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A MD study of low energy boron bombardment on silicon
Nuclear Instruments and Methods in Physics Research B 164±165 (2000) 431±440
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A MD study of low energy boron bombardment on silicon A. Mari Carmen Perez-Martõn *, Javier Domõnguez-V azquez, Jose J. Jimenez-Rodrõguez Departamento de Electricidad y Electr onica. Facultad de Ciencias Fõsicas, Universidad Complutense, E-28040 Madrid, Spain
Abstract Low energy boron bombardment of silicon has been simulated at room temperature by means of molecular dynamics (MD). Terso potential T3 was used in the simulation smoothly linked to the universal potential. The boron± silicon interaction was simulated following the ideas of Terso for the SiC potential but modi®ed to take into account, in the B±Si interaction, whether or not the neighbours of either of both are entirely or partially boron or silicon atoms. (0 0 1) Si±C with (2 1) reconstruction surface was bombarded with boron at energies of 200 and 500 eV, which were initially chosen as good representative values of the low energy range of interest. Reliable results require a reasonable good statistic so that 100 impact points were chosen which were uniformly distributed over a representative area of a (2 1) surface. Special care was taken to determine the kind of damage produced in a Si crystal by the slowing down of boron. It is described in detail the way to determine vacancies and interstitials. The damage produced can be classi®ed in regions were the accumulation of damage does not allow to identify properly the type of defects produced and regions in which defects are isolated and can be beautifully identi®ed in terms of the potential energy variation and the displacements of their neighbours. Clusters of vacancies and interstitials are determined. Mean number of interstitials, vacancies, adatoms, sputtering, etc. are summarised in a table. Range distributions of boron are also determined. Ó 2000 Elsevier Science B.V. All rights reserved. Keywords: Molecular dynamics; Defects in semiconductors; Low energy ion bombardment; Boron on silicon
1. Introduction The stopping of energetic particles in matter induces in general structural damage. Over the last few decades special attention has been paid to the damage induced by ion bombardment at medium and high energy regimens [1±5]. It has been only
*
Corresponding author. Fax: +34-91-394-5196. E-mail address: [email protected] (A.M.C. PeÂrezMartõÂn).
more recently when, triggered perhaps by the challenge to obtain very shallow junctions, studies have begun to focus on the very low energy regimen [6,7]. The growing of microstructures using well-known techniques such as low energy ion beam deposition or low energy ion beam assisted deposition is also of great interest [8±10]. Ion beam damage processes in silicon have been studied intensively in the past [11±14] with emphasis in the reconstruction of the crystal after annealing [13,14] and also more recently [15,16] in the energy regimen of a few tens of eV of silicon
0168-583X/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 9 9 ) 0 1 1 6 6 - 0
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self-bombardment. In most of these cases the Stillinger±Weber interaction potential was employed [17]. The aim of this work is to characterise as best as possible the damage induced in silicon by low energy ion bombardment. To start with, special attention has been paid to determine an empirical interatomic potential for the usual case of boron bombarding on silicon. We were rather interested in an interatomic potential, which is able to take account of the B±Si interaction at all distances and also able to discriminate whether or not the neighbours of silicon were either boron or silicon. All along this work we shall be able to distinguish regions, in which there is an accumulation of damage, from others where the defects produced are more isolated. In the former regions it is consequently dicult to characterise the type of defects while in the latter defects are perfectly characterised in terms of the potential energy variation and the displacements of their neighbours. A detailed description of the method of calculation is given in Section 2. The results obtained are described and analysed in Section 3 and ®nally, Section 4 is devoted to the summary and conclusions. 2. Simulation conditions The interatomic potential used in our calculations is based on Terso potential [18,19] developed to work with binary structures in tetrahedral con®guration (SiC or SiGe). Under equilibrium conditions and the limit of low B concentration, B atoms occupy substitutional sites in a Si diamond crystal lattice [20]. Although B has only three valence electrons, experimental results analysing surface reconstruction for d-doped boron, suggest that the fourth sp3 -hybridized boron dangling bond which, normally empty, could be partially ®lled due to charge transfer from Si [21]. These authors also observe that boron substitution at the Si (0 0 1) surface distorts the Si lattice to achieve a By ab initio short B±Si bond, of about 2.0±2.1 A. calculations, Zhu et al. [20] have demonstrated
that the B±Si bond length is not the same when comparing a substitutional B in the limit of low concentration with a ®ctitious SiB compound with b-SiC structure. In the former they found a 12% bond distance reduction in the substitutional B case, a compromise between SiB (15%) and zero stress to the neighbouring Si lattice (0%). They also show that the second-shell Si atoms relax by 3% of the bond distance. In diluted systems, therefore, in which one of the components is in a rather small concentration, some modi®cations need to be done to take into account the fractional atomic concentration in the neighbourhood of the atoms involved in the interaction. As the interaction also depends on the neighbours of the interacting atoms, modi®cations have also been done to make the interaction depend on the atomic specie of the neighbours. Concisely, following the same notation as that in Refs. [18,19], the repulsive and attractive interactions between an i-atom and a j-atom are A
B
fR rij AA exp ÿkA rij Xij AB exp ÿkB rij Xij ; A
B
fA rij ÿBA exp ÿlA rij Xij BB exp ÿlB rij Xij ; 1 where XijA and XijB 1 ÿ XijA are fractional atomic concentrations of the A and B atomic species around the i and j atoms, respectively (Terso, in his work, takes XijA 0:5). In our particular case, A refers to silicon and B to boron. This interaction has been approximated to be like the Si±C interaction, assuming a fourth saturated bond for B. So the values of the parameters describing the Si±B interaction have been taken the same as those proposed in Refs. [18,19] for Si and C, respectively. Fractional concentrations, however, are calculated from the expression XijA
wAi
wAi wAj ; wAj wBi wBj
2
where the weight functions, wAi , are given by A
wAi
Ni X nik ; k6i
3
