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Langmuir 1999, 15, 5876-5882
Diffusion on Simple Heterogeneous Surfaces: Bivariate Trap Model with Random Topography† F. Nieto*,‡,§ and C. Uebing‡,| Max-Planck-Institut fu¨ r Eisenforschung, D-40074 Du¨ sseldorf, Germany, Departamento de Fı´sica and Centro Latinoamericano de Estudios Ilya Prigogine, Universidad Nacional de San Luis, CONICET, Chacabuco 917, 5700 San Luis, Argentina, and Lehrstuhl fu¨ r Physikalische Chemie II, Universita¨ t Dortmund, D-44227 Dortmund, Germany Received August 24, 1998. In Final Form: February 22, 1999 The well-known bivariate trap model with random topography is used to investigate the effects of surface heterogeneities on surface diffusion. Monte Carlo simulations are performed in order to characterize how the tracer, jump, and chemical diffusion coefficients depend on temperature and concentration of deep traps, Θ. The chemical diffusion coefficient is calculated by applying the fluctuation and the Kubo-Green methods. Both methods are in good agreement at low values of Θ only. The discrepancies are discussed and explained. At low surface coverages and low temperatures the effects of heterogeneities are largely pronounced since most of the adatoms are trapped by the deep traps. At high coverages the mobility of adatoms adsorbed on shallow traps dominates the diffusion behavior.
1. Introduction In the last decade several attempts have been undertaken to describe the microscopical mechanisms of adatom migration on idealized homogeneous and more realistic heterogeneous surfaces. It is quite obvious that even single crystal surfaces are not perfect and contain structural and electronical heterogeneities. Many publications have demonstrated very clearly the importance of including the energetic surface topography in the statistical description of heterogeneity.1-9 In recent years, the rapid development and improvement of new powerful experimental techniques for surface analysis on the atomic scale has clearly indicated the necessity to develop more refined atomistic models for heterogeneous surfaces. These models should be capable of including the characteristics of the adsorptive energy surface. The elementary steps of surface diffusion are jumps of adatoms to vacant sites in the vicinity. The correct description of such jumps is a quite delicate matter. It is intuitively obvious that the energy barrier between two adsorption sites is an important quantity that decisively controls adatom jumps. In essence, this energy barrier; † Presented at the Third International Symposium on Effects of Surface Heterogeneity in Adsorption and Catalysis on Solids, held in Poland, Auguest 9-16, 1998. * Corresponding authors. Departamento de Fı´sica, Universidad Nacional de San Luis, Chacabuco 917, 5700 San Luis, Argentina. Phone: 54-2652-425109. Fax: 54-2652-430224. E-mail: fnieto@ linux0.unsl.edu.ar. ‡ Max-Planck-Institut fu ¨ r Eisenforschung. § Universidad Nacional de San Luis. | Universita ¨ t Dortmund.
(1) Gomer, R. Rep. Prog. Phys. 1990, 53, 917. (2) Zgrablich, G. In Equilibria and dynamics of gas adsorption on heterogeneous solid surfaces; Rudzinski, W. W., Zgrablich, G., Eds.; Elsevier: Amsterdam, 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, V. P. Surf. Sci. Lett. 1985, 149, L13. (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.
