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Coalescence of B ions during high-fluence implantation into a Si target
Nuclear Instruments and Methods in Physics Research B 180 (2001) 91±98
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Coalescence of B ions during high-¯uence implantation into a Si target S.T. Nakagawa a
a,*
, G. Betz
b
Simulation Science Center, Okayama University of Science, 700-0005 Okayama, Japan b Institut fur Allgemeine Physik, Technical University, A-1040 Wien, Austria
Abstract We have investigated the question, what is the driving force causing coalescence among implanted ions. A previous explanation was that the criterion of clustering is an electrostatic interaction. That is if the formation enthalpy is positive, clustering starts. This is the case of B into Si. Moreover experimentally it was observed, that real clustering started, if the B concentration reached 2 at.%. Nevertheless this is too low a concentration to assume an alloy model, because the encountering probability for B ions is unreasonably small. In fact using molecular dynamic (MD) simulations, we found that without energetic ion impacts and the consequent mixing, no clustering is observed at room temperature (RT), even if we start with a 4 at.% concentration of B in Si. This means that the electrostatic explanation seems to support experiments, however, the encountering probability is elevated by heavy ion bombardment as MD proved. From our calculations we conclude, that ion impact and the consequent mixing processes in the lattice can trigger B clustering, while the enthalpy criterion of clustering is not always ful®lled. Ó 2001 Elsevier Science B.V. All rights reserved.
1. Introduction High-¯uence ion implantation is a promising technique to fabricate highly functional materials, like optical or electronic devices. There are two kinds of trends. One type of ions form solid solution involving host atoms. For example, C beams produce SiC and oxygen beams SiO2 . There is another type of ions that coalesce among themselves to form nanocrystals or just clusters. We
*
Corresponding author. Fax: +81-86-25-69-458. E-mail address: [email protected] (S.T. Nakagawa).
have observed Au clustering in Si [1] and Xe bubbles [2] in Fe. In the former case, this happened at liquid-nitrogen temperature under high-energy implantation. We inferred that the strong in¯uence of electronic stopping might be the trigger of coalescence for Au in Si. In the latter case measured at room temperature (RT), we presumed the ``rolling model'' for Xe ions coupled with odd number of vacancies could probably explain the driving force. The in¯uence of the temperature is to control the size of clusters and the degree of crystallinity of both the coalesced nanocluster and host target, as was discussed in the case of Au into Si [3,4]. We have been investigating the question,
0168-583X/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 0 1 ) 0 0 4 0 1 - 3
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S.T. Nakagawa, G. Betz / Nucl. Instr. and Meth. in Phys. Res. B 180 (2001) 91±98
why in some cases the implanted ions agglomerate and in other cases they do not. In this paper we will discuss the case of B into Si. Ryssel [5] speculated, the implanted B ions will cluster as quasi-molten B ions in the vicinity of the mean projected range (Rp ). In addition Mizushima et al. [6] found B12 molecules under high-¯uence (1017 cm 2 ) implantation of 35 keV B into Si, making use of XPS and FT±IR. At the same time they also found that there is a critical concentration of 2 at.%, for B clustering. This corresponds average separation between two B atoms, to 10 A if they are distributed randomly. This is too far apart to adopt a regular solution model, as discussed below. There was a qualitative criterion to judge whether or not a binary-alloy (AB) could be formed from two metals A and B. This is the socalled ``regular solution model''. Sometimes this is used qualitatively even for cases including nonmetallic elements. It predicts the tendency of solid solution depending on the sign of formation enthalpy. As will be discussed in Section 2.1, the positive sign of the formation enthalpy predicts B clustering in Si. In order to resolve how B clusters are produced, a key to the solution might be in the reaction diusion equation with presuming some free parameters. Thus a realistic picture of the relaxation processes resulting in nanoclusters does not exist. We have to look at the physicochemical change occurring in the intermediate processes. Therefore we started to use molecular dynamics (MD) calculations to get additional insight into the processes during agglomeration. We studied the mixing and consequent relaxation process under and after energetic ion impacts, which we believe will control the growth of nanoclusters. Indeed under repeated B ion impacts formation of B nanoclusters could be observed in our calculations. Empirical potentials were used to describe the atomic interactions. In Section 2.1 we discuss the empirical criterion. In Section 2.2, we take advantage of the Monte-Carlo (MC) simulation to supply the input data for our MD simulation. In Section 3 we summarize MD results: we discuss the in¯uence of attractive potential in Section 3.1, the in¯uence of ion impact in Section 3.2, then the critical concentration in Section 3.3.