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NiA being the total number of neighbours of the i-atom belonging to the A atomic species. While this is an integer number, wAi is not, provided that nik rik is given by 8 1 h for rik6 rc ÿ 2 Drc ; > i > < p r ÿ r ÿ Dr 0:5 1:0 sin 2Dr ik c c c nik rik > for rc ÿ 2 Drc 6 rik 6 rc ; > : 4 0 for rc 6 rik ; which is a cut function that goes from unity to zero in a distance of 2Drc . This function smoothly takes an atom in or out of the neighbour list of the iatom, as a function of the distance rik . nik rik takes values as a function of the distance, rik , of the iatom to the k-atom, but does not depend on the atomic species of the atoms. In other words, the weight function, wAi , takes into account only katoms of the A-species to determine the fractional concentration of A-type neighbours. The values for the cut o radius, rc and taken were: 3 A 2Drc 0:3 A. The link between the binary part of Terso potential and the universal binary potential [22] has been performed by means of a continuous transition in the force. The B±Si and the Si±Si universal interaction potentials have been linked into all possible con®gurations which give rise to the above explained B±Si interaction, depending on the neighbours of either of both. The cohesive energy in SiC is of 6.165 eV/atom [18,19]. The and the bond length is of 1.87 A suggested modi®cations on the interatomic potential lead to a potential energy of B-substitutional in silicon of 5.39 eV, which is slightly higher than the 4.63 eV corresponding to silicon in silicon. The bond length of B-substitutional in Si is of slightly shorter than for Si in Si (2.35 A). 2.22 A This bond length is slightly longer than the prediction in Ref. [21]. It should be considered, however, while we have only one B-substitutional in an otherwise perfect Si-crystal, they were dealing with a d-doped B layer close to the surface. The values for B-substitutional in Si are, in all the cases, between the values corresponding to Si±Si and Si±C interactions. The crystal for the simulation was a (0 0 1) Si±C of dimensions 16 16 14a0 , being a0 5:432 A
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the lattice parameter. Periodic conditions along the x- and y-directions and free wall at the surface were considered; the z-axis is taken perpendicular to the surface and positive in the inward direction, being the origin at the centre of the surface plane. Surrounding the crystal in ®ve of the six faces that delimit the crystal, a thickness of one lattice cell was taken as a thermal bath to control the temperature of the system. On this region the temperature is kept constant by assigning to the atoms in the thermal bath a Maxwell±Boltzmann velocity distribution according to the desired temperature. The algorithm followed is called Brownian Dynamics [23] and it essentially consists of replacing the velocities of the atoms every 0.1 ps in order to keep constant the temperature; by this way resonant eects that may appear by simply rescaling the velocities are avoided. The 2 1 surface reconstruction was built by setting the atoms in the ®rst four atomic layers following the work of Roberts and Needs [24]. The system was allowed to relax, afterward, for 3 ps. The equations of motion were integrated step by step by means of the ``Two Step A'' method [25]. Subsequently the temperature of the system was slowly increased up to 300 K, in a period of time of 19 ps in which, obviously the system was also self-evolving dynamically. This will constitute the starting point for boron bombardment. The collisional phase was simulated in all cases for a period of time of 3 ps. This elapsed time was long enough to ensure that no atom in the crystal moves with a kinetic energy of above 1 eV. The damage induced in the solid by ion bombardment is determined by comparing this crystal with a reference crystal. The reference was a crystal prior to any relaxation process. Fluctuations due to the lack of relaxation of the reference crystal are expected to be less signi®cant than those others that arise from the fact to perform the analysis of the bombarded crystal at room temperature. Some analyses were done after cooling down the bombarded crystal till 0 K. No signi®cant dierences were observed except for the fact that all the distributions, we shall talk about below, were narrower due precisely to the lack of the thermal vibrations.
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3. Results The results showed hereafter are the average over 100 impacts of B on Si 2 1 at room temperature. Two sets of results have been obtained for 200 and 500 eV energies of boron, at a polar angle of incidence of 7° to avoid channelling and a random azimuthal angle in the range [0, 2p]. Most of the detailed analysis performed in this work to determine vacancies, interstitials etc. is energy independent so that the distributions shown below correspond to the case of 500 eV of B simply since the statistic is better, provided that more atoms are set in motion. Our ®rst objective is to determine precisely what we mean by a vacancy in a damaged region after ion bombardment. In order to do that we plot in Fig. 1 distributions of lattice points as a function of the distance d, to their ®rst neighbour. The ®rst neighbour of a lattice point is the closest atom to it. We consider, in Fig. 1, only those atoms that were initially situated in a lattice point, and were set in motion and are now ®rst neighbours of another lattice point. The lattice point distribution, without any extra restriction, is what
Fig. 1. Lattice points distribution vs distance d to the closest atom.
has been plotted as a solid line in Fig. 1. Three regions are clearly delimited by the solid line in corresponds to all Fig. 1. Region I, d < 0:8 A, those lattice points for which original atoms were set in motion leaving vacancies behind them which were occupied later by other atoms. In other words, it corresponds clearly to replacements. It instead of to a smaller value due peaks at 0.25 A mainly to thermal vibrations and distortion of the lattice by damage nearby. The third maximum, is due to those lattice points whose around 2.45 A, closest atoms are atoms usually associated to another lattice point (an atom is said to be associated to a lattice point when it is considered to be occupying that lattice point). All those lattice points are empty, so they are clearly identi®ed as vacan may be cies. The position of this peak, 2.45 A, interpreted as the mean distance from a vacancy to its ®rst neighbours. This is slightly longer than the due to an expansion of atoms bond length, 2.35 A, nearby a vacancy. From the analysis so far, region I can be identi®ed with replacements and region III with vacancies. In other words, whenever a recoil atom gets at rest at a distance d < 0:8 A, from a vacancy, it can be said, without any doubt, that this atom is occupying that place and consequently it might be said that a vacancy has been annihilated by an atom. In the other limit, whenever the closest atom to a lattice point is at a dis it can be assured that this lattice tance d > 2 A, point is a vacancy. Region II, between them, is an overlap of both, replacements (or annihilation) and vacancies regions. Typical con®gurations where an atom lies from a lattice point to distances in the range belonging to region II are illustrated in Fig. 2. These are split vacancies where an atom is between two lattice points. It is clear that at least a vacancy results. As we shall see later, one of the lattice points will be considered as occupied by the atom. In this region some criterion must be imposed in order to associate a lattice point with an atom. It is clear that an atom may be the closest atom to more than one lattice point. The scheme in Fig. 1 illustrates this case: atoms labelled 1 and 2 are around lattice points A, B and C to distances d1 < d2 < d3 < d4 . Atom 1 contributes as ®rst neighbour of A to the distance d1 and also to the
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thick slice of silicon target Fig. 2. Contour mapping of a 6 A after 500 eV boron bombardment. Grey-scale indicates depth: the lighter the colour the shallower the atom is. Various atomic con®gurations are displayed.
