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Lattice gas model for O on Ni(110)
Surface Science 0 North-Holland
83 (1979) L335-L338 Publishing Company
SURFACE SCIENCE LETTERS LATTICE GAS MODEL FOR 0 ON Ni( 110)
Peter H. KLEBAN Department Received
of Physics and Astronom_v, University of Maine, Orono, Maine 04469,
2 January
1979; manuscript
received
in final form 26 January
USA
1979
We consider the unusual behavior of the chemisorption system 0 on Ni( 110). At a coverage of about l/3, as the temperature is increased from room temperature to about 3OO”C, a wellordered (3 X 1) structure changes reversibly into a poorly organized (2 X 1) structure. This has been observed for a Ni crystal saturated with oxygen in the interior. We propose a two-dimensional lattice gas model with anisotropic and competing interactions between the adsorbed atoms to explain this behavior. Monte Carlo calculations on a 30 X 30 lattice show a transition of this type for coverages near l/3.
Our purpose in this work is to show that an unusual feature of the behavior [l] of the chemisorption system 0 on Ni(ll0) may be explained by means of a twodimensional lattice gas model. We are concerned here with the Low Energy Electron Diffraction (LEED) features observed at adatom coverage 0 = l/3. A wellordered (3 X 1) structure forms at room temperature and changes to a poorly organized (2 X 1) structure at about 300°C. This transition is reversible and the observed LEED patterns are stable. These facts imply [1] that the adatoms are in thermal equilibrium and there is no significant diffusion into or out of the substrate. It should also be mentioned that the interior of the Ni crystal used was saturated with oxygen [2]. We are not aware of any evidence for a reconstruction of the clean (110) face of Ni in this temperature range. Hence, it is reasonable to assume that one can describe this behavior using equilibrium statistical mechanics at fixed coverage. Some of the other assumptions involved in using a lattice gas model are reviewed by Domany et al. [3]. This type of model can predict the LEED pattern (or equivalently, the pair correlation function) as a function of temperature T and 0 for a given set of adatom-adatom interactions. Thus one can gain information about the signs and strengths of these important interactions by comparing theory and experiment. It is worth noting that this approach is entirely independent of the difficult question of where the adatoms are located relative to the substrate. It requires only that they be in registry with it. An interesting one-dimensional lattice gas model for this system has been examined by Carroll [4]. This has the advantage of allowing an exact solution. However, the thermodynamic behavior of lattice systems is strongly dimension dependent. L335
P.H. Rfeban /Lattice
L336
gas model for 0 on Ni(Il0)
For instance, diffraction maxima are limited in one dimensional lattices since long range order cannot occur for T > 0.Thus while one might have a well defined structure at low temperatures, in general one would not expect a transition to a second structure at higher temperature. This is apparent here on comparison of Carroll’s results with those reported below. In the former case, the ratio of LEED intensity appropriate to the (3 X 1) structure to that for the (2 X 1) structure is about 4.5/ 3.5 at 300 K and about 3.5/4.5 at 573 K (see fig. 1 of ref. [4]). In our results, the ratio of LEED intensity for the (l/3, 0) reciprocal lattice beam to that for the (l/2, 0) beam is 9 at T/E = 0.2 and l/29 at T/e = 0.4 (see fig. 1). Thus the ordered structures are better differentiated in a two-dimensions model, as is in fact suggested in ref. [4]. This makes for better agreement with experiment. Another problem with working in one dimension is that the complete behavior of this system certainly depends on the adatom-adatom interactions in both directions (along the surface and there is no way to parameterize the perpendicular interactions in a onedimensional model. Therefore, to make further progress it is necessary to consider the two-dimensional case. The available experimental data is not sufficient to determine the interactions uniquely. Hence we assume a simple model for them. It is clear from the observed diffraction patterns that the interaction in the [OOl] direction must be predominantly attractive. We parameterize it by a nearest nei~bor attraction E. To recover the (2 X 1) structure we assume a very strong nearest neighbor repulsion in the [ 1 iO] direction. This was taken to be infinite in the calculation reported below, On the real surface it must be finite, but this discrepancy will cause no problems as long as the temperature remains sufficiently low. To stabilize the (3 X 1) structure at low T, we assume a weak third neighbor attraction e3 in the [ Xi01 direction. The second neighbor interaction was taken to be zero. A second neighbor interaction e2
030
LEED intensity
Fig. 1. LED intensity for the (l/2, 0) (circles) and (l/3, 0) (crosses) reciprocal lattice beams as a function of T/E for 0 = 0.372. The vertical scale is normalized so that the maximum possible value is o2 = 0.138. The points at T/e = 0.2 were averaged over 9 initial configurations, those at 0.25 and 0.3 over 6, and the rest over 3. Larger values of the (l/3,0) intensity (up to 0.067) were observed at T/e = 0.2 and smaller values of 8.
