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Monte Carlo study of diffusion on stepped surfaces
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Surface
ELSEVIER
Science 306 (1994) 419-426
Monte Carlo study of diffusion on stepped surfaces I. Step attraction or repulsion only C. Uebing * Institut fiir FestMrperforschung,
Forschungszentrum Jiilich, GmbH, 52425 Jiilich, Germany
R. Gomer Department of Chemistry and James Franck Institute, The University of Chicago, Chicago, IL 60637, USA (Received
2 August
1993; accepted
for publication
15 November
1993)
Abstract Diffusion of adsorbates on a square lattice containing steps every fourth row was investigated by Monte Carlo simulations. For the case where steps are attractive, i.e. step sites bind more tightly than terrace sites, there is very little diffusion anisotropy at low coverage, since steps act as traps about equally for diffusion parallel and perpendicular to steps. As coverage increases steps become highly populated and then act as barriers for diffusion across them so that anisotropy increases, particularly as temperature is lowered. Diffusion parallel to steps then becomes temperature independent and diffusion perpendicular to steps remains activated. For diffusion coefficients obtained by the fluctuation autocorrelation method the inability to leave terraces at high .!-Iand low T introduces artifacts for measurements perpendicular to steps. For repulsive steps, constituting diffusion barriers but not serving as adsorption sites anisotropy is pronounced at all temperatures; diffusion parallel to steps is temperature independent, and diffusion perpendicular to steps is activated at all coverages.
1. Introduction
Steps and kinks play an important role in many surface phenomena from crystal growth [ll through chemisorption [21 through diffusion of adsorbed atoms or molecules. Steps are necessarily present to varying degrees on all macroscopic surfaces as well as on microscopic ones such as field or ion microscope tips. Recently some new techniques for measuring surface diffusion coeffi-
* Corresponding
author.
0039-6028/94/$07.00 0 1994 Elsevier SSDI 0039-6028(93)E0977-9
Science
cients of adsorbates on macroscopic surfaces have been developed [3] and it is obviously of interest to have some understanding of the possible role of steps and to a lesser extent kinks (which constitute a much smaller number of sites in most cases> in diffusion. Some experimental studies have been carried out for metal diffusion on macroscopic stepped metallic and semiconductor surfaces [4]. Some of the studies discussed in Ref. [4] suggest that diffusion along step edges may be more rapid in some cases than on flat terraces. Detailed atomistic experiments have been carried out by field ion microscopy and indicate that for
B.V. All rights reserved
420
C. Ue&ng, R. Gamer /&face
metal atom diffusion at least the p~euomena occurring at step edges can be very complex. For instance Wang and Ehrlich [S] find that diffusion in the immediate vicinity of the “down side” of a step, i.e. on a terrace toward an ascending step can be faster than on a terrace far from a step. These authors also found that diffusion descending a step seems to involve in some cases exchange with a step edge atom, rather than direct diffusion over the step edge 161. Field ion microscopy also indicates in many instances reflection of atoms on a terrace at its edge suggesting an extra barrier at the step edge, an effect also postulated by Schwoebel [I], Work from this laboratory by the field emission ffuctuation method indicates that some steps constitute very Iittie by way of diffusion barriers, while others are virtually insurmountable 17-Q]. Thus H and W atoms seem to climb and descend (111) oriented closepacked tungsten step edges with ease, while (100) oriented edges are insurmountable for W atoms and also affect H diffusion in complex ways. There has also been considerably theoretical work on the energetics of adsorption at steps [lo-131 and in some cases on diffusion itself. The idea of inco~oration of Cu atoms at descending steps on Cu{lll) has been supported by recent calculations by Stoltze and N@rskov f12f using the embedded atom method. The latter has also been used by numerous other workers. For instance Liu and Adams [13] calculate formation energies of ledges and kinks on various Ni planes as well as activation energies of diffusion on steps (upper edges) and ledges, reaching the conclusion that Ni self-diffusion along ledges can in some cases occur more easily than on terraces. Single atom diffusion in the zero coverage limit has recently been studied analytically on stepped surfaces by Natori and Godby 141 for a potential energy profile assuming a tighter well at the bottom of a step edge, an extra activation barrier at the top edge (i.e. for descending the step, and lower diffusion barriers for motion along the step ledges than on terraces (lower edges)). Adsorption, but not diffusion on stepped surfaces of square lattices for various adsorbate-step and adsorbate-adsorbate interactions via Monte Carlo simuiations was carried out by Albano et
Science 306 (19941419-426
al. [141 with emphasis on wetting and phase transitions. The aim of this paper is to examine via Monte Carlo simulations an extremely simple case in which only attraction or repulsion of adsorbed atoms by steps (in effect ledges) is considered, but in which the effect of coverage is included. In a second paper we also examine the effect of adsorbate-adsorbate repulsion. Clearly these simple models do not consider many of the effects just alluded to but for chemisorbed foreign atoms and molecules like H, 0, or CQ on metal surfaces probably incorporate the most important effects, with the possible exception of significantly changed diffusion rates along a ledge or step.
