was given recently”. The defect state of the surface during the irradiation was already observed earlier by LEED”‘. Depending on the projectile to target mass ratio, the major source of damage can be the displaced atoms or the implanted impurities. These types of damage can be quite different. Also, when insulators are irradiated, clectron excitations may induce additional atomic displacements. When the annealing time is small enough (which is generally the case for metals), the use of low dose rates may guarantee the state of the surface to remain undisturbed on small scales, as far as the surface structure is stable and the incident ions are heavy enough so that their penetration is small enough to allow their spontaneous dcsorption. When the probe ions are light, they penetrate deeper and are not eliminated by sputtering or desorption. Since their mean penetration depth is of several tenths of 312
Angstroms in conditions typical to ISS and that their spatial distribution is rather widespread. a usual heat treatment between measurements may not be sufficient to desorb them entirely. The helium diffusion, which is enhanced by high temperatures, favours the clustering of the impurities, their trapping in damage sites, with high binding energies. A review of possible surface damage consequent to helium injection is given in ref98, although higher energies than these typical to ISS arc considered. When the growth of overlayers is studied by means of ISS with heavy ions, recoil mixing and induced segregation are. in principle. possible. The consequences are not commented in the literature. Such processes can be studied by computer simulation. which should allow a better control of ovcrlayers during their irradiation. 4. Summary In this review, I have attempted to describe the interaction between the ISS and computer simulation methods. with emphasis on recently tackled questions. Although limited, the bibliography provides keys for further documation. General reviews about ISS and surface techniques are suggested in the introduction, others arc given which involve computer simulation. Literature relevant to computer modelling is mentioned with the description of the basic principles. The third section aims to show the reciprocal usefulness of ISS and computer simulation. Several topics arc taken up. An example is given where one specific experiment was simulated both in the binary collision approximation and by molecular dynamics. suggesting that the former is sufficient. However, it breaks down at very low energies and a limit to the application of molecular dynamics to hyperthermal ion scattering is pointed out. Progress in interatomic potential determination by means of the comparison between ISS and simulation is described. Some data could be provided that arc helpful to adjust the Moliere potential parameter. Dithculties in the simulation of charge exchange between an ion and a surface are emphasized. They suggest that. owing to the various possible charge exchange mechanisms, specific modelling might be necessary. On the other hand, when the model parameters are reasonably settled, comparison with real experiments helps to learn about neutralization. When systems arc used in which neutralization has no influence on scattering energy distributions, additional features appear which may be complex. Computer simulation helps in their interpretation. Once the relation between these features and scattering mechanisms are identified and a reasonable model potential is found. computei simulation, in combination with experiment can be used efficiently for surface structure characterization. Several steps thus appear necessary before surface structure analysis is possible: to control the neutralization mechanisms, to identify the relevant surface scattering processes and to find good potential parameters. Still. important questions arc shown to remain open, concerning the physical interpretation of the background in cxpcrimental energy distributions, the damage produced by low energy light and heavy ions and the modelling of inelastic energy losses. It is suggested that further computer simulations in combination with experiment should help in their better understanding. Acknowledgements
It is a pleasure
to thank P Bertrand. W Eckstein. W Heiland and R Ghrayeb for their discussion, their help in preparing the bibliography and their critical reading of the manuscipt.
M Hou:
Computer simulation
of ISS
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A4 Hou: Computer simulation
of ISS
“” L K Verheij, B Poelsema and A L Boers, Nucl Instrum Meth, 132, 565 (1971). ” D P Woodruff, Nucl Ozstrwn Mrtlr, 194, 639 (I 982). “‘T M Thomas, H Neumann, A W Czanderna and J R Pitts, Surfuc~e Sci, 175, L737 (1986). ‘l*R Souda, M Aono, C Oshima. S Otani and Y Ishizawa, SU+CP Sci, 176,657 (1986). ‘)-IR Saouda and M Aono, Nucl Instrum Meth, 815, I 14 (1986).
314
“T Von dcm Hagen, thesis, Clausthal-Zellerfeld, Germany (1982). u6S H Overbury, Nucl Instrum Merh, B2, 448 (1984). ‘)’ W Heiland and E Taglauer, Radiat EJ&nrs, 19, 1 (I 973). ‘)’ B M U Scherzer, in Sputtering h.v Particle Bomhardmenl II. Sputtering of Alloys and Compounds, Neutron Spulteriny, Surfhcc Topcqraphy. Topics in Applied Physics (Edited by R Behrisch), ~0152. p 27 1.SpringerVcrlag, Berlin (1983).
