Roughening of $k$-mer–growing interfaces in stationary regimes

We discuss the steady-state dynamics of interfaces with periodic boundary conditions arising from body-centered solid-on-solid growth models in $1+1$ dimensions involving random aggregation of extended particles (dimers, trimers, $...,k$-mers). Roughening exponents as well as width and maximal height distributions can be evaluated directly in stationary regimes by mapping the dynamics onto an asymmetric simple exclusion process with $k$-type of vacancies. Although for $k\ge 2$ the dynamics is partitioned into an exponentially large number of sectors of motion, the results obtained in some generic cases strongly suggest a universal scaling behavior closely following that of monomer interfaces.

Figure 1

(a) Schematic view of a dimer-growing BCSOS interface (formed by slopes $s_n\equiv h_{n+1}-h_n=\pm 1$) and its equivalent driven lattice gas of reconstituting particles. At each step of the former, a dimer aggregation on local minima corresponds in the latter to an exchange of $A_2$ particles with either of the $A_0$ or $A_1$ vacancies referred to in the text. Note that under $A_1{A}_2\to A_2{A}_1$ exchanges the identity of $A_2$ dimers is not preserved. (b) Sublattice currents contributing to the growth velocity of $h_n$ as discussed through Eqs. (5, 6, 7, 8).

Figure 2

(a) Finite-size growth of average widths, (b) height difference correlation functions ($L=4.2\times {10}^5$), and (c) growth of average maximum heights (measured with respect to spatially averaged ones), for various string sectors. In main panels, circles, squares, and rhomboids stand, respectively, for the first three sectors of Table 2 ($k=2$), whereas its following ones correspond to triangles, downwards triangles, and plus signs ($k=3$). Insets denote in turn cases of null string sectors with $k\ge 4$. All data in main panels (insets) were normalized by $\mathcal{A}$ amplitudes given in the rightmost columns of Table 2 (Table 3), and are consistent with a common roughening exponent $\zeta =1/2$.

Figure 3

Scaling of (a) width distributions, and (b) maximal height distributions for the $k$-mer sectors of Table 2 using the symbols of Fig. 2. Here, these stand for sizes $L=3\times {10}^4$ (circles), $6\times {10}^4$ (rhomboids), ${10}^5$ (squares), $2.4\times {10}^4$ (triangles), $4.8\times {10}^4$ (plus signs), and $9\times {10}^4$ (downwards triangles). For comparison, in (a) and (b) solid lines correspond respectively to the exact scaling functions referred to in Eq. (21a) (Ref. [13]), and Eq. (21b) (Ref. [15]). The insets exhibit the tails of those scaling distributions, which also follow our data in the null string sectors of $k=4,5,6,7,8$ with $L=(3.6,4.2,3.3,2.8,4)\times {10}^4$ respectively.

Figure 4

Width distributions of faceting sectors (a) ${\left[A_1\right]}^{L/4}{\left[A_0\right]}^{L/4}$ ($\mathcal{L}/L=3/4$), and (b) ${\left[A_2\right]}^{L/10}{\left[A_1\right]}^{L/10}{\left[A_0\right]}^{3L/10}$ ($\mathcal{L}/L=4/5$) for $L={10}^4,3\times {10}^3,$ and ${10}^3$ (topmost solid lines in downward direction). In contrast to roughening sectors (lowermost universal distribution), fluctuations around average widths become negligible as $L$ increases. Snapshots of the resulting interfaces ($L={10}^4$) are displayed by the insets. The slopes of their facets (indicated by dashed lines) are evaluated in the text. For comparison, snapshots of the roughening sectors ${\left[A_1{A}_0\right]}^{L/4}$, and ${\left[A_2{A}_1{A}_0^3\right]}^{L/10}$ are also exhibited in (a) and (b) respectively.