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Thermodynamic properties of surface steps
Journal of Crystal Growth 43(1978) 385—387 © North-Holland Publishing Company
THERMODYNAMIC PROPERTIES OF SURFACE STEPS John D. WEEKS and George H. GILMER Bell Laboratories, Murray Hill, New Jersey 07974, USA Received 6 December 1977
This letter compares mean field, pair approximation, and Monte Carlo calculations of the excess energy of a step. The pair approximation offers good improvement over the mean field theory.
Recently we derived pair approximation equations for the solid-on-solid model of the crystal—vapor interface [11, and also for the interface in a more general anisotropic Ising model [2]. The thermodynamic properties given by the pair method proved to be considerably more accurate than those of the mean field approximation. The excess energy, for example, was in excellent agreement with Monte Carlo data at all temperatures. As mentioned in refs. [1] and [2], we also calculated some properties associated with steps on the crystal surface. These results are of practical significance since steps are involved in many of the mechanisms of adsorption, catalysis, and crystal growth. Also, they are a sensitive indicator of the interface roughening transition, since all step excess quantities should vanish at the roughening temperature [3—5]. Mean field calculations of some thermodynamic properties of an isolated step on the simple cubic (100) face were reported by Swendsen [61. Orientations along the (001) and the (011) (45°) directions were considered, and Swendsen reported several anomalous properties for the 45° step at low temperatures. Unfortunately, these anomalies are caused by an incorrect solution of the mean field equations for the 45°step [7]. (His solution for the (001) step is correct.) Here we compare the mean field energy for both step orientations with the pair approximation theory and the Monte Carlo simulations of Leamy and Gilmer [31.As before, the pair approximation offers good quantitative improvement over the mean field theory. Our method is not limited to the case of the
single isolated step; properties of periodic arrays of steps and the interactions between these steps are also easily treated. Details of the method and many additional step properties (gamma plots, the distribution of kink sites, and other thermodynamic properties) are included in a forthcoming report [81. The mean field and pair equations for steps are analogous to those employed in the interface calculations of ref. [1]. The average occupation of a site depends on its height h above or below a (100) reference surface as before, but in addition there is a dependence on the lateral displacement of the site from the edge of the step. Hence, we must solve for the self-consistency parameters of a two-dimensional array of lattice sites, and more computational effort is required. Steps are inserted by a vertical displacement at the boundary similar to that used in the Monte Carlo studies [3]. The system considered is equivalent to an array of parallel and equidistant steps. The results given here correspond to steps separated by forty lattice spacings, where interactions between adjacent steps are negligible. Step excess quantities are defined as the difference between these vicinal surface quantities and those corresponding to the flat surface. The temperature dependence of the excess energy of a step oriented along the close-packed (001) direction is indicated in fig. 1. At zero degrees Kelvin, the energy per edge position is J, the energy of one broken bond. The initial rise in the energy in the vicinity of kBT/J 0.4 corresponds to the excitation of kink sites in the edge of the step; the number of ‘-‘
385
386
J.D. Weeks, G.H. Gilmer
/ Thermodynamic properties of surface steps
these positions has a direct effect on certain sitespecific processes of crystal growth and catalysis. A rapid decrease in the energy is observed at high temperatures. This occurs when the surface becomes very rough on an atomic scale, and the step has a negligible influence on the system. Note that this region is steeper as calculated by the pair than by the mean field approximation. The Monte Carlo data of ref. [31is included for comparison. This should provide accurate results at the temperatures considered, since cooperative interactions are fully represented. (A comparison with the two-dimensional finite systems studied by Ferdinand and Fisher [9] suggests that finite system rounding effects are small.) At low ternperatures all excitations occur in a single plane contaming the step edge, and the system can be mapped directly onto an interface in the two-dimensional Ising model. Then the theories can be compared with the exact results of Onsager [10] and Fisher and Ferdinand [11]. Clearly the pair approximation offers a great improvement over the mean field theory, and a rapid convergence toward the Monte Carlo results is observed. Although the Monte Carlo data and general theoretical considerations [3—5] indicate that the
step energy vanishes at a finite temperature (the roughening temperature), both approximate methods yield a finite step energy at high temperatures. This effect is a result of the assumption of long-ranged interactions between atoms that is implicit in these methods. However, the very large reduction in the high-temperature energy on going from the mean field to the pair method strongly indicates a convergence toward zero as more cooperative effects are included. A step that is not oriented along a close-packed (001) direction must possess a higher energy and contam kink sites, even at low temperatures. The maximum occurs at the 45° orientation, and the excess energy of this step is plotted in fig. 2. The zero Kelvin energy is larger and the increase in the energy with temperature is smaller than in the previous case, since a high density of kink sites is present even at low ternperatures. This contradicts Swendsen’s finding of an anomalously rapid increase of step energy at low ternperatures. The specific heat and entropy anomalies he reports are also spurious. Note that these curves are almost identical to those of the (001) step at temperatures higher than the point of maximum energy; the step properties become isotropic at lugh temperatures.