distance d2 as ®rst neighbour of B. This situation a half of the bond occurs for distances d > 1:17 A, length. To associate an atom to a lattice point two criteria are shown in Fig. 1. Dot line is obtained by imposing on solid line that each atom is counted only once as a neighbour of the closest lattice point. This means, on the scheme of Fig. 1, that atom 1 is only associated with the lattice point A, while B is associated with its second neighbour, the atom 2 and no atom contributes to the distance d4 to the lattice point C. The extra condition, that an atom is associated with a lattice point only if it is its ®rst neighbour, leads to the dash line in Fig. 1. This is a more restrictive condition that also implies a slight rearrangement in the association of the atoms with the lattice points. For instance, atom 2 is now associated with point C, in Fig. 1. Therefore, atom-lattice point association depends on the criterion where both unless we take distances d < 1:6 A criteria make curves to coincide so, we shall take this distance to de®ne a vacancy. On summary, a vacancy is a lattice point that, up to a distance of
435
has no atoms associated. Consequently, an 1.6 A, interstitial is any atom that has not been associated with any lattice point. The distance between a vacancy and an interstitial is therefore at least of 1.6 A. It is important to remark that it may happen to an interstitial (atom) to be from a lattice point (vacancy) to a distance in the range between 1.2 For instance, whenever an atom had and 1.6 A. not been the closest to a lattice point (2nd neighbour) and beside that the closest atom had already been associated with another lattice point. This is illustrated in Fig. 2 where split vacancies and dumbbell con®gurations are plotted. This ®gure is thick a contour mapping of a damaged region 6 A of silicon. Lattice points are marked as open circles and among them those that result to be vacancies are labelled as V. This ®gure shows in a rather small area three con®gurations I, II and III, which we describe as follows: con®guration I is a dumbbell. Atoms are along the (0 1 0) direction and the interstitial is labelled as D. The interstitial from the lattice point D is at a distance of 1.6 A II and III areas corwhile the atom is at 1.16 A. respond to split vacancy con®gurations [26]. The atom between two lattice points has six neighbours to a distance slightly longer than the bond length reminding a ¯oating bond. Some of (2.6 A), these atoms are at closer distances to the vacancy than expected (distances range in the interval from while the potential energy variation of 1.4±2.1 A) any of them is about 0.5 eV. Con®guration III defers from II in which the former has also an interstitial involved in the con®guration. Apart from that, it is very similar and characteristic distances and potential energy variation are in same range. The distributions of vacancies as a function of depth are shown in Fig. 3a. It can be seen, besides the expected depth distributions, an important production of vacancies just at the surface (®rst monolayer). Fig. 3a also discerns the number of vacancies constituting a cluster (cluster order). No signi®cant features are observed and all curves appear to show the same tendency. Concerning the number of clusters it is worthwhile to remark, although it is not shown here, that most of the large vacancy clusters are not longitudinal. Other
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Fig. 3. Depth distribution of (a) vacancies and number of clusters, (b) interstitials and number of clusters vs cluster order.
feature, not shown here, but very well known [27,28] is the vacancies' tendency to distribute themselves closer to the z-axis than the interstitials (z corresponds to the impact direction). Depth distributions of interstitials are plotted in Fig. 3b. The peak at the surface is less signi®cant than the corresponding vacancies. The order of the interstitial clusters is rather smaller than for vacancies and their arrangement is not observed so clearly. Depth distributions are not sensible to the order of the cluster although interstitials tend to be further transversally than vacancies to the zaxis. A tail in the distribution seems to indicate an appreciable presence of channelling. Channelling has not been quanti®ed but there has been observed a great facility to channel along the (0 1 1) directions. These features have been reported previously [12]. An interstitial has been considered to belong to a cluster when it was at a distance from at least other belonging smaller than 3.1 A to the cluster. A similar criterion was followed to look for vacancy clusters but up to a distance of the bond length.
An indirect method to quantify the damage produced in silicon after boron bombardment is the analysis of the surroundings of vacancies, interstitials and defects in general. This analysis has been achieved by determining, for the neighbours of a defect, their potential energy variation and their distances to the corresponding defects (see Figs. 4 and 5). In general, we shall distinguish between the case in which any atom (lattice atom, interstitial or projectile) is at least neighbour of a defect (solid line) from the case where lattice atoms are exclusively neighbours of one defect (dash line). Fig. 4a shows a clear peak around 1 eV that corresponds to those neighbours of a vacancy that have consequently lost a bond. This energy is slightly smaller than the expected value of 1.2 eV, due to the lattice relaxation. The second maximum centred about 0.5 eV (dashed line) corresponds mainly to situations as described in Fig. 2 (split vacancy). The solid line also shows the same maximum which corresponds, besides those described and considered in the dashed line, to
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Fig. 4. Neighbours of vacancies vs their (a) potential energy variation, (b) distance to the vacancy.
Fig. 5. Neighbours of interstitials vs their (a) potential energy variation, (b) distance of the interstitial.
neighbours of vacancies which are also neighbours of other defects such as interstitials, a con®guration also shown in Fig. 2. Fig. 4b illustrates that atoms expand around a vacancy. This expansion is
437
slightly greater when atoms are neighbours of only one vacancy (dash line). Distances to a vacancy are only shorter than the bond length (2.35 A) possible, as mentioned above, in a stressed lattice or in split vacancies. Fig. 5a depicts the potential energy variation of the neighbours of an interstitial up to a radius of The dashed line refers to better accommo3.2 A. dated atoms, in a less damaged region, which are exclusively neighbours of one (and only one) interstitial. The distance to their corresponding atoms is plotted in Fig. 5b. Please note that the curves in Figs. 5a and 5b are not normalized to the interval and consequently the areas do not coincide. The double peak, in both graphs, corresponds to ®rst and second neighbours of the interstitials. The higher peak is due to second neighbours, which are more abundant. This ®gure shows that the potential energy variation corresponding to ®rst (0.5 eV) is more appreciable than that corresponding to second neighbours (0.1 eV). The displacements, however, undergone by the second neighbours are signi®cantly greater than the displacements of the closest atoms to an interstitial. The reason is that, just at the centre of the lattice cell, there is enough room to accommodate an atom (T-type interstitial) with the four bonds to the right distances and in the right directions. The tiny amount of atoms that have had a negative potential energy variation, see Fig. 5a, either are neighbours of boron or are atoms coming from the surface. Fig. 5b also shows that there is a considerable amount of atoms, which are closer to an We have interstitial than the bond length (2.35 A). investigated this point and we have found, for instance, that partner atoms in a split interstitial con®guration (dumbbell) are at distance around The rest of neighbours were between 2.5 and 2.3 A. from the interstitial. Distances to the inter2.6 A mean that the interstitial stitial smaller than 2.2 A is the boron atom. Neighbours of two interstitials or neighbour atoms of both an interstitial and a vacancy also contribute to the solid line curve. Fig. 6a and b summarise our study about interstitials produced in Si by B bombardment. Fig. 6a depicts the number of interstitials distributed as a function of their potential energy variation while Fig. 6b displays the distance travelled by
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Fig. 6. Number of interstitials vs their (a) potential energy variation, (b) travelled distance.
them. T-type interstitials and dumbbell con®guration are dierentiated from all the other interstitials (solid line) that include these as well. The criterion followed to identify a T-type interstitials was to have four neighbours along the [1 1 1] directions with an uncertainty of 10°. Dumbbell structures were also observed: an interstitial and a lattice atom, to both sides of a lattice point, bound along the [1 1 0] or [1 0 0] directions [29]. The ¯uctuations allowed around the typical directions for dumbbells were 18°, greater than for interstitials provided that dumbbell con®guration requires in general a more distorted surroundings. Fig. 6a shows that most of T-type interstitial con®gurations have an almost null potential energy variation. Dumbbell interstitial con®guration requires, however, a mean contribution of energy of 0.5 eV. Fluctuations around the maxima are partially due to structural damage but thermal vibrations have an important contribution to the spread out of all the distributions shown in this work. (This analysis has also been performed to 0 K, not shown here, and corroborates this point.) Again negative potential energy variation, DEp < 0, means either originally the atom came from the surface or the interstitial is neighbour of the projectile.