which needs to be overcome by diffusing particles, is given by the energy difference between saddle point energy (i.e., the maximum potential energy along the trajectory of a jumping adatom) and the initial adsorption energy. This simple picture has motivated the development of an alternative description for heterogeneous surfaces in which the energy distributions for the two important elements, i.e., the adsorption sites and the saddle point energies (or bonds) of the periodic potential, are both taken into account.10-13 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 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 latticegas scheme.14-16 In the present work we aim to characterize surface diffusion for one of the simplest models of a heterogeneous surface, the well-known bivariate trap model. In this model it is assumed that the heterogeneous surface is formed by two different adsorption sites, deep and shallow traps, which are arranged at random. It is trivially expected that the presence of deep traps must influence surface diffusion. The intention of this work is to identify and characterize the most prominent features of surface diffusion processes for the simplest heterogeneous surface and to draw general conclusions on the effects of heterogeneities on surface diffusion that are helpful for the evaluation of experimental diffusion studies on heterogeneous surfaces. For this purpose we consider a highly idealized model, which is not meant to reproduce a particular experimental system. This lattice gas model (10) Riccardo, M. C. J. L.; Pereyra, V.; Zgrablich, G. Langmuir 1992, 8, 1518. (11) Mayagoitia, V.; Rojas, F.; Riccardo, J. L.; Pereyra, V.; Zgrablich, G. Phys. Rev. B 1990, 41, 7150. (12) Mussawisade, T. W. K.; Kehr, K. J. Phys. C 1997, 9, 1181. (13) Limoge, Y.; Boucquet, J. L. Phys. Rev. Lett. 1990, 65, 60. (14) Sapag, K.; Pereyra, V.; Riccardo, J. L.; Zgrablich, G. Surf. Sci. 1993, 295, 433. (15) Uebing, C. Phys. Rev. B 1994, 49, 13913. (16) Uebing, C.; Gomer, R. Surf. Sci. 1994, 306, 419.
10.1021/la9810962 CCC: $18.00 © 1999 American Chemical Society Published on Web 05/07/1999
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Langmuir, Vol. 15, No. 18, 1999 5877
and the Monte Carlo simulational technique used to obtain the desired diffusion quantities is described in section 2. Results are presented and discussed in section 3. Finally, we give our conclusions in section 4. 2. Basic Definitions and Simulational Details In this work we will consider a square lattice that is built up by only two types of adsorption sites, namely shallow and deep traps with adsorption energies S and D, respectively. The concentration of the deep traps is given by Θ. Thus, 1 - Θ is the concentration of the shallow traps. It is assumed that the deep traps are randomly distributed. All adsorption sites are separated by wells of the periodic potential. The saddle point energies (describing the wells which need to be overcome by diffusion adatoms) are uniformly given by a common value, SP, throughout the whole lattice. The relative minima of the periodic potential, i.e., the adsorption site energies, are given by
{
, i ) D, S
for deep traps for shallow traps
}
(1)
In the absence of ad-ad interactions the lattice-gas Hamiltonian can be written as N
H)-
∑i cii
(2)
The occupation of lattice sites by adsorbates is described by local occupation variables ci defined as
ci )
{
1, 0,
if site i is occupied if site i is vacant
}
(3)
Double occupancy of lattice sites is excluded. The Monte Carlo simulation of surface diffusion is performed in the canonical ensemble applying the Metropolis importance sampling algorithm.17,18 In our MC algorithm, the bivariate trap surface given by eq 2 is realized by a two-dimensional array of L × L sites with periodic boundary conditions. The site specific adsorption energies, i, are assigned at random according to the desired concentration Θ of deep traps. Initial lattice-gas configurations are generated by throwing θL2 particles at random on the surface. Here θ denotes the adsorbate coverage. We assume that the elementary steps of surface diffusion are jumps of adsorbed particles from occupied initial sites i to empty nearest neighbor sites j. In essence, the energy barrier that needs to be overcome by diffusing particles is given by the energy difference between saddle point energy (i.e., the maximum potential energy along the trajectory of a jumping adatom) and the initial adsorption energy,
