Step fluctuations

On a so-called vicinal face there are many steps running in parallel with almost the same separation or terrace width in between. At a finite temperature, these steps also fluctuate. But due to the high energy cost for the formation of overhangs on the crystal surface, steps cannot cross each other. This non-crossing condition suppresses the step fluctuation. 

For a step train, two neighboring steps collide when the fluctuation Wgq determined in (35) becomes the average step separation i. Therefore, the collision takes place for each step length Lcow 7 /T. To realize a large fluctuation, all the steps meander simultaneously. The step fluctuation for a long step (L > F gon) is reduced to the logarithmic form [3]... 

An elastic interaction between steps can also be approximated by a harmonic potential when the deviation of the steps from a straight line is small [18]. Even though steps fluctuate with a diverging width, Eq. (36), the separation between neighboring steps or the terrace width fluctuates a little... 

Another effect of elastic deformation is that it causes a long-range interaction between steps. From the continuum elasticity theory, two steps sepa-rated by a distance have a repulsive interaction proportional to l for homo- and to In i for hetero-epitaxial cases, respectively [84]. This interaction plays an important role, for example, in step fluctuations, terrace width distribution, step bunching, and so forth [7,85-88]. 

N. C. Bartelt, T. L. Einstein, E. D. Williams. Measuring surface mass diffusion coefficients by observing step fluctuation. Surf Sci 572 411, 1994. 

FIG. 21. The influence of potential on step fluctuations, x(t), may be described by means of a time correlation function F(t) = ((x(t) - x(0) ). At negative potentials, fluctuations are due solely to mass transport along the steps, while at more positive potentials the magnitude of the fluctuations increases rapidly. This is attributed to the onset of adatom exchange with terraces as well as the electrolyte, which occurs even at the potential well below the reversible value for Ag/Ag+. (From Refs. 207, 208.)... 

Usually, experimentalists quantify step fluctuations by averaging the data to find the correlation function G(t) = 0.5 < (h(x,i) - h(x,0)Y >, where h x,t) specifies the step position at time t and the average is over many sample points, x. G(f) measures how far a position on a step wanders with time. If that position were completely free to wander, it would obey a diffusive law G(t) t. However, its motion is restricted by the fact that it is connected to the other parts of the step. For that reason G(t) is sub-diffusive. The detailed law which G(f) obeys is dependent on the atomic processes which mediate step motion. For example, if the step edge is able to freely exchange... 

Figure 1. Three limiting mechanisms for atomic processes which mediate step fluctuations, a) Step-edge diffusion b) Evaporation-recondensation c) Terrace diffusion with diffusion kernel P(). By appropriate choice of P(J), this case can reduce to cases a) and b) (see text).

The relative lifetimes of the two terraee types at any one saddle point location has been measured[31] to differ by a factor of 6 at 1060C. The change in terrace type occurs by the bridging of the short dimension by step fluctuations. Since the probability of a fluctuation of a particular amplitude depends linearly on the step stiffness[8] the observed lifetime ratio is consistent with measured step stiffnesses[37] and the geometrical picture given above[38]. 

Fig. 3 Part of a semi-infinite step fluctuating away from an attracting line at x=0...

STEP FLUCTUATIONS FROM EQUILIBRIUM ANALYSIS TO STEP UNBUNCHING AND CLUSTER DIFFUSION IN A UNIFIED PICTURE... 

In future work we plan to extend this approach to consider the effect of external fields due to applied potentials or adsorbed species, as well as the modifications when surface islands can change their mean size (ripening or decaying). Correspondingly, there is noteworthy current work on the effect of sublimation or deposition on the step fluctuations of a vicinal surface (E.g. Pierre-Louis and Misbah, 1996). It would also be interesting to consider the effects of weak pinning potentials. 