<001> STEP ENERGY
<011> STEP ENERGY ~1)
IL
0.0
0.4
0.8
1.2
.6
2.0
2.4
2.0
32
kT/1 Fig. 1. A comparison ofthe excess energy of a (001) step calculated by the various approximations. The Monte Carlo data of ref. [3] are indicated by the open squares, and the dashed line represents the interface energy of a two-dimensional Ising model [10].
~
F~~) LI//
Fig. 2. The excess energy of a 45°step. The Monte Carlo data indicated by the open squares are unpublished results calculated by H.J. Leamy. The energy of the two-dimensional Ising model interface along the 45° orientation was calculated in ref. [11].
J.D. Weeks, G.H. Gilmer
/ Thermodynamic properties of surface steps
We gratefully acknowledge many helpful discussions with Harry J. Leamy. References [1] J.D. Weeks and G.H. Gilmer, J. Crystal Growth 33 (1976) 21. 12] J.D. Weeks and G.H. Gilmer, J. Chem. Phys. 63 (1975) 3136. [3] H.J. Leamy and G.H. Gilmer, J. Crystal Growth 24/25 (1974) 499. See also ref. [5]. [4] W.K. Burton, N. Cabrera and F.C. Frank, Phil. Trans. Roy. Soc. London 243A (1951) 299. [5] For a recent review see: H.J. Leamy, G.H. Gilmer and K.A. Jackson, in: Surface Physics of Crystalline Materials (Academic Press, New York, 1975) p. 121. [6] R.H. Swendsen, J. Crystal Growth 36 (1976) 11.
387
[7] That Swendsen’s results must be in error is most easily seen from his entropy data for the 45° step. This drops precipitously to zero at low temperatures. A simple combinatorial argument shows that the correct mean 49kBat zero temperature i.e., prefield entropy is ‘~O. cisely one-half the exact step entropy of ‘,4k~ ln 2. This residual mean field zero point entropy corresponds
[8] [9] [10] [11]
to a half-filled row along the step edge. Swendsen used the boundary conditions appropriate for the (100) step and hence constrained this row to be completely filled (R. Swendsen, private communication). G.H. Gilmer and J.D. Weeks, to be published in J. Chem. Phys. A.E. Ferdinand and M.E. Fisher, Phys. Rev. 185 (1969) 832. L. Onsager, Phys. Rev. 65 (1944) 117. ME. Fisher and A.E. Ferdinand, Phys. Rev. Letters 19 (1967) 169.
THERMODYNAMIC PROPERTIES OF SURFACE STEPS John D. WEEKS and George H. GILMER Bell Laboratories, Murray Hill, New Jersey 07974, USA Received 6 December 1977
This letter compares mean field, pair approximation, and Monte Carlo calculations of the excess energy of a step. The pair approximation offers good improvement over the mean field theory.
Recently we derived pair approximation equations for the solid-on-solid model of the crystal—vapor interface [11, and also for the interface in a more general anisotropic Ising model [2]. The thermodynamic properties given by the pair method proved to be considerably more accurate than those of the mean field approximation. The excess energy, for example, was in excellent agreement with Monte Carlo data at all temperatures. As mentioned in refs. [1] and [2], we also calculated some properties associated with steps on the crystal surface. These results are of practical significance since steps are involved in many of the mechanisms of adsorption, catalysis, and crystal growth. Also, they are a sensitive indicator of the interface roughening transition, since all step excess quantities should vanish at the roughening temperature [3—5]. Mean field calculations of some thermodynamic properties of an isolated step on the simple cubic (100) face were reported by Swendsen [61. Orientations along the (001) and the (011) (45°) directions were considered, and Swendsen reported several anomalous properties for the 45° step at low temperatures. Unfortunately, these anomalies are caused by an incorrect solution of the mean field equations for the 45°step [7]. (His solution for the (001) step is correct.) Here we compare the mean field energy for both step orientations with the pair approximation theory and the Monte Carlo simulations of Leamy and Gilmer [31.As before, the pair approximation offers good quantitative improvement over the mean field theory. Our method is not limited to the case of the
single isolated step; properties of periodic arrays of steps and the interactions between these steps are also easily treated. Details of the method and many additional step properties (gamma plots, the distribution of kink sites, and other thermodynamic properties) are included in a forthcoming report [81. The mean field and pair equations for steps are analogous to those employed in the interface calculations of ref. [1]. The average occupation of a site depends on its height h above or below a (100) reference surface as before, but in addition there is a dependence on the lateral displacement of the site from the edge of the step. Hence, we must solve for the self-consistency parameters of a two-dimensional array of lattice sites, and more computational effort is required. Steps are inserted by a vertical displacement at the boundary similar to that used in the Monte Carlo studies [3]. The system considered is equivalent to an array of parallel and equidistant steps. The results given here correspond to steps separated by forty lattice spacings, where interactions between adjacent steps are negligible. Step excess quantities are defined as the difference between these vicinal surface quantities and those corresponding to the flat surface. The temperature dependence of the excess energy of a step oriented along the close-packed (001) direction is indicated in fig. 1. At zero degrees Kelvin, the energy per edge position is J, the energy of one broken bond. The initial rise in the energy in the vicinity of kBT/J 0.4 corresponds to the excitation of kink sites in the edge of the step; the number of ‘-‘
