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Long range adatom diffusion mechanism on fcc (100) EAM modeled materials
surface science letters ELSEVIER
Surface Science Letters 306 (1994) L545-L549
Surface Science Letters
Long range adatom diffusion mechanism on fee (100) EAM modeled materials Jennifer M. Cohen * Physics department, New Mexico htitute omening and Tec~~o~, Socorro, NM 87801, USA Theoretical Division, T-12, Los Alamos National Laboratory, Los Aiamos, NM 87545, USA
(Received 15 November 1993; accepted for publication 3 January 1994)
Abstract
A long range adatom diffusion mechanism has been observed in simulations on the fee (100) surfaces of embedded atom method (EAM) modeled materials. This mechanism primarily involves the adatom and two atoms in the top substrate layer. After passing through the saddle configuration the adatom is found two rows down and one row over from its original position. The movement of the adatom is like that of a knight on a chess board. The activation barrier of this mechanism is high, and it begins to appear at temperatures about half the melting temperature of the material. Due to the long range nature of this mechanism it could significantly affect the adatom
diffusion rate as compared to typical hopping between adjacent minima or diffusion via the exchange mechanism.
In a recent publication Black and Tian examined the self-diffusion of several adatoms on a Cu(100) surface at high temperature El] modeled by the Cu EAM potential of Daw and Foiles [2]. They report observing times when the adatom was embedded in the top layer of the substrate, followed by the emergence of the adatom in a site several binding sites away from the adatom’s originaf position. This observation seems likely to be related to the knight’s move event reported by Lynden-Bell during a high temperature molecular dynamics (MD) simulation of a single Pt atom on Pt@O) [3] using a semi-empirica potential by Sutton and Chen [4].
* Corresponding author. Fax: + 1 (50.5) 83.5-2982. E-mail: jenc~ivaIdi.IanI.gov. ~39-6028/94/$07.~
We have also observed such behavior for the diffusion of a single AI adatom on the AI(100) surface modeled with an AI EAM potential IS] beginning at temperatures of about 400-500 K. Upon further examination of the crystal configuration when the adatom is apparently embedded in the surface, we find that this is due to the system passing through a single transition surface. The procedure for finding the saddle point was as follows. During an MD simulation at 500 K of AI/AI(lOO), we observed that the adatom travelled two binding sites down the fee (100) row and over one site. We began from a configuration in the interim when the adatom seemed embedded in the surface. Performing a Newton-Raphson search for stationary points, it was determined that the system was passing through a saddle configuration like that depicted in Fig. 1. By definition, a saddIe point exists when diago-
0 1994 Elsevier Science B.V. All rights reserved
SSDI 0167-2584(94)00267-C
J.M. Cohen /Surface
Science Letters 306 (1994) L545-L549
nalization of the system’s 3N X 3N dynamical matrix yields a set of normal modes having exactly one imaginary frequency. The simulation crystal typically had layers consisting of between 10 x 10 and 12 x 12 atoms with periodic boundary conditions parallel to the surface. The top four layers and the adatom were free to move. These were attached to six to eight layers of atoms restricted to the bulk configuration, which is deep enough such that an atom in the deepest moving layer does not detect the finite depth. We have previously addressed the significance of crystal size in simulations involving EAM potentials and exchange-type mechanisms [71. It is advisable to use layers that are at least 8 x 8 or 9 x 9 atoms in size for MD simulations of this type. This is demonstrated in Fig. 1, where the apparent displacement of the atoms from their equilibrium sites extends beyond that expected for a bridge site saddle. A subsection of the crystal is shown in the depiction of the overall mechanism in Fig. 2. Motion through the saddle point for the three atoms principally involved in the mechanism is illustrated in the inset. The direction of the system through the saddle point is described by the eigenvector corresponding to the imaginary frequency of the saddle. This diffusion path is consistent with our simulation results.
7
30
20 w E 10
J
0
Fig. 2. Adatom diffusion mechanism found in simulations on the fee (100) surface of several transition metals. The original adatom (shaded) and two surface atoms act together so that the adatom moves on the surface like a knight in chess. The inset depicts the movement of the three primary atoms through the saddle point.
