A Model for Texture Evolution in a Growing Film
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Mat. Res. Soc. Symp. Proc. Vol. 389 © 1995 Materials Research Society
If one considers atomic jumps between some column number i and its neighbouring columns, one can derive the following expression for the net number of atoms dni jumping towards column i during a time interval dt 10) dni = 2rtLv 0(VJ)-
ri dt < exp( - Ej /kT) - exp( -Eij /kT) >j
(1)
In this equation L is the thickness of a thin layer on the surface, where atomic diffusion is possible; v0 is an attempt frequency for atomic jumps and Vat the atomic volume. Further ri is the radius of a circle with the same surface as the top region of column i. The brackets in eqn. (1) represent the mean value (taken over all neighbouring columns j) of the difference between two Boltzmann factors. Eji is the activation energy for surface atomic jumps from column j to column i ; Ej is the activation energy for jumps in the opposite direction and T the absolute temperature. These activation energies may be viewed as activation energies for atomic diffusion in the surface layer, where the atoms cross the grain boundary between two columns. Each activation energy may be considered as a sum of a first energy, describing the local thermal equilibrium concentration of vacancies in the surface layer, and a second energy which represents the barrier for an atomic jump across the grain boundary separating the columns. It can be argued 10)that Eij depends on the surface energies of column i and j; it is indeed plausible that an atomic jump to a situation of lower energy will involve a lower activation energy too. As a realistic approximation one finds 10) Eii = _E + (1/2) ( y - yi )(VJ,)3 . E is a mean activation energy and y the surface energy for column i. If yi for some column is smaller than the mean y,of its neighbours, Er will be large and E.p small. This results in a positive value of the brackets in the r.h.s. of eqn. (1), so that dni > 0. This means that a column with a small surface energy will show a growing top surface, in agreement with the principle stated at the beginning of this section. From dni the change dri of the radius of the column can easily be calculated. If the average of the y values of the neighbouring columns is substituted by an overall y value for the whole surface, equation (1) easily leads to 10) dri / dt = K(-y(t) - yi). In this equation for the growth velocity of the radius of column i, K equals the constant v0L(kT)-(Vat)'exp(-E/kT) , and -y is the mean surface energy for all columns. The calculations are performed as follows. In the starting condition (t=0, situation near the substrate), the number of columns for each angular orientation, and the corresponding column radii are imposed by experiment. Then the value of dri / dt is calculated for all orientations from the above formula, so that the new r1 value for a time dt later can be calculated. This procedure is repeated for the times dt, 2dt, 3dt, etc. so that the texture evolution can be followed. For simplicity one ri value (instead of a statistical distribution) is assigned to
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