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50 G. Antczak, G. Ehrlich / Surface Science Reports 62 (2007) 39–61 Fig. 30. Temperature dependence of length of correlated jumps in diffusion of adatom on Lennard-Jones fcc(100) surface [107]. Fig. 32. Adatom trajectories on bcc(110) plane modelled with Price potential [42,43] during 200 ps at ∼0.4T m [41]. (a) Single jumps. (b) Double jumps. (c) Complicated trajectories. 3. Long and rebound jumps 3.1. Theoretical work Fig. 31. Adatom trajectories on Lennard-Jones (100) surface at T = 0.34T m , revealing non-nearest neighbour transitions [11]. The study by Evangelakis and Papanicolaou [82] has been the most detailed work on multiple exchange events, but these processes were rediscovered several years later by Xiao et al. [105,106]. Using EAM potentials they found multiple atom exchange events for strained Cu(100) and Pt(100), which were designated as crowdions. Although as yet there are no experiments to demonstrate such large scale exchange processes, there is little doubt that they will be found at elevated temperatures. However, detailed theoretical studies of such complicated processes will have to wait until procedures for investigating simple exchanges become reliable. One thing is certain: much more extended cell sizes will be needed for such calculations. From the experimental point of view, STM should be the most suitable tool to uncover such transitions. The view that in diffusion over a surface, atoms jump at random between nearest-neighbor sites, remained widespread until the end of the seventies. At that time a number of simulations appeared which suggested a more complicated picture of atomic events. Tully et al. [107] in 1979 carried out ghost particle simulations for a (100) surface of a Lennard- Jones crystal at different temperatures below the melting point T m . They discovered that, as shown in Fig. 30, the average jump length more than doubled as the temperature increased from 0.2T m to 0.6T m . More extensive molecular dynamics were carried out at much the same time by DeLorenzi et al. [10,11], again on an fcc Lennard-Jones crystal. Most interesting was the fact that on the (100) plane at ∼0.3T m long jumps were observed, between sites as much as three to four spacings apart. This is clear from the atom trajectories in Fig. 31. Corrections to diffusion rates predicted by transition-state theory for Rh(100) stemming from transitions to sites farther away than a nearest-neighbour distance were done by Voter and Doll [108], again with Lennard-Jones interactions. Only in the vicinity of 1000 K were jumps to other than nearestneighbour positions found. In the same year, 1985, DeLorenzi and Jacucci [41] published molecular dynamics simulation on various bcc surfaces modelled with a metallic potential developed by Price [42,43] for sodium. On the most densely packed surface, the (110), at a temperature of 0.4T m , they found not only jumps between nearest-neighbour sites, as in Fig. 32(a), but also double and more elaborate transitions, shown in Fig. 32(b) and (c). Evangelakis and Papanicolaou [82] also saw that double jumps started to be active on the (100) plane of copper, an fcc metal, at temperatures above 750 K. Above 800 K, more complicated processes, such as quadruple exchange began playing a role as well.
G. Antczak, G. Ehrlich / Surface Science Reports 62 (2007) 39–61 51 the length of atomic jumps can be obtained, as the probability that after a time t an atom will be at a displacement x from the origin was given by Wrigley et al. [114] as Fig. 33. Schematic of single, double and triple transitions in one-dimensional atom motion. In 1996, Ferrando [104] looked at self-diffusion on Ag(110) and found that single jumps represented ∼90% of the total, the rest were double or more complicated transitions. Investigations of long jumps were done by Montalenti and Ferrando [32] who looked at self-diffusion on the (110) planes of gold, silver and copper. Although the activation energy for movement on gold and silver is almost the same, and for copper slightly lower, their behaviour with respect to long jumps differed greatly. At 450 K, long jumps were absent on gold, there were 3% long jumps on silver, and 6% on copper. For copper, the fraction of long jumps increased to 15% at 600 K, but never got to this value on the Ag(110) surface. Ferrón et al. [113] saw long jumps as well as rebound jumps in self-diffusion on Cu(111). They claimed that at 500 K, 95% of jumps were correlated; at 100 K this decreased to 50%. What is quite clear from these simulations is that at elevated temperatures, larger than 0.2T m , long jumps should participate significantly in surface diffusion. The question still remained, however, how to detect the transitions in an experiment. 3.2. Experimental studies Information about the characteristics of surface diffusion has usually been obtained from measurements of the meansquare displacement. However, these measurements provide no simple connection to the types of transitions carried out by atoms diffusing over the surface. Usually an increase in the diffusivity is expected due to long transitions, since the diffusivity in Eq. (4) depends on the jump length squared, but so far this expectation has not been realized in experiments. More information is definitely needed. From measurements of the distribution of displacements in one dimension insight into p x (t) = exp[−2(α + β + γ )t] ∞∑ ∞∑ × I k (2γ t) k=−∞ j=−∞ I j (2βt)I x−2 j−3k (2αt). (8) Here it is assumed that single jumps at the rate α, double jumps at the rate β, and triple jumps at the rate γ take place on the surface, as is illustrated in Fig. 33; I m (u) is the modified Bessel function of the first kind, of order m and argument u. The jump rates clearly affect the probability of finding a set of atom displacements, and that is more immediately evident from the plots in Fig. 34, where single and double transitions participate in diffusion. All that needs to be done is to carry out an adequate number of observations of atomic displacements, and from these deduce the probability p x (t). To derive the individual jump rates, Eq. (8) is not all that useful, however, as it holds for diffusion on an infinite plane. For experiments in the field ion microscope the individual planes are quite small, and may extend over only ∼20 spacings, so that Monte Carlo simulations must be employed to extract rates. The first attempt to derive jump rates from the measured distribution of displacements was made in 1989 by Wang et al. [115] in one-dimensional diffusion on W(211). Tested were Re, Mo, Ir, and Rh atoms. As shown for rhenium in Fig. 35, the best fit to the distribution measured at 300 K was obtained assuming only single jumps. Even though long jumps were not found, for the first time the picture of diffusion as occurring by random jumps between nearest-neighbor sites had been demonstrated. Much the same also held for the other atoms studied, although for iridium and rhodium there were negligibly small contributions from double jumps. The next step was taken by Senft [116–118], who surmised that energy transfer between the moving adatom and the lattice would affect the participation of long jumps in surface diffusion. She therefore elected to study palladium as well as nickel, with low values of the diffusion barriers amounting to 0.314±0.006 eV for Pd and 0.46±0.06 eV for Ni, which imply rather poor energy transfer. The displacement distribution was tested for self-diffusion of tungsten on W(211) at 307 K, and as Fig. 34. Effect of double jumps on the distribution of atomic displacements in diffusion with a mean-square displacement of two.
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