2. Initial considerations not using MD 2.1. Empirical criterion: formation enthalpy The formation enthalpy is used to judge whether an alloy AB is produced or not. This quantity is proportional to the dierence of the binding energies for three cases A±A, A±B and B±B. This discussion presumes pairwise potentials V R for atoms separated by a distance R, for each case. The clustering occurs when the binding energy of EAB is higher than EAA EBB =2. The binding energies are known from interaction potential used for MD calculation. We use the Terso potential [7] for Si±Si, the Smith potential [8] for B±Si, and present potential for B±B. The latter two potentials are both hybrid potentials composed of a ZBL [9] and Morse potentials. For the last Morse potential we partly adopt the result of an ab inito calculation for B2 molecule obtained by Ray [9]. Both of Morse potentials were splined to the purely repulsive ZBL [10] potential at short atomic distances. The Morse type B±Si potential after Smith potential assumes a dissociation-energy De 0:25 (eV) and an equilibrium distance for a dimer of In the case of B±B, the Morse parc 2:351 (A). rameters used were De 1 (eV), b 1:5, rc 0:16 The binding energy ( De ) and the equilib(A). rium distance ( rc ) were obtained from ab initio calculation for a B2 molecule [9]. Therefore in our calculation EAA 1:14 eV per bond, EAB 0:25 eV and EBB 1 eV (A@Si, B@B). The formation enthalpy is DH is proportional to the energy gained, if solid-solution AB is formed. That is, DH / fEAB
EAA EBB =2g:
1
Substituting the binding energies mentioned above, we get a positive sign of DH . Thus the regular solution model predicts B ions will stick together if two B ions approach so close, rather than mixing with other Si atoms. This seems to support the experiments. Considering the average between B atoms for internuclear distance of 10 A B clustering in Si, however, we are not sure this criterion is really valid or not.
S.T. Nakagawa, G. Betz / Nucl. Instr. and Meth. in Phys. Res. B 180 (2001) 91±98
93
2.2. Monte-Carlo simulation using TRIM 35 keV B bombardment of a Si crystal using MD is much too cpu-time consuming and cannot be done realistically. So in our MD calculations we do not simulate over the whole crystal. We know clustering occurs only at the vicinity of the mean projected range, Rp , where the impurity concentration reaches its maximum. We thus will look in our MD calculations only at the relaxation processes in a cubic MD box located at a depth where impurity concentration is high enough to expect clustering. In order to estimate the entire pro®le of collision process we take advantage of a MC simulation. Making use of the well-known TRIM code, we can determine the depth pro®le, Rp and other quantities of interest for the bombarding B ions. In addition from these MC we obtain information on the energy, direction of motion, and stopping probability of the B ions at a certain depth. The TRIM reproduced reasonably the experimentally measured pro®le of 35 keV B ions implanted into Si, at ion ¯uences of 3 1016 and 1017 cm 2 . These data obtained from the TRIM calculations will use as input data for the following MD calculations. The maximum of the distribution is between 120 and 130 nm. Therefore our MD cubic box is set at a depth of 130 nm. The entrance face is Si(1 0 0), assuming a Si(1 0 0) crystal. The dimensions of our MD box are 3 3 3 nm3 in most cases 3 nm and we assumed it extends from 130±133 nm in depth. Such a box contains 1728 Si atoms. As was mentioned in Section 1, the critical concentration for coalescence was 2 at.%, which means that 34 B atoms are in the MD box. From the TRIM calculations we ®nd, that about 30 B atoms are stopped between a depth of 130 and 133 nm for 1000 B ion impacts. This means for 1017 cm 2 ion impacts we will obtain a B concentration of about 2% at 130 nm depth, where MD cubic box is located. We have modi®ed the TRIM code, so we could record all B atoms, which penetrate further than 130 nm and record their energy at this depth. In this way we ®nd from TRIM that for 1000 impacts (which gives a 2% concentration of stopped B atoms in our MD crystal) 266 B ions (26%) penetrate deeper than 130 nm. The energy distribution of
Fig. 1. The energy distribution of B ions at a depth of 130 nm. The two lines are drawn to de®ne the incident energy for B ions entering the MD box. This is obtained by TRIM.
these B ions penetrating below 130 nm is shown in Fig. 1. Most ions are still energetic with energies up to about 9 keV, and the average was 4.5 keV. The three zones indicated by shade distinguish the strength of ion impact, as will be discussed later. Another information we can obtain from the TRIM calculations is the direction of the B ions at 130 nm with respect to the depth axis. Only the angle with respect to the depth axis (z) is important, as the azimuthal angle will be equally distributed from 0±2p. The distribution for the tilted angle with the z-axis can be determined. The directions of most ions are still forward when they reach at the depth of Rp . Again we have used this dependence as input in our MD calculations. 3. MD results To study agglomeration we have performed MD simulations with the potentials outlined before. The Si single crystal fragment was terminated by (1 0 0) planes in the depth direction. Periodic boundary conditions were assumed at all sides of the box and for all atoms inside the box. For a ®nal B impurity concentration of 2% or 3% the MD cell should ®nally contain 34 or 51 B atoms in the box, respectively.
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S.T. Nakagawa, G. Betz / Nucl. Instr. and Meth. in Phys. Res. B 180 (2001) 91±98
An energetic B ion in the course of implantation will enter the MD cell. We assume that for such an implanted B ion no periodic boundary conditions apply in the depth direction. Thus it can enter the cell from the outside, and if it does not loose sucient energy in collisions it will penetrate the cell and leave the box at the other side (at 133 nm depth). However, if it is stopped inside the box, it will become part of the atoms in the box, and periodic boundary conditions will also apply for it, when the next energetic B ion impinges. We have performed three dierent types of calculations, as will be seen in Sections 3.1±3.3. Therein we look at the process of B coalescence, the distribution of cluster size, and the time-development of growing clusters as ion impact continues.
Fig. 2. The distribution of Bn clusters after equilibrated for 200 fs at 300 K, for cases with dierent intial concentration of B ions.
