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In this article we investigate the complex 1D mesoscopic model of adatom diffusion and the evolution of an ordered phase on the substrate surface. The analysis of the theoretical model is compared with the experimental results of the spreading of Au adatoms on Si(111)-(7×7) surface. The steady state solutions and their stability conditions are determined within the concept of the traveling-wave solution. It is shown that the formation of the ordered phase (5×2) and the difference in the diffusion of Au on (7×7) and on (5×2) structure results in a sharp edge of diffusion front which corresponds to the coverage of a saturated (5×2) phase. This edge moves linearly in time and α can be determined by experiment. The system of model equations enables the damped waves solution or temporary evolution of two steps.

We use Monte Carlo simulations to study the mobility and collective diffusion of mutually interacting adparticles on stepped surfaces. We consider the case of nearest–neighbor repulsive interactions together with different binding and activation energies at the step edges in the disordered phase. To complement previous work that focused on diffusion across steps, here we concentrate on diffusion along the steps in the case of strong step binding. We find that the collective diffusion behavior is dictated by a complex interplay between enhanced or suppressed diffusion along the lower step edge positions, binding at step edges, and interactions between adparticles. Diffusion along the lower step edge positions plays a particularly important role at low coverages, while at high coverages diffusion is governed mainly by adparticle interactions.

We study non-equilibrium effects in spreading and collective diffusion of adatoms on stepped surfaces through Monte Carlo simulations of a lattice-gas model. The spreading density profiles are analyzed by the Boltzmann–Matano method to determine the temporal behavior of the effective collective diffusion coefficients. We find that the presence of steps induces considerable non-equilibrium effects in diffusion. For spreading along the steps, we find that these deviations can be explained by the slow approach of the different adparticle concentrations on terraces and at step edges towards equilibrium. For spreading across the steps, however, we find no such dependence, indicating the breakdown of the linear response theory at early times.

Surface diffusion is studied in a two barrier system, i.e. a system with two different activation energies. The dynamic and thermodynamic properties of the system of interacting ad-atoms on an fcc(1 1 1) surface are studied using Monte-Carlo simulations, the Bortz–Kalos–Lebowitz algorithm and the Metropolis algorithm. The results are compared with our previous analytical calculations in the same system. Numerical results fit well to the analytical results for high temperatures. At low temperatures, contrary to previous analytical calculation (based on quasi-chemical approximation), we observe a sharp local minimum at a coverage Θ =0.5 in the case with repulsive interaction and a large difference between activation energies. The Monte-Carlo simulations result in a higher occupation of deep sites, formation of an ordered phase and in lower diffusion coefficient in comparison with the analytical results.

The collective diffusion coefficient is studied for a system of particles adsorbed on a surface with two non-equivalent adsorption sites. The non-equilibrium conditions are characterized by the non-equilibrium distribution of atoms between adsorption sites with different energy of adsorption. The determination of the diffusion coefficient is based on the general theory published in [J. Phys. Condens. Matter. 12 (2000) 6343] and links to the former studies of the diffusion coefficient and phase diagrams of the same systems in equilibrium. Our analysis shows that non-equilibrium conditions strongly influence the step-like dependence of diffusion coefficient on coverage, as determined in equilibrium conditions for non-interactive adparticles, and that the diffusion coefficient can even have an opposite behavior. Far from equilibrium the appearance of a local maximum is observed. These deviations reflect the influence of the state of the adparticle system on their collective behavior.

Despite its importance in theoretical studies of surface diffusion there has been limited experimental progress in developing methods to measure the dynamic structure factor S(q,t). In this work we study gases adsorbed on single-crystal surfaces and show that their equilibrium dynamics is measurable in a diffraction experiment through density fluctuations described by the dynamic correlation function . Using a lattice-gas model with nearest-neighbor repulsive interaction, we demonstrate how to separate out the microscopic dynamics caused by diffusion and adsorption-desorption from the form of . Our analytical results show that decays as an exponential with a time constant {tau}(q). Significant deviations from the hydrodynamic dependence of 1/{tau}(q)=D{sub c}q{sup 2} are found with and without desorption. These deviations have important consequences in extracting diffusion constants from experiments. (c) 2000 The American Physical Society.

