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Surface diffusion on heterogeneous surfaces: Competition between ordering and heterogeneity effects
Vacuum 54 (1999) 119 — 124
Surface diffusion on heterogeneous surfaces: Competition between ordering and heterogeneity effects F. Nieto , C. Uebing * Max-Planck-Institut fu¨ r Eisenforschung, Max-Planck-Strasse 1, D-40074 Du¨ sseldorf, Germany Departamento de Fı& sica and Centro Latinoamericano de Estudios Ilya Prigogine, Universidad Nacional de San Luis, 5700 San Luis, Argentina Lehrstuhl fu¨ r Physikalische Chemie II, Universita¨ t Dortmund, D-44227 Dortmund, Germany
Abstract The Monte Carlo method is used to investigate diffusion of interacting particles adsorbed on an energetically heterogeneous bivariate trap surface. Repulsive interactions between nearest-neighbour adatoms are considered. These interactions induce c(2;2) ordering of the adatoms at low temperatures. Simulations in the canonical ensemble are used to calculate the tracer, jump- and chemical-diffusion coefficients. The effects of surface heterogeneities and adsorbate-adsorbate interactions are largely pronounced at low temperatures. The existence of heterogeneities disturbs c(2;2) ordering at h+0.5. It is shown that this breakdown of the order substantially affects surface diffusion. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Heterogeneous surface; Surface diffusion; Computer simulations
1. Introduction In recent years, the progress in the experimental techniques for surface analysis at atomic scales has improved our knowledge about the energetic surface topography. As a consequence refined atomistic models for heterogeneous surfaces have been developed which in principle are capable to include the energetic surface topography in the statistical description [1—13]. The mobility of adsorbed particles on such heterogeneous surfaces is a topic of growing interest in the literature. This is partly due to its importance in the understanding and improvement of several technical devices such as gas separation and purification tubes, automotive catalysts, just to name a few. On the other hand, the detailed understanding of this process appears to be also very important from the fundamental aspects of basic science. Surface diffusion is a many particle process. The exact analytical calculation of diffusion coefficients is possible only for a few exceptional cases (e.g. for noninteracting
* Correspondence address: Max-Planck-Institut fu¨r Eisenforschung Max-Planck-Stra{e 1, D-40237 Du] sseldorf, Germany. Tel.: 0049 211 6792 290; fax: 0049 211 6792 268; E-mail: uebing@mpie-duesseldorf. mpg.de
lattice gases). However, in more realistic cases analytical expressions cannot be derived and Monte Carlo simulations have proven to be an adequate and powerful tool to study surface diffusion in the framework of the lattice-gas scheme [14—16]. The aim of the present work is to investigate the mutual influence of surface heterogeneities and adsorbate—adsorbate interactions on surface diffusion. For this purpose we apply the well-known bivariate trap model, which probably is one of the simplest models of a heterogeneous surface. In our lattice model it is assumed that the heterogeneous surface is formed by two different adsorption sites, deep and shallow traps, which are arranged at random. In previous work it has been shown that surface diffusion is significantly influenced (i) by surface heterogeneities [17—19] and (ii) by the presence of interactions between adsorbed particles [8, 20]. The intention of this work is to investigate the surface diffusion of interacting particles on a simple heterogeneous surface and to characterize the possible competition between these two factors. A detailed comprehension of this competition could be helpful for the evaluation of experimental diffusion studies on heterogeneous surfaces. However, it is important to note that we consider a highly idealized model here, which is not meant to reproduce a particular experimental system.
