Instituto de Física, Universidade Federal da Bahia, Campus Universitário da Federação, Rua Barão de Jeremoabo s/n, 40170-115, Salvador, BA, Brazil.
Instituto de Física, Universidade de Brasília, 70910-900, Brasília, DF, Brazil.
Phys Rev E. 2023 Mar;107(3-1):034802. doi: 10.1103/PhysRevE.107.034802.
Fractal properties on self-affine surfaces of films growing under nonequilibrium conditions are important in understanding the corresponding universality class. However, measurement of the surface fractal dimension has been intensively investigated and is still very problematic. In this work, we report the behavior of the effective fractal dimension in the context of film growth involving lattice models believed to belong to the Kardar-Parisi-Zhang (KPZ) universality class. Our results, which are presented for growth in a d-dimensional substrate (d=1,2) and use the three-point sinuosity (TPS) method, show universal scaling of the measure M, which is defined in terms of discretization of the Laplacian operator applied to the height of the film surface, M=t^{δ}g[Θ], where t is the time, g[Θ] is a scale function, δ=2β, Θ≡τt^{-1/z}, β, and z are the KPZ growth and dynamical exponents, respectively, and τ is a spatial scale length used to compute M. Importantly, we show that the effective fractal dimensions are consistent with the expected KPZ dimensions for d=1,2, if Θ≲0.3, which include a thin film regime for the extraction of the fractal dimension. This establishes the scale limits in which the TPS method can be used to accurately extract effective fractal dimensions that are consistent with those expected for the corresponding universality class. As a consequence, for the steady state, which is inaccessible to experimentalists studying film growth, the TPS method provided effective fractal dimension consistent with the KPZ ones for almost all possible τ, i.e., 1≲τ<L/2, where L is the lateral size of the substrate on which the deposit is grown. In the growth of thin films, the true fractal dimension can be observed in a narrow range of τ, the upper limit of which is of the same order of magnitude as the correlation length of the surface, indicating the limits of self-affinity of a surface in an experimentally accessible regime. This upper limit was comparatively lower for the Higuchi method or the height-difference correlation function. Scaling corrections for the measure M and the height-difference correlation function are studied analytically and compared for the Edwards-Wilkinson class at d=1, yielding similar accuracy for both methods. Importantly, we extend our discussion to a model representing diffusion-dominated growth of films and find that the TPS method achieves the corresponding fractal dimension only at steady state and in a narrow range of the scale length, compared to that found for the KPZ class.
在非平衡条件下生长的薄膜的自仿射表面上的分形性质对于理解相应的普遍性类别很重要。然而，表面分形维数的测量已经得到了深入的研究，但仍然存在很大的问题。在这项工作中，我们报告了在涉及晶格模型的薄膜生长中有效分形维数的行为，这些模型被认为属于 Kardar-Parisi-Zhang (KPZ) 普遍性类别。我们的结果是在 d 维衬底（d=1,2）上的生长情况下得到的，并使用三点正弦法（TPS），表明在离散化应用于薄膜表面高度的拉普拉斯算子的度量 M 上存在普适标度，M=t^{\delta}g[\Theta]，其中 t 是时间，g[\Theta]是一个标度函数，(\delta=2\beta)，(\Theta\equiv\tau t^{-1/z})，(\beta)和 z 分别是 KPZ 生长和动力学指数，(\tau)是用于计算 M 的空间标度长度。重要的是，如果(\Theta\lesssim0.3)，我们表明有效分形维数与 d=1,2 时预期的 KPZ 维数一致，这包括薄膜提取分形维数的范围。这确定了 TPS 方法可以用来准确提取与相应普遍性类别一致的有效分形维数的标度限制。因此，对于实验研究人员难以获得的稳态，TPS 方法提供的有效分形维数与 KPZ 一致，几乎适用于所有可能的(\tau)，即 1(\leq\tau)(\lt L/2)，其中 L 是沉积生长的衬底的横向尺寸。在薄膜生长中，真实分形维数可以在(\tau)的一个狭窄范围内观察到，其上限与表面的相关长度具有相同的数量级，这表明了在实验可访问的范围内表面自相似性的限制。对于 Higuchi 方法或高度差相关函数，上限相对较低。对度量 M 和高度差相关函数的标度修正进行了分析，并在 d=1 时对 Edwards-Wilkinson 类进行了比较，两种方法的准确性相似。重要的是，我们将讨论扩展到代表薄膜扩散主导生长的模型，并发现与 KPZ 类相比，TPS 方法仅在稳态和标度长度的狭窄范围内实现相应的分形维数。