3.1. Without ion bombardment
3.2. The in¯uence of energetic ion impacts on a existing random B concentration
B atoms, which are randomly distributed inside to MD cell, will give information on the statistical probability of the occurrence of B cluster, without any mixing processes taking place. Thus such data can be used to compare these results with distributions if B ions are implanted energetically, i.e. if mixing will occur. For this purpose we have calculated the initial distribution of B atoms if they are randomly deposited inside the MD cell. The temperature of the cell was set to 300 K. B atoms were randomly distributed with one restriction: only positions of B atoms were accepted if the distance to the closest Si atom was greater than 0.144 nm, to prevent strong repulsive forces between a B and a Si atom. Afterwards the system was equilibrated for 2 ps to reach equilibrium at 300 K. The normalized distributions formed are shown in Fig. 2, where 17 atoms are in the MD box at 1%, 34 atoms at 2%, 51 atoms at 3% and 69 atoms at 4%. As can be seen even at the highest B concentration of 4% more than 80% of the B atoms exist as single atoms and only about 5% of the B atoms are in the form of dimers and on the average only one trimer is formed. We can say no clustering is observed. This is against the conventional explanation based on the regular solution model that explains such phenomena only from an electrostatic viewpoint.
In this case we study the eect of energetic B ion bombardment. B atoms up to a given concentration are distributed randomly in the MD cell and after equilibrium at 300 K is reached we start with energetic B ion bombardment. B ion energies were selected in this case randomly between 1 and 5 keV, so that the energy distribution reproduces the previous MC calculation, Fig. 1. The B ion starts outside the Si crystal (normal ion incidence) and penetrates through. Periodic boundary conditions only apply for the Si and for the pre-implanted B atoms, but not for the incoming B ion in the z direction. If the B ion becomes trapped, at the next B ion impact, this trapped B atom is added to the impurity B atoms and periodic boundary conditions apply for it also in the z direction. The calculation for such an energetic B ion impact is ®nished, as soon as the temperature of the crystal is down to 400 K. In addition we use a maximum calculation time of 2 ps for such an impact, but in all cases the MD cell has cooled down to 400 K before 2 ps. In doing such energetic B bombardment 2 eects become clear: (a) These B ions penetrate through the crystal and the probability of losing all their energy and getting trapped inside the 3 nm of Si is extremely small and is less than 2%, i.e. for 100 bombardments maybe 1±2 B atoms become trapped, which
S.T. Nakagawa, G. Betz / Nucl. Instr. and Meth. in Phys. Res. B 180 (2001) 91±98
means a negligible increase in the total B concentration. This was the reason for choosing 1 keV as the lower limit in energetic ion bombardment, where we wanted to investigate their eect on an already existing random B distribution. (b) There is a clear tendency for B cluster being formed or increasing in size due to the mixing processes initiated by the B ions passing through the MD cell, as is shown in Fig. 3(a). On the average about 30±40% or 50±60% of all impacts cause a change in the cluster distribution for an initial B concentration of 2% or 3%, respectively. Fig. 3(a) presents the results for a MD cell (1728 atoms) with an originally statistically distributed B impurity concentration of 2%. Two
Fig. 3. (a) The distribution of Bn cluster after ion impacts at 300 K, with an initial concentration of 2 at.% B. (b) The time development of B clustering in the case of 91 impacts.
95
calculations with 91 and 126 B ion impacts are shown and compared to the original random B concentration at 2% (34 atoms). The original B concentration increased 1 and 2 B atom, respectively. The development of cluster formation for the case of a B concentration of 2% during 91 energetic impacts is shown in Fig. 3(b). Larger cluster can grow by agglomeration. In this way the cluster of size 6 is formed by adding monomers to a trimer 3 ! 4 ! 5 ! 6. The cluster of size 9 is formed during one impact in which 7 dimers 1 trimer and one cluster of size 6 are reorganized during one impact to 4 dimers, 2 trimers and one cluster of size 9. The cluster of size 9 is formed from the cluster of size 6 by addition of a dimer and a monomer. In this impact a B ion with almost 5 keV underwent a close collision with a Si atom and was re¯ected. For an initial B concentration of 3% we performed a series of calculations, with about 100 energetic impacts each to obtain statistically signi®cant information. The cluster size distribution varies signi®cantly among cases. However, we always observed B2 ±B8 clusters, largest one was B11 cluster. For example, Fig. 4 shows B8 cluster near the surface of the MD box. The development of cluster formation for two cases with 100 impacts each is shown in Fig. 5, where the initial B concentration was 3%. Again we would like to point out that due to this 100 energetic B impacts the concentration remains almost unchanged, as at most 2 atoms become trapped. These two cases are identical except for the seed number for random-number-generator that de®ne the impact points and the energy of the B ions. In Fig. 5(a), a big cluster is formed from an early stage, and a smaller cluster will be formed much later. On the other hand, in the other case shown by Fig. 5(b), the clusters look to grow as ion impact accumulates the damage energy inside the box. Both cases might show the realistic phenomena that happen in dierent places in the same box. At a place big cluster may be born, but in an another place cluster grows almost regularly as the total damage energy is increased. To check if the number of energetic B ion impacts is realistic we can use again our TRIM results. We found that to achieve a 2% B con-
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S.T. Nakagawa, G. Betz / Nucl. Instr. and Meth. in Phys. Res. B 180 (2001) 91±98
Fig. 4. B8 cluster formed near the surface of the MD box, after 42 impacts on a Si target with 3 at.% B.
centration we need about 1000 ion impacts and 26% of those penetrate below 13 nm. From the energy distribution at 13 nm (Fig. 1) we can also obtain the number of B impacts with energies between 1 and 5 keV as we have used and ®nd 64% or 130 impacts. So the number of B impacts we have used (100 and below) is quite realistic and actually at 3% we could increase the number to almost 200 impacts. Thus we can expect that an even stronger agglomeration will occur as compared to what we have observed so far. However, in these calculations we have only investigated the mixing eect and the agglomeration due to it. The eect of the actual implantation (stopping of B ions between 13 and 13.3 nm) which will arrive at 13 nm depth with a low energy (below 1 keV) is not considered in these calculations. In addition the energetic B ions bombarded the MD cell at normal incidence, but in a real experiment there will be scattering inside the target and the B ions will enter the MD cell at dierent angles of incidence. Therefore we performed another set of calculations discussed below.
