Diffusion Processes and Pre-Exponential Factors in Homo-Epitaxial Growth on Ag(100)
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263 Mat. Res. Soc. Symp. Proc. Vol. 492 01998 Materials Research Society
b
a
Figure 1: (a) topview for diffusion on flat Ag(100) via hopping and exchange: adatom in the four-fold (minimum energy) position (left) and in the transition state (right) for the hopping process (above: two-fold bridge position) and for the exchange process (below); (b) and (c) initial geometry for descent from a < 110 > step; a < 100 > step, respectivly. The cross represents the final position the adatom would occupy if it hops from the respective step edge to the terrace below. For the exchange process the step atom labeled (2) moves towards the position marked by the cross and the adatom labeled (1) moves into the step edge position of atom (2). In (d) and (e) the arrows I to IV indicate paths for adatom diffusion via hopping. diffusion coefficient D for a single adatom on a surface may be written as D
with
D0 (T)
= -
Do(T)exp (kT)
(1)
2
kBT n1 ASvib )exp( ) -AUib, ) h -aexp(
ha
kB T
kB
where, AS,,ib, AU,,ib and AM) are the differences in the vibrational entropy, internal energy and static (structural) energy, respectively, calculated with the relevant atoms (adatom for hopping, two moving atoms for exchange), in the transition state and at the minimum point on the potential energy surface. Here n is the number of jump-equivalent possibilities. The thermodynamical quantities appearing in the above equations can be obtained from the vibrational partition function calculated within the harmonic and quasiharmonic approximation of lattice dynamics, as:. x+
U,,ib
=
S
=ib = kB j0.... N(v)(-ln(1-e-)+
kBT
N(v)(
)dv
ex x
)dv.
(2)
where, x = hi' N(v) is the density of phonon states as a function of frequency v, which can be written as N(v) = ZE n,(v), where n((v) is the local density of states in region 1. Depending on the location of the adatom (or the two moving atoms for exchange) along the reaction coordinate (minimum energy site and transition state) it encounters particular LDOS leading to differences in the local thermodynamic functions in these two regions. To calculate the LDOS we diagonalise the force constant matrix which yields the eigenvalues
264
and eigenvectors, from which the LDOS at the site 1 in the direction 3 is calculated. The force constant matrices for atomic configurations along the diffusion path (minimum energy and transition state) were evaluated with all atoms relaxed in their minimum energy configuration. The substrate is build by (10 x 10) atoms in the x-y plane, where periodic boundary conditions were applied, and 10 layers, which is large enough for the LDOS not to exhibit significant finite size effects. For the calculations in the quasiharmonic approximation, we included thermal expansion in the bulk, and in the surface plane, but not in the direction normal to the surface. The lattice constant at 600 K was derived from molecular dynamics simulations [8]. RESULTS AND DISCUSSION In Figs. 2a and 2b we show the LDOS and their x-,y-, and z-components for the moving atoms for step descent
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