2. Method Simulations were carried out on a square lattice of 64 X 64 sites with periodic boundary conditions. Steps were introduced by adding an attractive adsorption energy .& to every site in every 4th row #I. This amounts to increasing the activation energy of jumps from step sites by an amount JsteP for the case of attractive steps. Most of the work presented here was carried out for step attraction, but simulations were also performed for repulsion by step sites by adding an extra activation energy E,,,, to jumps in rows adjacent to step rows for jumps directed toward steps. Such jumps did not terminate in step sites, however, but would start say, in the third row and end in the fifth, and so on, the fourth row being a step row. As in our previous simulations [16-IQ] calculations were carried out in terms of Do, the chemical diffusion coefficient for zero interactions between adsorbates with each other or with steps, so that all calculated energies are differences between those for the interacting and noninter-
‘* The notation J for j~tera~t~~n energies is not universal but has been used by us previously [l&19] and we retain this notation.
C. Uebing, R. Gomer / Swface Science 306 (1994) 419-426
acting systems. The algorithms and notations used have been described in detail previously [16] and will not be repeated here. The tracer diffusion coefficient defined by D* =
; i
((Ar)2> 1
was separated into its components parallel and perpendicular to steps by calculating the averages of (AxI and (Ayj2 separately, i.e.
The perpendicular and parallel (relative to step directions) components of D can also be determined by the fluctuation autocorrelation method [21]. For this purpose a probe 2a X 2b = 4 X 32 sites is placed on the surface and the autocorrelation function measured. It was shown previously [21] that for a probe oriented along the principal axes of the diffusion tensor the autocorrelation function f,(t) decomposes into a product of onedimensional functions
fn =fxwJfy(f/~y)~
Pb) This was also done when calculating the jump diffusion and chemical diffusion coefficients Dj and D respectively by means of the Kubo-Green equation [3] which can be shown to reduce to [201
(34 (3b) and
(N) D. Dll= ((SN)2) “I’
(4a)
(N) Dj,.
(4b)
D,=
<(SN)2>
Here N is the number of adatoms and ((SN)2) the mean square fluctuation in the particle number for a sample of area A in which the mean number is (N). ((SN>2) can be determined in principle in the grand canonical ensemble directly but is more conveniently found from isotherms (again in the grand canonical ensemble) and the relation [3]
(5) where p is chemical potential and 0 (absolute or relative) coverage.
421
(6)
where x stands (for instance) for the direction perpendicular and y for that parallel to steps. 7x and T,, are given respectively by rx=a2/Dl
Va)
T,, = b2/D,,
(7b)
if the narrow dimension (2~) is parallel to steps; if the probe is rotated 90” the D components in Eqs. (7a) and (7b) must be interchanged. Thus for short times the decay of fy(f/Ty)is much slower than that of f&/T,), unless D I -=xD,, and the observed decay measures D I . Simulations for both probe orientations were carried out and compared with the theoretical one-dimensional curves, given previously [31. Average values of D by the fluctuation method were also obtained by using a square 16 X 16 site probe and the two-dimensional form of f,(t/T) which is obtained from Eq. (6) with T~= ry. All simulations were carried out on Cray XMP and YMP-M94 super computers at Jiilich. Equilibration procedures and other details were the same as in our previous simulations [16]. Average values of D* and Dj were also calculated as the arithmetic means of perpendicular and parallel components.
3. Step population It will be useful to consider first the population of steps est as function of overall coverage 8 and reduced inverse temperature, Jstep/kBT. In the case of no ad-ad interactions this can be done analytically by equating chemical potentials
C. Uebing, R. Gamer /Surface
422
Science 306 (1994) 419-426
at very low total coverage, where step saturation leads nearly to terrace depletion.
4. D* and OS
o
I
2
3
4
5
6
Fig. 1. Step coverage tJ,,,, (dashed lines) and terrace coverage elelraCe(dotted lines) versus Jstep/knT for different total coverages etota,= 0.2, 0.5 and 0.8 as indicated. The caiculations are based on Eqs. (9) and (10) for attractive step interactions Jstep > 0.
Fig. 2 shows both components of D*/D’ and their average versus Jstep/kBT for 0 = 0.2, 0.5 and 0.8 respectively. At low coverage there is very little diffusion anisotropy, except at low temperature. Anisotropy goes in the intuitively expected direction: It is slightly easier to diffuse parallel to rather than across steps. The reason for the small anisotropy can be understood from Fig. 1 which shows step populations versus J_/kBT. For low total coverage Bstepis also fairly low except at low T so that steps trap diffusing adatoms when they happen to hit steps and thus affect diffusion equally parallel and perpendicular to steps, since randomly moving adatoms encounter steps repeatedly. The latter thus affect mean square displacements parallel and perpendicular to steps about equally. At low T step populations in-
for step and terrace sites and remembering that the total number of adparticles is conserved: P,tep =
BO -
Jstep + k BTIn-1 -“e,
pt = ~~ + k,T
@t
In1 - 8, )
’
(8b)
or -J,,,,/k,T=
1-
ln&
t
4,
es, ’
(9)
and e=@,+fe,,,
(10)
where Bt is the terrace coverage. Combination of Eqs. (9) and (10) then yields 6,,. It was verified that this gave the same result as direct simulation in the canonical ensemble. estep and 8, versus Jstep/kBT for total coverages 0 = 0.2, 0.5 and 0.8 are shown in Fig. 1. As expected e,,,, approaches maximum allowed values at low T in all cases, but does so at highest T for highest 8. By conservation Bt decreases accordingly, but the effect is smaller since there are more terrace sites, except
8
0.1
8
Et;
0.01 0123456 Jste&oT
Fig. 2. Normalized tracer diffusion coefficients,
D* versus
Js,,,/kBT for step density :. (0) D*/D"; (+) Dt /Do; (X) DIf/Do. D* is average, DT diffusion perpendicular, D; parallel to steps. Do is the chemical diffusion coefficient
in the absence of steps.