The adatom diffusion barrier for this mechanism was computed as the energy difference between the crystal’s saddle configuration and its minimum. This barrier is much higher than that of the exchange mechanism, which is preferred at low temperatures for Al, Au, Pd, and Pt in this set of model potentials. Fig. 3 shows the calculated barriers to diffusion for the exchange mechanism, typical hopping on the surface between adjacent minima, and this new saddle configuration versus lattice constant for Al/Al(lOO). If the energy barrier were the only consideration in the escape rate we should not have observed the system passing through this saddle. Although this is found in high temperature simulations, one may gain some insight into why the new mechanism is observed by examining the harmonic approximation to the pre-exponential factor in the escape rate. The escape rate may be written as k ‘I k, e-Eaa/kBT,
Fig. 1. Projections of the new saddle configuration found in high temperature molecular dynamics runs on the fee (100) surface. The figure shows the Cu/Cu(lOO) case. This type of saddle also exists on the Al, Ag, Au, Pd, Pt, and Ni surfaces modeled using the EAh4 potentials by Voter [S] and Voter and Chen [6].
k, = n;,
(1)
where E,,, is the activation barrier, k, is Boltzman’s constant, and T is the temperature. The pre-exponential factor in Eq. (1) is given by the product of the number of directions of escape, IZ, from the minimum for the diffusion mechanism
-
J.M. Cohen /Surface
Science Letters 306 (1994) L545-L549
0.6
0.6
s
2. 0.5
5 ‘i: 2
0.4
m
2 3
9
l_.. 0.2
- 0.2
-.....
0.1 -
-.._ Exchange _.-.._.. -..-..-.--.._.._.._.._,. o,l
A2E.O
O.O 4.05
4.10
Lattice
Constant
(Ang.)
Fig. 3. Activation barrier height versus lattice constant for three adatom diffusion mechanisms in the Al/Al(lOO) model system. The exchange and new mechanism barriers decrease, while the bridge site hopping barrier increases with increasing lattice constant.
and the attempt frequency 5. We use the Vineyard ratio [8] to approximate the attempt frequency. For a system of N atoms this is the ratio of the 3N normal mode frequencies of the minimum y”rN to the 3N - 1 real frequencies of the saddle configuration ysAD. That is, z; v,MIN v=
*
3N-1 n
(2)
vi””
j=l
There are eight ways for an adatom to escape from a binding site via the new mechanism, two for each of the four corner surface atoms surrounding the adatom at a minimum. This yields a factor of two compared to the number of escape directions for either the exchange or typical hopping mechanism. However, it is the attempt frequency that governs the increase in the escape rate such that a countable number of events occur in our simulations. Indeed, the attempt frequency for the new mechanism is a factor of lo-100 times higher than that of either the ex-
change or hopping mechanism throughout the temperature range considered. Using the harmonic approximation for Ag/ Ag(lOO), we find the pre-exponential factor to be about 60-80 times higher than that of the hopping mechanism favored at low to moderate temperatures. The ratio of the pre-exponential factor for the new mechanism to that of exchange is of a similar order of magnitude. It should be noted that while the attempt frequency given by Eq. (2) exhibits some temperature dependence, that change does not affect the escape rate as much as the change in the diffusion barrier heights. In the case of Ag/Ag(lOO) the ratio of the attempt frequencies of the new to either the hopping or exchange mechanisms, and thus the exponential pre-factor for the escape rate, ranges from a low of about 60 at 50 K to a high of roughly 80 at temperatures above about 500 K. We can compare the number of events observed for the various mechanisms in our Al/ Al(100) MD run at 500 K to the numbers given by the static crystal approximation. In that case the new mechanism appears with two to three times the frequency in direct MD, with respect to the number of successful escapes via exchange, than is predicted within the harmonic approximation. Given the distances the adatom diffuses for the hopping, exchange, and new mechanisms, the influence of this new mechanism on the total adatom diffusion constant could be great. Since one expects that anharmonic terms will become important at high temperatures [9], deviations from the harmonic approximation are not surprising at the relatively high temperature of 500 K for Al. Any detailed study of this mechanism should employ methods that account for the anharmonic nature of these potentials at high temperatures. One such method is the dynamical corrections formalism for transition state theory [lo] that is valid over the extended temperature range for which the solid crystal is well defined