Roughly speaking solid line curves, in Fig. 6, reproduce magni®ed sum of the other two curves. This is because interstitials in all kind of situations contribute to these curves. For instance, the full line maximum in Fig. 6a has been corroborated to be due to interstitials, which are neighbours of other interstitials or vacancies. In other words, interstitials, in a much more damaged region are not so well classi®ed. Fig. 6b shows the distribution of interstitial produced by B (500 eV) bombarding Si, distributed as a function of the travelled distance. First T-type interstitials travelled a distance of 2.35 A (bond length). This peak has to come from a replacement collision. T-type interstitials are only identi®ed having four neighbours so that to reproduce this situation any of the four atoms near the centre of the silicon cell have had to be kicked o in a replacement collision and sent to the centre of the cell. We have constantly that most of interstitials have their origin in replacement collision; this fact has also been corroborated by other represent authors [30]. Peaks at 2.7 and 4.7 A travelled distances of half a lattice parameter and half a diagonal of the cubic cell, respectively. We have plotted in Fig. 6b, travelled distances up to a lattice unit. It has been observed (not shown here) the peak structure corresponding to travelled distances beyond the unit cell where the heights of the peaks diminish rapidly with distance. Dumbbell type shows a maximum at travelled as it is depicted in Fig. 6b. It distances of 1.6 A, may occur that an atom, coming from far away, gets at rest closer to a lattice point than the atom that was originally there and it is pushed out afterwards to form a split interstitial; this explains the appearance of points for travelled distances smaller than 1.6 A. Again solid line reproduces roughly the shape of the other two curves. This means that solid lines account for interstitials that have not been well characterized for being in a more damaged area. A summary of the results obtained for the 200 and 500 eV energies of boron is displayed in Table 1. The total numbers of interstitials and vacancies are in agreement with very well-established models [31,32]. The Kinchin Pease model, with the replacement correction, predicts, respectively for
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439
Table 1 Summary of the results obtained for 200 and 500 eV boron bombarding on silicon at room temperature. The results are the average over 100 impact points B-energy (eV) 200 500
Interstitials
V
I
IT (%)
ID (%)
6.3 13.8
23 18
13 13
7.1 14.2
200 and 500 eV, a number of interstitials of 5.6 and 14, where the displacement energy has been assumed to be of 15 eV. Other similar calculations [15,16] predict similar values although the comparison is more questionable because they bombard with silicon at smaller energies. While the total number of interstitials I, and vacancies V, increase with the energy, there are no signi®cant changes in the relative number of dumbbell con®gurations. The relative number of T-type interstitials increase with the energy probably because replacement collisions also increase and these collisions are the main origin of the T-type interstitials. Adatoms are identi®ed as those atoms set at above the surface. The number of more than 0.7 A adatoms diminish with energy provided that the cascades are deeper. Mean projected range, Rp , and standard deviation, rp , are in good agreement with experimental measurement [6], where they are de termined by SIMS, a mean projected range of 30 A for the case of 500 eV of boron bombarding silicon. Boron atoms in substitutional positions, PSub , are found to be around 70% and energy independent while the re¯ection probability, R, slightly decreases with the energy. With respect to channelling we have found that it is quite possible to ®nd atoms (either the ion or a recoil) in channelling along the (0 1 1) directions, according to previous results [12]. This tendency increases with the energy but it has not been quanti®ed because it was out of the scope of this work. 4. Summary and conclusions Damage eects and range distributions have been studied for boron bombarding silicon, at energies of 200 and 500 eV. Molecular dynamics
Adat
1.7 1.2
Sputt
0.2 0.3
Projectile R (%)
Psub (%)
Rp (A)
rp (A)
8 6
69 65
8.0 27.6
6.4 15.4
calculations have been performed and the B±Si interaction has been approximated to the Si±C interaction proposed by Terso [18,19]. Some modi®cations have been introduced in this manybody potential in order to distinguish not only in number but also in atomic species of the neighbours of the interacting atoms. This yields a cohesive energy for boron in silicon of 5.39 eV, slightly higher than for silicon in silicon (4.63 eV) but smaller than for carbon in silicon (6.165 eV). Consequently, the bond length for boron embed is also between the ded in silicon which is 2.22 A, and bond lengths of silicon in silicon (2.35 A) carbon in silicon (1.87 A). Vacancies have been de®ned as a lattice point, has not got any which up to a distance of 1.6 A associated atom. An interstitial is therefore any atom that has not been associated to any lattice point. A vacancy and an interstitial are conse These quently at a minimum distance of 1.6 A. de®nitions lead to a mean number of interstitials and vacancies in good agreement with well-established models and similar calculation (see Table 1). A detailed description of the surroundings of interstitials and vacancies is given in terms of the potential energy variation and displacements of the corresponding neighbours of them. Interstitials are classi®ed in terms of their structure. T-type and dumbbell are well determined. A less strict procedure to determine them will probably result in a higher number of them. T-type interstitial results to be a more frequent structure than dumbbell provided that the energy con®guration is lower. It has also been found that split vacancy is quite a frequent con®guration. No changes have been observed in the characterisation of defects and damage produced for
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bombarding energies of 200 and 500 eV. The range distribution of B in Si follows the shapes expected. Mean values and standard deviations are in agreement with values experimentally determined. Boron atoms have a probability of 0.7 to settle down in substitutional positions.
Acknowledgements This work has been partially supported by DGICYT (Project No. MAT98-0859).