0042-207X/99/$ — see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 2 - 2 0 7 X ( 9 8 ) 0 0 4 4 6 - 1
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F. Nieto, C. Uebing / Vacuum 54 (1999) 119—124
2. The lattice gas model The characteristic features of the bivariate trap model used in the present work are schematically outlined in Fig. 1. We consider a square lattice which is built up by shallow and deep adsorption sites. The corresponding adsorption energies, i.e. the relative minima of the twodimensional periodic potential, are given by e and e ac1 " cording to
e for deep traps, e" " (1) G e for shallow traps. 1 The relative minima of the periodic potential are modified by pairwise nearest-neighbour interaction energies, u. The concentration of the randomly distributed deep traps is given by #. Thus, the concentration of the shallow traps is given by 1!#. All adsorption sites are separated by wells of the periodic potential, which need to be overcome by diffusion adatoms. The saddle-point energies are uniformly given by a common value, e , throughout the whole lattice. 1. For the sake of simplicity, we assume that the saddlepoint energies are not affected by particle-particle interactions. The lattice—gas hamiltonian can be written as , (2) H"! c e # c c u, G H GG G G(H where the second sum runs over the nearest neighbours of a given site i and describes the influence of lateral interactions. c denote local occupation variables, G H 1 if site i is occupied, c" (3) G 0 if site i is vacant,
Fig. 1. Schematic drawing of the periodic potential used for simulation of surface diffusion on heterogeneous surfaces. The minima of the periodic potential constitute the adsorption sites, namely shallow and deep traps with adsorption energies given by (e ) and (e ), respectively. 1 " Nearest-neighbour interaction energies are assumed to modify the adsorption energies (dashed lines) but not the saddle point energy, (e ). 1.
which describe the occupation of lattice sites by adsorbed atoms. Double occupancy of lattice sites is excluded. All calculations to be discussed below are performed for the highly idealized situation (e !e )""u", i.e. the 1 " site energy difference of both types of traps is numerically equal to the absolute value of the nearest-neighbour repulsion. Diffusion simulations are carried out in the canonical ensemble. The bivariate trap surface given by Eq. (2) is realized by a two-dimensional array of ¸;¸ sites with periodic boundary conditions. Throughout the present work we use ¸"64. The site specific adsorption energies, e "e , e , are assigned at random according to the G " 1 desired concentration # of deep traps. Initial lattice gas configurations are generated by throwing h¸ particles at random on the surface. Here h denotes the adsorbate coverage. The elementary steps of diffusion are jumps of adsorbed particles from occupied initial sites i to adjacent empty sites j. The associated jump probabilities, P are G given by [8]
*e P Jexp ! G , G k ¹
(4)
where *e denote the activation energies for such jumps. G *e can be calculated as the energy difference between G saddle point, e , and the energy of the initial site e , the 1. G latter being influenced by the nearest-neighbour interactions as already mentioned:
*e "e ! e #u c . G 1. G H H
(5)
Details of the MC algorithm have been presented in recent publications [8, 21] and will not be repeated here. In essence, the jump algorithm consists of the following steps: first, an initial site i of the ¸;¸ lattice is picked at random, if filled, an adjacent final site j is randomly selected. If the destination is vacant, a jump can occur with the probability given by Eq. (4), otherwise no jump occurs. One Monte Carlo step (MCS) corresponds to ¸ interrogations (in random order) of lattice sites. Before starting a diffusion run, a large number of initial MCSs were performed to establish thermodynamic equilibrium at the desired temperature ¹. The simulations were carried out using the supermassive parallel Intel Paragon supercomputer of the Ju¨lich research center. The chemical diffusion coefficient can be determined either via the Kubo—Green formalism or via the fluctuation method. Both methods have been discussed in detail in [8]. In essence, the Kubo—Green method
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121
measures the chemical diffusion coefficient (which we denote D in the following) according to [22] )% *k/k ¹ D " . (6) )% * ln h
Here k is the chemical potential. The first factor represents the thermodynamic factor, and D is the jump diffu( sion coefficient given by [1]
, "R (t)!R (0)" , (7) G G G where d is the Euclidean dimension (in the case of surface diffusion d"2). The vector R(t) determines the position of a tagged particle at time t. The simulation of surface diffusion in the framework of the Kubo—Green methodology consists of two well-defined steps: (i) the calculation of the thermodynamic factor by simulating the adsorption equilibrium in the grand-canonical ensemble, and (ii), the calculation of the jump diffusion coefficient D according to Eq. (7) in the canonical ensemble. The ( fluctuation method measures the autocorrelation function of the particle density, f (t)/f (0), for a small square L L probe region embedded in the whole two-dimensional lattice. The ratio f (t)/f (0) is then compared with the L L theoretical correlation function [23, 24] in order to obtain the chemical diffusion coefficient D (in the following this quantity is denoted D ). Thus, this method is a com$ puter simulation of the field emission fluctuation method [23] used experimentally to determine adsorbate diffusion coefficients. Details of this method are presented in Refs. [16, 25]. In the present work we use a 8;8 and 16;16 probes for the determination of D . $ The diffusion algorithm also determines the tracer diffusion coefficient, D* defined by
1 D " lim ( 2dNt P R
1 , D*" lim "R (t)!R (0)" . (8) G G 2dNt P R G It should be noted that D* is a single particle diffusion coefficient, in contrast to D and D . H As in previous simulations [8] calculations are carried out in terms of D , the chemical diffusion coefficient for zero interactions between adsorbates on a homogeneous lattice (Langmuir gas).