24/pages
309 to 314/I
0042-207X/89$3.00+.00 Pergamon Press plc
989
The computer simulation MHou. Universite Libre de Bruxelles CP234, Campus
of ISS de la Plaine, Bd du Triomphe,
B- 7050 Brussels, Belgium
A review is presented of the use of ion surface spectroscopy (ES) in combination with computer simulation in the last half decade. The basic principles of the ISS method and the computer simulation are presented, using either the binary collision approximation or molecular dynamics. Recent progress in modelling that was made possible by a systematic comparison with experiment is then described. Potential parameters are given for some systems. Conversely, examples are outlined that show how experimental data can be interpreted with the support of computer simulation. They concern the study of neutralization, the analysis of ISS energy and angular distributions and surface structure characterization. The references quoted should be sufficient to start a more extensive study of the topics presented in this rather short overview.
1. Introduction According to the suggestion of the American Society for Testing and Materials, the following definition of ‘ion scattering spectroscopy’ (US) will be adopted here’ : ‘a technique to elucidate composition and structure of the outermost atomic layers of a solid material, in which principally monoenergetic, singly charged, low energy (less than 10 keV) probe ions are scattered from the surface and are subsequently detected and recorded as a function of energy or scattering angle or both’. This definition provides no low energy limit to the method. In what follows, however, only high enough energies will be considered so that quantum effects can reasonably be neglected in the elastic scattering. Many other techniques are available for surface analysis and comparison is provided in several reviewszm5. General aspects of atomic surface scattering have been considered in refs 6 and 7 and reviews are also available concerning the general principles of ISS”~“. More specific aspects of ISS are the subject of many recent reviews. Discussions of methods for surface structure studies are provided in refs 12-17. The use of light ion scattering at higher energies for structure analysis is described, among other, in refs 5, 18-20. Charge exchange processes related to ISS between the probe ions and solid surfaces are reviewed in refs 15524. Some reviews are available concerning the use of ISS for the study of atomic adsorption and desorptior?” and, to our knowledge, none is available about the computer simulation of ISS, nor about the influence of ion-induced surface damage on ISS data. It is the purpose of the present paper to review progress performed in the computer simulation of ISS, with special emphasis on the last half decade. Some short comments about the production of surface damage induced by the ions used as a probe will be given, suggesting the plausible relevance of the phenomenon. 2. Basic principles 2.1. The experimental ISS method. The method is extensively described in the reviews quoted above and will only be briefly outlined here. A parallel ion beam with a well defined energy is directed onto a solid surface. As a consequence of the ion-solid interaction, a fraction of the ions may emerge from the irradiated
surface. The ISS technique aims to deduce information about the surface from the angular and energy distributions of the emerging scattered ions. In general, most attention is paid to the structures observed in the energy distributions of ions scattered within a limited solid angle. As the detection of neutral atoms is difficult (but possible) at the low energies involved in ISS, the energy distributions are often measured with an electrostatic prism and only the charged particles are then analysed. These electrostatic analysers may be mobile, then providing, in addition, angular distributions. The collected scattered angular and energy distributions depend on the lattice structure in the vicinity of the surface, the atomic masses, electronic configurations, but also on the instrumental transmission function, which is unfortunately not well described in the literature. Computer simulation significantly contributed to the characterization of surface scattering mechanisms27~34, surface structure?(in refs 39 and 40, higher energies than conventionally used for ISS are used), charge exchange processes37,3y-47 and thermal vibrations 35.46.48 In addition, the comparison between simulated and experimental ion distributions are sometimes used to discuss the validity of computer models’5.37.49m55, or experimental uncertainties4’.*6.57. Comparison between different experimental methods and different simulation codes is given in ref 58. 2.2. Computer simulation in the binary collision approximation. In such an approximation, the ion-solid interaction is described by a sequence of binary collisions between the incident ion and the target atoms. In general, only their asymptotical motion is considered and the scattering and recoiling momentum are determined from the scattering integrals :
s 73
O=n-2h
r2g(r) RO
z = (R; -by”2
_ s
’ dr
m R, g(r)-
’ - (1 -~‘/r’)~
‘!’ dr,
(2)
with
g(r) = [l -b2/r2
- (1 +A)V(r)/(AE,)]“2
and where 0 is the centre of mass scattering
angle, b the impact 309