Fig. 5. Time development of B cluster formation for two cases, when bombardered with high energy of 1 < E keV < 5, and an intial B concentration of 3 at.%. Upward arrows labeled by number n in parenthesis, indicate the moment when a Bn cluster is formed.
3.3. Low-energy bombardment to build up the B concentration In this case we have bombarded the MD cell (1729 atoms) without any pre-implanted B atoms (concentration 0%) with low energy B ions of energy randomly selected between 0 and 1000 eV according to the energy distribution of B ions at a depth of 13 nm (see Fig. 1). In addition also in this case the angle of incidence was chosen according to our TRIM results. Under these conditions we observe that about every third B ion will be trap-
S.T. Nakagawa, G. Betz / Nucl. Instr. and Meth. in Phys. Res. B 180 (2001) 91±98
97
Fig. 6. Time development of B cluster formation for an initial 0% B concentration and low-energy (<1 keV) bombardment.
ped inside our MD cell (1729 atoms). Thus we need roughly 100 impacts to reach a B concentration of 2% or 150 impacts for 3%, respectively.
Thus in this case we can investigate the eect of the actual implantation as compared to a random B atom distribution. However, the eect of addi-
Fig. 7. The change in the number of monomers as is shown from the three timelines. The notation (2) indicates the moment when a dimer was ®rst born. The moment when the largest cluster in respective cases are shown by (6), (7) and (8).
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S.T. Nakagawa, G. Betz / Nucl. Instr. and Meth. in Phys. Res. B 180 (2001) 91±98
tional mixing induced by more energetic B ions (up to 9 keV, see Fig. 1) as studied in our calculations Section 3.2 is excluded. The cluster development with time (number of impacts) is shown in Fig. 6, where we have also indicated after how many impacts the B concentration has reached 1%, 2% and 3%. It can be seen that up to a B concentration of 2% the monomer concentration increases, but after above 2% we observe a decrease and a simultaneously increase in larger cluster, as is seen clearly in Fig. 7. This ®gure indicates the change in the number of monomers in three cases. At ®rst until the B concentration reaches 1 at.%, monomer increases with involved B ions, while dimers (2) were born. Then the increase seems a little to be suppressed up to 2 at.%. Beyond 2 at.%, the monomers decreases evidently to form bigger clusters, while B6 B8 are observed here. This we believe is correlated with the fact that in the experiments agglomeration was only observed above 2% B concentration. 4. Conclusion In order to study, why B ions coalesce in Si, if they are implanted at a high-¯uence, we examined the relaxation processes making use of MD calculation. The general solution model predicts B clustering for the interaction potentials we used. However the critical concentration of 2 at.% B for clustering was too low to apply the solution model with con®dence. Using MC simulation (TRIM) as input data, we performed MD calculation inside a MD box located in the vicinity of the mean projected range. We found no signi®cant B clustering at RT for randomly deposited atoms. This means that electrostatic explanation alone is not ade-
quate, but mixing due to energetic ion bombardment was necessary to observe B clustering, as found in experiments. Under ion implantation, we found remarkable B clustering to occur, if the accumulated B concentration exceeded 2 at.%. We have distinguished in our calculations between low-energy and high-energy ion bombardment. Both will occur during ion implantation at a depth of the mean projected range. High-energy bombardment (above 1 keV) will produce clustering but not deposition, i.e. the probability that a B ion with an energy larger than 1 keV is trapped inside our MD box is negligibly small. Low-energy bombardment (below 1 keV) is the source of stopped atoms and contributes to agglomeration in a similar manner as high-energy bombardment. Thus both processes contribute to agglomeration. More qualitative investigation concerning the general solution model is progressing. References [1] S.T. Nakagawa, S. Nakano, H. Ogiso, M. Iwaki, W. Eckstein, Rev. Sci. Instrum. 71 (2000) 793. [2] E. Yagi, T. Sasahara, T. Joh, M. Hacke, T. Urai, T. Sasamoto, N. Tajime, T. Watanabe, S.T. Nakagawa, J. Phys. Soc. Jpn. (1999) 4037. [3] J.K.N. Lindner, N. Hecking, E. te Kaat, Nucl. Instr. and Meth. B 26 (1987) 551. [4] J.K.N. Lindner, R. Zuschlag, E.H. te Kaat, Nucl. Instr. and Meth. B 62 (1992) 314. [5] H. Ryssel, K. M oler, K. Haberger, R. Henkelmann, F. Jahnel, Appl. Phys. 22 (1980) 35. [6] I. Mizushima et al., Jpn. J. Appl. Phys. 33 (1994) 404. [7] J. Terso, Phys. Rev. B. 39 (1989) 5566. [8] R. Smith, M. Shaw, R.P. Webb, M.A. Foad, J. Appl. Phys. 83 (1998) 3148. [9] A.K. Ray, I.A. Howard, K.M. Kanal, Phys. Rev. B 45 (1992) 14247. [10] J.F. Ziegler, J.P. Biersack, U. Littmark, The Stopping and Ranges of Ions in Solids, Pergamon, New York, 1985.