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[10] M. Kitabatake, J.E. Greene, Thin Solid Films 272 (1996) 271. [11] K. G artner, M. Nitschke, W. Eckstein, Nucl. Instr. and Meth. B 83 (1993) 87. [12] K. G artner, D. Stock, C. Wende, M. Nitschke, Nucl. Instr. and Meth. B 90 (1994) 124. [13] T. Diaz de la Rubia, G.H. Gilmer, Phys. Rev. Lett. 74 (1995) 2507. [14] M.J. Caturla, T. Diaz de la Rubia, G.H. Gilmer, Nucl. Instr. and Meth. B 106 (1995) 1. [15] H. Hensel, H.M. Urbassek, Rad. E. (1998) 955. [16] H. Hensel, H.M. Urbassek, Phys. Rev. A 57 (1998) 4756. [17] F.H. Stillinger, Th.A. Weber, Phys. Rev. B 31 (1985) 5262. [18] J. Terso, Phys. Rev. B 39 (1989) 5566. [19] J. Terso, Phys. Rev. B 38 (1988) 9902. [20] J. Zhu, T. Diaz de la Rubia, L.H. Yang, C. Mailhiot, G.H. Gilmer, Phys. Rev. B 54 (1996) 4741. [21] Y. Wang, R.J. Hamers, E. Kaxiras, Phys. Rev. Lett. 74 (1995) 403. [22] W.S. Wilson, L.G. Haggmark, J.P. Biersack, Phys. Rev. B 15 (1977) 2458. [23] D.W. Heermann, Computer Simulation Methods in Theoretical Physics. Springer, 1990. [24] N. Roberts, R.J. Needs, Surf. Sci. 112 (1990) 236. [25] R. Smith, D.E. Harrison Jr., Comput. Phys. 3 (1989) 68. [26] H. Balamane, T. Halicioglu, W.A. Tiller, Phys. Rev. B 46 (1992) 2250. [27] A.M.C. Perez Martõn, J. Domõnguez V azquez, J.J. Jimenez Rodrõguez, R. Collins, Martõ Gras, Nucl. Instr. and Meth. B 90 (1994) 424. [28] A.M.C. Perez Martõn, J. Domõnguez V azquez, J.J. Jimenez Rodrõguez, R. Martõ Collins, Martõ Gras, J. Phys. A 5 (1993) 257. [29] I.P. Batras, F.F. Abraham, S. Ciraci, Phys. Rev. B 35 (1987) 9552. [30] M.J. Caturla, T. Diaz de la Rubia, L.A. Marques, G.H. Gilmer, Phys. Rev. B 54 (1996) 16683. [31] G.H. Kinchin, R.S. Pease, Rep. Prog. Phys. 18 (1955) 1. [32] P. Sigmund, Rev. Roum. Phys. 17 (1972) 969.
www.elsevier.nl/locate/nimb
A MD study of low energy boron bombardment on silicon A. Mari Carmen Perez-Martõn *, Javier Domõnguez-V azquez, Jose J. Jimenez-Rodrõguez Departamento de Electricidad y Electr onica. Facultad de Ciencias Fõsicas, Universidad Complutense, E-28040 Madrid, Spain
Abstract Low energy boron bombardment of silicon has been simulated at room temperature by means of molecular dynamics (MD). Terso potential T3 was used in the simulation smoothly linked to the universal potential. The boron± silicon interaction was simulated following the ideas of Terso for the SiC potential but modi®ed to take into account, in the B±Si interaction, whether or not the neighbours of either of both are entirely or partially boron or silicon atoms. (0 0 1) Si±C with (2 1) reconstruction surface was bombarded with boron at energies of 200 and 500 eV, which were initially chosen as good representative values of the low energy range of interest. Reliable results require a reasonable good statistic so that 100 impact points were chosen which were uniformly distributed over a representative area of a (2 1) surface. Special care was taken to determine the kind of damage produced in a Si crystal by the slowing down of boron. It is described in detail the way to determine vacancies and interstitials. The damage produced can be classi®ed in regions were the accumulation of damage does not allow to identify properly the type of defects produced and regions in which defects are isolated and can be beautifully identi®ed in terms of the potential energy variation and the displacements of their neighbours. Clusters of vacancies and interstitials are determined. Mean number of interstitials, vacancies, adatoms, sputtering, etc. are summarised in a table. Range distributions of boron are also determined. Ó 2000 Elsevier Science B.V. All rights reserved. Keywords: Molecular dynamics; Defects in semiconductors; Low energy ion bombardment; Boron on silicon
1. Introduction The stopping of energetic particles in matter induces in general structural damage. Over the last few decades special attention has been paid to the damage induced by ion bombardment at medium and high energy regimens [1±5]. It has been only
*
Corresponding author. Fax: +34-91-394-5196. E-mail address: [email protected] (A.M.C. PeÂrezMartõÂn).
more recently when, triggered perhaps by the challenge to obtain very shallow junctions, studies have begun to focus on the very low energy regimen [6,7]. The growing of microstructures using well-known techniques such as low energy ion beam deposition or low energy ion beam assisted deposition is also of great interest [8±10]. Ion beam damage processes in silicon have been studied intensively in the past [11±14] with emphasis in the reconstruction of the crystal after annealing [13,14] and also more recently [15,16] in the energy regimen of a few tens of eV of silicon
0168-583X/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 9 9 ) 0 1 1 6 6 - 0
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self-bombardment. In most of these cases the Stillinger±Weber interaction potential was employed [17]. The aim of this work is to characterise as best as possible the damage induced in silicon by low energy ion bombardment. To start with, special attention has been paid to determine an empirical interatomic potential for the usual case of boron bombarding on silicon. We were rather interested in an interatomic potential, which is able to take account of the B±Si interaction at all distances and also able to discriminate whether or not the neighbours of silicon were either boron or silicon. All along this work we shall be able to distinguish regions, in which there is an accumulation of damage, from others where the defects produced are more isolated. In the former regions it is consequently dicult to characterise the type of defects while in the latter defects are perfectly characterised in terms of the potential energy variation and the displacements of their neighbours. A detailed description of the method of calculation is given in Section 2. The results obtained are described and analysed in Section 3 and ®nally, Section 4 is devoted to the summary and conclusions. 2. Simulation conditions The interatomic potential used in our calculations is based on Terso potential [18,19] developed to work with binary structures in tetrahedral con®guration (SiC or SiGe). Under equilibrium conditions and the limit of low B concentration, B atoms occupy substitutional sites in a Si diamond crystal lattice [20]. Although B has only three valence electrons, experimental results analysing surface reconstruction for d-doped boron, suggest that the fourth sp3 -hybridized boron dangling bond which, normally empty, could be partially ®lled due to charge transfer from Si [21]. These authors also observe that boron substitution at the Si (0 0 1) surface distorts the Si lattice to achieve a By ab initio short B±Si bond, of about 2.0±2.1 A. calculations, Zhu et al. [20] have demonstrated
that the B±Si bond length is not the same when comparing a substitutional B in the limit of low concentration with a ®ctitious SiB compound with b-SiC structure. In the former they found a 12% bond distance reduction in the substitutional B case, a compromise between SiB (15%) and zero stress to the neighbouring Si lattice (0%). They also show that the second-shell Si atoms relax by 3% of the bond distance. In diluted systems, therefore, in which one of the components is in a rather small concentration, some modi®cations need to be done to take into account the fractional atomic concentration in the neighbourhood of the atoms involved in the interaction. As the interaction also depends on the neighbours of the interacting atoms, modi®cations have also been done to make the interaction depend on the atomic specie of the neighbours. Concisely, following the same notation as that in Refs. [18,19], the repulsive and attractive interactions between an i-atom and a j-atom are A
B
fR rij AA exp ÿkA rij Xij AB exp ÿkB rij Xij ; A
B
fA rij ÿBA exp ÿlA rij Xij BB exp ÿlB rij Xij ; 1 where XijA and XijB 1 ÿ XijA are fractional atomic concentrations of the A and B atomic species around the i and j atoms, respectively (Terso, in his work, takes XijA 0:5). In our particular case, A refers to silicon and B to boron. This interaction has been approximated to be like the Si±C interaction, assuming a fourth saturated bond for B. So the values of the parameters describing the Si±B interaction have been taken the same as those proposed in Refs. [18,19] for Si and C, respectively. Fractional concentrations, however, are calculated from the expression XijA
