A method for the calculation of the damping rate due to electron-hole pair excitation for atomic and molecular motion at metal surfaces is presented. The theoretical basis is provided by Time Dependent Density Functional Theory (TDDFT) in the quasi-static limit and calculations are performed within a standard plane-wave, pseudopotential framework. The artificial periodicity introduced by using a super-cell geometry is removed to derive results for the motion of an isolated atom or molecule, rather than for the coherent motion of an ordered over-layer. The algorithm is implemented in parallel, distributed across both ${\bf k}$ and ${\bf g}$ space, and in a form compatible with the CASTEP code. Test results for the damping of the motion of hydrogen atoms above the Cu(111) surface are presented.
Considerable progress has been made in recent years in understanding the fundamental processes involved in gassurface interactions. This has been based on the parallel developments of large-scale electronic structure calculations based on density functional theory, combined with multi-dimensional quantum and classical analysis of the dynamics [1]. Despite these advances there remains one key area that is still largely unexplored and poorly understood; the process of energy dissipation into substrate degrees of freedom. Although this is known to be of central importance in many situations [2], there exist no 'real' calculations to date for the energy loss to either phonons or electrons in the surface.
In particular there have been a number of recent experiments that have shown convincing evidence that energy dissipation by the creation of electron-hole pairs is a significant effect in gas-surface dynamics. Gostein et al [3] carried out a detailed state-to-state analysis of H 2 scattering from Pd(111) and showed that, for example, in the vibrational relaxation of (ν = 1, J = 1) to (ν = 0, J = 5) an average of 120 meV is lost to the substrate during the scattering event, presumably to electron-hole pair formation. Nienhaus and co-workers [4] measured directly the hot electrons and holes created at Ag and Cu surfaces by the adsorption of thermal hydrogen and deuterium in the form of ‘chemicurrents’ in a Schottky diode. Finally, Huang et al [5] have studied NO scattering from Au(111) and have concluded that the main sink of energy for the vibrational relaxation of ν = 2 molecules is the surface, with the strong dependence of the de-excitation probability on incident energy providing evidence that an electron-hole pair mechanism is the dominant factor.
We carry out a calculation of the ground state properties of an interacting surface/molecule system using a planewave basis and a super-cell geometry, and use these results to evaluate the friction coefficient associated with the motion of a molecule at a chosen position and in a direction of choice. This is achieved using the well established ‘Golden Rule’ expression [6,7,8] that may be obtained by applying Time Dependent Density Functional Theory (TDDFT) together with a quasi-static limit [7], or less stringently by applying the Golden Rule directly to the available Kohn-Sham states [6]. Essentially the theory is as described by Hellsing and Persson [7] or Liebsch [8]. The super-cell method has the advantage that it retains the continuous spectrum of one-electron excitations, unlike cluster models [9], and this is important for the interactions considered here. In first principles calculations of moleculesurface systems a super-cell of sufficient size is usually chosen to prevent any significant interaction between the adsorbates in neighbouring super-cells. When considering electron-hole pair excitation a slightly more subtle effect must be taken into account, arising from the enforced periodicity of the perturbation that produces the electron-hole pairs. We are primarily interested in the energy loss by electron-hole pair excitation due to the motion of an isolated molecule interacting with the surface, whereas the super-cell geometry will naturally describe the damping of an ordered over-layer. For the periodic system, conservation of crystal momentum prevents transitions occurring that will occur for an isolated molecule interacting with the surface. Results for an isolated molecule are derived from the available periodic perturbation, and a significant difference is found between the energy loss behaviour of the periodic and isolated systems.
To test our method we investigate the friction coefficient of an H atom above the hcp hollow site of a Cu(111) surface. Spin is included explicitly in the Kohn-Sham theory using the gradient corrected local spin density approximation for exchange-correlation (LSDA-GC). In the next section the evaluation of the dynamic self-energy and friction coefficient from Kohn-Sham results using a plane-wave basis is described. In section III the implementation of this as a parallel algorithm is described, along with a brief description of the performance of the algorithm. Results for H/Cu(111) are discussed in section IV, and section V is the conclusion.
The experimentally measurable energy loss spectrum for a particular mode is directly related to the dynamic selfenergy, Λ(ω). Since we are interested in the energy loss, the imaginary part of this self-energy is the required quantity and this can be expressed using [7] Im
where ψ n k and ǫ n k are the Kohn-Sham wavefunctions and energies resulting from a density-functional description of the ground state, and f (ǫ) is the Fermi-Dirac occupation function. Spin degeneracy is assumed in Eq. ( 1), hence the factor of 2; the extension to spin polarised systems is straightforward. The effective field, φ eff , is the TDDFT effective field with contributions from the field of the disp
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