2424 K.NORDLUND AND R.S.AVERBACK 56 ment with the experimental value 1685 K.36 Details of this 10 ps (this includes all of the final interstitials)is shown. calculation will be given elsewhere.37 Most of the interstitials seen in the figure actually existed A short-range repulsive part of the potential was deter- throughout the simulation.The open markers indicate the mined from density-functional theory calculations.38 It was initial positions and the solid ones the final positions.The smoothly fitted to the Tersoff potential using a Fermi func- defects move very little during the simulation,even though tion F(r)=(1+eb)-1 with the values 12 A-1 and they clearly have been exposed to the pressure and heat 1.6 A for b and r,respectively.40 These values provide a waves from the cascade.Noteworthy is that the interstitials smooth fit between the two potentials both between two at- move slightly inwards on average. oms in a dimer and two nearby atoms in bulk silicon. These conclusions were verified by a calculation of the The point defect recognition procedure in silicon was total movement of interstitials with respect to the center of similar to the one used in gold.The Pst structure factor was mass of the liquid zone throughout the simulation.The inter- evaluated for the four nearest neighbors of each atom,and stitials which remained after the cascade had on average used to recognize both interstitials and vacancies.Liquid at- moved inwards 1.0,1.5,and 1.6 A in the three cascade oms were recognized using a combined P and kinetic en- events containing preexisting defects.There was,however,a ergy criterion. considerable spread in the distribution of the movement:The We carried out 5 simulations for silicon:one reference maximum inwards movement seen in the three events was 10 run with no initial defects,one run with 50 initial intersti- A and the maximum outward movement 14 A.On average, tials.and three with 50 initial interstitials and 50 initial va- the interstitials outside the liquid region performed about cancies. three lattice site jumps.From Fig.I we see that the overall effect of the interstitial movement on the defect distribution E.Copper,aluminum,and platinum is quite small. For copper,aluminum,and platinum we employed EAM In the events with 50 initial interstitials and vacancies. potentials4142 onto which the universal repulsive potentiall about 40-45 interstitial structures remained after the cascade had been fitted to realistically describe strong collisions2 event.A few of the interstitials produced in our simulations We carried out 3 simulations for these metals.I without formed small clusters,but the majority remained as single initial defects and 2 with 50 initial interstitials and vacancies. dumbbell interstitials.Some of the initially existing intersti- The initial defect distributions were the same as for the gold tials have recombined with vacancies during the event.Be- simulations.The analysis of defects was performed with the cause this is more likely to occur close to the cell center,the Wigner-Seitz method for about 20 selected time steps. root-mean-square distance of interstitials from the cell center may actually be larger for the final than the initial defect distribution (cf.Table D). IIL.RESULTS AND DISCUSSION Vacancies were identified geometrically using Wigner- A.Gold Seitz cells;otherwise,the vacancy analysis employed the same principles as those described above for interstitials.The The gold results will be presented in four subsections.The vacancy movement results are illustrated in Fig.4.We see first deals with defects outside the liquid core of a cascade, that the vacancies outside the liquid region do not move at all the second with defects in the core,the third with uniform defect distributions,and the last one with high-temperature or move very little.The average number of lattice site jumps per vacancy was 0.2.This is readily understood in terms of runs. their large migration energy,0. Inspection of animations of the collision cascades showed 1.Defects outside the liquid core that most of the interstitial movement occurred when the Results of one simulation with 50 initial vacancies and liquid zone was shrinking in size or had already vanished.To interstitials are illustrated in Figs.2 and 3.Figure 2 shows understand the reason for the migration,we calculated tem- snapshots of the positions of all atoms in the system pro- perature and pressure distributions in the cell,which are jected onto the xy plane at 0,1.5,and 50 ps.At 0 ps,the shown in Figs.5 and 6.High temperatures are present in the initial interstitials and lattice relaxation caused by them are cell for a relatively long period of time after the collision visible among the otherwise undisturbed fcc lattice atoms.At cascade,suggesting this might cause thermal migration of 1.5 ps,we see the liquid cascade center and the pressure the interstitials.The pressure decreases towards the center of wave emanating from it.Some of the interstitials can be the cell before and after the cascade due to the presence of discerned even through the pressure wave,showing that they vacancies near the cell center and the presence of interstitials are not destroyed by it.At 50 ps,spike effects are no longer in the region near the borders(see Fig.6).We believe this visible,and the lattice has relaxed back close to its initial contributes to the inward motion of the interstitials. state.Comparison of the 0 and 50 ps figures shows that most In the event which had 50 initial interstitials and no va- interstitials are positioned at roughly the same sites before cancies,a similar pressure gradient was seen,which indicates and after the cascade.In examining the figures,one should the interstitials are the predominant cause of it.In the cas- keep in mind that when a dumbbell interstitial is oriented cade event with no initial defects,in which only four inter- along the =axis,it will not be visible when projected on the stitials existed after the cascade,no reduction of the pressure xy plane. towards the center occurred. Our analysis of interstitial movement enabled us to relate To test whether the defect movement could be explained initial and final interstitials to each other.In Fig.3,the as a thermal migration process,we simulated the same initial movement of all interstitials which have existed for at least defect distribution as in the cascade events but with no cas-ment with the experimental value 1685 K.36 Details