56 POINT DEFECT MOVEMENT AND ANNEALING IN 2423 By analyzing the structure factor of each atom and its neighbors,combined with a kinetic energy criterion,we were Initial int. able to recognize interstitials and the liquid region during the 35 ◆ Final int. simulation. c 30 Initial vac. The motion of interstitials throughout a simulation was Final vac. followed using an interstitial database which contains histo- 25 ries of all interstitials.When an interstitial structure is recog- 6 nized during one time step,its position is compared to the 20 position of all interstitial structures recognized during previ- 15 ous time steps.In the new time step,if the new interstitial atom is closer than one lattice constant from one of the pre- wnN 10 viously recognized interstitial structures,it is interpreted to 5 be part of the same interstitial.Otherwise,a new interstitial database entry is created. 20 40 60 80 100 The position of each interstitial structure during any given time step is calculated as the average position of all atoms Distance from center(A) which are part of it.If an interstitial structure becomes part of the liquid zone or has not been seen for at least three FIG.1.Distribution of interstitials (int.)and vacancies (vac.)as interstitial analysis steps,it is discarded from the list of cur- a function of the distance from the center of the simulation cell rent interstitial structures.Thus,after the simulation the in- before and after the cascade event.The numbers are the sum over terstitial database will contain information on the positions, the three cascade events discussed in the text.The low number of lifetime,and movement of all interstitial recognized during defects is the reason for the "roughness''of the distributions. the simulation. Independently of the structure-factor analysis,we per- vacancies and interstitials as a function of the distance from formed an analysis of the population of the Wigner-Seitz cell the cell center.summed over the three cascade events. of each perfect lattice atom position for a few time steps in each simulation.Wigner-Seitz cells with more than one atom were interpreted as interstitials and empty cells as vacancies. D.Silicon The numbers of defects obtained for any given time in the For realistic MD simulation of covalently bonded materi- Wigner-Seitz and structure-factor methods were in good als like silicon,it is important to use an interatomic potential agreement with each other. which adequately describes the bonding interactions in the C.Gold material.We used the Tersoff(C)three-body interatomic potential which gives a good description of several proper- For the MD simulations of gold we employed the gold ties of Si,including the different bonding types and elastic embedded-atom method(EAM)potential used previously at moduli.27.28 The potential also gives a reasonable description this laboratory in studies of collision cascades.720 The uni- of point defect energies,although the order of the point de- versal repulsive potentials has been fitted to the potential to fect energies is not correct.2829 Since we are primarily inter- realistically describe strong collisions. ested in the possibility of defect motion,the migration ener- The interstitials introduced into the cell were given the gies of point defects are more significant in any case than (100)fcc dumbbell interstitial structure,which is the lowest- their equilibrium structure.Experimental and density- energy interstitial of fcc metals.This interstitial is known functional results suggest the vacancy migration energy is to migrate very easily.Experimentally no determination of -0.3 eV.30,31 Recent tight-binding molecular-dynamics re- the migration energy in gold has been achieved.19 The EAM sults give a migration energy of 1.4 eV for the dumbbell potential predicts a migration energy of 0.06 eV 20 which is interstitial2 (however,in real Si the interstitial migration realistic for high-temperature migration where quantum ef- may be enhanced by defect charge state processes30).For the fects are no longer important. Tersoff potential,many different values for the interstitial In the automatic interstitial recognition procedure,we and vacancy migration energies have been reported in the evaluated the structure factor P for the 12 nearest neighbors literature,probably due to the difficulty of recognizing the of each atom.We found that different values of P could be lowest-energy migration path.28,34 For the vacancy,the used to distinguish between crystalline atoms,interstitials, lowest migration activation energy calculated for an actual and liquid atoms.By comparing the Pst analysis to visual migration path is 1.6 ev,33 and for the tetrahedral interstitial inspection of moving defects,we ensured that the analysis 1.1 ev.34 Thus,it appears that the Tersoff potential clearly did not lose track of moving interstitials and fine-tuned the overestimates the vacancy migration energy,but gives a recognition criteria to recognize interstitials even in interme- fairly reasonable value for interstitials. diate configurations during migration. Frequently,the high melting point predicted by the Ter- We performed four simulations of 5 keV recoil events in soff potential is used as an argument against its suitability for which our interstitial analysis method was used:a reference collision cascade studies.However,using a new reliable ap- run in an undamaged gold crystal,two simulations with 50 proach to calculate the melting point3s we obtained a value interstitials and 50 vacancies,and one with only 50 intersti- of 2300+100 K,which is much less than the conventionally tials.Figure 1 shows the initial and final distributions of cited value of 3000 K (Refs.27 