P.H. Kleban /Lattice gas model for 0 on Ni(I IO)
I.337
of either sign should not alter our results as long as le2 I& leg I. In the calculation described below we took e3 = e/10. A nearest neighbor pair in the [ 1TO] direction is the smallest adatom separation allowed in this system and a strong repulsion is consistent with a “direct” interaction between the 0 atoms [5,6]. The magnitude and anisotropy of the other interactions we use are consistent with what is known about indirect electronic interactions on this surface [5,6]. The one-dimensional model examined by Carroll [4] could be generalized by using repulsive second and third neighbor [l iO] interactions, but we have not done this. One should also see a (3 X 1) structure at low T and 0 near l/3 in this case. Whether the further neighbor interactions are predominantly repulsive or attractive could be determined by detailed LEED studies at low 19and T. If they are attractive, (3 X 1) islands will form. In the experiment of ref. [l], at room temperature and starting with a clean Ni(ll0) surface, exposure to oxygen led to streaks connecting the (h, k) and (h t 1, k) reciprocal lattice beams. With further exposure these spots coalesced into the (3 X 1) pattern observed at 8 = l/3. There was apparently [2] some evidence of blobs forming around the (3 X 1) spot positions for 6’ < l/3. This might indicate the presence of (3 X 1) islands, which would imply an attractive further neighbor interaction. For 19near l/3, as the temperature is lowered, one will have long chains of adatoms in the [OOl] direction (perpendicular to the “troughs” on the Ni(ll0) surface). For T ,< e3, the system will show strong (3 X 1) ordering. As the temperature is raised, the rows will begin to “melt”. One can imagine a lattice gas of rows and isolated adatoms being formed. If 0, T and the row lengths are right, a weak (2 X 1) ordering can be formed by the gas pressure of the isolated adatoms packing the rows together into ordered (2 X 1) regions. Note that a fixed 8, one will have relatively more isolated adatoms (or short rows) as T increases. In order to verify this idea, we have performed Monte Carlo calculations using the interactions described above. These were done on a 30 X 30 lattice, a size that approximates the typical coherence length of LEED equipment [7]. The FORTRAN code was very similar to that described by Landau [8]. We used at least three runs of 900 Monte Carlo steps/site for each data point. The initial configurations were chosen randomly. Our results are shown in fig. 1, where we have plotted the (2 X 1) and (3 X 1) LEED spot intensities as a function of T for e = 0.372. These intensities were calculated for the (l/2, 0) and (l/3, 0) reciprocal lattice beams assuming single scattering only. Similar behavior was observed for a range of coverages 0.36 5 0 5 0.38. Note that the (2 X 1) LEED spot never attained more than about 20% of its maximum value (0’). At T/E= 0.5 we also calculated the LEED intensities for the (0.533, 0) and (0.5, 0.033) reciprocal lattice beams. The former was about half as large as the (l/2, 0) intensity and the latter about l/4 as large. These features are consistent with the shape of the rather diffuse (2 X 1) spots observed experimentally [ 11.