∆i ) SP - i
(4)
Pi )
(5)
with κ as a normalization factor. κ essentially determines the time in which an adsorbed atom is allowed to attempt a jump, as explained in detail in ref 3. In order to optimize the computational time of a Monte Carlo algorithm, a suitable choice of κ is indispensable. An obvious choice would be
(
κ ) κmax ) exp -
)
∆i(min) kBT
(6)
Here ∆i(min) represents the activation energy for the most favorable physically realizable jump.8 This choice avoids jump events with Pi > 1 and has been used throughout this work. The jump algorithm used in the present work has been discussed in detail in ref 8 and will be summarized only briefly: first, an initial site i of the L × L lattice is picked at random; if it is filled, an adjacent final site j is randomly selected. If the destination is vacant, a jump can occur with the probability given by eq 5; otherwise no jump occurs. One Monte Carlo step (MCS) corresponds to L2 interrogations (in random order) of lattice sites. Before a diffusion run was started, a large number of initial MCS’s were performed to establish a desired temperature T and to reach thermodynamic equilibrium. As in ref 8, approach to equilibrium is monitored by following the total energy and is assumed to occur when this quantity fluctuates about an average value. The time (in units of MCS’s) needed for equilibration depends on lattice size, temperature, and coverage. Typically, 5 × 103 MCS are required to establish equilibrium in lattices containing up to 64 × 64 sites. In order to obtain accurate values of the desired surface diffusion coefficients (to be discussed below), diffusion runs of up to 6 × 104 MCS’s for up to 136 different initial configurations were performed. These simulations were carried out using the supermassive parallel Intel Paragon supercomputer of the Ju¨lich research center. After approaching thermodynamic equilibrium, we have measured the tracer surface diffusion coefficient D* by following the noncorrelated random-walk of N ) θL2 tagged particles. D* is defined through the generalized definition
D* ) lim tf∞
[2dt1 〈|RB (t) - RB (0)| 〉] 2
i
(7)
i
where d is the Euclidean dimension (in the case of surface diffusion d ) 2); the vector R B (t) determines the position of a tagged particle at time t, and (R B (t) - R B (0))2 is its mean square displacement, which is expressed in units of the lattice constant. The tracer diffusion coefficient is a single particle diffusion coefficient. However, in the course of Monte Carlo simulations it is quite useful to average over all N particles according to
The associated jump probabilities, Pi are given by8 (17) Metropolis, N.; Rosenbluth, A. W.; Rosenbluth, M. N.; Teller, A. H.; Teller, E. J. Chem. Phys 1953, 21, 1087. (18) Binder, K.; Stauffer, D. A simple Introduction to Monte Carlo Simulation and some specialized Topics. In Applications of the Monte Carlo Method in Statistical Physics; Binder, K., Ed.; Springer-Verlag: Berlin, 1984.
[ ]
∆i 1 exp κ kBT
D* ) lim tf∞
[
1
N
∑〈|RB i(t) - RB i(0)|2〉 2dNti)1
]
(8)
We note that the tracer diffusion coefficient can be defined as the product of a tracer correlation factor f,19,20 a vacancy
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Nieto and Uebing
availability factor V, and an average jump probability (Pj),21,22
D* ) fV〈Pj〉
(9)
The chemical diffusion coefficient D, which is a many particle diffusion coefficient, is determined via two different approaches, the fluctuation method and the KuboGreen method. In essence, the fluctuation method measures the particle number autocorrelation function, fn(t)/fn(0), for a small probe region embedded in the whole two-dimensional lattice. The ratio fn(t)/fn(0) is then compared with the theoretical curve,23,24 yielding D, which we call DF. Thus, this method is a computer simulation of the field emission fluctuation method23 used experimentally to determine adsorbate diffusion coefficients. For the autocorrelation function, we can write
fn(t) fn(0)
〈δN(t) δN(0)〉
)
(10)
〈(δN2)〉
Here N is the number of adatoms in the probe area. Details of this method are presented in refs 16 and 25. In the present work we use 8 × 8 and 16 × 16 square probes for the determination of DF. The second method for determining the chemical diffusion coefficient is based on the Kubo-Green equation, which we write here as26
