Coupling among bulk, surface and edge diffusion

Simple atomic jumps transform thermal defects on crystalline surfaces, in the form of adatoms and advacancies, reversibly into interstitials and vacancies that constitute thermal point defects of the bulk. This paper treats the consequent coupling of bulk, surface, and edge diffusion. It is demonstrated, as an exact result, how bulk, surface, and edge processes factor into a single eigenvalue problem that describes, by orthogonal eigenvectors, the relaxation of the bulk, surface and edge defects, together, towards the (nonuniform) defect distribution for thermal equilibrium. The way long-range surface flow mixes degenerately with bulk modes is explained, and the discussion is extended to local surface modes in which fast surface processes separate off from slower bulk modes. Bulk diffusion modes bound to surfaces, and driven by surface energetics, are discussed to recover results of Mullins for surface smoothing by surface and bulk diffusion processes. Analogous results for coupling of surface and bulk diffusion to edge diffusion along surface steps are also described. The significance of the results is illustrated by numerical examples that employ a recently proposed universal modeling of diffusion in metals and on metal surfaces.

Figure 1

Creation and interconversion of defects. (a) A fluctuation transforms an interstitial (left) into an adatom (right). (b) A fluctuation converts a vacancy (left) into an advacancy (right). (c) Fluctuations create a vacancy-interstitial pair (top) and an advacancy-adatom pair (bottom) from perfect lattice. (d) Fluctuations form mixed surface-bulk antidefect pairs from perfect lattice. Each transformation can occur spontaneously in reverse also. Broken lines marked $s$ indicate surface planes.

Figure 2

Relaxation rate ${\tau }^{-1}$ of eigenfunction shown as a function of real part $k$ and imaginary part $\zeta$ of the wave vector. The branch with $\zeta$ positive near $k=0$ corresponds to distributions bound to the surface, and the branch near $k=\pi ∕a$ corresponds to localized surface processes that lie outside the continuum of bulk modes.

Figure 3

The character of an atomic hop between specific lattice sites changes as defects accumulate. (a) The atomic hop from $i$ to $j$ is an atom jumping into a neighboring advacancy site (top). As advacancies accumulate (bottom), the same atomic hop from $i$ to $j$ becomes an adatom hop on a surface terrace. That the two occur with differing rates can be regarded as a consequence of defect interactions (see text). (b) At higher defect concentrations, the presence of neighboring defects on sites $i$ and $j$ blocks hops of both to the other occupied site (top), illustrated here for interstitials. In real cases, the interactions also modify the transition rates of both defects to unoccupied sites $k$ also, identified at the bottom of the sketch. Both (a) and (b) can alter the evolution of a system.

Figure 4

Model potential showing defect binding to surface (layer 1) and consequently change to $w^{\prime }$ of the hopping rate from the surface to the bulk (layer 2). The hopping rates from all bulk sites are $w$.

Figure 5

Admixture at $T=3T_{\mathrm{m}}∕4$ of the surface state of wave vector $q$, into bulk modes, shown for cases that result in admixture at $\kappa ∕3$, $\kappa ∕30$, and $\kappa ∕300$, with $\kappa$ as the Brillouin zone boundary. The mixing occurs over an increasingly large fractional bandwidth the lower the resonance lies in the bulk band.

Figure 6

Temperatures $T$ for (111) surfaces of fcc metals at which surface diffusion with wave vector $q$ undergoes transitions. $T_l$ is the temperature below which mode $q$ is localized at the surface itself. $T_{\mathrm{x}}$ is the temperature below which surface diffusion exceeds bulk diffusion as a means for surface smoothing. The temperature $T_{\mathrm{x}}{}^{\prime }$ below which surface diffusion exceeds bulk diffusion in promoting step edge relaxation scarcely differs from $T_{\mathrm{x}}$. All temperatures are shown as fractions of $T_{\mathrm{m}}$.