ml. For the EAM potential models investigated thus far, we have found this type of saddle configuration for adatom self-diffusion on Al, Ag, Au, Cu, Pd, Pt, and Ni. The rough harmonic approximations done to this point indicate that this long
J.M. Cohen /Surface
Science Letters 306 (1994) L545-LS49
range mechanism would be observed above a temperature about half that of melting except perhaps for Ni. In the Ni/Ni(lOO) system at 1000 K, the barrier height for the new mechanism is about 1.42, 0.74 eV for typical hopping, and 0.85 eV for exchange. Since the ratio between the effective frequencies of the new mechanism and typical hopping at moderate to high temperatures is about 10, we would not expect it to appear in our simulations. Direct MD simulations have not been attempted to confirm this. Nevertheless, the exchange mechanism may begin to play a role on NXlOO) at high temperatures since its activation barrier differs from standard hopping by about 0.1 eV at 1000 K, while the two attempt frequencies are similar. The EAM model potentials are more noted for their qualitative accuracy than precise activation barrier values. The Pt potential used here correctly predicts that the exchange mechanism will be favored for Pt/Pt(lOO) at field ion microscopy (FIM) temperatures [12]. Still, the predicted value of 0.64 eV is considerably higher than the FIM value of 0.47 eV [13]. Recent experimental evidence also suggests that the exchange favored by these potentials for Au/ Au(100) may in fact be the preferred mechanism in the physical system [14]. Thus, while the models indicate that the mechanism may exist in the actual physical system, the precise temperature of onset and the numerical values we have presented are expected to differ from experimental values. This mechanism, or one similar to it, may be related to the simulation results for the diffusion of Pt/Pt(115) [15l. Further calculations suggest that it may be possible to control the preference in diffusion mechanisms for some real systems by varying the substrate temperature or other experimental parameters. Finally, we offer an example of the diffusion mechanism temperature dependence found in the Ag/Ag(lOO) EAM modeled system. The harmonic approximation of adatom escape rates by hopping, exchange, and diffusion of Ag/Ag(lOO) versus temperature are shown in Fig. 4. There we find the hopping mechanism is preferred at low temperatures. At somewhat higher temperatures, diffusion by exchange is expected to rival that by
15
\ ,
10
15 104/T
0
20 (T
\,
25
in K)
Fig. 4. The harmonic approximation of escape rate versus inverse temperature for three diffusion mechanisms in the Ag/Ag(100) model system. While typical hopping (Hop) is predicted for low temperatures, the exchange (Ex.) may be favored at high T. The new mechanism (New) is expected to become a factor in the diffusion rate at temperatures just lower than the crossover point between the hopping and exchange mechanisms.
hopping. Increasing the temperature of our EAM model system still further, we would anticipate diffusion via the new mechanism to play an important role and diffusion by hopping to become less significant.
1. References 111J.E. Black and Zeng-Ju Tian, Phys. Rev. Lett. 71 (1993) 2445.
121This type of potential is described in: SM. Foiles, M.I. Baskes and M.S. Daw, Phys. Rev. B 33 (1986) 7983. [31R.M. Lynden-Bell, Surf. Sci. 259 (1991) 129. [41A.P. Sutton and J. Chen, Philos. Mag. Lett. 61 (1990)
139. [51A.F. Voter,
in: Modeling of Optical Thin Films, Ed. M.R. Jacobson, SPIE 821 (1987) p. 214; A.F. Voter, Los Alamos National Laboratory Report, LAUR-93-3901. 161A.F. Voter and S.P. Chen, in: Mater. Res. Sot. Symp. Proc. 82 (1987) 175.
J.M. Cohen /Surface
Science Letters 306 (1994) L545-L549
[7] J.M. Cohen and A.F. Voter, Surf. Sci., submitted. [8] G.H. Vineyard, J. Phys. Chem. Solids 3 (1957) 121. [9] A recent study of anharmonic effects on diffusion rates for bridge site hopping and the exchange mechanism for Cu/Cu(lOO) is found in: L.B. Hansen, P. Stolze, K.W. Jacobson and J.K. Norskov, Surf. Sci. 289 (1993) 68. [lo] A.F. Voter and J.D. Doll, J. Chem. Phys. 82 (1985) 80. [ll] A.F. Voter, J.D. Doll and J.M. Cohen, J. Chem. Phys. 90 (1989) 2045; A.F. Voter, Phys. Rev. Lett. 63 (1989) 167.
[12] C.L. Liu, J.M. Cohen, J.B. Adams and A.F. Voter, Surf. Sci. 253 (1991) 334. [13] G.L. Kellogg and P.J. Feibelman, Phys. Rev. Lett. 64 (1990) 3143; G.L. Kellogg, Surf. Sci. 246 (1991) 31. [14] X. Gao, G.J. Edens, A. Hamelin and M.J. Weaver, Surf. Sci. 296 (1993) 333. [15] K.D. Hammonds and R. Lynden-Bell, Surf. Sci. 278 (1992) 437.
