Stability of vicinal surfaces and role of the surface stress

Abstract

Vicinal surfaces of type (0, 1, M) are investigated and compared with surfaces with opposite steps, M being an integer larger than 2. When admitting that the step behaves as a dipole force f → = ( 0 , f y , f z ) , f y and f z being respectively parallel and normal to the surface, the Marchenko–Parshin ( MP) model gives the surface displacement due to one step equal to - Λ f → / y , Λ being an elastic constant and y the position from the step. On vicinals, the MP model indicates that the interaction energy between steps varies as Λ f → 2 / L 2 , L being the step–step distance. For Cu(0, 1, M) and Au(0, 1, M), the components f y and f z are deduced from surface displacements obtained by molecular dynamics at T = 0 K. Due to the minimization of the terrace stress, we confirm that the terrace is more contracted in the direction parallel to the surface by a factor ( 1 + X ) > 1 with respect to the MP model where X is recursively proportional to the parallel deformation. This leads to an interstep interaction energy increased by a factor ( 1 + 2 X ) 2 (instead of ( 1 + X ) 2 with respect to both the terrace deformation and the MP model). This effect due to the terrace stress is larger for Au. We note that f y ( 1 + X ) is close to f z , opposite in sign to the surface stress of the nominal surface and to the isolated step stress. By comparison with surfaces with opposite steps, the parallel deformation at the step position, ϵ yy ( 0 ) , includes a term in L - 1 in addition to the term in L - 2 deduced from the MP model. The term in L - 1 corresponds to a weak displacement parallel to the surface towards the descending steps. From the step energy, the first order and second order energies as function of the relaxation deformations can be subtracted. In the MP model, the first order one is opposite in sign and twice in magnitude the second order one. For Au, we observe a deviation from this equality due to the minimization of the terrace stress. In the last part and for vicinals, we confirm that the step–step interaction stress varies as L - 1 and results from the component f z multiplied by minus the derivative of the normal displacement due to neighbouring lines of monopole forces, the forces being parallel to the surface and proportional to the homogeneous deformation. In the literature, it is known that the stability of vicinal surfaces results essentially from the repulsive interstep interaction energy. Concerning the surface stress, the isolated step stress attenuates the surface stress of the nominal surface. Because the interaction stress decreases this attenuation, we note that the steps are repulsive also by the surface stress.