L338
P.H. Kleban / Lattice gas model for 0 on Ni(1 IO)
The lattice gas model proposed here must be modified to explain the experimental results for 8 > l/2. First the infinite nearest neighbor repulsion in the [ 1 iO] direction must be replaced by a large but infinite value e2 to allow coverages greater than l/2 at all. Next one must deal with the well-known fact that in general lattice gases are symmetric about 0 = l/2 (this corresponds to the up-down symmetry of the Ising model). The observed behavior [l] of 0 on Ni(ll0) does not exhibit this symmetry. Carroll has suggested [4] that this lack of symmetry may be due to a surface reconstruction at higher coverages. Another possibility is the influence of non-pairwise three adatom (or higher order odd) forces, often called “trio” interactions. These are believed to account for the corresponding lack of symmetry in the chemisorption system 0 on W(110) [9]. There are general theoretical arguments for their existence on transition metal surfaces [S,lO]. In the case of 0 on W(110) the inclusion of trio interactions does not change the low coverage behavior qualitatively. In future work we will compute the LEED pattern in greater detail and gain more information about the adatom interactions. This will require considerably more computation than the results reported here, which does not seem justified in the absence of experimental data for comparison. A new experimental study of this chemisorption system has been initiated by Professor William N. Unertl. It is a pleasure to acknowledge useful and stimulating Unertl, J. Zollweg, D. Jasnow, R.B. Griffiths, S. Ostlund, Comins.
discussions with W.N. T.L. Einstein, and N.
References [l] [2] (31 [4] [S] [6] [7]
L.H. Germer, J.W. May and R.J. Szostak, Surface Sci. 7 (1967) 430. E. Domany, M. Shick, J.S. Walker and R.B. Griffrths, Phys. Rev. B18 (1978) 2209. J.W. May, private communication. C.E. Carroll, Surface Sci. 32 (1972) 119. T.L. Einstein, CRC Crit. Rev. Solid State Mater. Sci. 7 (1978) 261. T.L. Einstein, private communication. There are other situations in which this criterion may be insufficient, such as in computing LEED intensities near phase transitions where the adlayer coherence length may exceed the LEED coherence length (L.D. Roelofs, R.L. Park and T.L. Einstein, J. Vacuum Sci. Technol., submitted); but this is probably not a significant effect here. [8] D.P. Landau, Phys. Rev. 13 (1976) 2997. [9] W.Y. Ching, D.L. Huber, M.G. Lagally and G.-C. Wang, Surface Sci. 77 (1978) 550. [lo] T.L. Einstein, preprint.
83 (1979) L335-L338 Publishing Company
SURFACE SCIENCE LETTERS LATTICE GAS MODEL FOR 0 ON Ni( 110)
Peter H. KLEBAN Department Received
of Physics and Astronom_v, University of Maine, Orono, Maine 04469,
2 January
1979; manuscript
received
in final form 26 January
USA
1979
We consider the unusual behavior of the chemisorption system 0 on Ni( 110). At a coverage of about l/3, as the temperature is increased from room temperature to about 3OO”C, a wellordered (3 X 1) structure changes reversibly into a poorly organized (2 X 1) structure. This has been observed for a Ni crystal saturated with oxygen in the interior. We propose a two-dimensional lattice gas model with anisotropic and competing interactions between the adsorbed atoms to explain this behavior. Monte Carlo calculations on a 30 X 30 lattice show a transition of this type for coverages near l/3.