The interrelation between the surface symmetry and diffusion of adsorbed atoms is studied theoretically on a model face-centred cubic (fcc) (111) surface within the lattice-gas approximation. Diffusion proceeds through two non-equivalent sites on the fcc(111) surface. Short-range repulsive interaction between particles in the nearest-neighbour non-equivalent sites is considered. This repulsion leads to a modification of the occupation numbers of both kinds of site and to changes in the energy barrier height for diffusion. The chemical surface diffusion coefficient, D, has been found to be strongly dependent on coverage, Θ, the interaction strength between adatoms, and the difference between the activation energies, ΔE. In the case of a small ΔE value, the diffusion coefficient D increases with Θ, as was expected for repulsive interaction. In our model the increase of D at Θ=0.5 is reduced for higher values of ΔE and a local maximum of D(Θ) is observed.

We study the temperature and coverage dependence of an effective diffusion barrier assuming an Arrhenius shape of the chemical diffusion coefficient for a system of interacting particles. The previously published model of the diffusion of an fcc (111) surface with bivariate trap is used. The presence of two nonequivalent occupation sites and interaction result in a non-Arrhenius shape of the diffusion coefficient and a coverage- and temperature-dependent effective diffusion barrier. The temperature dependence of effective energy Ea shows a minimum at low temperatures (at approximately 150 K) for strong interactions. The coverage dependence of Ea has a deep minimum in the vicinity of Θ=1/2, even in a system without interaction.

Diffusion on a fcc(111) surface will be studied theoretically. Diffusion proceeds through non-equivalent hcp and fcc sites with different adsorption energy. Repulsive interaction between particles in the first neighbour sites and change of saddle point energy due to interaction are included. In this model the repulsion to the next neighbour site results in the inadequately increased diffusion coefficient. The large difference of adsorption energy between hcp and fcc sites depresses diffusion at higher coverages and diffusion coefficient has maximum at Θ=1/2. Inclusion of the change of the saddle point energy reduces increase of D and results in a tame local minimum, emphasizing a local maximum.

This paper deals with the determination of the tracer (DT ) and chemical (Dch ) surface diffusion coefficients in the presence of a gradient of the coverage . The lattice gas model and quasi-chemical approximation are taken as accepted in the theory. The results are discussed with respect to the interaction energy of particles, which influences the equilibrium energy of atoms as well as the saddle-point energy. Such interactions break the symmetry of jumps in the systems with gradient grad . This model predicts a decrease of DT with the square of the gradient of the coverage .Dch depends on the coefficient of proportionality of the difference between the mean jump rate in the direction of grad and that in the opposite direction. It has been found that, in the case of repulsive interaction, the coverage dependencies of DT and Dch have local maxima, whose positions depend on the rates of change of the saddle-point energy and of the equilibrium energy of the atoms due to the interaction. For attractive interaction, DT either decreases with or increases depending on the saddle-point energy changes. At low temperatures our results differ substantially from those of the calculations made within dynamical mean-field theory.

The effective energy barrier for diffusion, E a , has been previously determined as a function of coverage and temperature within a two-barrier model of diffusion, assuming an Arrhenius form of chemical diffusion coefficient. To determine the diffusion coefficient at a given temperature and coverage, we used previously published results based on the lattice gas model of non-interacting adatoms. The presence of two non-equivalent sites for adsorption and diffusion of atoms on a surface (with energy barriers E 1 , E 2 ) results in substantial changes to E a within coverage Θ , with a step-like shape having a value close to Θ =1/3. The temperature dependence of E a shows a slight minimum, and the Arrhenius shape (with constant E a ) seems to be good approximation of the relatively small differences in energy barriers E 1 , E 2 . We obtain considerable deviation from such an Arrhenius behaviour of diffusion when there is a large difference in the energy barriers.

