B. Salanon

Roughening of vicinal surfaces and step—step interactions: application to Cu(1,1,11) by means of statistical analysis of STM images

Abstract The surface morphology of vicinal surfaces is governed by thermal kink creation energies and step—step interactions. Kink creation energies Ek were measured first by means of the statistical analysis of STM images and within one-dimensional or free-step approximations. In the present paper a two-dimensional model is investigated where in addition to the simple no-crossing condition for steps, elastic like step—step interactions with A L 2 dependence are considered. In the framework of this model, novel analytical expressions are obtained for the correlation function, G(0,y) parallel to steps and G(m,0) perpendicular to steps, in terms of microscopic energetic parameters Ek and A. The good agreement observed between our approach, exact results (available for A = 0 and one peculiar A value) and Monte Carlo (MC) simulations confirms its validity. By statistical analysis of STM images of Cu(1,1,11), we have measured both G(0,y) and G(m,0) together with the terrace width distribution (TWD). All these measurements evidence a step—step interaction stronger than the mere no-crossing condition. This set of data can be consistently fitted within our approach with one couple of microscopic parameters Ek = 1430 K and A = 70 K.

Diffraction of helium from Cu(110), (113), (115) and (117); Interaction potential and surface crystallography

We present data obtained by helium scattering on different faces of copper: (110), (113), (115) and (117). It is shown that they are well fitted by model potentials, a corrugated Morse (CMP) for all faces and a modified form (MCMP) for the low corrugated (110) and (113) faces. Then, the isopotentials ZE(R), representing the effective corrugation for an incident helium atom of energy E are compared to those resulting from a method of superposition of atomic densities calculated in two different ways. So, starting from the tables of atomic wave functions given by Clementi and Roetti, a good agreement is obtained with experiment, provided that the proportionality constant between the potential and the electronic density is taken to be α = 600 eV03.

Diffraction of He from Cu(110) in the 20–240 meV range; comparison with a realistic model potential

Abstract The results of the diffraction of helium from Cu(110) are presented in the 21–240 meV energy range. The corrugated Morse potential gives a good fit but its corrugation amplitude increases with incident energy. A new form of potential, the Modified Corrugated Morse Potential, is presented and its main properties are examined. With this potential, a good fit of the experimenal results for various incidence angles is obtained; no change of the parameters is necessary in the whole range of incident energies. So one gets a realistic picture of the He/Cu(110) interaction.

Vicinal surfaces: free energy, terrace width distribution and step correlation functions

Abstract The equilibrium morphology of vicinal surfaces is governed by thermal kink creation energies and step–step interactions. Such parameters can be determined within microscopic step models by means of the statistical analysis of surface images such as those provided by STM. A versatile multi-step transfer matrix (strip) method is introduced to calculate thermodynamical averages. The correlation function of step displacements along step edges, the terrace width distribution and the free energy are thus obtained. The applicability of various one-step and many-step formulae is discussed, and properties of the capillary wave approximation are given. Within this approximation, the width of the terrace width distribution at the roughening temperature is found to be universal.

Step interactions and surface stability for Cu vicinals

Abstract The surface energies are computed for vicinals of Cu(100), Cu(110) and Cu(111) using a molecular dynamics technique modified to minimize the surface energy at zero temperature. The steps are simple, double or triple. A many-body, semi-empirical potential derived from tight-binding models is used. The step energy and the step-step interaction are deduced. The energy of multiple steps is found to be roughly proportional to the step height, except for vicinals of (111) with (111)-type step edges. On Cu(115) the energy for displacing a single step is found to be very small. As a general rule, the energy of perturbed configurations cannot be deduced from the surface energy of macroscopic facets.

Statistical analysis of STM images and measurement of the step-step interaction on Cu vicinal surfaces

Abstract The time and time-space correlation functions of the step positions as well as the terrace-width distribution are measured on STM pictures of the Cu(1111) surface. It is shown how the space-time correlation function can be interpreted. The kink energy together with the hopping frequency for atom diffusion along step edges are deduced. From measurements of the terrace-width distribution we estimate the strength of the repulsion between steps. The quantitative analysis is made in the framework of a two-dimensional model with an A / L 2 step-step repulsion.

Temperature dependence of the atomic relaxations and vibrations on a stepped surface: a molecular dynamics study of Cu(1,1,19)

We have studied the relaxations and vibrations of atoms near the surface of a (1,1,19) copper crystal. For this purpose, we have performed molecular dynamics simulations using a semi-empirical many-body potential derived from tight binding models. The total displacement field can be described as the sum of a mean surface relaxation and a specific contribution of the steps, which is maximal for step edge atoms (0.08 A) and corner atoms (0.06 A). Whereas step edge atoms relax towards the inner terrace and towards the bulk, corner atoms relax in the opposite direction, leading to vortex-like structures in the displacement field. We demonstrate that, as predicted by continuous elasticity, the displacement field induced by steps is equivalent to the one created by a line of dipoles on a flat surface. In the particular case studied here, the equivalent dipole density is 3.3×10−10 N. The specific relaxations of kink atoms have been calculated. We have also studied the variation of the relaxations as a function of temperature (T). A strong effect is predicted for inner terrace atoms: when increasing T, the contraction of the first interplanar distance, with respect to the bulk value, progressively cancels and turns to an expansion at high T. This is not the case for the specific contraction of step edge atoms that is nearly temperature independent. This latter behaviour is related to very strong longitudinal correlation between vibrations of the step edge atom and of its nearest neighbour inside the terrace. In the same time, whereas the vibrations of inner terrace atoms are found to be isotropic, the ones of step edge atoms are anisotropic, with a larger component along the direction parallel to the terrace plane and perpendicular to the step edge, the other components being the same as for inner terrace atoms.

Scattering from a hard corrugated wall: An exact solution in the two-dimensional case

Abstract We present a method for computing exactly the intensities diffracted by a hard corrugated wall potential. We use a two-dimensional corrugation of any amplitude and calculate the source function by means of special points in the lattice unit cell. Our procedure is shown to be convergent, it is compared with the NN ′ method and with the Fourier expansion method. The use of special points, about which a theoretical summary is given, allows us to consider rather small matrices to invert (64 × 64 or 128 × 128). Symmetrical cases are discussed and are shown to yield reduced matrix sizes. Our method is applied to the calculation of the intensity disturbances induced by the emergence of a beam. The validity of the eikonal approximation is also discussed.

Stress on vicinal surfaces

Abstract We develop an understanding of the variation of the surface stress tensor with facet orientation within the framework of elasticity theory. The surface stress behaves very differently from the surface energy. It is known that the elastic repulsion between steps behaves as d −2 and contributes a tan 3 θ term to the surface energy, with d the step–step distance and θ the vicinal angle. We show that the presence of steps induces a strong tan 2 θ term in the surface stress. Atomistic calculations confirm this behavior.

Influence of an adsorbate on the morphology of a vicinal surface: sulfur on Cu(1 1 11), a STM and helium diffraction study

Abstract The adsorption of sulfur atoms on Cu(1 1 11) is shown to alter the step mobility and morphology. S atoms are adsorbed preferentially on kink sites, on the upper side of steps. The kink density increases with sulfur coverage and S-decorated kinks are immobile. The mobility of steps is correspondingly reduced although we have seen that the density of bare kinks is also enhanced for low S coverage.