/VlHou: Computer simulation of ISS parameter, R,, the distance of closest approach, T the time integral, E,, the incident energy, A the ratio of the mass of the recoiling atom to that of the scattered one and V(r) the central pairwise repulsive potential between atoms separated by a distance r. Methods for the numerical integration of equations (I) and (2) can be found in ref 59 and the relation with the asymptotical motion in the laboratory system is described in detail in rcf 60. The role of the time integral in the estimation of the position of the trajectories is most important for the lowest cnergctic collisions and is negligible in the cast of light particles at high energies, when the momentum approximation applies. Since the exact position of the asymptotes of the trajectories arc dependent on the time integral, its role is significant in the calculation of correlated binary collisions, as they take place in single crystals. In order to simulate a single crystal. its translational symmetry is used to determine successive atomic target positions. Collision partners arc chosen on the basis of criteria of which details depend upon the code. A maximum impact parameter is. however, always assumed, which limits the number of collisions to be calculated. The trajectories of recoiling atoms do not need to bc calculated for the purpose of ISS and those of the scattered atoms are cut off either if they escape from the target or if their kinetic energy falls below some threshold value. Several sophistications to this model exist which may, in principle, improve the relation between simulated and real experiments. For instance, in many situations, several candidates as target atoms can meet the criteria to be collision partners at the same step of the trajectory calculation. The quasi-simultaneous interaction between three or more atoms is not handled correctly in the binary collision approximation. Such interactions are however of significant importance in some scattering processes such as the focusing of trajectories and rainbow scattering. An approximate treatment of quasi simultaneous events is given in ref 61. The treatment preserves the total momentum but overestimates somewhat the energy lost by the scattered particle. Provision can be made for inelastic energy loss to electronic excitation, currently based on the Lindhard theory”’ or, as local inelastic energy loss is considered, the Firsov model”’ or the OenRobinson formulah4. Both local and nonlocal inelastic losses may have weighted contributions, the consequences of which are not yet thoroughly studied on the simulation of ISS. A modification of the Firsov model accounting for was found reasonable in several prclarge mass difference? dictionshh but is not very convenient for numerical calculations. The possibility of specific surface modelling”-” allows defect states, reconstruction and relaxation to be taken into account. Finally, the characterization of surface thermal vibrations requires their modelling as well. The description of uncorrelated thermal displacements is conveniently provided by the Debyc Waller modelh7 with the Debye temperature as single parameter. A convenient procedure for sampling displacements with avoiding occasional very large amplitudes is given in ref 65. The treatment of correlated thermal vibrations is discussed in ref 6X. A Fortran algorithm based on a simple correlation model is published in ref 70. An ISS experiment may require more than I nA during 60 s. This represents a dose of 3.75 x IO” incident ions. Experimentalists consider this number necessary in order to get sufficient statistics in energy distributions collected within a limited solid angle. The order of magnitude of the computer time required to fully simulate an experiment on one of the most powerful computers in scalar mode can be evaluated to be about 310
one century, which may reasonably be considered as prohibitive. Therefore, the statistics in experimental data are always bettct than in computer simulations. In this respect, computer simulation does not allow experimental data to bc analyscd in full detail. The comparison between experiment and simulation has thus to be limited to the most salient features. Even though. required computer times may be prohibitive and several procedures to speed up calculations are suggested : significant ones concern the calculation of the scattering integrals which can bc replaced by a table look up procedure as in ref 47. This may represent an improvement of the order of 30% in computing time; in case of axial channelled trajectories, the target search can bc most simplified”, allowing a further reduction of 30% of the required computer time. For the study of some specific scattering mechanisms, two dimensional trajectory simulations may hc sufficient. Two dimensional calculations require about two orders of magnitude less calculation time than three dimensional ones. A special algorithm was recently developed. associating impact arcas to surface scattering mechanisms”.“. which allows cffectivc three dimensional trajectory calculations related to specific scattcring mechanisms. To our knowledge. it requires the smallest computer time and predicts ISS intensities associated to a well defined scattering mechanism with an accuracy better than USLIally reached in experiment. The estimation of one impact arca requires about IO s on an IBM 3090 machine. The validity of the binary collision approximation for the simulation of ISS is assessed by many comparisons between simulation and experiment~7.