3. Results and discussion In this section the focus is on the analysis of the coverage dependence of the tracer and chemical surface diffusion coefficients and related quantities. We will start with an analysis of the coverage dependence of the normalized tracer diffusion coefficient D*/D . Fig. 2 shows corres ponding Monte Carlo results for different values of ¹ and #, respectively. It is clear that at large surface
Fig. 2. Normalized tracer diffusion coefficient D*/D for three different values of the deep traps concentration # versus the total coverage h; (a) #"0.1, (b) #"0.5 and (c) #"0.9. Shown are results for different temperatures expressed in terms of (e !e )/k ¹. As in previous studies 1 " [8, 18], the diffusion coefficients are normalized with respect to D , the chemical diffusion coefficient of Langmuir gas. The dashed line corresponds to the Langmuir case. The solid line (without symbols) represents the diffusive behavior of repulsively interacting particles adsorbed on a homogeneous surface for the lowest temperature considered here, (e !e )/k ¹"4.82). 1 "
coverages, h'0.5, surface diffusion is strongly accelerated relative to the Langmuir case (dashed line), and this is due to the presence of nearest-neighbour repulsive interactions. However, the measured values of D*/D are significantly reduced relative to the homogeneous case (solid curve without symbols), and this is attributable to the presence of deep traps. At low coverages (where the adatoms are far apart on average) the accelerating effect of the repulsive interactions is much less pronounced, and the behavior of the normalized tracer diffusion coefficient is essentially governed by the heterogeneities, i.e. by the deep traps. Its decelerating influence becomes more pronounced upon increasing the deep-trap concentration, #. Upon decreasing the temperature, the tracer diffusion coefficient exhibits a well pronounced minimum at half coverage. This behavior is clearly attributed to the c(2;2) ordering. This minimum (Fig. 2) is less pronounced and substantially broadened compared with the
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F. Nieto, C. Uebing / Vacuum 54 (1999) 119—124
homogeneous case, indicating that the ordering phenomenon is disturbed by the presence of heterogeneities and is smeared out over a certain range of surface coverages, respectively. Fig. 3 presents the coverage dependence of the jump diffusion coefficient D , which is given by Eq. (7). It is ( quite obvious that D* and D behave in a strikingly ( similar way, despite their substantially different meanings (see Eqs. (7) and (8)). Figs. 4—6 compare the coverage dependence of the chemical diffusion coefficient calculated via fluctuation (open symbols) and Kubo—Green method (filled symbols), respectively. At high values of ¹ both methods show an excellent agreement. However, small discrepancies appear for lower values of ¹ (note the logarithmic scale). Discrepancies between D and D have already )% $ been reported in the literature for different cases. The most striking finding probably is that the minimum of D* corresponds to a sharp maximum of D which is )% attributed to a maximum of the thermodynamic factor (Fig. 7). This can be understood by recalling that the thermodynamic factor is inversely proportional to the two-dimensional compressibility of the adlayer which accordingly is smallest for the well-ordered c(2;2) lattice gas phase. In a similar way to the normalized tracer diffusion coefficient, it can be concluded that the maximum of D is less pronounced with respect to the )% homogeneous case.
