No abstract

We address the problem of MBE growth in one horizontal and one vertical direction in the presence of Schwoebel barriers. The time-independent growth equation introduced previously is shown to be identical to that for a classical particle in a potential well. We solve this equation using periodic boundary conditions and find time-independent solutions consisting of a periodic array of mounds. We derive the .dispersion relation., i.e. the amplitude as a function of wavelength for these mounds. The equation of motion is derivable from a free energy indicating that there is a most stable ground state, which is independent of the initial conditions. The mounds are marginally unstable and there is a minimum wavelength below which no mounds exist. The wavelength of the mounds coarsens slowly in time according to At " , with a = 114.

PHYSICAL RYan et al. Reply: In his Comment [1] Kim calculates the quantity Ar and claims that a "sudden jump" in Av is evidence that the phase transition that we found [2,3] is trivial in 2-hi dimensions, much like the transition of Amar and Family [4] where the coupling X, in the Kardar-Parisi-Zhang equation, changes sign as a function of control parameter [5]. Ar is the average over the surface of X(p)\\h\ 2 .Kim is correct that this function varies rapidly. The corresponding X, if we assume that (|V/j| 2 > is a constant of order unity, is small for /?=^0.3, where we locate the transition. Thus we might be in a crossover regime from the X =0 fixed point. This is Kim's argument as we understand it.We first note that we do not think that Ar is the best quantity for assessing our model. We will focus in what follows on X, as is conventional in this field. The ratio between Ar and A., (|V7?| 2 >, is not a constant and depends on p. We show this by calculating X itself [5] by forcing an average tilt 0 on the surface and setting X=jd 2 v/ d0 2 |0=o-We present our results in Fig. 1 plotted together with Kim's results. Kim is correct that X(p) is nonlinear; in our paper we were wrong on this point. However, an examination of the figure will show that X is smooth, has no sudden jump, and is fairly large in the important region 0.2

for 0.2

is small for small p because the surface is smoother.Of course, the fact that X(p) is nonlinear is not a problem, in principle. It is sufficient for our purposes that X(p) is continuous, monotonic, and zero only where p =0, just as a nonlinear thermometer can indicate an ordinary thermal transition. X certainly does not change sign as in Ref.[4]. We correctly found the (noncontroversial) transition in 34-1 dimensions using the same thermometer and the absence of a transition in 1 -I-1 dimensions.There remains only a numerical question: Is A,(0.2) ^ 0.12 so small that we could not distinguish it from 0? Kim's suggestion is that X =0.12 appears to give weak coupling because our sample sizes are too small and we have a finite-size crossover. To answer this sort of criticism, the proper tool is finite-size scaling. This is exactly what we did [2] in our paper. In 3+ 1 dimensions we got expected and reasonable results. In 2+1 dimensions, below the putative transition, the effective roughness exponent a became smaller for large systems. In the absence of a phase transition this is very hard to understand in finite-size scaling, which says that a should cross over to larger values. The behavior that we found is clearly not a simple crossover as Kim implies and has been suggested [6] elsewhere. It may be a phase transition or a more complicated type of crossover [3j, but that is another issue, not addressed by Kim's Comment.

No abstract

We give a mean-field, continuum treatment of ballistic aggregation on a seed and a line. The treatment is deterministic, except for one statistical assumption, the so-called tangent rule which determines the mean direction of growth. Our treatment represents progress toward the explanation of the columnar microstructure.PACS nUmbers: 68.55. + b, 05.70.Ln, 81.15.Jj In recent years much interest has focused on nonequilibrium aggregation processes, that is, the formation of structures by the irreversible addition of subunits from outside. An example of such a process is diffusion-limited aggregation (DLA) where fractals are formed. A simpler problem than the diffusionlimited case (where the aggregating particles perform random walks) is ballistic aggregation. In this process particles moving in straight lines are added to a structure whenever they touch a previously added particle. Early work on this problem seemed to show that fractals were produced, but it is now believed both on the basis of more detailed numerical studies and from analytical results that ballistic aggregates are amorphous solids of fixed density. Nevertheless, the patterns formed in this simple problem are both intriguing theoretically and technologically interesting. In Fig. 1 we show two types of ballistic aggregates: one (a "fan") formed by attachment to a seed4 and another by attachment to a plane of a beam of nonnormal incidence. 5 In both cases the particles all move in parallel straight lines, as shown, from random launching points. The similarities of the patterns are striking.The peculiar long open streaks are the unexpected feature.For the case of attachment to a plane these patterns are known as the columnar micro structure. The columns form both in computer simulations (as shown in Fig. 1) and in the real world in vapor-deposited thin films of both metals and insulators. For example, aluminum films deposited on cold substrates often show this morphology. An understanding of this structure is of particular technological interest as the surface properties, notably electrical and optical, are substantially modified from the bulk properties of the material by the surface microstructure. See Ref. 5 for more detail and actual experimental results. Even in the presence of short-range attractive forces among the atoms, which curve the straight-line trajectories, columns still form in numerical simulations. The streaks and columns have not heretofore been explained.At first glance, it seems that any such explanation would be very complicated because the voids clearly arise from shadowing of one part of the structure by another. In fact, this is an interesting feature of the system; precisely the same sort of nonlocal shadowing produces the fractals of the DLA problem. However, we will show here that many features of the structures can be explained in a remarkably simple way.In the next section we propose a kind of mean-field treatment for aggregation both on a point and on a line. We will always consider a situation in which the particles ...