C. Uebing, R. Gomer /Surface
423
Science 306 (1994) 419-426
5. Chemical diffusion coefficients
o.oL 0123456
JSteplkBT Fig. 3. Normalized tracer diffusion coefficients versus JStep/kBT for step density $ and Otata, = 0.8: (0) D*/D’; (+) 0: /D’;(x) D,T/D’.
crease, so that diffusion parallel to steps is enhanced, since full step sites lead essentially to reflection, with parallel diffusion now possible within terraces without crossing steps. Perpendicular diffusion still requires crossing of steps, however, and is therefore decreased. As 8 increases steps become nearly saturated at low T and stay so to increasingly higher T, the higher the coverage. Consequently diffusion parallel to steps is now largely confined to within terraces and hence virtually temperature independent (relative to Do>. At high 8 and low T diffusion perpendicular to steps is also confined to terraces and thus apparently temperature independent with ((Ax)*> maximally the square of the terrace width. As T increases, however, activated diffusion across steps occurs, mean square displacements increase, and 0: increases with T. This explanation is confirmed by some runs for step densities of f rather than b. Results for 8 = 0.8 are shown in Fig. 3 and indicate that there is again a temperature independent regime for 0; but with higher values. “Temperature independence” here means constant ratios of D/Do. The behavior and numerical values of Dj are very similar to those for D*, which is not surprising, since both represent somewhat differently averaged single particle microscopic random walk records.
Fig. 4 shows the chemical diffusion coefficient for 0 = 0.2 and 0.5 obtained from fluctuation measurements. The curves for 0 = 0.8 are very similar to those for 8 = 0.5. The label D, is used to distinguish D from that obtained from Dj and (N>/((SN)*>, i.e. from Eq. (41, which is designated D,,. In order to compare D, and D,, ((SAQ2)/(N) is needed. Values of this quantity for various coverages as a function of Jstep/kBT and as functions of 0 for two temperatures are shown in Figs. 5 and 6 respectively. Fig. 5 indicates that ((SN2>/(N> is only slightly temperature dependent at all coverages, and least so at 0 = 0.8, since at this coverage steps are already largely saturated even at high T. Fig. 6 indicates that the 8 dependence at high T is nearly that of a Langmuir gas, i.e. 1 - 8, while at low temperature this is only true at high coverages, for the reason just mentioned. The main point is that deviations from 1 - 0 are relatively small, so that D KG is close to (1 - O)-‘Dj. We are now in a position to compare D,, and
Js~&LIT Fig. 4. Normalized chemical diffusion coefficients obtained from fluctuation measurements D, versus Jster, /k,T for step density 4. (01 D,/D’; (+) DFI /Do; (XI D,,,/D’. DF obtained on square probe, DFL and D,,, via slit probe.
C. Uebing, R. Gomer /Surface
424
Science 306 (1994) 419-426
O.lU 0123456
2 3E
JSteplkBT Fig. 5. Mean square JSleP /k,T for different
fluctuations ((SN)*>/(N) versus total coverages etota, as indicated.
D,, as function of Jstep/kBT. This is shown for 0 = 0.8 in Fig. 7. At lower coverages, not shown, there is good agreement, particularly for D,, over the entire temperature range. At 8 = 0.8 this is still true for D,, but DFI falls below D,, I for Jstep/bT a 2.5. D,,, corresponds to fluctuations into and out of the probe in a direction parallel to steps and thus is not affected much by steps when these are largely saturated, so that agreement with D,,,, is to be expected. It should be
1 0.8 0.6 0.4 0.2 n “0
0.2
0.4
0.6
Coverage
0.8
1
f3
Fig. 6. Mean square fluctuations ((&N)‘)/(N) versus total and (0) Js,,,/kBT coverage etota, for (0) Jstep/knT=3.44 = 1.60. The dotted line represents the high temperature limit (Langmuir gas) ((SN)‘>/(N> = 1- etota,.
0.6.
4/D”
0
1
2
3
4
JSteplkBT Fig. 7. Comparison of D, /Do (thin symbols, dotted line) and D,o/D’ (thick symbols, solid line) for step density $. (0) D, /Do> D,,/D”; (+) &,/Do, D,,./D’; (x) D,,,/D’, D,,,,/D’. 0 = 0.8.
emphasized that ((SN>2>/(N) is a thermodynamic quantity and not affected by the fact that diffusion may be difficult in a particular direction. The situation is different for D I . Mean square displacements within a terrace can contribute to Dj and hence D,, I but not to D, since the probe covers a terrace and a step. Consequently at low T the (rare) activated jumps across a step dominate D, I . This could be confirmed by changing the probe location or making the step density f, keeping the probe in the center of a terrace. For the latter case the values of D,, are an order of magnitude higher. For a step density of f but with the probe containing a . . . single step m its middle, DFL also rose dramatically. Thus at low T the results for D i depend critically on the method of measurement and the probe location.