Fig. 3. As Fig. 3, for the normalized jump diffusion coefficient, D /D . (
4. Conclusions In the present work we have used the bivariate trap model in order to study how surface heterogeneities and the nearest-neighbour repulsive interactions affect surface diffusion. In this model two kinds of traps are distributed randomly. The Monte Carlo method has been utilized to simulate surface diffusion and to calculate the tracer, jump and chemical diffusion coefficients. The chemical diffusion coefficient was calculated via two different approaches: the fluctuation and the Kubo—Green method. The work presented here has clearly shown that the adsorption and the motion of adsorbed particles are strongly affected by both heterogeneities and repulsive An overall consistent explanation of these findings has been given in Ref. [19]: the fluctuation method fails when the applicable length scale of the lattice gas system becomes comparable to the probe dimension, or in other words, when the probe misses the long wavelength fluctuations of the particle density. Then, it is clear that the discrepancies between D and D decrease when the size of the probe area used for )% $ the calculation of D is increased, (however, an increase in the probe $ area produces a costly increase in computing time). Therefore, we conclude that the Kubo—Green method for determining the chemical diffusion coefficient is more appropriate in our case and we will proceed to analyze this quantity.
Fig. 4. Chemical diffusion coefficient calculated by the fluctuation method, D (open symbols), and by the Kubo—Green method, D (fil$ )% led symbols), as a function of coverage for #"0.1. Temperature is expressed in terms of (e !e )/k ¹. At high temperatures, i.e. 1 " (e !e )/k ¹"1.20 both methods give almost identical results and the 1 " corresponding graphs are indistinguishable. The dashed line corresponds to the Langmuir case. The solid line (without symbols) represents the diffusive behavior of repulsively interacting particles adsorbed on a homogeneous surface for the lowest temperature considered here, (e !e )/k ¹"4.82). 1 "
F. Nieto, C. Uebing / Vacuum 54 (1999) 119—124
123
Fig. 5. As Fig. 4, for #"0.5.
Fig. 7. Thermodynamic factor d(k/k ¹ )/d ln h as a function of coverage for three different values of the deep trap concentration # as indicated. The thermodynamic factor is obtained via the differentiation of adsorption isotherms calculated in the grand canonical ensemble. Temperature is expressed in terms of (e !e )/k ¹. 1 "
Fig. 6. As Fig. 4, for #"0.9.
interactions and, as is expected, this effect becomes more pronounced as the temperature decreases. The adsorption isotherms present several plateaus which are the result of the competitive effects between the preferential occupation of deep trap sites and the repulsive interactions. The behavior of the different diffusion coefficients reflect how the heterogeneities affect the c(2;2) ordering at h&0.5 and low temperatures. In fact, the tracer and the jump diffusion coefficient present a sharp minimum at h&0.5, which is less pronounced and substantially broadened as # increases. These findings indicate that the ordering phenomenon is disturbed by the presence of heterogeneities and is smeared out over a certain range of surface coverages respectively. The chemical diffusion coefficient presents a maximum at h&0.5, and this is a consequence of the maximum in the thermodynamic factor which probably is the most important controlling
factor in the collective motion of adsorbed particles. This maximum is reduced and broadened as # increases due to the surface heterogeneities.
Acknowledgements It is a pleasure to acknowledge many helpful and stimulating discussions with K. Kehr, V. Pereyra, J. Riccardo and G. Zgrablich. This work was made possible by the Heisenberg program of the Deutsche Forschungsgemeinschaft (DFG).
References [1] Gomer R. Rep Prog Phys 1990;53:917. [2] Zgrablich G. In Rudzinski W, Zgrablich G, editors. Equilibria and dynamics of gas adsorption on heterogeneous solid surfaces, Amsterdam: Elsevier, 1996. [3] Tringides M., Gomer R. Surf Sci 1984;145:121. [4] Tringides M., Gomer R. Surf Sci 1985;155:254. [5] Sadiq A, Binder K. Surf Sci 1983;128:350. [6] Zhdanov VP. Surf Sci Lett 1985;149:L13.
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[7] Uebing C, Gomer R. Surf Sci 1995;331—333:930. [8] Uebing C, Gomer R. J Chem Phys 1991;95:7626,7636,7641, 7648. [9] Mayagoitia V, Rojas F, Pereyra V, Zgrablich G. Surf Sci 1989; 221:394. [10] Riccardo JL, Pereyra VD, Zgrablich G. Langmuir 1992;8:1518. [11] Mayagoitia V, Rojas F, Riccardo JL, Pereyra V, Zgrablich G. Phys Rev B 1990;41:7150. [12] Mussawisade K, Wichmann T, Kehr KW. J Phys C 1997;9: 1181. [13] Limoge Y, Boucquet JL, Phys Rev Lett 1990;65:60. [14] Sapag K, Pereyra V, Riccardo JL, Zgrablich G. Surf Sci 1993;295: 433.