(
)
∂[µ/kBT] DJ ∂ ln θ
DKG )
As in previous simulations7,8 calculations are carried out in terms of D0, the chemical diffusion coefficient for zero interactions between adsorbates on a homogeneous lattice (Langmuir gas). 3. Results and Discussion In this section we will describe how the presence of surface heterogeneity affects the thermodynamic and kinetic properties of our lattice-gas model. As already mentioned, the characteristic feature of the bivariate trap model to be considered here is the existence of two kinds of traps, namely shallow and deep traps, where the concentration of the latter is Θ. For such a lattice it is interesting to know how the adatoms are distributed over the deep and shallow trap sites. In the absence of ad-ad interactions the exact solution of this purely thermodynamical problem is trivially known,29 In fact, according to eq 2, we have considered a two-dimensional lattice gas where each site can be either occupied or empty with a probability governed by the adsorption energy. Thus, the model can be interpreted as two independent Langmuir gases, each one characterized by its own adsorption energy. Let us define θS and θD as the fraction of occupied shallow and deep trap sites, respectively. The total coverage θ is then given by
θ ) ΘθD + (1 - Θ)θS
The chemical potential of adatoms on the deep and shallow trap sites, µS and µD can be expressed as
(11) µS ) µo - S + kBT ln
Here µ is the chemical potential, 〈(δN) 〉 is the mean square number fluctuation in an area A containing 〈N〉 particles and DJ is the jump diffusion coefficient given by1,2,24,27,28 2
DJ ) lim tf∞
[
N
1
2dNt
〈(
|R B i(t) - R B i(0)|)2〉 ∑ i)1
]
(12)
The jump diffusion coefficient (sometimes also referred to as kinetic factor) is a many particle diffusion coefficient. The thermodynamic factor of eq 11 is obtained in either one of its two equivalent forms
(
) [
∂[µ/kBT] ∂ ln θ
]
〈(δN)2〉 ) 〈N〉 T
-1
(13)
either via the differentiation of adsorption isotherms obtained in the grand canonical ensemble or via the normalized mean square fluctuations 〈(δN)2〉/(N) obtained in the canonical ensemble. (19) LeClaire, A.D. In Physical Chemistry - An Advanced Treatise; Eyring, H., Henderson, D., Jost, W., Eds.; Academic Press: New York, 1970; Vol. 10. (20) Murch, G. E. Phil. Mag. 1981, A43, 871. (21) Murch, G. E.; Thorn, R. J. J. Phys. Chem. Solids 1977, 38, 789. (22) Kehr, K.; Binder, K. Simulation of Diffusion in Lattice Gases and Related Kinetic Phenomena. In Applications of the Monte Carlo Method in Statistical Physics; Binder, K., Ed.; Topics in Current Physics; Springer-Verlag: Berlin, 1987; Vol. 36, p 181. (23) Gomer, R. Surf. Sci. 1973, 38, 373. (24) Mazenko, G.; Banavar, J. R.; Gomer, R. Surf. Sci. 1981, 107, 459. (25) Uebing, C.; Gomer, R. Surf. Sci. 1994, 317, 165. (26) Reed, D. A.; Ehrlich, G. Surf. Sci. 1981, 102, 588. (27) Reed, D.A.; Ehrlich, G. Surf. Sci. 1981, 105, 603. (28) Mazenko, G. Surface mobilities on solid materials; Plenum Publishing Corp.; New York, 1983.
(14)
( ) ( )
µD ) µo - D + kBT ln
θS 1 - θS
(15)
θD 1 - θD
(16)
Here µo is the chemical potential of the noninteracting Langmuir gas. At thermodynamic equilibrium both chemical potentials are equal (µD ) µS) and we have
ln
[
]
θ D 1 - θS (D - S) ) 1 - θD θS kBT
(17)
Solving eqs 14 and 17 yields the coverage dependence of the site specific surface coverages, θS(θ) and θD(θ). These quantities are shown in Figure 1 for three representative temperatures expressed in units of (S - D)/kBT and three different values of Θ. There is an excellent agreement between Monte Carlo results (symbols) and theoretical values calculated according to eqs 14 and 17. This agreement is trivially expected and the only justification to run Monte Carlo simulations is to test the correctness of the algorithm and the statistical accuracy of the sampling. For a given total coverage θ the coverage of the deep traps, θD, is always substantially larger than θS (Figure 1). Especially at low temperatures and low adatom concentrations, most of the adatoms are preferably located at deep trap sites. At low temperatures, i.e., (S D)/kBT ) 4.82, the saturation of the deep traps, θD ≈ 1, is established at θ J Θ, while the shallow traps remain empty, θS ≈ 0, for θ j Θ. (29) Hill, T. L. An Introduction to Statistical Thermodynamics; Addison-Wesley Series in Chemistry; Addison-Wesley; Reading, MA, 1960.