6. Activation energies Fig. 8 shows activation energies E * for tracer diffusion (relative to the non-interacting case) at high T (0) and at J_,/k,T = 5 (X , + 1. At high = 0.4-0.3, decreasing slightly with inl- WJse,
C. Uebing, R. Gomer /Surface Science 306 (1994) 419-426
creasing 8. The fact that E < Jstep for high T shows that steps do not control diffusion completely since many atoms are “free”. At low T E = Jstep at low coverage since almost all adatoms are now stuck at steps which therefore exert a much bigger effect. At low T and 8 2 0.4 E,, =0 since parallel diffusion is now confined to terraces and involves neither crossing of or trapping by steps. Perpendicular diffusion still involves some step crossing; this decreases with increasing coverage. Therefore E I > 0 but decreases with increasing 8, as ((Ax)~) is confined more and more to within terraces. (This also affects the value of II,*/@ for perpendicular diffusion.) For instance for 0 = 0.8. (O~/D$I = 0.04 at low temperature, because of the limitation of ((Ax)~)). Results for Ej are virtually identical to those for E *.
7. Step repulsion We turn finally to the case of step repulsion. Fig. 9 shows D* and II, respectively as function of IJ,t,p I/k,Tfor 0 = 0.5. Dj is very close to D* and not shown. As expected D,, is temperature independent and has values of 1 - 8 = 0.5 for tracer and jump diffusion, and 1 for D,,,. Diffusion perpendicular to steps is activated in all cases. At very low T a tailing off can be seen, for
425
8
b
Fig. 9. Normalized diffusion coefficients D*/D” and D, /Do for the case of step repulsion versus IJ,,,, I/k,T for step density $ and etota, = 0.5. (0) average; (+) perpendicular; (x) parallel to steps.
the same reasons involved for step attraction when steps become saturated.
8. Conclusion
The work presented here has shown that steps can affect diffusion anisotropy. In the case of step attraction this effect makes itself felt only at high coverage and becomes more pronounced the lower the temperature. For repulsive steps anisotropy increases with decreasing temperature but is coverage independent, since the model precludes actual adsorption at step sites. -0.24 0
: 0.2
0.4
0.6
0.8
4 1
9. Acknowledgments
Coverage 0 Fig. 8. Normalized activation energies for tracer diffusion versus total coverage 0tota,. (0) E, I w* = I?* - EO)/.&, = EAll for J,,,,/kaT+ 0; (+) EAI for Jstep/kBT= 5; and (x1 I?,+,,for J,,,,/kaT= 5.
This work was supported in part by NSF Grant CHE9222051. We have also benefited from the Materials Research Laboratory of the NSF at the University of Chicago.
426
C. Uebing, R. Gomer / Surface Science 306 (1994) 419-426
10. References [ll See for instance R.L. Schwoebel, J. Appl. Phys. 40 (1969) 614. 121M.A. Henderson, A. Szabo and J.T. Yates, Jr., J. Chem. Phys. 91 (1989) 7245, and references therein. [31 See for instance R. Gomer, Rep. Prog. Phys. 53 (1990) 917. [41 A. Natori and R.W. Godby, Phys. Rev. B 47 (1993) 15816, and references therein. [51 SC. Wang and G. Ehrlich, Phys. Rev. Lett. 70 (1993) 41. [61 SC. Wang and G. Ehrlich, Phys. Rev. Lett. 67 (1991) 2509. [71 D.-S. Choi, SK. Kim and R. Gomer, Surf. Sci. 234 (1990) 262. 181 D.-S. Choi, C. Uebing and R. Gomer, Surf. Sci. 259 (1991) 139. [91 C. Uebing and R. Gomer, Surf. Sci. 259 (1991) 151. mJ1 S.P. Chen, A.F. Voter and D.J. Srolovitz, Phys. Rev. Lett. 57 (1986) 1308. [ill J.S. Nelson and P.J. Feibelman, Phys. Rev. Lett. 68 (1992) 2188.
[12] P. Stoltze and J.K. Norskov, Phys. Rev. B 48 (1993) 5607. [13] C.-L. Liu and J.B. Adams, Surf. Sci. 265 (1992) 262. [14] E.V. Albano, K. Binder, D.W. Heermann and W. Paul, Surf. Sci. 223 (1989) 151. [15] C. Uebing and R. Gomer, Surf. Sci. 306 (1994) 427. [16] C. Uebing and R. Gomer, J. Chem. Phys. 95 (1991) 7626. [17] C. Uebing and R. Gomer, J. Chem. Phys. 95 (1991) 7636. [18] C. Uebing and R. Gomer, J. Chem. Phys. 95 (1991) 7641. [19] C. Uebing and R. Gomer, J. Chem. Phys. 95 (1991) 7648. [20] In Ref. [6] a typographical error occurred in Eq. (2): D was written as
instead of using the correct form of the bracket namely
. Calculations used the correct form, however. [21] D.R. Bowman, R. Gomer, K. Muttalib and M. Tringides, Surf. Sci. 138 (19841 581.
cka
.......
:x+:.:.:..:.:0>:.:
. . . . .. :.: .,......
:.:.:.:.L
.:,:,
“““:‘::“‘. I.$:::.:.:.: . . .............i. ....i................. x::::::$:::;.:~$$ ... ‘.
<.*:p ..,:,.. Q:z: .,...
‘surface .....
....