Our purpose in this work is to show that an unusual feature of the behavior [l] of the chemisorption system 0 on Ni(ll0) may be explained by means of a twodimensional lattice gas model. We are concerned here with the Low Energy Electron Diffraction (LEED) features observed at adatom coverage 0 = l/3. A wellordered (3 X 1) structure forms at room temperature and changes to a poorly organized (2 X 1) structure at about 300°C. This transition is reversible and the observed LEED patterns are stable. These facts imply [1] that the adatoms are in thermal equilibrium and there is no significant diffusion into or out of the substrate. It should also be mentioned that the interior of the Ni crystal used was saturated with oxygen [2]. We are not aware of any evidence for a reconstruction of the clean (110) face of Ni in this temperature range. Hence, it is reasonable to assume that one can describe this behavior using equilibrium statistical mechanics at fixed coverage. Some of the other assumptions involved in using a lattice gas model are reviewed by Domany et al. [3]. This type of model can predict the LEED pattern (or equivalently, the pair correlation function) as a function of temperature T and 0 for a given set of adatom-adatom interactions. Thus one can gain information about the signs and strengths of these important interactions by comparing theory and experiment. It is worth noting that this approach is entirely independent of the difficult question of where the adatoms are located relative to the substrate. It requires only that they be in registry with it. An interesting one-dimensional lattice gas model for this system has been examined by Carroll [4]. This has the advantage of allowing an exact solution. However, the thermodynamic behavior of lattice systems is strongly dimension dependent. L335
P.H. Rfeban /Lattice
L336
gas model for 0 on Ni(Il0)
For instance, diffraction maxima are limited in one dimensional lattices since long range order cannot occur for T > 0.Thus while one might have a well defined structure at low temperatures, in general one would not expect a transition to a second structure at higher temperature. This is apparent here on comparison of Carroll’s results with those reported below. In the former case, the ratio of LEED intensity appropriate to the (3 X 1) structure to that for the (2 X 1) structure is about 4.5/ 3.5 at 300 K and about 3.5/4.5 at 573 K (see fig. 1 of ref. [4]). In our results, the ratio of LEED intensity for the (l/3, 0) reciprocal lattice beam to that for the (l/2, 0) beam is 9 at T/E = 0.2 and l/29 at T/e = 0.4 (see fig. 1). Thus the ordered structures are better differentiated in a two-dimensions model, as is in fact suggested in ref. [4]. This makes for better agreement with experiment. Another problem with working in one dimension is that the complete behavior of this system certainly depends on the adatom-adatom interactions in both directions (along the surface and there is no way to parameterize the perpendicular interactions in a onedimensional model. Therefore, to make further progress it is necessary to consider the two-dimensional case. The available experimental data is not sufficient to determine the interactions uniquely. Hence we assume a simple model for them. It is clear from the observed diffraction patterns that the interaction in the [OOl] direction must be predominantly attractive. We parameterize it by a nearest nei~bor attraction E. To recover the (2 X 1) structure we assume a very strong nearest neighbor repulsion in the [ 1 iO] direction. This was taken to be infinite in the calculation reported below, On the real surface it must be finite, but this discrepancy will cause no problems as long as the temperature remains sufficiently low. To stabilize the (3 X 1) structure at low T, we assume a weak third neighbor attraction e3 in the [ Xi01 direction. The second neighbor interaction was taken to be zero. A second neighbor interaction e2
030
LEED intensity
Fig. 1. LED intensity for the (l/2, 0) (circles) and (l/3, 0) (crosses) reciprocal lattice beams as a function of T/E for 0 = 0.372. The vertical scale is normalized so that the maximum possible value is o2 = 0.138. The points at T/e = 0.2 were averaged over 9 initial configurations, those at 0.25 and 0.3 over 6, and the rest over 3. Larger values of the (l/3,0) intensity (up to 0.067) were observed at T/e = 0.2 and smaller values of 8.
P.H. Kleban /Lattice gas model for 0 on Ni(I IO)
I.337