[15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
Uebing C. Phys Rev B 1994;49:13 913. Uebing C, Gomer R. Surf Sci 1994;306:419. Viljoen E, Uebing C, Surf Sci 1996;352—354:1007. Viljoen E, Uebing C. Langmuir 1997;13:1001. Uebing C, Pereyra V, Zgrablich G. Surf Sci 1996;366:185. Uebing C, Gomer R. Ber Bunsenges Phys Chem 1996;100: 1138. Nieto F, Uebing C. Ber Bunsenges Phys Chem 1998;102:156. Reed DA, Ehrlich G. Surf Sci 1981;102:588. Gomer R. Surf Sci 1973;38:373. Mazenko G, Banavar JR, Gomer R. Surf Sci 1981;107:459. Uebing C, Gomer R. Surf Sci 1994;317:165.
Surface diffusion on heterogeneous surfaces: Competition between ordering and heterogeneity effects F. Nieto , C. Uebing * Max-Planck-Institut fu¨ r Eisenforschung, Max-Planck-Strasse 1, D-40074 Du¨ sseldorf, Germany Departamento de Fı& sica and Centro Latinoamericano de Estudios Ilya Prigogine, Universidad Nacional de San Luis, 5700 San Luis, Argentina Lehrstuhl fu¨ r Physikalische Chemie II, Universita¨ t Dortmund, D-44227 Dortmund, Germany
Abstract The Monte Carlo method is used to investigate diffusion of interacting particles adsorbed on an energetically heterogeneous bivariate trap surface. Repulsive interactions between nearest-neighbour adatoms are considered. These interactions induce c(2;2) ordering of the adatoms at low temperatures. Simulations in the canonical ensemble are used to calculate the tracer, jump- and chemical-diffusion coefficients. The effects of surface heterogeneities and adsorbate-adsorbate interactions are largely pronounced at low temperatures. The existence of heterogeneities disturbs c(2;2) ordering at h+0.5. It is shown that this breakdown of the order substantially affects surface diffusion. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Heterogeneous surface; Surface diffusion; Computer simulations
1. Introduction In recent years, the progress in the experimental techniques for surface analysis at atomic scales has improved our knowledge about the energetic surface topography. As a consequence refined atomistic models for heterogeneous surfaces have been developed which in principle are capable to include the energetic surface topography in the statistical description [1—13]. The mobility of adsorbed particles on such heterogeneous surfaces is a topic of growing interest in the literature. This is partly due to its importance in the understanding and improvement of several technical devices such as gas separation and purification tubes, automotive catalysts, just to name a few. On the other hand, the detailed understanding of this process appears to be also very important from the fundamental aspects of basic science. Surface diffusion is a many particle process. The exact analytical calculation of diffusion coefficients is possible only for a few exceptional cases (e.g. for noninteracting
* Correspondence address: Max-Planck-Institut fu¨r Eisenforschung Max-Planck-Stra{e 1, D-40237 Du] sseldorf, Germany. Tel.: 0049 211 6792 290; fax: 0049 211 6792 268; E-mail: uebing@mpie-duesseldorf. mpg.de
lattice gases). However, in more realistic cases analytical expressions cannot be derived and Monte Carlo simulations have proven to be an adequate and powerful tool to study surface diffusion in the framework of the lattice-gas scheme [14—16]. The aim of the present work is to investigate the mutual influence of surface heterogeneities and adsorbate—adsorbate interactions on surface diffusion. For this purpose we apply the well-known bivariate trap model, which probably is one of the simplest models of a heterogeneous surface. In our lattice model it is assumed that the heterogeneous surface is formed by two different adsorption sites, deep and shallow traps, which are arranged at random. In previous work it has been shown that surface diffusion is significantly influenced (i) by surface heterogeneities [17—19] and (ii) by the presence of interactions between adsorbed particles [8, 20]. The intention of this work is to investigate the surface diffusion of interacting particles on a simple heterogeneous surface and to characterize the possible competition between these two factors. A detailed comprehension of this competition could be helpful for the evaluation of experimental diffusion studies on heterogeneous surfaces. However, it is important to note that we consider a highly idealized model here, which is not meant to reproduce a particular experimental system.