We calculated the surface diffusion coefficient D as a function of coverage in the quasi-chemical approximation of a system of particles on a square lattice with two non-equivalent binding sites characterized by different activation energies E1 and E2 for diffusion. A short-range repulsive interaction between particles in the nearest neighbour non-equivalent sites is considered, resulting in the modification of these energies by δE. The thermodynamic and kinetic parts of D have been calculated separately, in order to estimate their contribution to the diffusivity at different concentrations. It has been found that D exhibits a step-like increase at a critical coverage Θc which varies from Θc=0.33 for δE=0 to Θc=0.5 for E1=E2. The origin of the concentration dependence of D is discussed in terms of the repulsion strength between particles and their jump-rate through the sites existing on the surface.

The chemical surface diffusion coefficient D has been determined for a lattice-gas model where diffusion proceeds through two non-equivalent lattice sites with different activation energies, E1 and E2. By calculating the thermodynamic factor as well as jump diffusion rate we obtain the coverage dependence of D as a function of the energy difference E1 − E2. No lateral interactions are considered. Depending on the sign of E1 − E2, the diffusion coefficient shows either a maximum or a step-like increase at a particular coverage, both of which develop into discontinuities in the limits of either E1 − E2 → ± ∞. In the case of the step-like behaviour of D, the variation of D can be classified as being due to a kind of diffuse phase transition between a low and a high diffusivity state, which in the limits is shown to be of first order.

By calculating the jump diffusion rate (Г) and the thermodynamical factor (Kd) for a layer of non-interacting particles on a lattice with the non-equivalent bond sites we show that the diffusion coefficient D = ГK d exhibits specific features as a function of coverage, such as steps or maxima at critical coverages, which correspond to phase transformations in the adlayer. The chemical potential of the adlayer is derived analytically from the free energy. The difference of binding energy between the two kinds of sites is described by a parameter k and it is shown that the chemical potential (or the thermodynamic factor) at the critical coverage indicates the existence of a diffuse phase transition which develops for k → 0 into a first-order point phase transition.

The capability of the laser-induced thermal desorption technique (LITD) to detect the migration of adsorbed oxygen on a semiconductor surface is demonstrated. The surface diffusion coefficient of oxygen on Ge{100}(2 × 1) has been measured for a constant initial coverage in a wide temperature range (220–650 K). The experimental results show that after an initial stage of fast refilling of the depleted area the diffusion process strongly slows down. In the first stage the data can be fitted by the solution of Fick's second law. Within this approximation of a constant diffusion coefficient, an extremely low activation energy (∼ 0.04 eV) is obtained. The observations suggest that the second stage in the refilling process is connected with an irreversible chemical reaction between adsorbate and substrate.

An analytical expression for the surface diffusion coefficient is derived by summing the fluxes due to single jump events between adjacent sites on a simple square lattice. All microscopic configurations of a layer of interacting particles are included at particular coverages. In a quasi-chemical approach, the nearest-neighbour particle-particle interactions are introduced by adjusting the jump rate as well as by adapting the probability of occurrence of particular configurations consistent with the concentration. The diffusion coefficient is found to be dominated by the latter contribution at medium and high concentrations. A comparison with the results of Monte Carlo simulations shows that an improved approximation can be achieved if the interactions are accounted for not only by changes of the activation energy of diffusion but also by their effect on the equilibrium distribution of the particles.

A dynamical description of surface diffusion is presented for a surface containing two non-equivalent bonding sites with different binding energies. Diffusion proceeds via single hopping events between adjacent sites which can be of both kinds. Dynamical coupling is introduced by assuming that the hopping rates into and out of one type of site depend on whether the adjacent site of the other kind is occupied or empty. The derivation starts out from the microscopic difference equations accounting for the single jump events for all possible configurations. These are then transformed into differential equations with continuous functions and solved by the method of finite differences. The activation energies for individual jumps serve as adjustable parameters. It is shown that the widely observed dependence of the diffusion coefficient on coverage also follows naturally from this model, although direct particle-particle interactions in the usual sense are specifically excluded. The influence of the substrate-adsorbate interaction and of the surface geometry on the diffusion process is also discussed.