“..::: :.:.:,:: :j:,:,, :.,:”
;:,:,:.::,,,
. . . . . . . . . . . ../....~.
i:;:i;z;,
science
..,.:.:.: ‘.:.:.:.::::::~~::::~.~:~:::i ““” ““.‘.‘.‘. “‘...‘......‘.‘. ...:,: ...,:_,,,_,,,_,, .z. ,,,,., ,,(,., ,,,, ..,,,, (,, ..............:,: ‘;‘:‘:‘“‘~:.6...... 3;,:,:.~.C,:,;,:,~,;,, “~“““~“““‘...n.,.,....,... .....i. .........v..... .._ “~“~‘+.~.~.~. ,/,._ S.,,:..:.::::::~.::::~,~:::::::::: :.:, x.:.:,.:...‘, :::.
Surface
ELSEVIER
Science 306 (1994) 419-426
Monte Carlo study of diffusion on stepped surfaces I. Step attraction or repulsion only C. Uebing * Institut fiir FestMrperforschung,
Forschungszentrum Jiilich, GmbH, 52425 Jiilich, Germany
R. Gomer Department of Chemistry and James Franck Institute, The University of Chicago, Chicago, IL 60637, USA (Received
2 August
1993; accepted
for publication
15 November
1993)
Abstract Diffusion of adsorbates on a square lattice containing steps every fourth row was investigated by Monte Carlo simulations. For the case where steps are attractive, i.e. step sites bind more tightly than terrace sites, there is very little diffusion anisotropy at low coverage, since steps act as traps about equally for diffusion parallel and perpendicular to steps. As coverage increases steps become highly populated and then act as barriers for diffusion across them so that anisotropy increases, particularly as temperature is lowered. Diffusion parallel to steps then becomes temperature independent and diffusion perpendicular to steps remains activated. For diffusion coefficients obtained by the fluctuation autocorrelation method the inability to leave terraces at high .!-Iand low T introduces artifacts for measurements perpendicular to steps. For repulsive steps, constituting diffusion barriers but not serving as adsorption sites anisotropy is pronounced at all temperatures; diffusion parallel to steps is temperature independent, and diffusion perpendicular to steps is activated at all coverages.
1. Introduction
Steps and kinks play an important role in many surface phenomena from crystal growth [ll through chemisorption [21 through diffusion of adsorbed atoms or molecules. Steps are necessarily present to varying degrees on all macroscopic surfaces as well as on microscopic ones such as field or ion microscope tips. Recently some new techniques for measuring surface diffusion coeffi-
* Corresponding
author.
0039-6028/94/$07.00 0 1994 Elsevier SSDI 0039-6028(93)E0977-9
Science
cients of adsorbates on macroscopic surfaces have been developed [3] and it is obviously of interest to have some understanding of the possible role of steps and to a lesser extent kinks (which constitute a much smaller number of sites in most cases> in diffusion. Some experimental studies have been carried out for metal diffusion on macroscopic stepped metallic and semiconductor surfaces [4]. Some of the studies discussed in Ref. [4] suggest that diffusion along step edges may be more rapid in some cases than on flat terraces. Detailed atomistic experiments have been carried out by field ion microscopy and indicate that for
B.V. All rights reserved
420
C. Ue&ng, R. Gamer /&face
metal atom diffusion at least the p~euomena occurring at step edges can be very complex. For instance Wang and Ehrlich [S] find that diffusion in the immediate vicinity of the “down side” of a step, i.e. on a terrace toward an ascending step can be faster than on a terrace far from a step. These authors also found that diffusion descending a step seems to involve in some cases exchange with a step edge atom, rather than direct diffusion over the step edge 161. Field ion microscopy also indicates in many instances reflection of atoms on a terrace at its edge suggesting an extra barrier at the step edge, an effect also postulated by Schwoebel [I], Work from this laboratory by the field emission ffuctuation method indicates that some steps constitute very Iittie by way of diffusion barriers, while others are virtually insurmountable 17-Q]. Thus H and W atoms seem to climb and descend (111) oriented closepacked tungsten step edges with ease, while (100) oriented edges are insurmountable for W atoms and also affect H diffusion in complex ways. There has also been considerably theoretical work on the energetics of adsorption at steps [lo-131 and in some cases on diffusion itself. The idea of inco~oration of Cu atoms at descending steps on Cu{lll) has been supported by recent calculations by Stoltze and N@rskov f12f using the embedded atom method. The latter has also been used by numerous other workers. For instance Liu and Adams [13] calculate formation energies of ledges and kinks on various Ni planes as well as activation energies of diffusion on steps (upper edges) and ledges, reaching the conclusion that Ni self-diffusion along ledges can in some cases occur more easily than on terraces. Single atom diffusion in the zero coverage limit has recently been studied analytically on stepped surfaces by Natori and Godby 141 for a potential energy profile assuming a tighter well at the bottom of a step edge, an extra activation barrier at the top edge (i.e. for descending the step, and lower diffusion barriers for motion along the step ledges than on terraces (lower edges)). Adsorption, but not diffusion on stepped surfaces of square lattices for various adsorbate-step and adsorbate-adsorbate interactions via Monte Carlo simuiations was carried out by Albano et
Science 306 (19941419-426
al. [141 with emphasis on wetting and phase transitions. The aim of this paper is to examine via Monte Carlo simulations an extremely simple case in which only attraction or repulsion of adsorbed atoms by steps (in effect ledges) is considered, but in which the effect of coverage is included. In a second paper we also examine the effect of adsorbate-adsorbate repulsion. Clearly these simple models do not consider many of the effects just alluded to but for chemisorbed foreign atoms and molecules like H, 0, or CQ on metal surfaces probably incorporate the most important effects, with the possible exception of significantly changed diffusion rates along a ledge or step.
