Accessibility of the surface fractal dimension during film growth

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, $M=t^{\delta }g\left[\mathrm{\Theta }\right]$, where $t$ is the time, $g\left[\mathrm{\Theta }\right]$ is a scale function, $\delta =2\beta , \mathrm{\Theta }\equiv \tau t^{-1/z}, \beta$, and $z$ are the KPZ growth and dynamical exponents, respectively, and $\tau$ 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 $\mathrm{\Theta }\lesssim 0.3$, 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 $\tau$, i.e., $1\lesssim \tau <L/2$, 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 $\tau$, 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.

Figure 1

Global roughness, $w(L,t)$, as a function of time, $t$, considering the etching, SS and RSOS models. (a) Results for $d=1$ with lateral size $L=2^{15}$ for an average over ${10}^3$ independent realizations. Dashed (red) lines indicate slopes equal to $1/3$. (b) Results for $d=2$ considering the lateral sizes $L=2^{11}$ for etching model and $L=2^{10}$ for the SS and RSOS models. Dashed (red) lines indicate slopes equal to $7/29$ (e.g., see Ref. [43]).

Figure 2

$\mathrm{\Gamma }(\mathbf{r},t)/\mathrm{\Gamma }\left(\mathbf{0}\right)$ as a function of $r$, for different times, for (left panels) $d=1$ and (right panels) $d=2$. (a) and (b), (c) and (d), and (e) and (f) correspond to the results for the SS, etching, and RSOS models, respectively.

Figure 3

Evolution of the correlation length, $\xi$ (see Table 1) for the etching, SS and RSOS models. (a) Results for $d=1$ are shown for the lateral size $L=2^{15}$ and were averaged over ${10}^3$ independent experiments. The dashed (blue) line shows a slope equal to $1/z$ with $z=3/2$. (b) Results for $d=2$ using the lateral sizes $L=2^{11}$ for the etching model ($4\times {10}^2$ independent experiments) and $L=2^{10}$ for the SS and RSOS models, averaged over ${10}^3$ independent experiments. The dashed (blue) line indicates a slope equal to $1/z$, where $z=29/18$ (e.g., see Ref. [43]).

Figure 4

Results for $d=1$. (Left panels) $M$ as a function of $\tau$. Vertical arrows indicate, for each time, the position $\tau =\xi$ (see Table 1). (Right panels) $d_{f_{\mathrm{eff}}}^{\mathrm{TPS}}$ as a function of $\mathrm{\Theta }\equiv \tau t^{-1/z}$. Vertical arrows indicate, for each time, the position at $\mathrm{\Theta }=\xi t^{-1/z}$. We note that $\xi t^{-1/z}$ is nearly constant for $t\lesssim {10}^4$ (the arrows overlap at these times). (a) and (b) for etching model; (c) and (d) for the SS model; (e) and (f) for the RSOS model. The results are presented for lateral size $L=2^{15}$, with averages calculated for ${10}^3$ independent realizations. In (a), (c), and (e), the insets show collapse achieved by replacing the variables $M\to M{t}^{-\delta }$ and $\tau \to \mathrm{\Theta }\equiv \tau t^{-1/z}$, where, for $d=1, \delta =2\beta =1/z$, with $z=1.5$. Dashed lines have slope 1. In (b), (d), and (f), the results are presented for $\tau \ge 3$ and dashed horizontal bottom and top lines represent the values of the KPZ fractal dimension, for $d=1$, using Eq. (5) $[d_f=2-\alpha =1.5]$, and the value 2, respectively. The pink area shows a limit for the variable $\mathrm{\Theta }\equiv \tau t^{-1/z}$, where the KPZ fractal dimension could be achieved.