385
386
J.D. Weeks, G.H. Gilmer
/ Thermodynamic properties of surface steps
these positions has a direct effect on certain sitespecific processes of crystal growth and catalysis. A rapid decrease in the energy is observed at high temperatures. This occurs when the surface becomes very rough on an atomic scale, and the step has a negligible influence on the system. Note that this region is steeper as calculated by the pair than by the mean field approximation. The Monte Carlo data of ref. [31is included for comparison. This should provide accurate results at the temperatures considered, since cooperative interactions are fully represented. (A comparison with the two-dimensional finite systems studied by Ferdinand and Fisher [9] suggests that finite system rounding effects are small.) At low ternperatures all excitations occur in a single plane contaming the step edge, and the system can be mapped directly onto an interface in the two-dimensional Ising model. Then the theories can be compared with the exact results of Onsager [10] and Fisher and Ferdinand [11]. Clearly the pair approximation offers a great improvement over the mean field theory, and a rapid convergence toward the Monte Carlo results is observed. Although the Monte Carlo data and general theoretical considerations [3—5] indicate that the
step energy vanishes at a finite temperature (the roughening temperature), both approximate methods yield a finite step energy at high temperatures. This effect is a result of the assumption of long-ranged interactions between atoms that is implicit in these methods. However, the very large reduction in the high-temperature energy on going from the mean field to the pair method strongly indicates a convergence toward zero as more cooperative effects are included. A step that is not oriented along a close-packed (001) direction must possess a higher energy and contam kink sites, even at low temperatures. The maximum occurs at the 45° orientation, and the excess energy of this step is plotted in fig. 2. The zero Kelvin energy is larger and the increase in the energy with temperature is smaller than in the previous case, since a high density of kink sites is present even at low ternperatures. This contradicts Swendsen’s finding of an anomalously rapid increase of step energy at low ternperatures. The specific heat and entropy anomalies he reports are also spurious. Note that these curves are almost identical to those of the (001) step at temperatures higher than the point of maximum energy; the step properties become isotropic at lugh temperatures.
<001> STEP ENERGY
<011> STEP ENERGY ~1)
IL
0.0
0.4
0.8
1.2
.6
2.0
2.4
2.0
32
kT/1 Fig. 1. A comparison ofthe excess energy of a (001) step calculated by the various approximations. The Monte Carlo data of ref. [3] are indicated by the open squares, and the dashed line represents the interface energy of a two-dimensional Ising model [10].
~
F~~) LI//
Fig. 2. The excess energy of a 45°step. The Monte Carlo data indicated by the open squares are unpublished results calculated by H.J. Leamy. The energy of the two-dimensional Ising model interface along the 45° orientation was calculated in ref. [11].
J.D. Weeks, G.H. Gilmer
/ Thermodynamic properties of surface steps
We gratefully acknowledge many helpful discussions with Harry J. Leamy. References [1] J.D. Weeks and G.H. Gilmer, J. Crystal Growth 33 (1976) 21. 12] J.D. Weeks and G.H. Gilmer, J. Chem. Phys. 63 (1975) 3136. [3] H.J. Leamy and G.H. Gilmer, J. Crystal Growth 24/25 (1974) 499. See also ref. [5]. [4] W.K. Burton, N. Cabrera and F.C. Frank, Phil. Trans. Roy. Soc. London 243A (1951) 299. [5] For a recent review see: H.J. Leamy, G.H. Gilmer and K.A. Jackson, in: Surface Physics of Crystalline Materials (Academic Press, New York, 1975) p. 121. [6] R.H. Swendsen, J. Crystal Growth 36 (1976) 11.
387
[7] That Swendsen’s results must be in error is most easily seen from his entropy data for the 45° step. This drops precipitously to zero at low temperatures. A simple combinatorial argument shows that the correct mean 49kBat zero temperature i.e., prefield entropy is ‘~O. cisely one-half the exact step entropy of ‘,4k~ ln 2. This residual mean field zero point entropy corresponds
[8] [9] [10] [11]
to a half-filled row along the step edge. Swendsen used the boundary conditions appropriate for the (100) step and hence constrained this row to be completely filled (R. Swendsen, private communication). G.H. Gilmer and J.D. Weeks, to be published in J. Chem. Phys. A.E. Ferdinand and M.E. Fisher, Phys. Rev. 185 (1969) 832. L. Onsager, Phys. Rev. 65 (1944) 117. ME. Fisher and A.E. Ferdinand, Phys. Rev. Letters 19 (1967) 169.