Surface diffusion of K atoms on Pd 111 has been measured by photoemission electron microscopy in a wide temperature range (150–450 K) for coverages η

We study the diffusion of Brownian particles with a short-range repulsion on a surface with a periodic potential through molecular dynamics simulations and theoretical arguments. We concentrate on the behavior of the tracer and collective diffusion coefficients DT(θ) and DC(θ), respectively, as a function of the surface coverage θ. In the high friction regime we find that both coefficients are well approximated by the Langmuir lattice-gas results for up to θ≈0.7 in the limit of a strongly binding surface potential. In particular, the static compressibility factor within DC(θ) is very accurately given by the Langmuir formula for 0⩽θ⩽1. For higher densities, both DT(θ) and DC(θ)show an intermediate maximum which increases with the strength of the potential amplitude. In the low friction regime we find that long jumps enhance blocking and DT(θ) decreases more rapidly for submonolayer coverages. However, for higher densities DT(θ)/DT(0) is almost independent of friction as long jumps are effectively suppressed by frequent interparticle collisions. We also study the role of memory effects for many-particle diffusion.

We study the submonolayer diffusion of hard disks and rodlike molecules on smooth surfaces through numerical simulations and theoretical arguments. We concentrate on the behavior of the various diffusion coefficients as a function of the two-dimensional (2D) number density rho in the case where there are no explicit surface-particle interactions. For the hard disk case, we find that while the tracer diffusion coefficient D(T)(rho) decreases monotonically up to the freezing transition, the collective diffusion coefficient D(C)(rho) is wholly determined by the inverse compressibility which increases rapidly on approaching freezing. We also study memory effects associated with tracer diffusion, and present theoretical estimates of D(T)(rho) from the mode-mode coupling approximation. In the case of rigid rods with short-range repulsion and no orientational ordering, we find behavior very similar to the case of disks with the same repulsive interaction. Both D(T)(rho) and the angular diffusion coefficient D(R)(rho) decrease with rho. Also in this case D(C)(rho) is determined by inverse compressibility and increases rapidly close to freezing. This is in contrast to the case of flexible chainlike molecules in the lattice-gas limit, where D(C)(rho) first increases and then decreases as a function of the density due to the interplay between compressibility and mobility.

I: Surface Diffusion at Equilibrium. Quasielastic Helium Scattering Studies of Surface Diffusion: The Diffusion Barrier A.P. Graham, J.P. Toennies. Experimental Aspects of Metal Particle Diffusion on a Silicon Surface J. Slezak, et al. Surface Dynamics of Stepped Si(001) Studied by Temporal LEED Spectroscopy K. Kammler. Electric Current Induced Asymmetry of Surface Diffusion - Instability of Vicinal Crystal Surfaces S.S. Stoyanov, et al. Memory Effects and Memory Functions in Surface Diffusion T. Hjelt, et al. Driven Diffusion in a Model of the O/W(110) System M.A. Zaluska-Kotur, et al. Surface Diffusion and Real-Space Renormalization Group A.A. Tarasenko, et al. Surface Diffusion Near Phase Transitions F. Nieto, et al. II: Surface Diffusion from Profile Evolution Methods. Surface Heterodiffusion in Adsorbed and Coadsorbed Overlayers of Li, Sr, and Cu on the W and Mo (112) Surfaces A.T. Loburets, et al. Metal Heterodiffusion on Metallic Surfaces: Case of Pb on Cu C. Cohen, et al. Step Effects on Surface Diffusion: A Comprehensive Study of CO on Pt(111) X. Xiao. Adsorption, Desorption, and Diffusion of Potassium on Metal and Oxide Surfaces G. Kerner, et al. III: Surface Diffusion Under Non-equilibrium Conditions. Dynamics of Interacting Adparticles Under Non-Equilibrium Conditions Z. Chvoj. Statistical-Mechanical Description of Nonequilibrium Processes in Interacting Lattice Gases V.S. Vikhrenko, et al. Model Studies of Collective Diffusion I. Vattulainen. Study of Non-Equilibrium Surface Diffusion From an Initial Step-Concentration Profile With a New Probability-Based Continuum Method M.R. Dudek, M.C. Tringides. IV: Diffusion Mechanisms in Growth Processes. Diffusion on and IN Surfaces: The Atomic Slide Puzzle J.W.M. Frenken, et al. Decay of Two-Dimensional Islands on Ag(110) K. Morgenstern, et al. Surface Morphology and dynamics: Using Ab-initio Total Energies to make the Most of STM Data P.J. Feibelman. Substrate-Mediated Interaction on Ag(111) Surfaces From First Principles K.A. Fichthorn, M. Scheffler. Diffusion of Adatoms and Small Clusters on Missing-Row-Reconstructed Surfaces F. Montalenti, R. Ferrando. Effects of Adsorbates on Submonolayer Growth M. Kotrla, et al. Surfactants in Semiconductor Heteroepitaxy: Thermodynamics and/or Kinetics? I. Markov. V: Substrate Effects and Surface Diffusion. Nonlinear Diffusion and Sliding Friction S.C. Ying, et al. Selective Activation Dynamics: A Minimal Path Approach L.Y. Chen, et al. Long Jumps in Surface Diffusion Investigated by Stochastic Equations R. Ferrando, et al. Analytical Kinetic Theory of Single-Particle and Collective Surface Diffusion S.Y. Krylov. Adatom Island Diffusion on Metal FCC(100) Surfaces O.S. Trushin, et al. Paths, Barriers, and Prefactors for Adatom Descent from Ag Clusters on Ag(111) T.S. Rahman, et al. Correlated Diffusion of Adatoms and Islands on the Low indexed Surfaces of Noble Metals and Alloys A.E. Evangelakis.

