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Extensive numerical simulations of surface growth with temporally correlated noise

Tianshu Song and Hui Xia
Phys. Rev. E 103, 012121 – Published 19 January 2021
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  • INTRODUCTION
  • MODELS AND METHODS
  • RESULTS AND DISCUSSIONS
  • CONCLUSION
  • ACKNOWLEDGMENTS
    • References

    Abstract

    Surface growth processes can be significantly affected by long-range temporal correlations. In this work, we perform extensive numerical simulations of a (1+1)- and (2+1)-dimensional ballistic deposition (BD) model driven by temporally correlated noise, which is regarded as the temporal correlated Kardar-Parisi-Zhang universality class. Our results are compared with the existing theoretical predictions and numerical simulations. When the temporal correlation exponent is above a certain threshold, BD surfaces develop gradually faceted patterns. We find that the temporal correlated BD system displays nontrivial dynamic properties, and the characteristic roughness exponents satisfy α≃αloc<αs in (1+1) dimensions, which is beyond the existing dynamic scaling classifications.

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    • Received 21 August 2020
    • Revised 22 December 2020
    • Accepted 23 December 2020

    DOI:https://doi.org/10.1103/PhysRevE.103.012121

    ©2021 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Growth processes
    1. Physical Systems
    Surface growth
    1. Techniques
    Kardar–Parisi–Zhang equation
    Statistical Physics & Thermodynamics

    Authors & Affiliations

    Tianshu Song1,2 and Hui Xia1,*

    • 1School of Materials Science and Physics, China University of Mining and Technology, Xuzhou 221116, China
    • 2School of Information and Control Engineering, China University of Mining and Technology, Xuzhou 221116, China

    • *Corresponding author: hxia@cumt.edu.cn

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    Issue

    Vol. 103, Iss. 1 — January 2021

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    Images

    • Figure 1
      Figure 1

      The interface width W(L,t) at different growth regimes with (a) θ=0.05; (b) θ=0.45. Results have been averaged over 1500 noise realizations, and these dotted lines are plotted to guide the eyes. The insets are the log-log plot of saturated interface width Wsat and system size L, and the solid lines are the fitting results with the values of global roughness exponent α.

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    • Figure 2
      Figure 2

      The log-log plot of the scaled interface width vs the scaled growth time: (a) θ=0.05; (b) θ=0.45. Results show data collapses with the chosen critical exponents: (a) α=0.52 and z=1.55; (b) α=0.92 and z=1.42.

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    • Figure 3
      Figure 3

      The log-log plot of the height-height correlation function G(l,t) vs l with L=4096: (a) θ=0.05; (b) θ=0.45. Results have been averaged over 1500 noise realizations. For clear comparison, each curve shifts accordingly along the vertical coordinate.

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    • Figure 4
      Figure 4

      The structure factor S(k,t) of the growth interface at different growth times for the BD system in presence of the correlated noises: (a) θ=0.05; (b) θ=0.45. Results have been averaged over different noise realizations: 1500 for θ=0.05 and 500 for θ=0.45. Insets show good data collapses with the chosen critical exponents: (a) α=0.518–0.538 and z=1.55; (b) α=0.921–0.931 and z=1.42.

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    • Figure 5
      Figure 5

      The scaling exponents as a function of θ for the correlated BD model: (a) α vs θ; (b) β vs θ; (c) z vs θ. The existing theoretical predictions are also provided for comparison quantitatively.

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    • Figure 6
      Figure 6

      The comparison of interface morphology in the (1+1)-dimensional correlated BD model with three growth regimes and two temporally correlated parameters: (a) t=1.2×103, (b) t=7.5×104, (c) t=4.8×106 for θ=0.05; and (d) t=1.2×103, (e) t=7.5×104, (f) t=4.8×106 for θ=0.45.

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    • Figure 7
      Figure 7

      The comparison of interface morphology in (1+1)-dimensional correlated BD model with three growth regimes and two temporally correlated parameters: (a) t=7.5×104, (b) t=4.8×106 for θ=0.20; and (c) t=7.5×104, (d) t=4.8×106 for θ=0.35.

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    • Figure 8
      Figure 8

      W(L,t) vs t in (2+1)-dimensional BD without temporal correlation. Inset exhibits Wsat against L.

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    • Figure 9
      Figure 9

      The log-log plot of surface width vs growth time in (2+1) dimensions: (a) θ=0.05; (b) θ=0.45. Results have been averaged over 500 noise realizations, and these dotted lines are plotted to guide the eyes. The insets are the log-log plot of saturated surface width Wsat and system size L, and the solid lines are the fitting results with the values of global roughness exponent α.

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    • Figure 10
      Figure 10

      The structure factor S(k,t) of the growth surface at different growth times for (2+1)-dimensional BD system in presence of the correlated noises: (a) θ=0.05; (b) θ=0.45. Insets show good data collapses with the chosen critical exponents: (a) α=0.415±0.010 and z≈1.60; (b) α=0.870±0.010 and z≈1.40. The system size L=1024×1024 is used, and data are averaged over 500 independent noise realizations.

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    • Figure 11
      Figure 11

      The values of critical exponents in (2+1) dimensions: (a) α and αs; (b) β, (c) z.

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    • Figure 12
      Figure 12

      The surface morphology in (2+1)-dimensional BD model with θ=0.05 at different growth times: (a) t=6.75×102; (b) t=4.32×104; (c) t=1.73×105; and with θ=0.45 at different growth times: (d)t=6.75×102; (e) t=4.32×104; (f) t=3.01×105. The subgraphs of each subfigure are height of cross section at X=1 (L) and Y=1 (R).

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    • Figure 13
      Figure 13

      The crossover from self-affine to mounded pattern in simulating BD model from small to large temporal correlations with different growth times: (a) t=3.529×104, (b) t=3.529×106 (θ=0.20); (c) t=3.529×104, (d) t=3.529×106 (θ=0.35). The subgraphs of each subfigure are height of cross section at X=1 (L) and Y=1 (R).

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