of this calculation will be given elsewhere.37 A short-range repulsive part of the potential was deter￾mined from density-functional theory calculations.38,39 It was smoothly fitted to the Tersoff potential using a Fermi func￾tion F(r)5(11e2b f(r2rf) )21 with the values 12 Å 21 and 1.6 Å for bf and rf , respectively.40 These values provide a smooth fit between the two potentials both between two at￾oms in a dimer and two nearby atoms in bulk silicon. The point defect recognition procedure in silicon was similar to the one used in gold. The Pst structure factor was evaluated for the four nearest neighbors of each atom, and used to recognize both interstitials and vacancies. Liquid at￾oms were recognized using a combined Pst and kinetic en￾ergy criterion. We carried out 5 simulations for silicon; one reference run with no initial defects, one run with 50 initial intersti￾tials, and three with 50 initial interstitials and 50 initial va￾cancies. E. Copper, aluminum, and platinum For copper, aluminum, and platinum we employed EAM potentials41,42 onto which the universal repulsive potential18 had been fitted to realistically describe strong collisions.26 We carried out 3 simulations for these metals, 1 without initial defects and 2 with 50 initial interstitials and vacancies. The initial defect distributions were the same as for the gold simulations. The analysis of defects was performed with the Wigner-Seitz method for about 20 selected time steps. III. RESULTS AND DISCUSSION A. Gold The gold results will be presented in four subsections. The first deals with defects outside the liquid core of a cascade, the second with defects in the core, the third with uniform defect distributions, and the last one with high-temperature runs. 1. Defects outside the liquid core Results of one simulation with 50 initial vacancies and interstitials are illustrated in Figs. 2 and 3. Figure 2 shows snapshots of the positions of all atoms in the system pro￾jected onto the xy plane at 0, 1.5, and 50 ps. At 0 ps, the initial interstitials and lattice relaxation caused by them are visible among the otherwise undisturbed fcc lattice atoms. At 1.5 ps, we see the liquid cascade center and the pressure wave emanating from it. Some of the interstitials can be discerned even through the pressure wave, showing that they are not destroyed by it. At 50 ps, spike effects are no longer visible, and the lattice has relaxed back close to its initial state. Comparison of the 0 and 50 ps figures shows that most interstitials are positioned at roughly the same sites before and after the cascade. In examining the figures, one should keep in mind that when a dumbbell interstitial is oriented along the z axis, it will not be visible when projected on the xy plane. Our analysis of interstitial movement enabled us to relate initial and final interstitials to each other. In Fig. 3, the movement of all interstitials which have existed for at least 10 ps ~this includes all of the final interstitials! is shown. Most of the interstitials seen in the figure actually existed throughout the simulation. The open markers indicate the initial positions and the solid ones the final positions. The defects move very little during the simulation, even though they clearly have been exposed to the pressure and heat waves from the cascade. Noteworthy is that the interstitials move slightly inwards on average. These conclusions were verified by a calculation of the total movement of interstitials with respect to the center of mass of the liquid zone throughout the simulation. The inter￾stitials which remained after the cascade had on average moved inwards 1.0, 1.5, and 1.6 Å in the three cascade events containing preexisting defects. There was, however, a considerable spread in the distribution of the movement: The maximum inwards movement seen in the three events was 10 Å and the maximum outward movement 14 Å. On average, the interstitials outside the liquid region performed about three lattice site jumps. From Fig. 1 we see that the overall effect of the interstitial movement on the defect distribution is quite small. In the events with 50 initial interstitials and vacancies, about 40–45 interstitial structures remained after the cascade event. A few of the interstitials produced in our simulations formed small clusters, but the majority remained as single dumbbell interstitials. Some of the initially existing intersti￾tials have recombined with vacancies during the event. Be￾cause this is more likely to occur close to the cell center, the root-mean-square distance of interstitials from the cell center may actually be larger for the final than the initial defect distribution ~cf. Table I!. Vacancies were identified geometrically using Wigner￾Seitz cells; otherwise, the vacancy analysis employed the same principles as those described above for interstitials. The vacancy movement results are illustrated in Fig. 4. We see that the vacancies outside the liquid region do not move at all or move very little. The average number of lattice site jumps per vacancy was 0.2. This is readily understood in terms of their large migration energy, 0.8 eV.19 Inspection of animations of the collision cascades showed that most of the interstitial movement occurred when the liquid zone was shrinking in size or had already vanished. To understand the reason for the migration, we calculated tem￾perature and pressure distributions in the cell, which are shown in Figs. 5 and 6. High temperatures are present in the cell for a relatively long period of time after the collision cascade, suggesting this might cause thermal migration of the interstitials. The pressure decreases towards the center of the cell before and after the cascade due to the presence of vacancies near the cell center and the presence of interstitials in the region near the borders ~see Fig. 6!. We believe this contributes to the inward motion of the interstitials. In the event which had 50 initial interstitials and no va￾cancies, a similar pressure gradient was seen, which indicates the interstitials are the predominant cause of it. In the cas￾cade event with no initial defects, in which only four inter￾stitials existed after the cascade, no reduction of the pressure towards the center occurred. To test whether the defect movement could be explained as a thermal migration process, we simulated the same initial defect distribution as in the cascade events but with no cas- 2424 K. NORDLUND AND R. S. AVERBACK 56