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Figure 1. Site specific coverage, θS (filled symbols) and θD (empty symbols), versus total coverage θ for three different concentrations of the deep traps Θ: (a) Θ ) 0.1, (b) Θ ) 0.5; (c) Θ ) 0.9. Results are shown for three characteristic temperatures, expressed in terms of (S - D)/kBT. Symbols denote Monte Carlo results, while solid lines represent thermodynamic calculations according to eqs 14 and 17.
Next we proceed to the analysis of the coverage dependence of the normalized tracer diffusion coefficient. Figure 2 shows our Monte Carlo results for D*/Do for different values of T and (Θ, respectively. From a first inspection of Figure 2 it is intuitively obvious that the effect of the surface heterogeneity is markedly pronounced at low temperatures and low surface coverages. At relatively high temperatures, (S - D)/kBT ) 1.20, the tracer diffusion coefficient decreases monotonically upon increasing the total surface coverage θ. Even at low concentrations of deep traps, i.e., Θ ) 0.1 (Figure 2a), the absolute values of D* are slightly reduced with respect to the Langmuir case (dashed line in Figure 2a), indicating that diffusion is slowed down as adatoms are adsorbed at deep traps. The deviations from Langmuir behavior become even more pronounced as the concentration of deep traps, Θ, is increased (Figure 2b,c). We note that for the Langmuir case, which is to be expected as T f ∞ (noninteracting limit), the tracer diffusion coefficient is given by
D* ) f(1 - θ) D0
(18)
Here f is the tracer correlation factor.19,20 It is f ) 1 for θ f 0. Upon decreasing the temperature, the effect of the surface heterogeneity becomes more pronounced as more and more adatoms are trapped at deep traps. At low coverages, θ j Θ, where most of the adatoms are trapped, the diffusion process is dominated by adatom jumps out of deep traps. Under this circumstances the average jump probabilities 〈PJ〉 are low and essentially determine the behavior of D*. After saturation of the deep traps (i.e., at θ ≈ Θ at low temperatures) the fraction of adatoms on shallow trap
Figure 2. Normalized tracer diffusion coefficient, D* for three different values of the deep traps concentration Θ versus the total coverage θ; (a) Θ ) 0.1; (b) Θ ) 0.5; (c) Θ ) 0.9. Shown are results for different temperatures expressed in terms of (S - D)/kBT. The insets show the curve corresponding to the lowest temperature value considered, (S - D)/kBT ) 4.82. The dashed line corresponds to the Langmuir case. As in previous studies,8.31 the diffusion coefficients are normalized with respect to D0, the chemical diffusion coefficient of a Langmuir gas.
