Modeling of nonequilibrium surface growth by a limited-mobility model with distributed diffusion length

Kinetic Monte Carlo (KMC) simulations are a well-established numerical tool to investigate the time-dependent surface morphology in molecular beam epitaxy experiments. In parallel, simplified approaches such as limited mobility (LM) models characterized by a fixed diffusion length have been studied. Here we investigate an extended LM model to gain deeper insight into the role of diffusional processes concerning the growth morphology. Our model is based on the stochastic transition rules of the Das Sarma–Tamborena model but differs from the latter via a variable diffusion length. A first guess for this length can be extracted from the saturation value of the mean-squared displacement calculated from short KMC simulations. Comparing the resulting surface morphologies in the sub- and multilayer growth regime to those obtained from KMC simulations, we find deviations which can be cured by adding fluctuations to the diffusion length. This mimics the stochastic nature of particle diffusion on a substrate, an aspect which is usually neglected in LM models. We propose to add fluctuations to the diffusion length by choosing this quantity for each adsorbed particle from a Gaussian distribution, where the variance of the distribution serves as a fitting parameter. We show that the diffusional fluctuations have a huge impact on cluster properties during submonolayer growth as well as on the surface profile in the high coverage regime. The analysis of the surface morphologies on one- and two-dimensional substrates during sub- and multilayer growth shows that the LM model can produce structures that are indistinguishable to the ones from KMC simulations at arbitrary growth conditions.

Figure 1

Illustration of particle deposition and surface relaxation in the system (in one dimension) that is used to simulate low-temperature MBE growth. The quantities $F$ and $D^0\left(T\right)$ refer to the KMC simulation, while the Gaussian distribution $P\left(l\right|l_n,{\sigma }^2)$ for the diffusion length $l$ is characteristic for the LM model with diffusional fluctuations whose strength is controlled via the variance ${\sigma }^2$.

Figure 2

KMC results for the nucleation length $l_n$ [see Eq. (10)] of particles adsorbed at the very early stage of submonolayer growth, and the geometrical distance $d_g$ [see Eq. (11)] between clusters, as a function of $R$. The dashed black line describes the dependency $\sim R^{\gamma }$, with $\gamma =1/4$ ($d=1$), and the dash-dotted line follows $\sim R^{1/6}$ ($d=2$).

Figure 3

Comparison of $N\left(\theta \right)$ in the KMC model for various values of $R$ (solid lines) with the LM model where $l=l_n\left(R\right)$ is used. The dashed lines represent results of the LM model with ${\sigma }^2=0$, and the dots correspond to $N\left(\theta \right)$ with optimal values of ${\sigma }^2$.

Figure 4

The quantity $\mathrm{\Delta }{N}_{\max }$ [see Eq. (12)] as a function of the variance ${\sigma }^2$ in the LM model for various values of the mean diffusion length $l_n\left(R\right)=l$. The black dashed line indicates $\mathrm{\Delta }{N}_{\max }=0$. The inset shows the optimal values of ${\sigma }^2$ as a function of $l$ fulfilling $\mathrm{\Delta }{N}_{\max }=0$.

Figure 5

Number of clusters $N$ and monomers $n$ as a function of coverage $\theta$ on a lattice of size $L=16384$ for the KMC model and the LM model with corresponding diffusion length and different values of the variance ${\sigma }^2$. The solid black line follows $\sim {\theta }^{1/4}$, and the dashed black line scales according to $\sim {\theta }^{-r}$, with $r\approx 0.64$ [53, 79].

Figure 6

Comparison of the cluster size distribution $P\left(S\right)$ during submonolayer growth at $\theta =0.5$ for (a) $l=12$, (b) $l=24$, and various values of the variance ${\sigma }^2$ that controls the strength of diffusional fluctuations.

Figure 7

(a) Evolution of the coverage in the first 10 layers in the KMC model at $R={10}^2$ and $l=5$ (${\sigma }^2=0.1$) in the LM model. (b) The same at $R=4\times {10}^4$ in the KMC model and $l=24$ (${\sigma }^2=7.0$) in the LM model. (c) Coverage evolution of the tenth layer at various growth conditions in both models. Solid lines represent KMC simulations, dotted lines are results from the LM model (with optimal variance ${\sigma }^2$).

