Motion of an adsorbed atom: theory and numerical results

Abstract

Abstract A theory for the motion of an atom adsorbed on a solid surface is developed, 1 based on the Zwanzig-Mori approach and on nonlinear mode coupling ideas. It generalizes the Fokker-Planck equation by explicitly taking into account the appropriate correlations at different times of the fluctuating force on the adatom. These depend on how the excitations in the substrate propagate and decay as well as on the motion of the adatom. They are included in an approximative way in a general frequency and wavevector dependent friction coefficient, called the memory function, which due to lack of translation invariance also depends on the position of the adatom. Besides this a static, conservative potential enters, which is periodic along the surface and is temperature dependent. Both are calculated, given the bare interaction potentials of the constituents. The general theory is applied to the motion of an atom adsorbed on a fcc (001) surface, where the bare interactions are Lennard-Jones 6–12 potentials and where the excitations of the substrate are phonons. 2 The adatom is restricted to follow a certain trajectory corresponding to a minimum energy path but the full three dimensional motion of the substrate atoms is retained. We do not believe that the conclusions we draw from our numerical results in any essential way depend on this restriction. The validity of earlier more approximative theories is examined and detailed numerical comparisons are made through calculating the velocity correlation function, its frequency spectrum and the dynamic structure factor for the ad-atom. It is found that, 1 when the adatom is similar to the substrate atoms with respect to mass and interaction, the frequency and wavevector dependence in the memory function can be ignored and one recovers the conventional Fokker-Planck equation with one important modification. The friction coefficient is strongly position dependent, giving in the present case support for the absolute rate theory for the escape rate. 4 The situation is very different if the adatom moves more rapidly than the substrate atoms, which happens for light adatoms. One then finds 3 that even the modified Fokker-Planck equation is inadequate, particularly for describing the local vibrations on the surface. Here the frequency dependence in the memory function becomes essential and this enters in our theory through a nonlinear coupling of the adatom to the density fluctuations in the substrate. Both the static potential and the memory function contribute to the width and to the shift of the oscillatory peak in the frequency spectrum of the velocity correlation function and a detailed analysis is made. 3 A comparison with the data for the diffusion constant obtained in the field ion microscope experiments is limited by the uncertainties in the basic interaction potentials but the calculated diffusion constant is found to agree 5 very well with available molecular dynamics data 6,7 on the same system. The static potential is then determined in a proper way by including the relaxation of the lattice in the presence of a fixed adatom and by taking into account the thermal fluctuations of the substrate atoms. Generalizations to include particle-hole and plasmon excitations for metals are straightforward and making some further approximations one can get contact with other wellknown treatments. 8 The present approach seems equally well adapted for treating the motion of a layer of adatoms, where the interaction between these must be included.