Vacancy diffusion in the Cu() surface II: Random walk theory

Abstract

We develop a version of the vacancy mediated tracer diffusion model, which follows the properties of the physical system of In atoms diffusing within the top layer of Cu(001) terraces. This model differs from the classical tracer diffusion problem in that (i) the lattice is finite, (ii) the boundary is a trap for the vacancy, and (iii) the diffusion rate of the vacancy is different, in our case strongly enhanced, in the neighborhood of the tracer atom. A simple continuum solution is formulated for this problem, which together with the numerical solution of the discrete model compares well with our experimental results.

Diffusion of vacancies in metal surfaces: theory and experiment

“…This is justified by numerical calculations presented elsewhere where this problem was addressed for the case of indium [33]. The calculations show that the fraction of jumps that have zero net length changes only marginally with temperature.…”

“…We first discuss experimental results of the In/Cu(0 0 1) [21][22][23] and Pd/Cu(0 0 1) [24] systems. After this we discuss the theoretical framework that was developed to analyze the diffusion measurements [33]. We then discuss how to use this framework to interpret the measurements and discuss the differences between these two material systems.…”

“…Both distributions can be fit very well with the modified Bessel function, again confirming the vacancy-mediated diffusion mechanism for both cases. The only free parameter used in the fits is the probabilityp rec for vacancies to recombine at steps, between subsequent encounters with the same embedded atom [33]. This probability is directly related to the average terrace width and variations in this number can be ascribed to the proximity of steps.…”

“…The mean square displacement r 2 is directly proportional to the square of the width of the Bessel solution. We can determine the width of the Bessel solution from the parameters D eff and c, which can be obtained from the return probabilities [33] …”

“…For the case of repulsion, on the other hand, the long-jump rate is limited by the vacancy overcoming the barrier to exchange with the tracer, presumably arriving at a nearest-neighbor site several times before doing so. EAM [33] and first-principles [24] calculations have shown that the exchange barrier for a vacancy in the Cu(0 0 1) surface with an In or a Pd atom is 243 meV and 466 meV respectively. First-principles calculations [24] showed that the vacancy diffusion barrier in the Cu(0 0 1) surface is equal to 426 meV.…”