www.elsevier.nl/locate/nimb
Coalescence of B ions during high-¯uence implantation into a Si target S.T. Nakagawa a
a,*
, G. Betz
b
Simulation Science Center, Okayama University of Science, 700-0005 Okayama, Japan b Institut fur Allgemeine Physik, Technical University, A-1040 Wien, Austria
Abstract We have investigated the question, what is the driving force causing coalescence among implanted ions. A previous explanation was that the criterion of clustering is an electrostatic interaction. That is if the formation enthalpy is positive, clustering starts. This is the case of B into Si. Moreover experimentally it was observed, that real clustering started, if the B concentration reached 2 at.%. Nevertheless this is too low a concentration to assume an alloy model, because the encountering probability for B ions is unreasonably small. In fact using molecular dynamic (MD) simulations, we found that without energetic ion impacts and the consequent mixing, no clustering is observed at room temperature (RT), even if we start with a 4 at.% concentration of B in Si. This means that the electrostatic explanation seems to support experiments, however, the encountering probability is elevated by heavy ion bombardment as MD proved. From our calculations we conclude, that ion impact and the consequent mixing processes in the lattice can trigger B clustering, while the enthalpy criterion of clustering is not always ful®lled. Ó 2001 Elsevier Science B.V. All rights reserved.
1. Introduction High-¯uence ion implantation is a promising technique to fabricate highly functional materials, like optical or electronic devices. There are two kinds of trends. One type of ions form solid solution involving host atoms. For example, C beams produce SiC and oxygen beams SiO2 . There is another type of ions that coalesce among themselves to form nanocrystals or just clusters. We
*
Corresponding author. Fax: +81-86-25-69-458. E-mail address: [email protected] (S.T. Nakagawa).
have observed Au clustering in Si [1] and Xe bubbles [2] in Fe. In the former case, this happened at liquid-nitrogen temperature under high-energy implantation. We inferred that the strong in¯uence of electronic stopping might be the trigger of coalescence for Au in Si. In the latter case measured at room temperature (RT), we presumed the ``rolling model'' for Xe ions coupled with odd number of vacancies could probably explain the driving force. The in¯uence of the temperature is to control the size of clusters and the degree of crystallinity of both the coalesced nanocluster and host target, as was discussed in the case of Au into Si [3,4]. We have been investigating the question,
0168-583X/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 0 1 ) 0 0 4 0 1 - 3
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S.T. Nakagawa, G. Betz / Nucl. Instr. and Meth. in Phys. Res. B 180 (2001) 91±98
why in some cases the implanted ions agglomerate and in other cases they do not. In this paper we will discuss the case of B into Si. Ryssel [5] speculated, the implanted B ions will cluster as quasi-molten B ions in the vicinity of the mean projected range (Rp ). In addition Mizushima et al. [6] found B12 molecules under high-¯uence (1017 cm 2 ) implantation of 35 keV B into Si, making use of XPS and FT±IR. At the same time they also found that there is a critical concentration of 2 at.%, for B clustering. This corresponds average separation between two B atoms, to 10 A if they are distributed randomly. This is too far apart to adopt a regular solution model, as discussed below. There was a qualitative criterion to judge whether or not a binary-alloy (AB) could be formed from two metals A and B. This is the socalled ``regular solution model''. Sometimes this is used qualitatively even for cases including nonmetallic elements. It predicts the tendency of solid solution depending on the sign of formation enthalpy. As will be discussed in Section 2.1, the positive sign of the formation enthalpy predicts B clustering in Si. In order to resolve how B clusters are produced, a key to the solution might be in the reaction diusion equation with presuming some free parameters. Thus a realistic picture of the relaxation processes resulting in nanoclusters does not exist. We have to look at the physicochemical change occurring in the intermediate processes. Therefore we started to use molecular dynamics (MD) calculations to get additional insight into the processes during agglomeration. We studied the mixing and consequent relaxation process under and after energetic ion impacts, which we believe will control the growth of nanoclusters. Indeed under repeated B ion impacts formation of B nanoclusters could be observed in our calculations. Empirical potentials were used to describe the atomic interactions. In Section 2.1 we discuss the empirical criterion. In Section 2.2, we take advantage of the Monte-Carlo (MC) simulation to supply the input data for our MD simulation. In Section 3 we summarize MD results: we discuss the in¯uence of attractive potential in Section 3.1, the in¯uence of ion impact in Section 3.2, then the critical concentration in Section 3.3.