wAi
wAi wAj ; wAj wBi wBj
2
where the weight functions, wAi , are given by A
wAi
Ni X nik ; k6i
3
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NiA being the total number of neighbours of the i-atom belonging to the A atomic species. While this is an integer number, wAi is not, provided that nik rik is given by 8 1 h for rik6 rc ÿ 2 Drc ; > i > < p r ÿ r ÿ Dr 0:5 1:0 sin 2Dr ik c c c nik rik > for rc ÿ 2 Drc 6 rik 6 rc ; > : 4 0 for rc 6 rik ; which is a cut function that goes from unity to zero in a distance of 2Drc . This function smoothly takes an atom in or out of the neighbour list of the iatom, as a function of the distance rik . nik rik takes values as a function of the distance, rik , of the iatom to the k-atom, but does not depend on the atomic species of the atoms. In other words, the weight function, wAi , takes into account only katoms of the A-species to determine the fractional concentration of A-type neighbours. The values for the cut o radius, rc and taken were: 3 A 2Drc 0:3 A. The link between the binary part of Terso potential and the universal binary potential [22] has been performed by means of a continuous transition in the force. The B±Si and the Si±Si universal interaction potentials have been linked into all possible con®gurations which give rise to the above explained B±Si interaction, depending on the neighbours of either of both. The cohesive energy in SiC is of 6.165 eV/atom [18,19]. The and the bond length is of 1.87 A suggested modi®cations on the interatomic potential lead to a potential energy of B-substitutional in silicon of 5.39 eV, which is slightly higher than the 4.63 eV corresponding to silicon in silicon. The bond length of B-substitutional in Si is of slightly shorter than for Si in Si (2.35 A). 2.22 A This bond length is slightly longer than the prediction in Ref. [21]. It should be considered, however, while we have only one B-substitutional in an otherwise perfect Si-crystal, they were dealing with a d-doped B layer close to the surface. The values for B-substitutional in Si are, in all the cases, between the values corresponding to Si±Si and Si±C interactions. The crystal for the simulation was a (0 0 1) Si±C of dimensions 16 16 14a0 , being a0 5:432 A
433
the lattice parameter. Periodic conditions along the x- and y-directions and free wall at the surface were considered; the z-axis is taken perpendicular to the surface and positive in the inward direction, being the origin at the centre of the surface plane. Surrounding the crystal in ®ve of the six faces that delimit the crystal, a thickness of one lattice cell was taken as a thermal bath to control the temperature of the system. On this region the temperature is kept constant by assigning to the atoms in the thermal bath a Maxwell±Boltzmann velocity distribution according to the desired temperature. The algorithm followed is called Brownian Dynamics [23] and it essentially consists of replacing the velocities of the atoms every 0.1 ps in order to keep constant the temperature; by this way resonant eects that may appear by simply rescaling the velocities are avoided. The 2 1 surface reconstruction was built by setting the atoms in the ®rst four atomic layers following the work of Roberts and Needs [24]. The system was allowed to relax, afterward, for 3 ps. The equations of motion were integrated step by step by means of the ``Two Step A'' method [25]. Subsequently the temperature of the system was slowly increased up to 300 K, in a period of time of 19 ps in which, obviously the system was also self-evolving dynamically. This will constitute the starting point for boron bombardment. The collisional phase was simulated in all cases for a period of time of 3 ps. This elapsed time was long enough to ensure that no atom in the crystal moves with a kinetic energy of above 1 eV. The damage induced in the solid by ion bombardment is determined by comparing this crystal with a reference crystal. The reference was a crystal prior to any relaxation process. Fluctuations due to the lack of relaxation of the reference crystal are expected to be less signi®cant than those others that arise from the fact to perform the analysis of the bombarded crystal at room temperature. Some analyses were done after cooling down the bombarded crystal till 0 K. No signi®cant dierences were observed except for the fact that all the distributions, we shall talk about below, were narrower due precisely to the lack of the thermal vibrations.
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3. Results The results showed hereafter are the average over 100 impacts of B on Si 2 1 at room temperature. Two sets of results have been obtained for 200 and 500 eV energies of boron, at a polar angle of incidence of 7° to avoid channelling and a random azimuthal angle in the range [0, 2p]. Most of the detailed analysis performed in this work to determine vacancies, interstitials etc. is energy independent so that the distributions shown below correspond to the case of 500 eV of B simply since the statistic is better, provided that more atoms are set in motion. Our ®rst objective is to determine precisely what we mean by a vacancy in a damaged region after ion bombardment. In order to do that we plot in Fig. 1 distributions of lattice points as a function of the distance d, to their ®rst neighbour. The ®rst neighbour of a lattice point is the closest atom to it. We consider, in Fig. 1, only those atoms that were initially situated in a lattice point, and were set in motion and are now ®rst neighbours of another lattice point. The lattice point distribution, without any extra restriction, is what
Fig. 1. Lattice points distribution vs distance d to the closest atom.
has been plotted as a solid line in Fig. 1. Three regions are clearly delimited by the solid line in corresponds to all Fig. 1. Region I, d < 0:8 A, those lattice points for which original atoms were set in motion leaving vacancies behind them which were occupied later by other atoms. In other words, it corresponds clearly to replacements. It instead of to a smaller value due peaks at 0.25 A mainly to thermal vibrations and distortion of the lattice by damage nearby. The third maximum, is due to those lattice points whose around 2.45 A, closest atoms are atoms usually associated to another lattice point (an atom is said to be associated to a lattice point when it is considered to be occupying that lattice point). All those lattice points are empty, so they are clearly identi®ed as vacan may be cies. The position of this peak, 2.45 A, interpreted as the mean distance from a vacancy to its ®rst neighbours. This is slightly longer than the due to an expansion of atoms bond length, 2.35 A, nearby a vacancy. From the analysis so far, region I can be identi®ed with replacements and region III with vacancies. In other words, whenever a recoil atom gets at rest at a distance d < 0:8 A, from a vacancy, it can be said, without any doubt, that this atom is occupying that place and consequently it might be said that a vacancy has been annihilated by an atom. In the other limit, whenever the closest atom to a lattice point is at a dis it can be assured that this lattice tance d > 2 A, point is a vacancy. Region II, between them, is an overlap of both, replacements (or annihilation) and vacancies regions. Typical con®gurations where an atom lies from a lattice point to distances in the range belonging to region II are illustrated in Fig. 2. These are split vacancies where an atom is between two lattice points. It is clear that at least a vacancy results. As we shall see later, one of the lattice points will be considered as occupied by the atom. In this region some criterion must be imposed in order to associate a lattice point with an atom. It is clear that an atom may be the closest atom to more than one lattice point. The scheme in Fig. 1 illustrates this case: atoms labelled 1 and 2 are around lattice points A, B and C to distances d1 < d2 < d3 < d4 . Atom 1 contributes as ®rst neighbour of A to the distance d1 and also to the
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thick slice of silicon target Fig. 2. Contour mapping of a 6 A after 500 eV boron bombardment. Grey-scale indicates depth: the lighter the colour the shallower the atom is. Various atomic con®gurations are displayed.