2. Method Simulations were carried out on a square lattice of 64 X 64 sites with periodic boundary conditions. Steps were introduced by adding an attractive adsorption energy .& to every site in every 4th row #I. This amounts to increasing the activation energy of jumps from step sites by an amount JsteP for the case of attractive steps. Most of the work presented here was carried out for step attraction, but simulations were also performed for repulsion by step sites by adding an extra activation energy E,,,, to jumps in rows adjacent to step rows for jumps directed toward steps. Such jumps did not terminate in step sites, however, but would start say, in the third row and end in the fifth, and so on, the fourth row being a step row. As in our previous simulations [16-IQ] calculations were carried out in terms of Do, the chemical diffusion coefficient for zero interactions between adsorbates with each other or with steps, so that all calculated energies are differences between those for the interacting and noninter-
‘* The notation J for j~tera~t~~n energies is not universal but has been used by us previously [l&19] and we retain this notation.
C. Uebing, R. Gomer / Swface Science 306 (1994) 419-426
acting systems. The algorithms and notations used have been described in detail previously [16] and will not be repeated here. The tracer diffusion coefficient defined by D* =
; i
((Ar)2> 1
was separated into its components parallel and perpendicular to steps by calculating the averages of (AxI and (Ayj2 separately, i.e.
The perpendicular and parallel (relative to step directions) components of D can also be determined by the fluctuation autocorrelation method [21]. For this purpose a probe 2a X 2b = 4 X 32 sites is placed on the surface and the autocorrelation function measured. It was shown previously [21] that for a probe oriented along the principal axes of the diffusion tensor the autocorrelation function f,(t) decomposes into a product of onedimensional functions
fn =fxwJfy(f/~y)~
Pb) This was also done when calculating the jump diffusion and chemical diffusion coefficients Dj and D respectively by means of the Kubo-Green equation [3] which can be shown to reduce to [201
(34 (3b) and
(N) D. Dll= ((SN)2) “I’
(4a)
(N) Dj,.
(4b)
D,=
<(SN)2>
Here N is the number of adatoms and ((SN)2) the mean square fluctuation in the particle number for a sample of area A in which the mean number is (N). ((SN>2) can be determined in principle in the grand canonical ensemble directly but is more conveniently found from isotherms (again in the grand canonical ensemble) and the relation [3]
(5) where p is chemical potential and 0 (absolute or relative) coverage.
421
(6)
where x stands (for instance) for the direction perpendicular and y for that parallel to steps. 7x and T,, are given respectively by rx=a2/Dl
Va)
T,, = b2/D,,
(7b)
if the narrow dimension (2~) is parallel to steps; if the probe is rotated 90” the D components in Eqs. (7a) and (7b) must be interchanged. Thus for short times the decay of fy(f/Ty)is much slower than that of f&/T,), unless D I -=xD,, and the observed decay measures D I . Simulations for both probe orientations were carried out and compared with the theoretical one-dimensional curves, given previously [31. Average values of D by the fluctuation method were also obtained by using a square 16 X 16 site probe and the two-dimensional form of f,(t/T) which is obtained from Eq. (6) with T~= ry. All simulations were carried out on Cray XMP and YMP-M94 super computers at Jiilich. Equilibration procedures and other details were the same as in our previous simulations [16]. Average values of D* and Dj were also calculated as the arithmetic means of perpendicular and parallel components.
3. Step population It will be useful to consider first the population of steps est as function of overall coverage 8 and reduced inverse temperature, Jstep/kBT. In the case of no ad-ad interactions this can be done analytically by equating chemical potentials
C. Uebing, R. Gamer /Surface
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at very low total coverage, where step saturation leads nearly to terrace depletion.
4. D* and OS
o
I
2
3
4
5
6
Fig. 1. Step coverage tJ,,,, (dashed lines) and terrace coverage elelraCe(dotted lines) versus Jstep/knT for different total coverages etota,= 0.2, 0.5 and 0.8 as indicated. The caiculations are based on Eqs. (9) and (10) for attractive step interactions Jstep > 0.