Surface Science Letters 306 (1994) L545-L549
Surface Science Letters
Long range adatom diffusion mechanism on fee (100) EAM modeled materials Jennifer M. Cohen * Physics department, New Mexico htitute omening and Tec~~o~, Socorro, NM 87801, USA Theoretical Division, T-12, Los Alamos National Laboratory, Los Aiamos, NM 87545, USA
(Received 15 November 1993; accepted for publication 3 January 1994)
Abstract
A long range adatom diffusion mechanism has been observed in simulations on the fee (100) surfaces of embedded atom method (EAM) modeled materials. This mechanism primarily involves the adatom and two atoms in the top substrate layer. After passing through the saddle configuration the adatom is found two rows down and one row over from its original position. The movement of the adatom is like that of a knight on a chess board. The activation barrier of this mechanism is high, and it begins to appear at temperatures about half the melting temperature of the material. Due to the long range nature of this mechanism it could significantly affect the adatom
diffusion rate as compared to typical hopping between adjacent minima or diffusion via the exchange mechanism.
In a recent publication Black and Tian examined the self-diffusion of several adatoms on a Cu(100) surface at high temperature El] modeled by the Cu EAM potential of Daw and Foiles [2]. They report observing times when the adatom was embedded in the top layer of the substrate, followed by the emergence of the adatom in a site several binding sites away from the adatom’s originaf position. This observation seems likely to be related to the knight’s move event reported by Lynden-Bell during a high temperature molecular dynamics (MD) simulation of a single Pt atom on Pt@O) [3] using a semi-empirica potential by Sutton and Chen [4].
* Corresponding author. Fax: + 1 (50.5) 83.5-2982. E-mail: jenc~ivaIdi.IanI.gov. ~39-6028/94/$07.~
We have also observed such behavior for the diffusion of a single AI adatom on the AI(100) surface modeled with an AI EAM potential IS] beginning at temperatures of about 400-500 K. Upon further examination of the crystal configuration when the adatom is apparently embedded in the surface, we find that this is due to the system passing through a single transition surface. The procedure for finding the saddle point was as follows. During an MD simulation at 500 K of AI/AI(lOO), we observed that the adatom travelled two binding sites down the fee (100) row and over one site. We began from a configuration in the interim when the adatom seemed embedded in the surface. Performing a Newton-Raphson search for stationary points, it was determined that the system was passing through a saddle configuration like that depicted in Fig. 1. By definition, a saddIe point exists when diago-
0 1994 Elsevier Science B.V. All rights reserved
SSDI 0167-2584(94)00267-C
J.M. Cohen /Surface
Science Letters 306 (1994) L545-L549
nalization of the system’s 3N X 3N dynamical matrix yields a set of normal modes having exactly one imaginary frequency. The simulation crystal typically had layers consisting of between 10 x 10 and 12 x 12 atoms with periodic boundary conditions parallel to the surface. The top four layers and the adatom were free to move. These were attached to six to eight layers of atoms restricted to the bulk configuration, which is deep enough such that an atom in the deepest moving layer does not detect the finite depth. We have previously addressed the significance of crystal size in simulations involving EAM potentials and exchange-type mechanisms [71. It is advisable to use layers that are at least 8 x 8 or 9 x 9 atoms in size for MD simulations of this type. This is demonstrated in Fig. 1, where the apparent displacement of the atoms from their equilibrium sites extends beyond that expected for a bridge site saddle. A subsection of the crystal is shown in the depiction of the overall mechanism in Fig. 2. Motion through the saddle point for the three atoms principally involved in the mechanism is illustrated in the inset. The direction of the system through the saddle point is described by the eigenvector corresponding to the imaginary frequency of the saddle. This diffusion path is consistent with our simulation results.
7
30
20 w E 10
J
0
Fig. 2. Adatom diffusion mechanism found in simulations on the fee (100) surface of several transition metals. The original adatom (shaded) and two surface atoms act together so that the adatom moves on the surface like a knight in chess. The inset depicts the movement of the three primary atoms through the saddle point.