0042-207X/99/$ — see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 2 - 2 0 7 X ( 9 8 ) 0 0 4 4 6 - 1
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F. Nieto, C. Uebing / Vacuum 54 (1999) 119—124
2. The lattice gas model The characteristic features of the bivariate trap model used in the present work are schematically outlined in Fig. 1. We consider a square lattice which is built up by shallow and deep adsorption sites. The corresponding adsorption energies, i.e. the relative minima of the twodimensional periodic potential, are given by e and e ac1 " cording to
e for deep traps, e" " (1) G e for shallow traps. 1 The relative minima of the periodic potential are modified by pairwise nearest-neighbour interaction energies, u. The concentration of the randomly distributed deep traps is given by #. Thus, the concentration of the shallow traps is given by 1!#. All adsorption sites are separated by wells of the periodic potential, which need to be overcome by diffusion adatoms. The saddle-point energies are uniformly given by a common value, e , throughout the whole lattice. 1. For the sake of simplicity, we assume that the saddlepoint energies are not affected by particle-particle interactions. The lattice—gas hamiltonian can be written as , (2) H"! c e # c c u, G H GG G G(H where the second sum runs over the nearest neighbours of a given site i and describes the influence of lateral interactions. c denote local occupation variables, G H 1 if site i is occupied, c" (3) G 0 if site i is vacant,
Fig. 1. Schematic drawing of the periodic potential used for simulation of surface diffusion on heterogeneous surfaces. The minima of the periodic potential constitute the adsorption sites, namely shallow and deep traps with adsorption energies given by (e ) and (e ), respectively. 1 " Nearest-neighbour interaction energies are assumed to modify the adsorption energies (dashed lines) but not the saddle point energy, (e ). 1.
which describe the occupation of lattice sites by adsorbed atoms. Double occupancy of lattice sites is excluded. All calculations to be discussed below are performed for the highly idealized situation (e !e )""u", i.e. the 1 " site energy difference of both types of traps is numerically equal to the absolute value of the nearest-neighbour repulsion. Diffusion simulations are carried out in the canonical ensemble. The bivariate trap surface given by Eq. (2) is realized by a two-dimensional array of ¸;¸ sites with periodic boundary conditions. Throughout the present work we use ¸"64. The site specific adsorption energies, e "e , e , are assigned at random according to the G " 1 desired concentration # of deep traps. Initial lattice gas configurations are generated by throwing h¸ particles at random on the surface. Here h denotes the adsorbate coverage. The elementary steps of diffusion are jumps of adsorbed particles from occupied initial sites i to adjacent empty sites j. The associated jump probabilities, P are G given by [8]
*e P Jexp ! G , G k ¹
(4)
where *e denote the activation energies for such jumps. G *e can be calculated as the energy difference between G saddle point, e , and the energy of the initial site e , the 1. G latter being influenced by the nearest-neighbour interactions as already mentioned:
*e "e ! e #u c . G 1. G H H
(5)
Details of the MC algorithm have been presented in recent publications [8, 21] and will not be repeated here. In essence, the jump algorithm consists of the following steps: first, an initial site i of the ¸;¸ lattice is picked at random, if filled, an adjacent final site j is randomly selected. If the destination is vacant, a jump can occur with the probability given by Eq. (4), otherwise no jump occurs. One Monte Carlo step (MCS) corresponds to ¸ interrogations (in random order) of lattice sites. Before starting a diffusion run, a large number of initial MCSs were performed to establish thermodynamic equilibrium at the desired temperature ¹. The simulations were carried out using the supermassive parallel Intel Paragon supercomputer of the Ju¨lich research center. The chemical diffusion coefficient can be determined either via the Kubo—Green formalism or via the fluctuation method. Both methods have been discussed in detail in [8]. In essence, the Kubo—Green method
F. Nieto, C. Uebing / Vacuum 54 (1999) 119—124
121
measures the chemical diffusion coefficient (which we denote D in the following) according to [22] )% *k/k ¹ D " . (6) )% * ln h
Here k is the chemical potential. The first factor represents the thermodynamic factor, and D is the jump diffu( sion coefficient given by [1]