2. Initial considerations not using MD 2.1. Empirical criterion: formation enthalpy The formation enthalpy is used to judge whether an alloy AB is produced or not. This quantity is proportional to the dierence of the binding energies for three cases A±A, A±B and B±B. This discussion presumes pairwise potentials V R for atoms separated by a distance R, for each case. The clustering occurs when the binding energy of EAB is higher than EAA EBB =2. The binding energies are known from interaction potential used for MD calculation. We use the Terso potential [7] for Si±Si, the Smith potential [8] for B±Si, and present potential for B±B. The latter two potentials are both hybrid potentials composed of a ZBL [9] and Morse potentials. For the last Morse potential we partly adopt the result of an ab inito calculation for B2 molecule obtained by Ray [9]. Both of Morse potentials were splined to the purely repulsive ZBL [10] potential at short atomic distances. The Morse type B±Si potential after Smith potential assumes a dissociation-energy De 0:25 (eV) and an equilibrium distance for a dimer of In the case of B±B, the Morse parc 2:351 (A). rameters used were De 1 (eV), b 1:5, rc 0:16 The binding energy ( De ) and the equilib(A). rium distance ( rc ) were obtained from ab initio calculation for a B2 molecule [9]. Therefore in our calculation EAA 1:14 eV per bond, EAB 0:25 eV and EBB 1 eV (A@Si, B@B). The formation enthalpy is DH is proportional to the energy gained, if solid-solution AB is formed. That is, DH / fEAB
EAA EBB =2g:
1
Substituting the binding energies mentioned above, we get a positive sign of DH . Thus the regular solution model predicts B ions will stick together if two B ions approach so close, rather than mixing with other Si atoms. This seems to support the experiments. Considering the average between B atoms for internuclear distance of 10 A B clustering in Si, however, we are not sure this criterion is really valid or not.
S.T. Nakagawa, G. Betz / Nucl. Instr. and Meth. in Phys. Res. B 180 (2001) 91±98
93
2.2. Monte-Carlo simulation using TRIM 35 keV B bombardment of a Si crystal using MD is much too cpu-time consuming and cannot be done realistically. So in our MD calculations we do not simulate over the whole crystal. We know clustering occurs only at the vicinity of the mean projected range, Rp , where the impurity concentration reaches its maximum. We thus will look in our MD calculations only at the relaxation processes in a cubic MD box located at a depth where impurity concentration is high enough to expect clustering. In order to estimate the entire pro®le of collision process we take advantage of a MC simulation. Making use of the well-known TRIM code, we can determine the depth pro®le, Rp and other quantities of interest for the bombarding B ions. In addition from these MC we obtain information on the energy, direction of motion, and stopping probability of the B ions at a certain depth. The TRIM reproduced reasonably the experimentally measured pro®le of 35 keV B ions implanted into Si, at ion ¯uences of 3 1016 and 1017 cm 2 . These data obtained from the TRIM calculations will use as input data for the following MD calculations. The maximum of the distribution is between 120 and 130 nm. Therefore our MD cubic box is set at a depth of 130 nm. The entrance face is Si(1 0 0), assuming a Si(1 0 0) crystal. The dimensions of our MD box are 3 3 3 nm3 in most cases 3 nm and we assumed it extends from 130±133 nm in depth. Such a box contains 1728 Si atoms. As was mentioned in Section 1, the critical concentration for coalescence was 2 at.%, which means that 34 B atoms are in the MD box. From the TRIM calculations we ®nd, that about 30 B atoms are stopped between a depth of 130 and 133 nm for 1000 B ion impacts. This means for 1017 cm 2 ion impacts we will obtain a B concentration of about 2% at 130 nm depth, where MD cubic box is located. We have modi®ed the TRIM code, so we could record all B atoms, which penetrate further than 130 nm and record their energy at this depth. In this way we ®nd from TRIM that for 1000 impacts (which gives a 2% concentration of stopped B atoms in our MD crystal) 266 B ions (26%) penetrate deeper than 130 nm. The energy distribution of
Fig. 1. The energy distribution of B ions at a depth of 130 nm. The two lines are drawn to de®ne the incident energy for B ions entering the MD box. This is obtained by TRIM.
these B ions penetrating below 130 nm is shown in Fig. 1. Most ions are still energetic with energies up to about 9 keV, and the average was 4.5 keV. The three zones indicated by shade distinguish the strength of ion impact, as will be discussed later. Another information we can obtain from the TRIM calculations is the direction of the B ions at 130 nm with respect to the depth axis. Only the angle with respect to the depth axis (z) is important, as the azimuthal angle will be equally distributed from 0±2p. The distribution for the tilted angle with the z-axis can be determined. The directions of most ions are still forward when they reach at the depth of Rp . Again we have used this dependence as input in our MD calculations. 3. MD results To study agglomeration we have performed MD simulations with the potentials outlined before. The Si single crystal fragment was terminated by (1 0 0) planes in the depth direction. Periodic boundary conditions were assumed at all sides of the box and for all atoms inside the box. For a ®nal B impurity concentration of 2% or 3% the MD cell should ®nally contain 34 or 51 B atoms in the box, respectively.
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An energetic B ion in the course of implantation will enter the MD cell. We assume that for such an implanted B ion no periodic boundary conditions apply in the depth direction. Thus it can enter the cell from the outside, and if it does not loose sucient energy in collisions it will penetrate the cell and leave the box at the other side (at 133 nm depth). However, if it is stopped inside the box, it will become part of the atoms in the box, and periodic boundary conditions will also apply for it, when the next energetic B ion impinges. We have performed three dierent types of calculations, as will be seen in Sections 3.1±3.3. Therein we look at the process of B coalescence, the distribution of cluster size, and the time-development of growing clusters as ion impact continues.
Fig. 2. The distribution of Bn clusters after equilibrated for 200 fs at 300 K, for cases with dierent intial concentration of B ions.