sites increases and D* goes through a relative maximum (at least for Θ ) 0.1 and 0.5 (Figure 2a,b) while for Θ ) 0.9 there is a cusp (Figure 2c)). It is clear that the occupation of shallow trap sites (after saturation of the deep ones) causes an increase of the average jump probability, 〈PJ〉, which accounts for the increase of D* at θ J Θ. The mobility of such adatoms is substantial larger, and therefore, it can be concluded that mobile adatoms on shallow trap sites dominate surface diffusion at intermediate coverages, θ ≈ Θ. For large values of θ, however, the vacancy availability factor, V, goes to zero and dominates the tracer diffusion coefficient as θ f 1. Figure 3 presents the coverage dependence of the jump diffusion coefficient, DJ, given by eq 12. It is quite obvious and expected that D* and DJ behave in a strikingly similar way. In particular, it is known that they are numerically equal if there are no velocity-velocity cross correlation terms.24 However, it is interesting to note that they represent different views of the diffusive phenomenon. In fact, the tracer diffusion coefficient describes the motion of tagged particles on the surface, while the jump diffusion coefficient represents the mobility of the center of mass of the system. In Figure 4 we present the coverage dependence of the chemical diffusion coefficient calculated via two different methods, the fluctuation (open symbols) and Kubo-Green method (filled symbols), respectively. At low values of Θ both methods show an excellent agreement. However, significant discrepancies appear for larger values of Θ at low temperatures (see the inset of Figure 4b,c). Discrepancies between DKG and DF have already been observed for diffusion on homogeneous surfaces in the presence of ad-ad interactions causing first-order phase transitions7 or upon inclusion of the effects of the energetic topography
5880 Langmuir, Vol. 15, No. 18, 1999
Figure 3. As Figure 2, for the jump diffusion coefficient, DJ.
Figure 4. Chemical diffusion coefficient calculated by the fluctuation method, DF (open symbols), and by the Kubo-Green method, DKG (filled symbols), as a function of coverage for three different values of the deep trap concentration Θ as indicated. Temperature is expressed in terms of (S - D)/kBT.
are included.30 In a recent work, where a random distribution of deep traps with Θ ) 0.2 was considered, such discrepancies were not observed.31 An overall consistent explanation of these findings has been given in ref 30: the fluctuation method fails when the applicable length scale of the lattice-gas system becomes comparable (30) Uebing, C.; Pereyra, V.; Zgrablich, G. Surf. Sci. 1996, 366, 185. (31) Viljoen, E.; Uebing, C. Langmuir 1997, 13, 1001.
Nieto and Uebing
Figure 5. Adsorption isotherms, surface coverage θ vs normalized chemical potential µ/kBT, for three different values of the deep trap concentration Θ as indicated. The different curves are labeled according to their temperature (expressed in terms of (S - D)/kBT).
to the probe dimension, or in other words, when the probe misses the long wavelength fluctuations of the particle density. This argument is reinforced by previous results32 indicating that discrepancies between DKG and DF in the presence of phase transitions decrease when the size of the probe area used for the calculation of DF 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. At high temperatures the chemical diffusion coefficient does not depend much on coverage (Figure 4). This behavior is similar to that of the homogeneous (i.e., Langmuir) case. However, the absolute values of DKG are reduced with respect to the Langmuir gas due to the presence of adatoms adsorbed at deep traps. A very different situation is observed when the temperature is decreased. For small values of θ the chemical diffusion coefficient is relatively low but exhibits a monotonic increase and finally approaches a saturation value as θ f 1. At (S - D)/kBT ) 4.82, the lowest temperature considered in the present work, an almost stepwise increase of the chemical diffusion coefficient DKG is seen at θ ≈ Θ. As already mentioned, the chemical diffusion coefficient can be expressed as a product of a kinetic (the jump diffusion coefficient) and a thermodynamic factor (eq 11). The latter can be obtained by the differentiation of adsorption isotherms such as shown in Figure 5. The result of this procedure is shown in Figure 6 and clearly demonstrates that the thermodynamic factor presents a sharp peak at θ ≈ θ, which is largely pronounced at low temperatures and represents the sequential occupation of the two different adsorption sites. This sharp peak at (32) Uebing, C.; Gomer, R. J. Chem. Phys. 1994, 100, 7759.
Diffusion on Simple Heterogeneous Surfaces
Figure 6. Thermodynamic factor as a function of coverage for three different values of the deep trap concentration Θ as indicated. Temperature is expressed in terms of (S - D)/kBT.
Langmuir, Vol. 15, No. 18, 1999 5881
Figure 8. Arrhenius plot of the surface diffusion coefficients at θ ) 0.5 for different values of the deep trap concentration Θ as indicated: (a) D*, (b) DJ and (c) DKG.