‘h-‘x,‘” “. The approximation is, however. expected to break down when a characteristic distance in binary collisions, e.g. the distance of closest approach in head on collisions. cannot bc considered as small when compared to the target atoms spacing. In such circumstances, the trajectories cannot be approximated by the asymptotic motion and an altcrnativc method is required. 2.3. Computer simulation with molecular dynamics. With the molecular dynamics method, the motion of a given number of particles is calculated as a function of time. This motion is governed by an interatomic pairwise potential. The method is based on the numerical integration of the Hamilton equations : dr,
c?H
dt =
$p,= 111,
+4
-2H
(jt
=
&.,
p, 6 V(r) =
(jr,
3
where
H =
:,T:+V(r)
is the Hamiltonian. The method was applied to solids for the first time in radiation damage studies by Gibson C/ ~1” and the integration of the equations of motion were integrated stepwise in time. Problems inherent to the method and characteristic of a statistical system evolving toward thermodynamical equilibrium arc very nicely reviewed in ref 74. Since. owing to computer limitations, the Hamilton equations can only be integrated for a limited number of particles. geometrical continuation may be required. Rcnormalization of velocities and/or atomic volumes may be necessary, according to the choice of boundary conditions. The situ-
M Hou; Computer simulation
of ISS
ation is simplified as far as ISS is concerneds4 since the trajectory segments relevant to surface scattering are short and the elapsed time between the first impact and the emergence of the scattered particle is small when compared to the time required to reach equilibrium. As a result, the technique is easier and appreciably less time consuming for ISS than for radiation damage studies. Nevertheless, the same evaluation as performed for the binary collision approximation would suggest that to simulate an experimental spectrum with molecular dynamics may require a couple of millennia. Therefore, the impact area method mentioned above, which is already used in conjunction with the binary collision approximation could be of great help when molecular dynamics is necessary. In the Hamilton equations, the potential is usually constructed as a sum of pairwise central potentials. This was suggested to be a limitation to the molecular dynamics method on the basis of a comparison between simulation and alkali-ion hyperthermal scattering experiments53. An additional repulsion of the alkali ion from a tungsten cluster was estimated to have a significant influence on the angular reflection distribution. As can be seen from the present section, no existing simulation model may presently be assessed as entirely workable for the discussion of ISS experiments although it is found helpful in the understanding of several experimental facts. Therefore, the next section is subdivided into two parts. In the first, it will be shown how ISS experiments help in improving the computer models. Some examples will show how computer simulation helped to understand experimental data in the second. The overview is quite brief, but relevant references are provided. 3. Selected progress in computer modelling and ISS 3.1. ISS for modelling. Several authors use available experimental ISS results in order to assess their computer model. This is the case in refs 45,49 and 52, which represent a sample of codes using the binary collision approximation49*52 and molecular dynamics45. All three codes are found to provide reasonable agreement with the experimental data selected by the authors. In ref 52, the same experiment 75 is simulated as in ref 45 and, in addition, both binary collision codes49,52 are compared with the experimental data presented in ref 76. These latter codes are based on quite similar approximations. Both neglect the time integral (see equation 2) in the trajectory calculation and only non-local inelastic energy losses are considered on the basis of Lindhard’s theoryh3. No special care is taken in the distinction between distance of closest approach and impact parameter. The former is underestimated in ref 49, the latter overestimated in ref 52. Despite these approximations, reasonable qualitative agreement is found with the results in ref 76, which reports energy distributions obtained using a time of flight method, thus allowing to detect and analyse both neutrals and ions. The number of trials seems sufficient in ref 49 to detect a discrepancy between the experimental and simulated peak to background ratios. Although the most salient experimental features are reproduced by the simulations, the nature of the background is less investigated”. A systematic study of its origin began recently’“. Neither the binary collision approximation, nor molecular dynamics are found sufficient to reproduce the experimental results75. The authors of both papers suggest this major discrepancy to be a consequence of neutralization, which is not accounted for in the simulations nor controlled in the experiment. The point is not definitively settled, however. Detailed trajectory
