Time-space height correlations of anisotropic surfaces at thermal equilibrium

Luc BARBIER, Eric Le Goff and Bernard Salanon

Fig.1: (--) *G(t)* for a rough anisotropic surface ( ). Parameters are chosen to get well-separated regimes. The various dashed and dotted lines indicates the t, t^{1/4}, t^{1/2} and ln(t) regimes

Scanning probe microcopies allow measuring fluctuations of surface heights. Correlation functions *G(r)* versus time (at one fixed point *r*) or space-time (across an entire image) can thus be measured at thermal equilibrium. What behavior can be expected? How to interpret them, beyond the very first 1-d approaches (90's)? What is universal, and what depends on the microscopic parameters?

Time-space fluctuations of 2-d systems may fall in various university classes depending on their symmetry and conservation laws. For metallic surfaces, at temperatures *T* well below the melting point, matter conservation applies for surface diffusion and thermal noise. Isotropic (dense faces) as well as highly anisotropic (vicinal surfaces) systems can be considered. *h*(*x, y*) being the surface height and a conservative noise term, we write the 2-d Langevin equation: , with the surface Hamitonian ,

where is the anisotropic Laplacian , the rate constants for diffusion (related to elementary activation energies for diffusion) and (*η _{x}, η_{y}*) the surface tensions (related to the kink energy and step-step interaction constant, Calogero-Sutherland model).

The Langevin equation so-defined belongs to the university class for conserved dynamics and conserved noise. Simple scaling arguments give for , and , with: *α*=0, *β*=0 and *z*=4 (in the present context, 0 exponent means saturation (*T*<*T _{R}*) or logarithmic divergence). It is found (Fig.1) that for anisotropic surfaces (like vicinals) extended intermediate , and

These results allowed the interpretation of measured space-time correlation functions of surface heights at thermal equilibrium. Measurements of surface tensions and diffusion rate constants follow and thus of atomic parameters.

**REFERENCE: **L. Barbier, E. Le Goff and B. Salanon, Surface Science 531(3) (2003) 337 and reference therein.

#767 - Last update : 01/22 2009

• Systèmes complexes et transition énergétique › Statistical physics in mechanics

• Service de Physique et Chimie des Surfaces et des Interfaces

• Laboratory of Nano-Objects and Complex Systems (LNOSC) • GMT-MSIN : Modélisation des Surfaces Interfaces et Nanostructures