A more interesting problem is that the Metropolis Monte Carlo studies used a different (physically simplified) kinetic rate law for atomic motion than the KMC work. That is, the rules governing the rate at which atoms jump from one configuration to the next were fundamentally different. This can have serious implications for such dynamic phenomena as step fluctuations, adatom mobility, etc. In this paper, we describe the physical differences between the rate laws used in the previous work, and then present results using just one of the simulation techniques, namely KMC, but comparing both kinds of rate laws. 

The implication of this behavior suggests that there will be a quantitative difference in the kind of step fluctuation dynamics observed for each kinetic law system. For i-kinetics, we expect steps to fluctuate by variations in the flux of adatoms hitting the step from a uniform, quickly moving sea of equilibrated adatoms on the adjoining terrace. In this case, the time to create a step fluctuation of amplitude y (perpendicular to both the surface normal and the average step direction) will be given by... 

For A/-kinetics, on the other hand, adatoms diffuse rapidly along steps, and we can expect a tendency to wash out the variation of the terrace adatom flux onto the step. Thus, at the very least, we can say that step fluctuations in this system should form more slowly than for /-kinetics, i.e.. 

The relaxation of isolated, pairs of and ensembles of steps on crystal surfaces towards equilibrium is reviewed, for systems both above and below the roughening transition temperature. Results of Monte Carlo simulations are discussed, together with analytic theories and experimental findings. Elementary dynaniical processes are, below roughening, step fluctuations, step-step repulsion and annihilation of steps. Evaporation kinetics arid surface diffusion are considered. 

Most cases are well understood by now. Additional work seems to be needed for clarifying the flattening of gratings below roughening in the case of surface diffusion.- It is also desirable to elucidate the effect of (long-range) step-step interactions, induced, for example, by elastic forces, on step fluctuations. 

Direct step-step interaction terms in the step energy ( direct interactions are entropic repulsion, strain terms, electronic structure effects etc.) do influence the step fluctuations, and they also drive the spreading of step trains, wires and bumps. Nevertheless, it is instructive to first ignore these direcf step-step repulsion, as is done in... 

The key physics of our model (see Eqs. (9) and (10)) is contained in the nonlocal diffusion kernels which occur after integrating over the atomic processes which produce step fluctuations. We have calculated these kernels for a variety of physically interesting cases (see Appendix C) and have related the parameters in those kernels to atomic energy barriers (see Appendix B). The model used here is close in spirit to the work of Pimpinelli et al. [13], who developed a scaling analysis based on diffusion ideas. The theory of Einstein and co-workers and Bales and Zangwill is based on an equihbrated gas of atoms on each terrace. The concentration of this gas of atoms obeys Laplace s equation just as our probability P does. To make complete contact between the two methods however, we would need to treat the effect of a gas of atoms on the diffusion probabilities we have studied. Actually there are two effects that could be included. (1) The effect of step roughness on P(J) - we checked this numerically and foimd it to be quite small and (2) The effect of atom interactions on the terrace - This leads to the tracer diffusion problem. It is known that in the presence of interactions, Laplace s equation still holds for the calculation of P(t), but there is a concentration... 

Giesen etal. reported an analysis of step fluctuations on a Cu(100) electrode in aqueous electrolyte using STM [19]. These fluctuations involve a rapid exchange of atoms along the steps, between neighboring steps, and on adjacent terraces. They determined the mass transport involved in the fluctuations as a function of the electrode potential. 

Step fluctuations have been observed for both Ag and Cu surfaces in both vacuum and electrolytes [8]. As shown in Fig. 11, the steps on an immersed Ag(lll) actually appear to be friz2y due to kink motion, which is rapid compared to the tip raster speed [8,91,92]. In x — t images, the fluctuations can be quantitatively analyzed by means of a step correlation function, G(t) = [x t) — x(0)] >, where x defines the step position at a particular time, t. If image drift is a problem, the step pair correlation function may be used [8, 93]. The evolution of the correlation function and its dependence on step spacing is a reflection of the mass transport mechanism, which is dependent on both the potential and electrolyte composition. Furthermore, an assessment of the temperature dependence of the fluctuations allows the activation energy of the rate-limiting process to be evaluated. As shown in Fig. 11,... 

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