analysis showed that the main features observed in molecular dynamics calculations are reproduced in the binary collision approximation, at least qualitatively. This may not be true, however, when very low (hyperthermal) energies are involved33,53.54. One of the most critical parameters in the simulation of ISS is the potential. The Moliere approximation to the Thomas-Fermi potential79 is quite often used in relation with the binary collision approximation. It is a screened Coulomb potential and can be written as
V(r) =
(z,z,P’/r)qr/a)
with CD(x) = 0.35 exp (-0.3x)+0.55
exp (-1.2x) f0.1
exp (-6.0x),
(8)
where Yis the interatomic separation and a is the screening length, This parameter was calculated by Firsov” a = 0.8853a,(Z;i2
+z;:‘)-
213,
where a, = 0.529 A is the Bohr radius. Equation (9) is known only to provide an approximate value of the screening length and ISS is most helpful to adjust its magnitude for specific systems. Several methods were used for its quantitative estimate. One is based on the matching of the so-called ‘loop diagrams’ representing the relation between scattering energies and angles for single and multiple scattering peaks observed in energy distributionsasE2. A second makes use of the estimation of shadow cones in the so-called ‘impact collision mode’50, a third adjusts simulated and experimental relative peak intensities3” and a fourth” adjusts the screening length by matching experimental and simulated peak intensity ratios in ISS energy distributions for systems where neutralization effects are negligibly small. Cross checks exist between estimates with these different methods. The results are reported in Table 1. It should be noticed that the values in Table 1 do not deviate systematically from the Firsov equation (9). Also, all these estimates were performed for incident energies between 200 eV and 1 keV. The values in Table 1 are not necessarily valid for significantly different interaction energies since the Moliere screening function (8) is only approximate. Further recent discussions of potential functions are given in refs 50 and 55 and for lower energies in refs 53, 54 and 83. No model potential suggested in the literature is found to be sufficiently accurate for ISS without any parameter adjustment. The Moliere potential has the advantage with regard to others to depend on only one parameter and to allow fast numerical estimate since it requires the calculation of only one exponential. The discussion of experimental data on the basis of simulation is often hampered by charge exchange processes that are not always well known enough to be modelled. Several are reviewed in ref 24. Extensive experimental data are however available concerning the charge fraction of scattered ion beam?‘, Auger Table 1. Screening lengths, a, for the Moliere potential, comparison between computer simulation and ISS
estimated
by
System
a (A)
ref
System
a (A)
ref
Ne-Ag Ne-Ni Ne-W LikMo NapMo
0.105 0.0945 0.094 0.1035 0.1023
82 80 82 50 50
K-Ag K-Au KG K-MO K-W
0.0795 0.0823 0.0902 0.0955 0.0984
55 38, 55, 80 55 50, 55 55
311
M Hou: Computer simulation of ISS ncutrdlization”‘~x’ “. resonant electron cxchange”Y.xx and reionization processes X9” The available data however spread over quite different interaction systems and no universal empirical rule embodies the various possible charge exchange mechanisms which could be incoporated into a general computer model. 3.2. Modelling for ES. Since no accurate general model is available for the description of charge exchange. its contribution to ISS can only be somewhat qualitatively modclled in computer simulationi7~“” “. On the other hand, insofar as a good potential model is used, to compare experimental data with computer simulation not accounting for charge exchange may bring some bcttcr quantitative information about neutralization, as suggested by the work in ref 34. One way to get rid of difficulties inherent to charge exchange is to choose systems where they don’t play a role. To use alkalis as probe ions on metallic surfaces is therefore very useful. For several systems. the rate of charge exchange is negligibly small. Experiment suggests that the charge state of the scattered alkali-ions is trajectory indcpendent’4. As a consequence, a proportionality is expected between simulation and experimental results. When only a small fraction of the incident ions arc neutralized, depending on the scattering geometry, the structure of ISS energy distributions is rather complex. The computer simulation is helpful to resolve this complexity. An extensive scattering mechanism analysis was pcrformed in ref 95. This work, which is essentially experimental was supported by computer simulation all along its progress, providing the necessary confidence in the interpretation of energy distributions”‘. Similar mechanism identification procedures were used by other authors as ~ell”~“.