The adatom diffusion barrier for this mechanism was computed as the energy difference between the crystal’s saddle configuration and its minimum. This barrier is much higher than that of the exchange mechanism, which is preferred at low temperatures for Al, Au, Pd, and Pt in this set of model potentials. Fig. 3 shows the calculated barriers to diffusion for the exchange mechanism, typical hopping on the surface between adjacent minima, and this new saddle configuration versus lattice constant for Al/Al(lOO). If the energy barrier were the only consideration in the escape rate we should not have observed the system passing through this saddle. Although this is found in high temperature simulations, one may gain some insight into why the new mechanism is observed by examining the harmonic approximation to the pre-exponential factor in the escape rate. The escape rate may be written as k ‘I k, e-Eaa/kBT,
Fig. 1. Projections of the new saddle configuration found in high temperature molecular dynamics runs on the fee (100) surface. The figure shows the Cu/Cu(lOO) case. This type of saddle also exists on the Al, Ag, Au, Pd, Pt, and Ni surfaces modeled using the EAh4 potentials by Voter [S] and Voter and Chen [6].
k, = n;,
(1)
where E,,, is the activation barrier, k, is Boltzman’s constant, and T is the temperature. The pre-exponential factor in Eq. (1) is given by the product of the number of directions of escape, IZ, from the minimum for the diffusion mechanism
-
J.M. Cohen /Surface
Science Letters 306 (1994) L545-L549
0.6
0.6
s
2. 0.5
5 ‘i: 2
0.4
m
2 3
9
l_.. 0.2
- 0.2
-.....
0.1 -
-.._ Exchange _.-.._.. -..-..-.--.._.._.._.._,. o,l
A2E.O
O.O 4.05
4.10
Lattice
Constant
(Ang.)
Fig. 3. Activation barrier height versus lattice constant for three adatom diffusion mechanisms in the Al/Al(lOO) model system. The exchange and new mechanism barriers decrease, while the bridge site hopping barrier increases with increasing lattice constant.
and the attempt frequency 5. We use the Vineyard ratio [8] to approximate the attempt frequency. For a system of N atoms this is the ratio of the 3N normal mode frequencies of the minimum y”rN to the 3N - 1 real frequencies of the saddle configuration ysAD. That is, z; v,MIN v=
*
3N-1 n
(2)
vi””
j=l
There are eight ways for an adatom to escape from a binding site via the new mechanism, two for each of the four corner surface atoms surrounding the adatom at a minimum. This yields a factor of two compared to the number of escape directions for either the exchange or typical hopping mechanism. However, it is the attempt frequency that governs the increase in the escape rate such that a countable number of events occur in our simulations. Indeed, the attempt frequency for the new mechanism is a factor of lo-100 times higher than that of either the ex-
change or hopping mechanism throughout the temperature range considered. Using the harmonic approximation for Ag/ Ag(lOO), we find the pre-exponential factor to be about 60-80 times higher than that of the hopping mechanism favored at low to moderate temperatures. The ratio of the pre-exponential factor for the new mechanism to that of exchange is of a similar order of magnitude. It should be noted that while the attempt frequency given by Eq. (2) exhibits some temperature dependence, that change does not affect the escape rate as much as the change in the diffusion barrier heights. In the case of Ag/Ag(lOO) the ratio of the attempt frequencies of the new to either the hopping or exchange mechanisms, and thus the exponential pre-factor for the escape rate, ranges from a low of about 60 at 50 K to a high of roughly 80 at temperatures above about 500 K. We can compare the number of events observed for the various mechanisms in our Al/ Al(100) MD run at 500 K to the numbers given by the static crystal approximation. In that case the new mechanism appears with two to three times the frequency in direct MD, with respect to the number of successful escapes via exchange, than is predicted within the harmonic approximation. Given the distances the adatom diffuses for the hopping, exchange, and new mechanisms, the influence of this new mechanism on the total adatom diffusion constant could be great. Since one expects that anharmonic terms will become important at high temperatures [9], deviations from the harmonic approximation are not surprising at the relatively high temperature of 500 K for Al. Any detailed study of this mechanism should employ methods that account for the anharmonic nature of these potentials at high temperatures. One such method is the dynamical corrections formalism for transition state theory [lo] that is valid over the extended temperature range for which the solid crystal is well defined