The direct evolution of submonolayer two-dimensional Au phases on the $\mathrm{Si}(111)\ensuremath{-}(7\ifmmode\times\else\texttimes\fi{}7)$ surface was studied in real time using the spectroscopic photoemission and low energy electron microscope located at the synchrotron radiation source ELETTRA. A finite area covered by 1 monolayer (ML) of gold with a steplike transition zone was prepared by evaporation in situ. Subsequent annealing resulted in the spread of the Au layer and the formation of laterally extended $\mathrm{Si}(111)\ensuremath{-}(5\ifmmode\times\else\texttimes\fi{}1)\ensuremath{-}\mathrm{Au}$ and $\mathrm{Si}(111)\ensuremath{-}(\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3})R30\ifmmode^\circ\else\textdegree\fi{}\ensuremath{-}\mathrm{Au}$ surface reconstructions. At a temperature around 970 K, the boundary of the gold-covered region propagates on the clean $\mathrm{Si}(111)\ensuremath{-}(7\ifmmode\times\else\texttimes\fi{}7)$ and exhibits a nonlinear dependence on time. The ordered $\mathrm{Si}(111)\ensuremath{-}(5\ifmmode\times\else\texttimes\fi{}1)\ensuremath{-}\mathrm{Au}$ plateau develops a separated front moving with constant velocity. Two values of the Au diffusion coefficients were estimated at a temperature of about 985 K: (1) ${D}_{7\ifmmode\times\else\texttimes\fi{}7}=5.2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}8}{\mathrm{cm}}^{2}{\mathrm{s}}^{\ensuremath{-}1}$ as the average diffusion coefficient for Au on a clean $\mathrm{Si}(111)\ensuremath{-}(7\ifmmode\times\else\texttimes\fi{}7)$ surface in the concentration range from 0.4 ML up to 0.66 ML and (2) ${D}_{5\ifmmode\times\else\texttimes\fi{}1}=1.2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}7}{\mathrm{cm}}^{2}{\mathrm{s}}^{\ensuremath{-}1}$ as the lower limit for the diffusion of single Au atoms on the $\mathrm{Si}(111)\ensuremath{-}(5\ifmmode\times\else\texttimes\fi{}1)\ensuremath{-}\mathrm{Au}$ ordered phase.

