Fluctuations of isolated and confined surface steps of monatomic height

The temporal evolution of equilibrium fluctuations for surface steps of monatomic height is analyzed studying one-dimensional solid-on-solid models. Using Monte Carlo simulations, fluctuations due to periphery diffusion (PD) as well as due to evaporation and condensation are considered, both for isolated steps and for steps confined by the presence of straight steps. For isolated steps, the dependence of the characteristic power laws, their exponents, and prefactors on temperature, slope, and curvature is elucidated, with the main emphasis on PD, taking into account finite-size effects. The entropic repulsion due to a second straight step may lead, among other things, to an interesting transient power-law-like growth of the fluctuations, for PD. Findings are compared to results of previous Monte Carlo simulations and predictions based, mostly, on scaling arguments and Langevin theory.

Figure 1

Sketch of the different initial configurations, (a)–(c), for isolated steps, with $L=21$ sites, and, in case (a), with a confining step (small squares) at distance $d=3$.

Figure 2

Initial, $h(i,t=0)$ (open symbols) and PD equilibrium, $h_{\text{eq}}\left(i\right)$ (solid symbols) positions of curved steps at $k_{B}T/J=3.0$ for $L=41$ and 61. The dashed lines denote circles.

Figure 3

Effective exponent $x_{\text{eff}}$ at $k_{B}T/J=3.0$ for PD, for flat (circle), tilted, with slope $s=1$ (square), and curved (diamond) steps, $L=101$, as well as for EC, for flat (triangle up) and tilted, with $s=1$ (triangle down) steps, $L=201$.

Figure 4

Effective prefactor $a_p$ vs inverse number of active sites $1/L_a$ for PD for flat (circle), tilted, with slope $s=1$ (diamond) and curved (square) steps at $k_{B}T/J=3.0$.

Figure 5

Prefactor $a$ of the step fluctuations for PD, at various temperatures $k_{B}T/J$, comparing Monte Carlo findings (circle), extrapolated to the thermodynamic limit, with predictions based on Langevin theory (square).

Figure 6

Slope, $s$, dependence of the (extrapolated) prefactor $a$ for tilted steps with PD at $k_{B}T/J=3.0$.

Figure 7

Typical Monte Carlo equilibrium configuration, PD, for a confined step with $d=4$ at $k_{B}T/J=1.0$.

Figure 8

Effective exponent $x_{\text{eff}}$ for confined steps of length $L=201$, with PD (open symbols) at $k_{B}T/J=1.0$, for $d=3$ (circle), 5 (square), and 16 (diamond), and with EC (shaded symbols) at $k_{B}T/J=3.0$, for $d=3$ (circle), 20 (square), and 300 (diamond).

Figure 9

Histograms of $\Delta h$ for pinned isolated (diamond) and confined steps with $d=3$ (squares) and 5 (circles) of length $L=201$ at $k_{B}T/J=1.0$, using PD, comparing simulation data (open symbols) at $t=15000$ MCS/S to Gaussian distributions (shaded symbols) with the standard deviation set equal to the fluctuation length.

Figure 10

Finite-size dependence of characteristic time $t_{2/11}$ for confined steps with PD for $d=4$ (square) and $d=5$ (circle) at $k_{B}T/J=1.0$.

Figure 11

EC equilibrium positions (circle), $h_{\text{eq}}\left(i\right)$, for confined step of length $L=201$ with $d=20$ at $k_{B}T/J=3.0$.