Figure 5

Results for $d=2$. (Left panels) $M$ as a function of $\tau$. Vertical arrows indicate, for each time, the position $\tau =\xi$ (see Table 1). (Right panels) $d_{f_{\mathrm{eff}}}^{\mathrm{TPS}}$ as a function of $\mathrm{\Theta }\equiv \tau t^{-1/z}$. Vertical arrows indicate, for each time, the position at $\mathrm{\Theta }=\xi t^{-1/z}$. We note that $\xi t^{-1/z}$ is nearly constant for $t\lesssim {10}^4$ (striking is the overlapping of the arrows at times ${10}^2\lesssim t\lesssim {10}^3$). (a) and (b) for the etching model; (c) and (d) for the SS model; (e) and (f) for the RSOS model. The results are presented for lateral sizes $L=2^{11}$ for etching with 400 independent realizations and $L=2^{10}$ for the SS and RSOS models with ${10}^3$ independent realizations. In (a), (c), and (e), the insets show the collapse achieved by replacing the variables $M\to M{t}^{-\delta }$ and $\tau \to \tau t^{-1/z}$, where $z=29/18$ (e.g., Ref. [43]). In (a), (c), and (e), the dashed lines have slopes $2\alpha =7/9$ and the values of the exponent $\delta$ used were $\delta =2\beta =14/29$ [43]. In (b), (d), and (f), the results are presented for $\tau \ge 3$ and the dashed horizontal top and bottom lines represent the value of the KPZ fractal dimension, that, for $d=2$, using Eq. (5), is $d_f=29/18$ (using $\alpha$ reported in Ref. [43]), and the value 2, respectively. The pink area shows an approximate limit for the variable $\mathrm{\Theta }\equiv \tau t^{-1/z}$, from which the KPZ fractal dimension could be achieved.

Figure 6

Results for $d=1$ in (a) and (b) $[d=2$ in (c) and (d)]. (Left panels) $L_k$ as a function of $k$. Vertical arrows indicate, for each time, the position $\tau =\xi$ (see Table 1). (Right panels) $d_{f_{\mathrm{eff}}}^{\mathrm{HM}}$ as a function of $k{t}^{-1/z}$. Vertical arrows indicate, for each time, the position at $\xi t^{-1/z}$. The results are presented for the RSOS model with lateral size $L=2^{15}$ ($L=2^{10}$) for $d=1$ ($d=2$), with averages calculated for ${10}^3$ independent realizations. In (b) and (d), dashed horizontal bottom and top lines represent the values of the KPZ fractal dimension for $d=1,2$, using Eq. (5) $[d_f=2-\alpha ]$, and the value 2, respectively. The green area shows a limit for the variable $k{t}^{-1/z}$, where the KPZ fractal dimension could be achieved.

Figure 7

Results for $d=2$ and $L=512$, considering the CRSOS model and an average over 200 different samples. (a) and (b) correspond to the results for the height difference correlation function, $G(r,t)$, and the effective fractal dimension $d_{f_{\mathrm{eff}}}^{\mathrm{G}}$, respectively. (c) and (d) correspond to the results for the measure ${\mathcal{L}}_k$ (Higuchi method), and the effective fractal dimension, $d_{f_{\mathrm{eff}}}^{\mathrm{HM}}$, respectively. (e) and (f) correspond to the results for the measure $M$ (TPS method), and the effective fractal dimension, $d_{f_{\mathrm{eff}}}^{\mathrm{TPS}}$, respectively. The inset of (e) shows the collapse obtained by replacing the variables $M\to M{t}^{-\delta }$ and $\tau \to \tau t^{-1/z}$, where $z$ is the dynamic exponent and $\delta =2\beta$ for the VLDS class (for a two-loop RG approximation). The slope shown in the inset has slope $2\alpha$, where $\alpha$ is given by Eqs. (31) and (32), considering a two loop RG approximation (see the text). In (b), (d), and (f), dashed horizontal bottom and top lines represent the values of the VLDS fractal dimension for $d=2$, using Eq. (5) $[d_f=2-\alpha ]$, and the value 2, respectively.