3.1. Without ion bombardment
3.2. The in¯uence of energetic ion impacts on a existing random B concentration
B atoms, which are randomly distributed inside to MD cell, will give information on the statistical probability of the occurrence of B cluster, without any mixing processes taking place. Thus such data can be used to compare these results with distributions if B ions are implanted energetically, i.e. if mixing will occur. For this purpose we have calculated the initial distribution of B atoms if they are randomly deposited inside the MD cell. The temperature of the cell was set to 300 K. B atoms were randomly distributed with one restriction: only positions of B atoms were accepted if the distance to the closest Si atom was greater than 0.144 nm, to prevent strong repulsive forces between a B and a Si atom. Afterwards the system was equilibrated for 2 ps to reach equilibrium at 300 K. The normalized distributions formed are shown in Fig. 2, where 17 atoms are in the MD box at 1%, 34 atoms at 2%, 51 atoms at 3% and 69 atoms at 4%. As can be seen even at the highest B concentration of 4% more than 80% of the B atoms exist as single atoms and only about 5% of the B atoms are in the form of dimers and on the average only one trimer is formed. We can say no clustering is observed. This is against the conventional explanation based on the regular solution model that explains such phenomena only from an electrostatic viewpoint.
In this case we study the eect of energetic B ion bombardment. B atoms up to a given concentration are distributed randomly in the MD cell and after equilibrium at 300 K is reached we start with energetic B ion bombardment. B ion energies were selected in this case randomly between 1 and 5 keV, so that the energy distribution reproduces the previous MC calculation, Fig. 1. The B ion starts outside the Si crystal (normal ion incidence) and penetrates through. Periodic boundary conditions only apply for the Si and for the pre-implanted B atoms, but not for the incoming B ion in the z direction. If the B ion becomes trapped, at the next B ion impact, this trapped B atom is added to the impurity B atoms and periodic boundary conditions apply for it also in the z direction. The calculation for such an energetic B ion impact is ®nished, as soon as the temperature of the crystal is down to 400 K. In addition we use a maximum calculation time of 2 ps for such an impact, but in all cases the MD cell has cooled down to 400 K before 2 ps. In doing such energetic B bombardment 2 eects become clear: (a) These B ions penetrate through the crystal and the probability of losing all their energy and getting trapped inside the 3 nm of Si is extremely small and is less than 2%, i.e. for 100 bombardments maybe 1±2 B atoms become trapped, which
S.T. Nakagawa, G. Betz / Nucl. Instr. and Meth. in Phys. Res. B 180 (2001) 91±98
means a negligible increase in the total B concentration. This was the reason for choosing 1 keV as the lower limit in energetic ion bombardment, where we wanted to investigate their eect on an already existing random B distribution. (b) There is a clear tendency for B cluster being formed or increasing in size due to the mixing processes initiated by the B ions passing through the MD cell, as is shown in Fig. 3(a). On the average about 30±40% or 50±60% of all impacts cause a change in the cluster distribution for an initial B concentration of 2% or 3%, respectively. Fig. 3(a) presents the results for a MD cell (1728 atoms) with an originally statistically distributed B impurity concentration of 2%. Two
Fig. 3. (a) The distribution of Bn cluster after ion impacts at 300 K, with an initial concentration of 2 at.% B. (b) The time development of B clustering in the case of 91 impacts.
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calculations with 91 and 126 B ion impacts are shown and compared to the original random B concentration at 2% (34 atoms). The original B concentration increased 1 and 2 B atom, respectively. The development of cluster formation for the case of a B concentration of 2% during 91 energetic impacts is shown in Fig. 3(b). Larger cluster can grow by agglomeration. In this way the cluster of size 6 is formed by adding monomers to a trimer 3 ! 4 ! 5 ! 6. The cluster of size 9 is formed during one impact in which 7 dimers 1 trimer and one cluster of size 6 are reorganized during one impact to 4 dimers, 2 trimers and one cluster of size 9. The cluster of size 9 is formed from the cluster of size 6 by addition of a dimer and a monomer. In this impact a B ion with almost 5 keV underwent a close collision with a Si atom and was re¯ected. For an initial B concentration of 3% we performed a series of calculations, with about 100 energetic impacts each to obtain statistically signi®cant information. The cluster size distribution varies signi®cantly among cases. However, we always observed B2 ±B8 clusters, largest one was B11 cluster. For example, Fig. 4 shows B8 cluster near the surface of the MD box. The development of cluster formation for two cases with 100 impacts each is shown in Fig. 5, where the initial B concentration was 3%. Again we would like to point out that due to this 100 energetic B impacts the concentration remains almost unchanged, as at most 2 atoms become trapped. These two cases are identical except for the seed number for random-number-generator that de®ne the impact points and the energy of the B ions. In Fig. 5(a), a big cluster is formed from an early stage, and a smaller cluster will be formed much later. On the other hand, in the other case shown by Fig. 5(b), the clusters look to grow as ion impact accumulates the damage energy inside the box. Both cases might show the realistic phenomena that happen in dierent places in the same box. At a place big cluster may be born, but in an another place cluster grows almost regularly as the total damage energy is increased. To check if the number of energetic B ion impacts is realistic we can use again our TRIM results. We found that to achieve a 2% B con-
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Fig. 4. B8 cluster formed near the surface of the MD box, after 42 impacts on a Si target with 3 at.% B.
centration we need about 1000 ion impacts and 26% of those penetrate below 13 nm. From the energy distribution at 13 nm (Fig. 1) we can also obtain the number of B impacts with energies between 1 and 5 keV as we have used and ®nd 64% or 130 impacts. So the number of B impacts we have used (100 and below) is quite realistic and actually at 3% we could increase the number to almost 200 impacts. Thus we can expect that an even stronger agglomeration will occur as compared to what we have observed so far. However, in these calculations we have only investigated the mixing eect and the agglomeration due to it. The eect of the actual implantation (stopping of B ions between 13 and 13.3 nm) which will arrive at 13 nm depth with a low energy (below 1 keV) is not considered in these calculations. In addition the energetic B ions bombarded the MD cell at normal incidence, but in a real experiment there will be scattering inside the target and the B ions will enter the MD cell at dierent angles of incidence. Therefore we performed another set of calculations discussed below.
