Prediction of Dynamic Scaling Parameters from Growth

Let us consider the scaling behavior of growth models under equilibrium conditions. It is clear that for those models involving a surface which remains flat, as is the case in the FM model and in the early stages of growth in the SK model, this results in a = 0 and p= 0 [13]. The situation is more complex for the VW and advanced stages of the SK models which yield values of a in the range 0.5 < a < 0.6 and time-dependent values of ySas the growth results in these cases from the coalescence of islands.

Surface scaling parameters for a number of nonequilibrium atomistic models have also been established [6, 10]. Continuum equations for the surface motion have to be used to find a solution for discrete models. Thus, for ballistic deposition [14] and the Eden model [15] the interface saturates, resulting in a = 1/2 and /? = 1/3 for Z>top = 2, and a « 0.35 and P~ 0.21 for Z>rop = 3. Conversely, from the random deposition model P= 1/2 and, since the correlation length is always zero, the interface does not saturate and, therefore, a is not defined. Depending on the rules used in the simulations, for the atomistic model including surface diffusion a = 3/2 and p = 3/7 [6], a = 3/2 and p = 0.34 [7] for Z>top = 2 or a = 1 and P = 0.25 [6] and a = 0.66 and P = 0.20 [7] for Drop = 3. Furthermore, when energy barriers at step edges are included in these models, the value of p increases from P= 0.2 to /?= 0.5 [16].

As far as continuous models are concerned, the simplest one is represented by the Edwards-Wilkinson equation [17] for the local rate of increase in column heights. The corresponding rate equation contains the contribution of a smoothening term and a roughening term, i.e.,

The first term on the r.h.s. of eq. (6) can be assimilated to a surface-tension-like smoothening term; t](x,t), the roughening term, represents the white noise in the flux of arriving particles and it would be responsible by itself for the production of a Poisson surface. For a correlated surface it results in <j](x,t)> 0, whereas for a surface formed by a random deposition mechanism it corresponds to <1j(x,t)> = 0. The solution of Eq. (6) yields a = (3-£>top)/2, P= (3-Z>rop)/4 andz = 2 [10].

To account for lateral effects in the solid film growth, the KPZ equation adds a new term to eq. (6), leading to [8]:

0 0

Post a comment