Figure 7. Chemical, DKG (open symbols), and jump, DJ (filled symbols), diffusion coefficients as a function of deep trap concentration, Θ, for three different values of the total coverage, θ: (a) θ ) 0.02, (b) θ ) 0.5; (c) θ ) 0.9. Results are shown for three characteristic temperatures expressed in terms of (S D)/kBT as indicated.
≈Θ in conjunction with the wide peak of the jump diffusion coefficient DJ at θ > Θ (Figure 3) results in the observed steplike behavior of the chemical diffusion coefficient, DKG (Figure 4). In order to rationalize this peculiar coverage dependence of the chemical diffusion coefficient, DKG(θ), note that this abrupt change separates two diffusion regimes that are characterized by different diffusion mechanisms. In fact, at low coverages the chemical diffusion coefficient is controlled by the diffusion of adatoms on deep traps, while at high coverages the behavior of this quantity is dominated by the diffusion of mobile adatoms on shallow traps. Figure 7 shows the behavior of the jump and the chemical diffusion coefficients as the deep trap concentration increases for a fixed value of the total coverage. In the limits of Θ ) 0 or 1 the surface is homogeneous and consists of either shallow or deep traps, respectively, and the diffusion behavior is trivially known. The curves show the diffusion coefficients of the heterogeneous system in
Figure 9. Effective activation energies for the chemical surface diffusion coefficient, DKG, normalized with respect to S - D. The effective activation energies are obtained by differentiating the Arrhenius plots of Figure 8. The dashed lines correspond to the mean field result given by eq 19.
which more and more impurities (deep traps) are added for a fixed value of total coverage θ. Three representative values of θ have been selected in order to characterize the behavior at different coverage regimes. The log scale was chosen in order to emphasize the difference between the jump and the chemical diffusion coefficient, which is given by the thermodynamic factor (eq 11). At high temperatures, i.e., (S - D)/kBT ) 1.20, there are linear depend-
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ences of DKG and DJ with Θ for all surface coverage θ. Upon a decrease in temperature, important deviations from the linear response are seen, which become more pronounced as the temperature is decreased. This idea is reinforced in Figure 8 where the Arrhenius behavior of D*, DJ, and DKG is shown for a fixed total coverage, θ ) 0.5. At high temperatures (1/kBT f 0) the general behavior is Arrhenian, but strong deviations appears as the temperature decreases. These deviations are due to the surface heterogeneity considered. In Figure 9 we show effective activation energies for the chemical diffusion coefficient DKG normalized with respect to S - D. At high temperatures, i.e., (S - D)/kBT ) 1.20, the activation energies vary almost linear according to
Eeff ) -Θ|D - S|
(19)
Deviations from this “mean field” result (dashed line in Figure 9) are present at low temperatures, especially for small concentrations of deep traps, i.e., Θ ) 0.02 (Figure 9a). 4. Conclusions In the present work we have used the bivariate trap model in order to study how surface heterogeneities 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 surface heterogeneities strongly influence the diffusive process. The effects are most pronounced at low temperatures. At low surface coverages, i.e., θ j Θ, the diffusion process is dominated by adatoms adsorbed on deep traps. Under these circumstances the measured values of D*, DJ and DKG are very low. The effect of deep traps becomes more and more negligible as the coverage increases. At high coverages, i.e., θ J Θ, the mobility of adatoms adsorbed on shallow traps dominates the diffusion behavior. The transition between these two different regimes is clearly reflected by the diffusion coefficients. The tracer and jump diffusion coefficients pass through a relative maximum and the chemical diffusion coefficient exhibits a steplike increase around θ ≈ Θ. Acknowledgments This work has been supported by the Deutsche Forschungsgemeinschaft (DFG) and by the International Association for the promotion of cooperation with scientists from the New Independent States of the former Soviet Union INTAS-96-0533. LA9810962