Fig. 5. Time development of B cluster formation for two cases, when bombardered with high energy of 1 < E keV < 5, and an intial B concentration of 3 at.%. Upward arrows labeled by number n in parenthesis, indicate the moment when a Bn cluster is formed.
3.3. Low-energy bombardment to build up the B concentration In this case we have bombarded the MD cell (1729 atoms) without any pre-implanted B atoms (concentration 0%) with low energy B ions of energy randomly selected between 0 and 1000 eV according to the energy distribution of B ions at a depth of 13 nm (see Fig. 1). In addition also in this case the angle of incidence was chosen according to our TRIM results. Under these conditions we observe that about every third B ion will be trap-
S.T. Nakagawa, G. Betz / Nucl. Instr. and Meth. in Phys. Res. B 180 (2001) 91±98
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Fig. 6. Time development of B cluster formation for an initial 0% B concentration and low-energy (<1 keV) bombardment.
ped inside our MD cell (1729 atoms). Thus we need roughly 100 impacts to reach a B concentration of 2% or 150 impacts for 3%, respectively.
Thus in this case we can investigate the eect of the actual implantation as compared to a random B atom distribution. However, the eect of addi-
Fig. 7. The change in the number of monomers as is shown from the three timelines. The notation (2) indicates the moment when a dimer was ®rst born. The moment when the largest cluster in respective cases are shown by (6), (7) and (8).
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tional mixing induced by more energetic B ions (up to 9 keV, see Fig. 1) as studied in our calculations Section 3.2 is excluded. The cluster development with time (number of impacts) is shown in Fig. 6, where we have also indicated after how many impacts the B concentration has reached 1%, 2% and 3%. It can be seen that up to a B concentration of 2% the monomer concentration increases, but after above 2% we observe a decrease and a simultaneously increase in larger cluster, as is seen clearly in Fig. 7. This ®gure indicates the change in the number of monomers in three cases. At ®rst until the B concentration reaches 1 at.%, monomer increases with involved B ions, while dimers (2) were born. Then the increase seems a little to be suppressed up to 2 at.%. Beyond 2 at.%, the monomers decreases evidently to form bigger clusters, while B6 B8 are observed here. This we believe is correlated with the fact that in the experiments agglomeration was only observed above 2% B concentration. 4. Conclusion In order to study, why B ions coalesce in Si, if they are implanted at a high-¯uence, we examined the relaxation processes making use of MD calculation. The general solution model predicts B clustering for the interaction potentials we used. However the critical concentration of 2 at.% B for clustering was too low to apply the solution model with con®dence. Using MC simulation (TRIM) as input data, we performed MD calculation inside a MD box located in the vicinity of the mean projected range. We found no signi®cant B clustering at RT for randomly deposited atoms. This means that electrostatic explanation alone is not ade-
quate, but mixing due to energetic ion bombardment was necessary to observe B clustering, as found in experiments. Under ion implantation, we found remarkable B clustering to occur, if the accumulated B concentration exceeded 2 at.%. We have distinguished in our calculations between low-energy and high-energy ion bombardment. Both will occur during ion implantation at a depth of the mean projected range. High-energy bombardment (above 1 keV) will produce clustering but not deposition, i.e. the probability that a B ion with an energy larger than 1 keV is trapped inside our MD box is negligibly small. Low-energy bombardment (below 1 keV) is the source of stopped atoms and contributes to agglomeration in a similar manner as high-energy bombardment. Thus both processes contribute to agglomeration. More qualitative investigation concerning the general solution model is progressing. References [1] S.T. Nakagawa, S. Nakano, H. Ogiso, M. Iwaki, W. Eckstein, Rev. Sci. Instrum. 71 (2000) 793. [2] E. Yagi, T. Sasahara, T. Joh, M. Hacke, T. Urai, T. Sasamoto, N. Tajime, T. Watanabe, S.T. Nakagawa, J. Phys. Soc. Jpn. (1999) 4037. [3] J.K.N. Lindner, N. Hecking, E. te Kaat, Nucl. Instr. and Meth. B 26 (1987) 551. [4] J.K.N. Lindner, R. Zuschlag, E.H. te Kaat, Nucl. Instr. and Meth. B 62 (1992) 314. [5] H. Ryssel, K. M oler, K. Haberger, R. Henkelmann, F. Jahnel, Appl. Phys. 22 (1980) 35. [6] I. Mizushima et al., Jpn. J. Appl. Phys. 33 (1994) 404. [7] J. Terso, Phys. Rev. B. 39 (1989) 5566. [8] R. Smith, M. Shaw, R.P. Webb, M.A. Foad, J. Appl. Phys. 83 (1998) 3148. [9] A.K. Ray, I.A. Howard, K.M. Kanal, Phys. Rev. B 45 (1992) 14247. [10] J.F. Ziegler, J.P. Biersack, U. Littmark, The Stopping and Ranges of Ions in Solids, Pergamon, New York, 1985.