of either sign should not alter our results as long as le2 I& leg I. In the calculation described below we took e3 = e/10. A nearest neighbor pair in the [ 1TO] direction is the smallest adatom separation allowed in this system and a strong repulsion is consistent with a “direct” interaction between the 0 atoms [5,6]. The magnitude and anisotropy of the other interactions we use are consistent with what is known about indirect electronic interactions on this surface [5,6]. The one-dimensional model examined by Carroll [4] could be generalized by using repulsive second and third neighbor [l iO] interactions, but we have not done this. One should also see a (3 X 1) structure at low T and 0 near l/3 in this case. Whether the further neighbor interactions are predominantly repulsive or attractive could be determined by detailed LEED studies at low 19and T. If they are attractive, (3 X 1) islands will form. In the experiment of ref. [l], at room temperature and starting with a clean Ni(ll0) surface, exposure to oxygen led to streaks connecting the (h, k) and (h t 1, k) reciprocal lattice beams. With further exposure these spots coalesced into the (3 X 1) pattern observed at 8 = l/3. There was apparently [2] some evidence of blobs forming around the (3 X 1) spot positions for 6’ < l/3. This might indicate the presence of (3 X 1) islands, which would imply an attractive further neighbor interaction. For 19near l/3, as the temperature is lowered, one will have long chains of adatoms in the [OOl] direction (perpendicular to the “troughs” on the Ni(ll0) surface). For T ,< e3, the system will show strong (3 X 1) ordering. As the temperature is raised, the rows will begin to “melt”. One can imagine a lattice gas of rows and isolated adatoms being formed. If 0, T and the row lengths are right, a weak (2 X 1) ordering can be formed by the gas pressure of the isolated adatoms packing the rows together into ordered (2 X 1) regions. Note that a fixed 8, one will have relatively more isolated adatoms (or short rows) as T increases. In order to verify this idea, we have performed Monte Carlo calculations using the interactions described above. These were done on a 30 X 30 lattice, a size that approximates the typical coherence length of LEED equipment [7]. The FORTRAN code was very similar to that described by Landau [8]. We used at least three runs of 900 Monte Carlo steps/site for each data point. The initial configurations were chosen randomly. Our results are shown in fig. 1, where we have plotted the (2 X 1) and (3 X 1) LEED spot intensities as a function of T for e = 0.372. These intensities were calculated for the (l/2, 0) and (l/3, 0) reciprocal lattice beams assuming single scattering only. Similar behavior was observed for a range of coverages 0.36 5 0 5 0.38. Note that the (2 X 1) LEED spot never attained more than about 20% of its maximum value (0’). At T/E= 0.5 we also calculated the LEED intensities for the (0.533, 0) and (0.5, 0.033) reciprocal lattice beams. The former was about half as large as the (l/2, 0) intensity and the latter about l/4 as large. These features are consistent with the shape of the rather diffuse (2 X 1) spots observed experimentally [ 11.
L338
P.H. Kleban / Lattice gas model for 0 on Ni(1 IO)
The lattice gas model proposed here must be modified to explain the experimental results for 8 > l/2. First the infinite nearest neighbor repulsion in the [ 1 iO] direction must be replaced by a large but infinite value e2 to allow coverages greater than l/2 at all. Next one must deal with the well-known fact that in general lattice gases are symmetric about 0 = l/2 (this corresponds to the up-down symmetry of the Ising model). The observed behavior [l] of 0 on Ni(ll0) does not exhibit this symmetry. Carroll has suggested [4] that this lack of symmetry may be due to a surface reconstruction at higher coverages. Another possibility is the influence of non-pairwise three adatom (or higher order odd) forces, often called “trio” interactions. These are believed to account for the corresponding lack of symmetry in the chemisorption system 0 on W(110) [9]. There are general theoretical arguments for their existence on transition metal surfaces [S,lO]. In the case of 0 on W(110) the inclusion of trio interactions does not change the low coverage behavior qualitatively. In future work we will compute the LEED pattern in greater detail and gain more information about the adatom interactions. This will require considerably more computation than the results reported here, which does not seem justified in the absence of experimental data for comparison. A new experimental study of this chemisorption system has been initiated by Professor William N. Unertl. It is a pleasure to acknowledge useful and stimulating Unertl, J. Zollweg, D. Jasnow, R.B. Griffiths, S. Ostlund, Comins.
discussions with W.N. T.L. Einstein, and N.
References [l] [2] (31 [4] [S] [6] [7]
L.H. Germer, J.W. May and R.J. Szostak, Surface Sci. 7 (1967) 430. E. Domany, M. Shick, J.S. Walker and R.B. Griffrths, Phys. Rev. B18 (1978) 2209. J.W. May, private communication. C.E. Carroll, Surface Sci. 32 (1972) 119. T.L. Einstein, CRC Crit. Rev. Solid State Mater. Sci. 7 (1978) 261. T.L. Einstein, private communication. There are other situations in which this criterion may be insufficient, such as in computing LEED intensities near phase transitions where the adlayer coherence length may exceed the LEED coherence length (L.D. Roelofs, R.L. Park and T.L. Einstein, J. Vacuum Sci. Technol., submitted); but this is probably not a significant effect here. [8] D.P. Landau, Phys. Rev. 13 (1976) 2997. [9] W.Y. Ching, D.L. Huber, M.G. Lagally and G.-C. Wang, Surface Sci. 77 (1978) 550. [lo] T.L. Einstein, preprint.