Fig. 2 shows both components of D*/D’ and their average versus Jstep/kBT for 0 = 0.2, 0.5 and 0.8 respectively. At low coverage there is very little diffusion anisotropy, except at low temperature. Anisotropy goes in the intuitively expected direction: It is slightly easier to diffuse parallel to rather than across steps. The reason for the small anisotropy can be understood from Fig. 1 which shows step populations versus J_/kBT. For low total coverage Bstepis also fairly low except at low T so that steps trap diffusing adatoms when they happen to hit steps and thus affect diffusion equally parallel and perpendicular to steps, since randomly moving adatoms encounter steps repeatedly. The latter thus affect mean square displacements parallel and perpendicular to steps about equally. At low T step populations in-
for step and terrace sites and remembering that the total number of adparticles is conserved: P,tep =
BO -
Jstep + k BTIn-1 -“e,
pt = ~~ + k,T
@t
In1 - 8, )
’
(8b)
or -J,,,,/k,T=
1-
ln&
t
4,
es, ’
(9)
and e=@,+fe,,,
(10)
where Bt is the terrace coverage. Combination of Eqs. (9) and (10) then yields 6,,. It was verified that this gave the same result as direct simulation in the canonical ensemble. estep and 8, versus Jstep/kBT for total coverages 0 = 0.2, 0.5 and 0.8 are shown in Fig. 1. As expected e,,,, approaches maximum allowed values at low T in all cases, but does so at highest T for highest 8. By conservation Bt decreases accordingly, but the effect is smaller since there are more terrace sites, except
8
0.1
8
Et;
0.01 0123456 Jste&oT
Fig. 2. Normalized tracer diffusion coefficients,
D* versus
Js,,,/kBT for step density :. (0) D*/D"; (+) Dt /Do; (X) DIf/Do. D* is average, DT diffusion perpendicular, D; parallel to steps. Do is the chemical diffusion coefficient
in the absence of steps.
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5. Chemical diffusion coefficients
o.oL 0123456
JSteplkBT Fig. 3. Normalized tracer diffusion coefficients versus JStep/kBT for step density $ and Otata, = 0.8: (0) D*/D’; (+) 0: /D’;(x) D,T/D’.
crease, so that diffusion parallel to steps is enhanced, since full step sites lead essentially to reflection, with parallel diffusion now possible within terraces without crossing steps. Perpendicular diffusion still requires crossing of steps, however, and is therefore decreased. As 8 increases steps become nearly saturated at low T and stay so to increasingly higher T, the higher the coverage. Consequently diffusion parallel to steps is now largely confined to within terraces and hence virtually temperature independent (relative to Do>. At high 8 and low T diffusion perpendicular to steps is also confined to terraces and thus apparently temperature independent with ((Ax)*> maximally the square of the terrace width. As T increases, however, activated diffusion across steps occurs, mean square displacements increase, and 0: increases with T. This explanation is confirmed by some runs for step densities of f rather than b. Results for 8 = 0.8 are shown in Fig. 3 and indicate that there is again a temperature independent regime for 0; but with higher values. “Temperature independence” here means constant ratios of D/Do. The behavior and numerical values of Dj are very similar to those for D*, which is not surprising, since both represent somewhat differently averaged single particle microscopic random walk records.
Fig. 4 shows the chemical diffusion coefficient for 0 = 0.2 and 0.5 obtained from fluctuation measurements. The curves for 0 = 0.8 are very similar to those for 8 = 0.5. The label D, is used to distinguish D from that obtained from Dj and (N>/((SN)*>, i.e. from Eq. (41, which is designated D,,. In order to compare D, and D,, ((SAQ2)/(N) is needed. Values of this quantity for various coverages as a function of Jstep/kBT and as functions of 0 for two temperatures are shown in Figs. 5 and 6 respectively. Fig. 5 indicates that ((SN2>/(N> is only slightly temperature dependent at all coverages, and least so at 0 = 0.8, since at this coverage steps are already largely saturated even at high T. Fig. 6 indicates that the 8 dependence at high T is nearly that of a Langmuir gas, i.e. 1 - 8, while at low temperature this is only true at high coverages, for the reason just mentioned. The main point is that deviations from 1 - 0 are relatively small, so that D KG is close to (1 - O)-‘Dj. We are now in a position to compare D,, and
Js~&LIT Fig. 4. Normalized chemical diffusion coefficients obtained from fluctuation measurements D, versus Jster, /k,T for step density 4. (01 D,/D’; (+) DFI /Do; (XI D,,,/D’. DF obtained on square probe, DFL and D,,, via slit probe.
C. Uebing, R. Gomer /Surface
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O.lU 0123456
2 3E
JSteplkBT Fig. 5. Mean square JSleP /k,T for different
fluctuations ((SN)*>/(N) versus total coverages etota, as indicated.
D,, as function of Jstep/kBT. This is shown for 0 = 0.8 in Fig. 7. At lower coverages, not shown, there is good agreement, particularly for D,, over the entire temperature range. At 8 = 0.8 this is still true for D,, but DFI falls below D,, I for Jstep/bT a 2.5. D,,, corresponds to fluctuations into and out of the probe in a direction parallel to steps and thus is not affected much by steps when these are largely saturated, so that agreement with D,,,, is to be expected. It should be
1 0.8 0.6 0.4 0.2 n “0
0.2
0.4
0.6
Coverage
0.8
1
f3
Fig. 6. Mean square fluctuations ((&N)‘)/(N) versus total and (0) Js,,,/kBT coverage etota, for (0) Jstep/knT=3.44 = 1.60. The dotted line represents the high temperature limit (Langmuir gas) ((SN)‘>/(N> = 1- etota,.
0.6.
4/D”
0
1
2
3
4
JSteplkBT Fig. 7. Comparison of D, /Do (thin symbols, dotted line) and D,o/D’ (thick symbols, solid line) for step density $. (0) D, /Do> D,,/D”; (+) &,/Do, D,,./D’; (x) D,,,/D’, D,,,,/D’. 0 = 0.8.
emphasized that ((SN>2>/(N) is a thermodynamic quantity and not affected by the fact that diffusion may be difficult in a particular direction. The situation is different for D I . Mean square displacements within a terrace can contribute to Dj and hence D,, I but not to D, since the probe covers a terrace and a step. Consequently at low T the (rare) activated jumps across a step dominate D, I . This could be confirmed by changing the probe location or making the step density f, keeping the probe in the center of a terrace. For the latter case the values of D,, are an order of magnitude higher. For a step density of f but with the probe containing a . . . single step m its middle, DFL also rose dramatically. Thus at low T the results for D i depend critically on the method of measurement and the probe location.