and 28)and in better agree-By analyzing the structure factor of each atom and its neighbors, combined with a kinetic energy criterion, we were able to recognize interstitials and the liquid region during the simulation. The motion of interstitials throughout a simulation was followed using an interstitial database which contains histo￾ries of all interstitials. When an interstitial structure is recog￾nized during one time step, its position is compared to the position of all interstitial structures recognized during previ￾ous time steps. In the new time step, if the new interstitial atom is closer than one lattice constant from one of the pre￾viously recognized interstitial structures, it is interpreted to be part of the same interstitial. Otherwise, a new interstitial database entry is created. The position of each interstitial structure during any given time step is calculated as the average position of all atoms which are part of it. If an interstitial structure becomes part of the liquid zone or has not been seen for at least three interstitial analysis steps, it is discarded from the list of cur￾rent interstitial structures. Thus, after the simulation the in￾terstitial database will contain information on the positions, lifetime, and movement of all interstitial recognized during the simulation. Independently of the structure-factor analysis, we per￾formed an analysis of the population of the Wigner-Seitz cell of each perfect lattice atom position for a few time steps in each simulation. Wigner-Seitz cells with more than one atom were interpreted as interstitials and empty cells as vacancies. The numbers of defects obtained for any given time in the Wigner-Seitz and structure-factor methods were in good agreement with each other. C. Gold For the MD simulations of gold we employed the gold embedded-atom method ~EAM! potential used previously at this laboratory in studies of collision cascades.7,20 The uni￾versal repulsive potential18 has been fitted to the potential to realistically describe strong collisions. The interstitials introduced into the cell were given the ~100! fcc dumbbell interstitial structure, which is the lowest￾energy interstitial of fcc metals.19 This interstitial is known to migrate very easily. Experimentally no determination of the migration energy in gold has been achieved.19 The EAM potential predicts a migration energy of 0.06 eV,20 which is realistic for high-temperature migration where quantum ef￾fects are no longer important. In the automatic interstitial recognition procedure, we evaluated the structure factor Pst for the 12 nearest neighbors of each atom. We found that different values of Pst could be used to distinguish between crystalline atoms, interstitials, and liquid atoms. By comparing the Pst analysis to visual inspection of moving defects, we ensured that the analysis did not lose track of moving interstitials and fine-tuned the recognition criteria to recognize interstitials even in interme￾diate configurations during migration. We performed four simulations of 5 keV recoil events in which our interstitial analysis method was used: a reference run in an undamaged gold crystal, two simulations with 50 interstitials and 50 vacancies, and one with only 50 intersti￾tials. Figure 1 shows the initial and final distributions of vacancies and interstitials as a function of the distance from the cell center, summed over the three cascade events. D. Silicon For realistic MD simulation of covalently bonded materi￾als like silicon, it is important to use an interatomic potential which adequately describes the bonding interactions in the material. We used the Tersoff (C) three-body interatomic potential which gives a good description of several proper￾ties of Si, including the different bonding types and elastic moduli.27,28 The potential also gives a reasonable description of point defect energies, although the order of the point de￾fect energies is not correct.28,29 Since we are primarily inter￾ested in the possibility of defect motion, the migration ener￾gies of point defects are more significant in any case than their equilibrium structure. Experimental and density￾functional results suggest the vacancy migration energy is ;0.3 eV.30,31 Recent tight-binding molecular-dynamics re￾sults give a migration energy of 1.4 eV for the dumbbell interstitial32 ~however, in real Si the interstitial migration may be enhanced by defect charge state processes30!. For the Tersoff potential, many different values for the interstitial and vacancy migration energies have been reported in the literature, probably due to the difficulty of recognizing the lowest-energy migration path.28,33,34 For the vacancy, the lowest migration activation energy calculated for an actual migration path is 1.6 eV,33 and for the tetrahedral interstitial 1.1 eV.34 Thus, it appears that the Tersoff potential clearly overestimates the vacancy migration energy, but gives a fairly reasonable value for interstitials. Frequently, the high melting point predicted by the Ter￾soff potential is used as an argument against its suitability for collision cascade studies. However, using a new reliable ap￾proach to calculate the melting point35 we obtained a value of 23006100 K, which is much less than the conventionally cited value of 3000 K ~Refs. 27 and 28! and in better agree￾FIG. 1. Distribution of interstitials ~int.! and vacancies ~vac.! as a function of the distance from the center of the simulation cell before and after the cascade event. The numbers are the sum over the three cascade events discussed in the text. The low number of defects is the reason for the ‘‘roughness’’ of the distributions. 56 POINT DEFECT MOVEMENT AND ANNEALING IN . . . 2423