‘~~” “. Such kind of analysis by computer simulation requires three dimensional calculations with a target at least two atomic layers thick and rather large computer times. In many cases, the statistics are not sufficient to extract information about the background. Nevcrtheless, the method proved to be successful when only a few scattering processes contribute to each peak in the energy distributions Therefore alkali-ion surface scattering could be used, in combination with computer simulation for structure analysis” 3X as well as adsorbate distributions”. It is worth noticing that when an adequate geometry is used, the structure of ISS energy distributions is simple enough to be understood without the help of computer simulation. This simplicity is at the basis of the ‘impact collision mode’ of ISS. as described in ref 16. In this context, simulation may however become necessary when the fluxes close to the edge of the shadow cones need to be known and that analytical estimates are not sufficient. ISS is a destructive technique. A short qualitative discussion of the influence of surface defects on ISS was given recently”. The defect state of the surface during the irradiation was already observed earlier by LEED”‘. Depending on the projectile to target mass ratio, the major source of damage can be the displaced atoms or the implanted impurities. These types of damage can be quite different. Also, when insulators are irradiated, clectron excitations may induce additional atomic displacements. When the annealing time is small enough (which is generally the case for metals), the use of low dose rates may guarantee the state of the surface to remain undisturbed on small scales, as far as the surface structure is stable and the incident ions are heavy enough so that their penetration is small enough to allow their spontaneous dcsorption. When the probe ions are light, they penetrate deeper and are not eliminated by sputtering or desorption. Since their mean penetration depth is of several tenths of 312
Angstroms in conditions typical to ISS and that their spatial distribution is rather widespread. a usual heat treatment between measurements may not be sufficient to desorb them entirely. The helium diffusion, which is enhanced by high temperatures, favours the clustering of the impurities, their trapping in damage sites, with high binding energies. A review of possible surface damage consequent to helium injection is given in ref98, although higher energies than these typical to ISS arc considered. When the growth of overlayers is studied by means of ISS with heavy ions, recoil mixing and induced segregation are. in principle. possible. The consequences are not commented in the literature. Such processes can be studied by computer simulation. which should allow a better control of ovcrlayers during their irradiation. 4. Summary In this review, I have attempted to describe the interaction between the ISS and computer simulation methods. with emphasis on recently tackled questions. Although limited, the bibliography provides keys for further documation. General reviews about ISS and surface techniques are suggested in the introduction, others arc given which involve computer simulation. Literature relevant to computer modelling is mentioned with the description of the basic principles. The third section aims to show the reciprocal usefulness of ISS and computer simulation. Several topics arc taken up. An example is given where one specific experiment was simulated both in the binary collision approximation and by molecular dynamics. suggesting that the former is sufficient. However, it breaks down at very low energies and a limit to the application of molecular dynamics to hyperthermal ion scattering is pointed out. Progress in interatomic potential determination by means of the comparison between ISS and simulation is described. Some data could be provided that arc helpful to adjust the Moliere potential parameter. Dithculties in the simulation of charge exchange between an ion and a surface are emphasized. They suggest that. owing to the various possible charge exchange mechanisms, specific modelling might be necessary. On the other hand, when the model parameters are reasonably settled, comparison with real experiments helps to learn about neutralization. When systems arc used in which neutralization has no influence on scattering energy distributions, additional features appear which may be complex. Computer simulation helps in their interpretation. Once the relation between these features and scattering mechanisms are identified and a reasonable model potential is found. computei simulation, in combination with experiment can be used efficiently for surface structure characterization. Several steps thus appear necessary before surface structure analysis is possible: to control the neutralization mechanisms, to identify the relevant surface scattering processes and to find good potential parameters. Still. important questions arc shown to remain open, concerning the physical interpretation of the background in cxpcrimental energy distributions, the damage produced by low energy light and heavy ions and the modelling of inelastic energy losses. It is suggested that further computer simulations in combination with experiment should help in their better understanding. Acknowledgements
It is a pleasure
to thank P Bertrand. W Eckstein. W Heiland and R Ghrayeb for their discussion, their help in preparing the bibliography and their critical reading of the manuscipt.
M Hou:
Computer simulation
of ISS
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