, "R (t)!R (0)" , (7) G G G where d is the Euclidean dimension (in the case of surface diffusion d"2). The vector R(t) determines the position of a tagged particle at time t. The simulation of surface diffusion in the framework of the Kubo—Green methodology consists of two well-defined steps: (i) the calculation of the thermodynamic factor by simulating the adsorption equilibrium in the grand-canonical ensemble, and (ii), the calculation of the jump diffusion coefficient D according to Eq. (7) in the canonical ensemble. The ( fluctuation method measures the autocorrelation function of the particle density, f (t)/f (0), for a small square L L probe region embedded in the whole two-dimensional lattice. The ratio f (t)/f (0) is then compared with the L L theoretical correlation function [23, 24] in order to obtain the chemical diffusion coefficient D (in the following this quantity is denoted D ). Thus, this method is a com$ puter simulation of the field emission fluctuation method [23] used experimentally to determine adsorbate diffusion coefficients. Details of this method are presented in Refs. [16, 25]. In the present work we use a 8;8 and 16;16 probes for the determination of D . $ The diffusion algorithm also determines the tracer diffusion coefficient, D* defined by
1 D " lim ( 2dNt P R
1 , D*" lim "R (t)!R (0)" . (8) G G 2dNt P R G It should be noted that D* is a single particle diffusion coefficient, in contrast to D and D . H As in previous simulations [8] calculations are carried out in terms of D , the chemical diffusion coefficient for zero interactions between adsorbates on a homogeneous lattice (Langmuir gas).
3. Results and discussion In this section the focus is on the analysis of the coverage dependence of the tracer and chemical surface diffusion coefficients and related quantities. We will start with an analysis of the coverage dependence of the normalized tracer diffusion coefficient D*/D . Fig. 2 shows corres ponding Monte Carlo results for different values of ¹ and #, respectively. It is clear that at large surface
Fig. 2. Normalized tracer diffusion coefficient D*/D for three different values of the deep traps concentration # versus the total coverage h; (a) #"0.1, (b) #"0.5 and (c) #"0.9. Shown are results for different temperatures expressed in terms of (e !e )/k ¹. As in previous studies 1 " [8, 18], the diffusion coefficients are normalized with respect to D , the chemical diffusion coefficient of Langmuir gas. The dashed line corresponds to the Langmuir case. The solid line (without symbols) represents the diffusive behavior of repulsively interacting particles adsorbed on a homogeneous surface for the lowest temperature considered here, (e !e )/k ¹"4.82). 1 "
coverages, h'0.5, surface diffusion is strongly accelerated relative to the Langmuir case (dashed line), and this is due to the presence of nearest-neighbour repulsive interactions. However, the measured values of D*/D are significantly reduced relative to the homogeneous case (solid curve without symbols), and this is attributable to the presence of deep traps. At low coverages (where the adatoms are far apart on average) the accelerating effect of the repulsive interactions is much less pronounced, and the behavior of the normalized tracer diffusion coefficient is essentially governed by the heterogeneities, i.e. by the deep traps. Its decelerating influence becomes more pronounced upon increasing the deep-trap concentration, #. Upon decreasing the temperature, the tracer diffusion coefficient exhibits a well pronounced minimum at half coverage. This behavior is clearly attributed to the c(2;2) ordering. This minimum (Fig. 2) is less pronounced and substantially broadened compared with the
122
F. Nieto, C. Uebing / Vacuum 54 (1999) 119—124
homogeneous case, indicating that the ordering phenomenon is disturbed by the presence of heterogeneities and is smeared out over a certain range of surface coverages, respectively. Fig. 3 presents the coverage dependence of the jump diffusion coefficient D , which is given by Eq. (7). It is ( quite obvious that D* and D behave in a strikingly ( similar way, despite their substantially different meanings (see Eqs. (7) and (8)). Figs. 4—6 compare the coverage dependence of the chemical diffusion coefficient calculated via fluctuation (open symbols) and Kubo—Green method (filled symbols), respectively. At high values of ¹ both methods show an excellent agreement. However, small discrepancies appear for lower values of ¹ (note the logarithmic scale). Discrepancies between D and D have already )% $ been reported in the literature for different cases. The most striking finding probably is that the minimum of D* corresponds to a sharp maximum of D which is )% attributed to a maximum of the thermodynamic factor (Fig. 7). This can be understood by recalling that the thermodynamic factor is inversely proportional to the two-dimensional compressibility of the adlayer which accordingly is smallest for the well-ordered c(2;2) lattice gas phase. In a similar way to the normalized tracer diffusion coefficient, it can be concluded that the maximum of D is less pronounced with respect to the )% homogeneous case.