ml. For the EAM potential models investigated thus far, we have found this type of saddle configuration for adatom self-diffusion on Al, Ag, Au, Cu, Pd, Pt, and Ni. The rough harmonic approximations done to this point indicate that this long
J.M. Cohen /Surface
Science Letters 306 (1994) L545-LS49
range mechanism would be observed above a temperature about half that of melting except perhaps for Ni. In the Ni/Ni(lOO) system at 1000 K, the barrier height for the new mechanism is about 1.42, 0.74 eV for typical hopping, and 0.85 eV for exchange. Since the ratio between the effective frequencies of the new mechanism and typical hopping at moderate to high temperatures is about 10, we would not expect it to appear in our simulations. Direct MD simulations have not been attempted to confirm this. Nevertheless, the exchange mechanism may begin to play a role on NXlOO) at high temperatures since its activation barrier differs from standard hopping by about 0.1 eV at 1000 K, while the two attempt frequencies are similar. The EAM model potentials are more noted for their qualitative accuracy than precise activation barrier values. The Pt potential used here correctly predicts that the exchange mechanism will be favored for Pt/Pt(lOO) at field ion microscopy (FIM) temperatures [12]. Still, the predicted value of 0.64 eV is considerably higher than the FIM value of 0.47 eV [13]. Recent experimental evidence also suggests that the exchange favored by these potentials for Au/ Au(100) may in fact be the preferred mechanism in the physical system [14]. Thus, while the models indicate that the mechanism may exist in the actual physical system, the precise temperature of onset and the numerical values we have presented are expected to differ from experimental values. This mechanism, or one similar to it, may be related to the simulation results for the diffusion of Pt/Pt(115) [15l. Further calculations suggest that it may be possible to control the preference in diffusion mechanisms for some real systems by varying the substrate temperature or other experimental parameters. Finally, we offer an example of the diffusion mechanism temperature dependence found in the Ag/Ag(lOO) EAM modeled system. The harmonic approximation of adatom escape rates by hopping, exchange, and diffusion of Ag/Ag(lOO) versus temperature are shown in Fig. 4. There we find the hopping mechanism is preferred at low temperatures. At somewhat higher temperatures, diffusion by exchange is expected to rival that by
15
\ ,
10
15 104/T
0
20 (T
\,
25
in K)
Fig. 4. The harmonic approximation of escape rate versus inverse temperature for three diffusion mechanisms in the Ag/Ag(100) model system. While typical hopping (Hop) is predicted for low temperatures, the exchange (Ex.) may be favored at high T. The new mechanism (New) is expected to become a factor in the diffusion rate at temperatures just lower than the crossover point between the hopping and exchange mechanisms.
hopping. Increasing the temperature of our EAM model system still further, we would anticipate diffusion via the new mechanism to play an important role and diffusion by hopping to become less significant.
1. References 111J.E. Black and Zeng-Ju Tian, Phys. Rev. Lett. 71 (1993) 2445.
121This type of potential is described in: SM. Foiles, M.I. Baskes and M.S. Daw, Phys. Rev. B 33 (1986) 7983. [31R.M. Lynden-Bell, Surf. Sci. 259 (1991) 129. [41A.P. Sutton and J. Chen, Philos. Mag. Lett. 61 (1990)
139. [51A.F. Voter,
in: Modeling of Optical Thin Films, Ed. M.R. Jacobson, SPIE 821 (1987) p. 214; A.F. Voter, Los Alamos National Laboratory Report, LAUR-93-3901. 161A.F. Voter and S.P. Chen, in: Mater. Res. Sot. Symp. Proc. 82 (1987) 175.
J.M. Cohen /Surface
Science Letters 306 (1994) L545-L549
[7] J.M. Cohen and A.F. Voter, Surf. Sci., submitted. [8] G.H. Vineyard, J. Phys. Chem. Solids 3 (1957) 121. [9] A recent study of anharmonic effects on diffusion rates for bridge site hopping and the exchange mechanism for Cu/Cu(lOO) is found in: L.B. Hansen, P. Stolze, K.W. Jacobson and J.K. Norskov, Surf. Sci. 289 (1993) 68. [lo] A.F. Voter and J.D. Doll, J. Chem. Phys. 82 (1985) 80. [ll] A.F. Voter, J.D. Doll and J.M. Cohen, J. Chem. Phys. 90 (1989) 2045; A.F. Voter, Phys. Rev. Lett. 63 (1989) 167.
[12] C.L. Liu, J.M. Cohen, J.B. Adams and A.F. Voter, Surf. Sci. 253 (1991) 334. [13] G.L. Kellogg and P.J. Feibelman, Phys. Rev. Lett. 64 (1990) 3143; G.L. Kellogg, Surf. Sci. 246 (1991) 31. [14] X. Gao, G.J. Edens, A. Hamelin and M.J. Weaver, Surf. Sci. 296 (1993) 333. [15] K.D. Hammonds and R. Lynden-Bell, Surf. Sci. 278 (1992) 437.