distance d2 as ®rst neighbour of B. This situation a half of the bond occurs for distances d > 1:17 A, length. To associate an atom to a lattice point two criteria are shown in Fig. 1. Dot line is obtained by imposing on solid line that each atom is counted only once as a neighbour of the closest lattice point. This means, on the scheme of Fig. 1, that atom 1 is only associated with the lattice point A, while B is associated with its second neighbour, the atom 2 and no atom contributes to the distance d4 to the lattice point C. The extra condition, that an atom is associated with a lattice point only if it is its ®rst neighbour, leads to the dash line in Fig. 1. This is a more restrictive condition that also implies a slight rearrangement in the association of the atoms with the lattice points. For instance, atom 2 is now associated with point C, in Fig. 1. Therefore, atom-lattice point association depends on the criterion where both unless we take distances d < 1:6 A criteria make curves to coincide so, we shall take this distance to de®ne a vacancy. On summary, a vacancy is a lattice point that, up to a distance of
435
has no atoms associated. Consequently, an 1.6 A, interstitial is any atom that has not been associated with any lattice point. The distance between a vacancy and an interstitial is therefore at least of 1.6 A. It is important to remark that it may happen to an interstitial (atom) to be from a lattice point (vacancy) to a distance in the range between 1.2 For instance, whenever an atom had and 1.6 A. not been the closest to a lattice point (2nd neighbour) and beside that the closest atom had already been associated with another lattice point. This is illustrated in Fig. 2 where split vacancies and dumbbell con®gurations are plotted. This ®gure is thick a contour mapping of a damaged region 6 A of silicon. Lattice points are marked as open circles and among them those that result to be vacancies are labelled as V. This ®gure shows in a rather small area three con®gurations I, II and III, which we describe as follows: con®guration I is a dumbbell. Atoms are along the (0 1 0) direction and the interstitial is labelled as D. The interstitial from the lattice point D is at a distance of 1.6 A II and III areas corwhile the atom is at 1.16 A. respond to split vacancy con®gurations [26]. The atom between two lattice points has six neighbours to a distance slightly longer than the bond length reminding a ¯oating bond. Some of (2.6 A), these atoms are at closer distances to the vacancy than expected (distances range in the interval from while the potential energy variation of 1.4±2.1 A) any of them is about 0.5 eV. Con®guration III defers from II in which the former has also an interstitial involved in the con®guration. Apart from that, it is very similar and characteristic distances and potential energy variation are in same range. The distributions of vacancies as a function of depth are shown in Fig. 3a. It can be seen, besides the expected depth distributions, an important production of vacancies just at the surface (®rst monolayer). Fig. 3a also discerns the number of vacancies constituting a cluster (cluster order). No signi®cant features are observed and all curves appear to show the same tendency. Concerning the number of clusters it is worthwhile to remark, although it is not shown here, that most of the large vacancy clusters are not longitudinal. Other
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Fig. 3. Depth distribution of (a) vacancies and number of clusters, (b) interstitials and number of clusters vs cluster order.
feature, not shown here, but very well known [27,28] is the vacancies' tendency to distribute themselves closer to the z-axis than the interstitials (z corresponds to the impact direction). Depth distributions of interstitials are plotted in Fig. 3b. The peak at the surface is less signi®cant than the corresponding vacancies. The order of the interstitial clusters is rather smaller than for vacancies and their arrangement is not observed so clearly. Depth distributions are not sensible to the order of the cluster although interstitials tend to be further transversally than vacancies to the zaxis. A tail in the distribution seems to indicate an appreciable presence of channelling. Channelling has not been quanti®ed but there has been observed a great facility to channel along the (0 1 1) directions. These features have been reported previously [12]. An interstitial has been considered to belong to a cluster when it was at a distance from at least other belonging smaller than 3.1 A to the cluster. A similar criterion was followed to look for vacancy clusters but up to a distance of the bond length.
An indirect method to quantify the damage produced in silicon after boron bombardment is the analysis of the surroundings of vacancies, interstitials and defects in general. This analysis has been achieved by determining, for the neighbours of a defect, their potential energy variation and their distances to the corresponding defects (see Figs. 4 and 5). In general, we shall distinguish between the case in which any atom (lattice atom, interstitial or projectile) is at least neighbour of a defect (solid line) from the case where lattice atoms are exclusively neighbours of one defect (dash line). Fig. 4a shows a clear peak around 1 eV that corresponds to those neighbours of a vacancy that have consequently lost a bond. This energy is slightly smaller than the expected value of 1.2 eV, due to the lattice relaxation. The second maximum centred about 0.5 eV (dashed line) corresponds mainly to situations as described in Fig. 2 (split vacancy). The solid line also shows the same maximum which corresponds, besides those described and considered in the dashed line, to
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Fig. 4. Neighbours of vacancies vs their (a) potential energy variation, (b) distance to the vacancy.
Fig. 5. Neighbours of interstitials vs their (a) potential energy variation, (b) distance of the interstitial.
neighbours of vacancies which are also neighbours of other defects such as interstitials, a con®guration also shown in Fig. 2. Fig. 4b illustrates that atoms expand around a vacancy. This expansion is
437
slightly greater when atoms are neighbours of only one vacancy (dash line). Distances to a vacancy are only shorter than the bond length (2.35 A) possible, as mentioned above, in a stressed lattice or in split vacancies. Fig. 5a depicts the potential energy variation of the neighbours of an interstitial up to a radius of The dashed line refers to better accommo3.2 A. dated atoms, in a less damaged region, which are exclusively neighbours of one (and only one) interstitial. The distance to their corresponding atoms is plotted in Fig. 5b. Please note that the curves in Figs. 5a and 5b are not normalized to the interval and consequently the areas do not coincide. The double peak, in both graphs, corresponds to ®rst and second neighbours of the interstitials. The higher peak is due to second neighbours, which are more abundant. This ®gure shows that the potential energy variation corresponding to ®rst (0.5 eV) is more appreciable than that corresponding to second neighbours (0.1 eV). The displacements, however, undergone by the second neighbours are signi®cantly greater than the displacements of the closest atoms to an interstitial. The reason is that, just at the centre of the lattice cell, there is enough room to accommodate an atom (T-type interstitial) with the four bonds to the right distances and in the right directions. The tiny amount of atoms that have had a negative potential energy variation, see Fig. 5a, either are neighbours of boron or are atoms coming from the surface. Fig. 5b also shows that there is a considerable amount of atoms, which are closer to an We have interstitial than the bond length (2.35 A). investigated this point and we have found, for instance, that partner atoms in a split interstitial con®guration (dumbbell) are at distance around The rest of neighbours were between 2.5 and 2.3 A. from the interstitial. Distances to the inter2.6 A mean that the interstitial stitial smaller than 2.2 A is the boron atom. Neighbours of two interstitials or neighbour atoms of both an interstitial and a vacancy also contribute to the solid line curve. Fig. 6a and b summarise our study about interstitials produced in Si by B bombardment. Fig. 6a depicts the number of interstitials distributed as a function of their potential energy variation while Fig. 6b displays the distance travelled by
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Fig. 6. Number of interstitials vs their (a) potential energy variation, (b) travelled distance.