The surface diffusion of potassium on Pd{111} has been studied with photoelectron emission microscopy (PEEM) for coverages up to one monolayer. The coverage dependence of the chemical diffusion coefficient is determined by analysis of the concentration profiles obtained from the PEEM images with the Boltzmann–Matano method. The diffusion coefficient, D, decreases with increasing coverage but a local maximum is found at a coverage of Θ≈0.5 ML. The values of D at low coverages (Θ

The authors consider the influence of steps and nonequilibrium conditions on surface diffusion in a strongly interacting surface adsorbate system. This problem is addressed through Monte Carlo simulations of a lattice-gas model of OW(110), where steps are described by an additional binding energy EB at the lower step edge positions. Both equilibrium fluctuation and Boltzmann-Matano spreading studies indicate that the role of steps for diffusion across the steps is prominent in the ordered phases at intermediate coverages. The strongest effects are found in the p(2x1) phase, whose periodicity Lp is 2. The collective diffusion then depends on two competing factors: domain growth within the ordered phase, which on a flat surface has two degenerate orientations [p(2x1) and p(1x2)], and the step-induced ordering due to the enhanced binding at the lower step edge position. The latter case favors the p(2x1) phase, in which all adsorption sites right below the step edge are occupied. When these two factors compete, two possible scenarios emerge. First, when the terrace width L does not match the periodicity of the ordered adatom layer (LLp is noninteger), the mismatch gives rise to frustration, which eliminates the effect of steps provided that EB is not exceptionally large. Under these circumstances, the collective diffusion coefficient behaves largely as on a flat surface. Second, however, if the terrace width does match the periodicity of the ordered adatom layer (LLp is an integer), collective diffusion is strongly affected by steps. In this case, the influence of steps is manifested as the disappearance of the major peak associated with the ordered p(2x1) and p(1x2) structures on a flat surface. This effect is particularly strong for narrow terraces, yet it persists up to about L approximately 25Lp for small EB and up to about L approximately 500Lp for EB, which is of the same magnitude as the bare potential of the surface. On real surfaces, similar competition is expected, although the effects are likely to be smaller due to fluctuations in terrace widths. Finally, Boltzmann-Matano spreading simulations indicate that even slight deviations from equilibrium conditions may give rise to transient peaks in the collective diffusion coefficient. These transient structures are due to the interplay between steps and nonequilibrium conditions and emerge at coverages, which do not correspond to the ideal ordered phases.

We study collective diffusion of adsorbed particles on stepped surfaces using analytic and numerical techniques. We employ the Langmuir lattice gas model where the distribution of adatoms on the surface is solely determined by the difference in adsorption energy of atoms on terraces and along step edges. For the system in equilibrium, we consider the master equation approach for collective diffusion across the steps within the dynamic mean field approximation. We demonstrate that results obtained for the collective diffusion coefficient Dc(Θ) are sensitive to the choice of relevant slow variables for inhomogeneous systems such as stepped surfaces. Next, we consider diffusion across steps in situations where the system is not in equilibrium such as during spreading or ordering. To this end, we consider a phenomenological theory using balance between particle fluxes across a stepped surface within the linear response theory. This allows us to derive expressions for effective diffusion coefficients in the limit of large and small coverages, where the results agree well with Dc(Θ) in the corresponding limits.

The diffusion of single Pb atoms and the formation of stable clusters containing two and three Pb atoms on a Si(111)−(7×7) reconstructed surface was studied with variable temperature scanning tunneling microscopy (STM). The numbers of single Pb atoms and Pb atoms in clusters were monitored as a function of time. A new and simple statistical model describing diffusion was developed. This model enables us to estimate the diffusion coefficient and the activation energy of diffusion. The main advantage is the avoidance of questionable counting of jump events in successive STM images. For substrate temperatures of 308, 311, and 313 K we obtained diffusion coefficients of 8.1×10−18, 1.2×10−17, and 1.9×10−17 cm2 s−1, respectively, and the activation energy of diffusion equaled E=0.56±0.07 eV. The energy difference ΔE between the binding energies of single Pb atoms at faulted and unfaulted halves of the Si(111)−(7×7) unit cells was estimated to be 27±9 meV.