Fig. 3. As Fig. 3, for the normalized jump diffusion coefficient, D /D . (
4. Conclusions In the present work we have used the bivariate trap model in order to study how surface heterogeneities and the nearest-neighbour repulsive interactions affect surface diffusion. In this model two kinds of traps are distributed randomly. The Monte Carlo method has been utilized to simulate surface diffusion and to calculate the tracer, jump and chemical diffusion coefficients. The chemical diffusion coefficient was calculated via two different approaches: the fluctuation and the Kubo—Green method. The work presented here has clearly shown that the adsorption and the motion of adsorbed particles are strongly affected by both heterogeneities and repulsive An overall consistent explanation of these findings has been given in Ref. [19]: the fluctuation method fails when the applicable length scale of the lattice gas system becomes comparable to the probe dimension, or in other words, when the probe misses the long wavelength fluctuations of the particle density. Then, it is clear that the discrepancies between D and D decrease when the size of the probe area used for )% $ the calculation of D is increased, (however, an increase in the probe $ area produces a costly increase in computing time). Therefore, we conclude that the Kubo—Green method for determining the chemical diffusion coefficient is more appropriate in our case and we will proceed to analyze this quantity.
Fig. 4. Chemical diffusion coefficient calculated by the fluctuation method, D (open symbols), and by the Kubo—Green method, D (fil$ )% led symbols), as a function of coverage for #"0.1. Temperature is expressed in terms of (e !e )/k ¹. At high temperatures, i.e. 1 " (e !e )/k ¹"1.20 both methods give almost identical results and the 1 " corresponding graphs are indistinguishable. The dashed line corresponds to the Langmuir case. The solid line (without symbols) represents the diffusive behavior of repulsively interacting particles adsorbed on a homogeneous surface for the lowest temperature considered here, (e !e )/k ¹"4.82). 1 "
F. Nieto, C. Uebing / Vacuum 54 (1999) 119—124
123
Fig. 5. As Fig. 4, for #"0.5.
Fig. 7. Thermodynamic factor d(k/k ¹ )/d ln h as a function of coverage for three different values of the deep trap concentration # as indicated. The thermodynamic factor is obtained via the differentiation of adsorption isotherms calculated in the grand canonical ensemble. Temperature is expressed in terms of (e !e )/k ¹. 1 "
Fig. 6. As Fig. 4, for #"0.9.
interactions and, as is expected, this effect becomes more pronounced as the temperature decreases. The adsorption isotherms present several plateaus which are the result of the competitive effects between the preferential occupation of deep trap sites and the repulsive interactions. The behavior of the different diffusion coefficients reflect how the heterogeneities affect the c(2;2) ordering at h&0.5 and low temperatures. In fact, the tracer and the jump diffusion coefficient present a sharp minimum at h&0.5, which is less pronounced and substantially broadened as # increases. These findings indicate that the ordering phenomenon is disturbed by the presence of heterogeneities and is smeared out over a certain range of surface coverages respectively. The chemical diffusion coefficient presents a maximum at h&0.5, and this is a consequence of the maximum in the thermodynamic factor which probably is the most important controlling
factor in the collective motion of adsorbed particles. This maximum is reduced and broadened as # increases due to the surface heterogeneities.
Acknowledgements It is a pleasure to acknowledge many helpful and stimulating discussions with K. Kehr, V. Pereyra, J. Riccardo and G. Zgrablich. This work was made possible by the Heisenberg program of the Deutsche Forschungsgemeinschaft (DFG).
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