them. T-type interstitials and dumbbell con®guration are dierentiated from all the other interstitials (solid line) that include these as well. The criterion followed to identify a T-type interstitials was to have four neighbours along the [1 1 1] directions with an uncertainty of 10°. Dumbbell structures were also observed: an interstitial and a lattice atom, to both sides of a lattice point, bound along the [1 1 0] or [1 0 0] directions [29]. The ¯uctuations allowed around the typical directions for dumbbells were 18°, greater than for interstitials provided that dumbbell con®guration requires in general a more distorted surroundings. Fig. 6a shows that most of T-type interstitial con®gurations have an almost null potential energy variation. Dumbbell interstitial con®guration requires, however, a mean contribution of energy of 0.5 eV. Fluctuations around the maxima are partially due to structural damage but thermal vibrations have an important contribution to the spread out of all the distributions shown in this work. (This analysis has also been performed to 0 K, not shown here, and corroborates this point.) Again negative potential energy variation, DEp < 0, means either originally the atom came from the surface or the interstitial is neighbour of the projectile.
Roughly speaking solid line curves, in Fig. 6, reproduce magni®ed sum of the other two curves. This is because interstitials in all kind of situations contribute to these curves. For instance, the full line maximum in Fig. 6a has been corroborated to be due to interstitials, which are neighbours of other interstitials or vacancies. In other words, interstitials, in a much more damaged region are not so well classi®ed. Fig. 6b shows the distribution of interstitial produced by B (500 eV) bombarding Si, distributed as a function of the travelled distance. First T-type interstitials travelled a distance of 2.35 A (bond length). This peak has to come from a replacement collision. T-type interstitials are only identi®ed having four neighbours so that to reproduce this situation any of the four atoms near the centre of the silicon cell have had to be kicked o in a replacement collision and sent to the centre of the cell. We have constantly that most of interstitials have their origin in replacement collision; this fact has also been corroborated by other represent authors [30]. Peaks at 2.7 and 4.7 A travelled distances of half a lattice parameter and half a diagonal of the cubic cell, respectively. We have plotted in Fig. 6b, travelled distances up to a lattice unit. It has been observed (not shown here) the peak structure corresponding to travelled distances beyond the unit cell where the heights of the peaks diminish rapidly with distance. Dumbbell type shows a maximum at travelled as it is depicted in Fig. 6b. It distances of 1.6 A, may occur that an atom, coming from far away, gets at rest closer to a lattice point than the atom that was originally there and it is pushed out afterwards to form a split interstitial; this explains the appearance of points for travelled distances smaller than 1.6 A. Again solid line reproduces roughly the shape of the other two curves. This means that solid lines account for interstitials that have not been well characterized for being in a more damaged area. A summary of the results obtained for the 200 and 500 eV energies of boron is displayed in Table 1. The total numbers of interstitials and vacancies are in agreement with very well-established models [31,32]. The Kinchin Pease model, with the replacement correction, predicts, respectively for
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439
Table 1 Summary of the results obtained for 200 and 500 eV boron bombarding on silicon at room temperature. The results are the average over 100 impact points B-energy (eV) 200 500
Interstitials
V
I
IT (%)
ID (%)
6.3 13.8
23 18
13 13
7.1 14.2
200 and 500 eV, a number of interstitials of 5.6 and 14, where the displacement energy has been assumed to be of 15 eV. Other similar calculations [15,16] predict similar values although the comparison is more questionable because they bombard with silicon at smaller energies. While the total number of interstitials I, and vacancies V, increase with the energy, there are no signi®cant changes in the relative number of dumbbell con®gurations. The relative number of T-type interstitials increase with the energy probably because replacement collisions also increase and these collisions are the main origin of the T-type interstitials. Adatoms are identi®ed as those atoms set at above the surface. The number of more than 0.7 A adatoms diminish with energy provided that the cascades are deeper. Mean projected range, Rp , and standard deviation, rp , are in good agreement with experimental measurement [6], where they are de termined by SIMS, a mean projected range of 30 A for the case of 500 eV of boron bombarding silicon. Boron atoms in substitutional positions, PSub , are found to be around 70% and energy independent while the re¯ection probability, R, slightly decreases with the energy. With respect to channelling we have found that it is quite possible to ®nd atoms (either the ion or a recoil) in channelling along the (0 1 1) directions, according to previous results [12]. This tendency increases with the energy but it has not been quanti®ed because it was out of the scope of this work. 4. Summary and conclusions Damage eects and range distributions have been studied for boron bombarding silicon, at energies of 200 and 500 eV. Molecular dynamics
Adat
1.7 1.2
Sputt
0.2 0.3
Projectile R (%)
Psub (%)
Rp (A)
rp (A)
8 6
69 65
8.0 27.6
6.4 15.4
calculations have been performed and the B±Si interaction has been approximated to the Si±C interaction proposed by Terso [18,19]. Some modi®cations have been introduced in this manybody potential in order to distinguish not only in number but also in atomic species of the neighbours of the interacting atoms. This yields a cohesive energy for boron in silicon of 5.39 eV, slightly higher than for silicon in silicon (4.63 eV) but smaller than for carbon in silicon (6.165 eV). Consequently, the bond length for boron embed is also between the ded in silicon which is 2.22 A, and bond lengths of silicon in silicon (2.35 A) carbon in silicon (1.87 A). Vacancies have been de®ned as a lattice point, has not got any which up to a distance of 1.6 A associated atom. An interstitial is therefore any atom that has not been associated to any lattice point. A vacancy and an interstitial are conse These quently at a minimum distance of 1.6 A. de®nitions lead to a mean number of interstitials and vacancies in good agreement with well-established models and similar calculation (see Table 1). A detailed description of the surroundings of interstitials and vacancies is given in terms of the potential energy variation and displacements of the corresponding neighbours of them. Interstitials are classi®ed in terms of their structure. T-type and dumbbell are well determined. A less strict procedure to determine them will probably result in a higher number of them. T-type interstitial results to be a more frequent structure than dumbbell provided that the energy con®guration is lower. It has also been found that split vacancy is quite a frequent con®guration. No changes have been observed in the characterisation of defects and damage produced for
440
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bombarding energies of 200 and 500 eV. The range distribution of B in Si follows the shapes expected. Mean values and standard deviations are in agreement with values experimentally determined. Boron atoms have a probability of 0.7 to settle down in substitutional positions.
Acknowledgements This work has been partially supported by DGICYT (Project No. MAT98-0859).
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