Universality of (2+1)-dimensional restricted solid-on-solid models

Jeffrey Kelling, Géza Ódor, and Sibylle Gemming

Phys. Rev. E 94, 022107 – Published 5 August 2016

Abstract

Extensive dynamical simulations of restricted solid-on-solid models in $D = 2 + 1$ dimensions have been done using parallel multisurface algorithms implemented on graphics cards. Numerical evidence is presented that these models exhibit Kardar-Parisi-Zhang surface growth scaling, irrespective of the step heights $N$ . We show that by increasing $N$ the corrections to scaling increase, thus smaller step-sized models describe better the asymptotic, long-wave-scaling behavior.

Received 13 May 2016

DOI:https://doi.org/10.1103/PhysRevE.94.022107

Physics Subject Headings (PhySH)

Growth processes Nonequilibrium statistical mechanics Roughness

Statistical Physics & ThermodynamicsCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Jeffrey Kelling^1,2, Géza Ódor³, and Sibylle Gemming^2,4

¹Department of Information Services and Computing, Helmholtz-Zentrum Dresden-Rossendorf, P. O. Box 51 01 19, 01314 Dresden, Germany
²Institute of Ion Beam Physics and Materials Research, Helmholtz-Zentrum Dresden-Rossendorf, P. O. Box 51 01 19, 01314 Dresden, Germany
³Institute of Technical Physics and Materials Science, Centre for Energy Research of the Hungarian Academy of Sciences, P. O. Box 49, H-1525 Budapest, Hungary
⁴Institute of Physics, TU Chemnitz, 09107 Chemnitz, Germany

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Vol. 94, Iss. 2 — August 2016

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Images

Figure 1
(a) Squared roughness ( $W^{2}$ ) of surfaces of size $V = 4096^{2}$ (256 realizations) in the scaling regime (error bars are smaller than symbols). (b) Local slopes analysis of roughness scaling for size $V = 8192^{2}$ (128 realizations). Straight lines are linear fits to the tail ( $t \geq 1260 MCS$ ), extrapolating to $t \to \infty$ , assuming $\sqrt[4]{t}$ corrections. Uncertainties given for $β_{N}$ are errors of the singular linear fits displayed in the plot. The black dashed line is the PL extrapolation for $N = 1$ . The dashed lines corresponding in color to the respective plots for $N > 1$ are fits of the form (11). All PL fits were performed for $t \geq 148 MCS$ . Both figures show $N = 1, 3, 5, 7$ (bottom to top).
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Figure 2
(a) Local slopes of finite-size scaling analysis with $N = 1, 3, 5, 7$ . Error bars are propagated $1 σ$ errors. Straight lines are linear fits to extrapolate to infinity, uncertainties given for $α_{N}$ are pure fit errors. Steady-state data taken for $t > t_{start} = 50 t_{{steady}^{*}}$ (see text). (b) Dependence of extrapolated $α$ on $t_{start}$ is weak. Both figures: Sample sizes are at least 1024–2048 realizations and $\geq 8192$ realizations for $L \leq 64$ . All system sizes taken into account for finite-size scaling are listed in Fig. 3, where the considered time scales can also be read off.
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Figure 3
Collapse of squared roughness in the steady state for $N = 1, 3, 5, 7$ (from bottom to top). Panel (a) shows a perfect collapse for $N > 1$ , using $α = 0.4$ and $β = 0.25$ ( $z = α / β = 1.6$ ). Panel (b) shows a collapse using $α = 0.389$ and $β = 0.241$ ( $z \approx 1.61$ ). This looks perfect for $N = 1$ but not for $N > 1$ .
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Figure 4
Skewness $S_{h}$ (a) and kurtosis $Q_{h}$ (b) of the height distribution in the steady state plotted over the inverse lateral system size. Values are plotted only for $N = 1$ (black) and $N = 7$ (green) for the sake of clarity. The straight lines are linear fits, included to guide the eye. Different symbols indicate different ratios $t_{start} / t_{steady *} \geq 2$ . A key is not provided for the symbols because there is no correlation with this parameter.
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Figure 5
Local slopes of finite-size scaling analysis for $N = 1, 3, 5, 7$ . Error bars are propagated $1 σ$ errors. Straight lines are linear fits to extrapolate to infinity, and uncertainties given for $α_{N}$ are pure fit errors. Steady-state data are taken for $t > t_{start} = 50 t_{{steady}^{*}}$ (see text). (a) DD domains containing $6 (+ 1) \times 10 (+ 1)$ sites. Sample sizes are at least 512 realizations; for $N = 5, 7$ and sizes $L = 64$ and 128, $n = 16384, n = 8192$ are used. (b) DD domains containing $16 \times 16$ sites. For $L = 512$ the sample contains 256 realizations; for other system sizes at least 512 samples are included.
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