6. Activation energies Fig. 8 shows activation energies E * for tracer diffusion (relative to the non-interacting case) at high T (0) and at J_,/k,T = 5 (X , + 1. At high = 0.4-0.3, decreasing slightly with inl- WJse,
C. Uebing, R. Gomer /Surface Science 306 (1994) 419-426
creasing 8. The fact that E < Jstep for high T shows that steps do not control diffusion completely since many atoms are “free”. At low T E = Jstep at low coverage since almost all adatoms are now stuck at steps which therefore exert a much bigger effect. At low T and 8 2 0.4 E,, =0 since parallel diffusion is now confined to terraces and involves neither crossing of or trapping by steps. Perpendicular diffusion still involves some step crossing; this decreases with increasing coverage. Therefore E I > 0 but decreases with increasing 8, as ((Ax)~) is confined more and more to within terraces. (This also affects the value of II,*/@ for perpendicular diffusion.) For instance for 0 = 0.8. (O~/D$I = 0.04 at low temperature, because of the limitation of ((Ax)~)). Results for Ej are virtually identical to those for E *.
7. Step repulsion We turn finally to the case of step repulsion. Fig. 9 shows D* and II, respectively as function of IJ,t,p I/k,Tfor 0 = 0.5. Dj is very close to D* and not shown. As expected D,, is temperature independent and has values of 1 - 8 = 0.5 for tracer and jump diffusion, and 1 for D,,,. Diffusion perpendicular to steps is activated in all cases. At very low T a tailing off can be seen, for
425
8
b
Fig. 9. Normalized diffusion coefficients D*/D” and D, /Do for the case of step repulsion versus IJ,,,, I/k,T for step density $ and etota, = 0.5. (0) average; (+) perpendicular; (x) parallel to steps.
the same reasons involved for step attraction when steps become saturated.
8. Conclusion
The work presented here has shown that steps can affect diffusion anisotropy. In the case of step attraction this effect makes itself felt only at high coverage and becomes more pronounced the lower the temperature. For repulsive steps anisotropy increases with decreasing temperature but is coverage independent, since the model precludes actual adsorption at step sites. -0.24 0
: 0.2
0.4
0.6
0.8
4 1
9. Acknowledgments
Coverage 0 Fig. 8. Normalized activation energies for tracer diffusion versus total coverage 0tota,. (0) E, I w* = I?* - EO)/.&, = EAll for J,,,,/kaT+ 0; (+) EAI for Jstep/kBT= 5; and (x1 I?,+,,for J,,,,/kaT= 5.
This work was supported in part by NSF Grant CHE9222051. We have also benefited from the Materials Research Laboratory of the NSF at the University of Chicago.
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10. References [ll See for instance R.L. Schwoebel, J. Appl. Phys. 40 (1969) 614. 121M.A. Henderson, A. Szabo and J.T. Yates, Jr., J. Chem. Phys. 91 (1989) 7245, and references therein. [31 See for instance R. Gomer, Rep. Prog. Phys. 53 (1990) 917. [41 A. Natori and R.W. Godby, Phys. Rev. B 47 (1993) 15816, and references therein. [51 SC. Wang and G. Ehrlich, Phys. Rev. Lett. 70 (1993) 41. [61 SC. Wang and G. Ehrlich, Phys. Rev. Lett. 67 (1991) 2509. [71 D.-S. Choi, SK. Kim and R. Gomer, Surf. Sci. 234 (1990) 262. 181 D.-S. Choi, C. Uebing and R. Gomer, Surf. Sci. 259 (1991) 139. [91 C. Uebing and R. Gomer, Surf. Sci. 259 (1991) 151. mJ1 S.P. Chen, A.F. Voter and D.J. Srolovitz, Phys. Rev. Lett. 57 (1986) 1308. [ill J.S. Nelson and P.J. Feibelman, Phys. Rev. Lett. 68 (1992) 2188.
[12] P. Stoltze and J.K. Norskov, Phys. Rev. B 48 (1993) 5607. [13] C.-L. Liu and J.B. Adams, Surf. Sci. 265 (1992) 262. [14] E.V. Albano, K. Binder, D.W. Heermann and W. Paul, Surf. Sci. 223 (1989) 151. [15] C. Uebing and R. Gomer, Surf. Sci. 306 (1994) 427. [16] C. Uebing and R. Gomer, J. Chem. Phys. 95 (1991) 7626. [17] C. Uebing and R. Gomer, J. Chem. Phys. 95 (1991) 7636. [18] C. Uebing and R. Gomer, J. Chem. Phys. 95 (1991) 7641. [19] C. Uebing and R. Gomer, J. Chem. Phys. 95 (1991) 7648. [20] In Ref. [6] a typographical error occurred in Eq. (2): D was written as
instead of using the correct form of the bracket namely
. Calculations used the correct form, however. [21] D.R. Bowman, R. Gomer, K. Muttalib and M. Tringides, Surf. Sci. 138 (19841 581.