Figure 8

Global interface width $W(L,\theta )$ as a function of coverage $\theta$ for four different values of $R$ in the KMC model (symbols) and corresponding diffusion length $l$ in the LM model (solid lines) with optimal values for ${\sigma }^2$ for system sizes from $L=64$ to $L=256$.

Figure 9

(a) Scaling of the saturation roughness $W_{\mathrm{sat}}$ as a function of system size $L$ for two values of $R$ in the KMC model and the corresponding values of $l$ and ${\sigma }^2$ in the LM model (dotted black line $\sim L^1$). (b) Roughness exponent $\alpha$, (c) growth exponent $\beta$, and (d) dynamic exponent $z$ for various growth conditions in both models. The lines in panels (b–d) represent the values of $\alpha ,\beta ,z$ according to the VLDS universality class in one dimension.

Figure 10

Relative error $\epsilon \left(W_{\mathrm{sat}}\right)$ of the saturation roughness $W_{\mathrm{sat}}\left(L\right)$ in the LM model compared to KMC simulations. Results for $\epsilon \left(W_{\mathrm{sat}}\right)$ without fluctuations in the diffusion length $l$ (i.e., ${\sigma }^2=0$) are given by hexagons, while $\epsilon \left(W_{\mathrm{sat}}\right)$ with optimal ${\sigma }^2$ are represented by triangles.

Figure 11

(a) Scaling of the saturation roughness $W_{\mathrm{sat}}\left(l\right)$ in the LM model as a function of the average diffusion length $l$ with optimal values for ${\sigma }^2$ for different system sizes $L$. The black dotted line follows $\sim l^{-3/2}$. (b) Scaling of $W_{\mathrm{sat}}\left(R\right)$ as a function of the growth parameter $R$. The black dotted line follows $\sim R^{-1/2}$. (c) Rescaled saturation roughness $W_{\mathrm{sat}}^{\mathrm{RE}}$ in the LM model and corresponding KMC simulations.

Figure 12

(a–f) Height profiles in the LM model at $\theta ={10}^6$ for two representative values of $l$ and different values of the variance ${\sigma }^2$ at coverage $\theta ={10}^6$ in systems of size $L=4096$. (g) Height-height correlation function $\mathrm{\Gamma }\left(r\right)/\mathrm{\Gamma }\left(0\right)$ for the depicted system settings with $l=5$ and different values of ${\sigma }^2$ at $\theta ={10}^4$. (h) The same as in panel (g) for $l=10$.

Figure 13

(a) Height-height correlation function $\mathrm{\Gamma }\left(r\right)/\mathrm{\Gamma }\left(0\right)$ for KMC simulations at $R=4\times {10}^2$ together with results of the LM model at $l=5$ and various values of the variance ${\sigma }^2$. (b) The same as in panel (a) for $R=4\times {10}^3$ and $l=12$. (c) Correlation length ${\xi }_0$ for different combinations of $l$ and ${\sigma }^2$ (lines with symbols) together with corresponding values of ${\xi }_0$ from KMC simulations (dashed lines). Stars in panel (c) represent the optimal values of ${\sigma }^2$.

Figure 14

$N_{\max }$ in the submonolayer growth regime on two dimensional substrates of lateral size $L=200$ for various growth conditions in the KMC and the LM model, respectively. The black dotted line follows $\sim R^{-1/3}$.

Figure 15

Cluster size distribution $P\left(S\right)$ during submonolayer growth at $\theta =0.3$ on two-dimensional substrates of lateral size $L=200$ for two values of $l$ and various strengths of diffusional fluctuations controlled via ${\sigma }^2$ in the LM model.

Figure 16

(a) Exemplary lattice structures (of size $100\times 100$ from lattice with $L=200$) from KMC simulations with $R=4\times {10}^2$ (a) and $R=2\times {10}^4$ (d) at $\theta =50$ along with results obtained from the LM model with $l=4$ and ${\sigma }^2=1.1$, (b) $l=11$ and ${\sigma }^2=2.8$ (e), at the same coverage, $\theta =50$. Panels (c) and (f) depict $\mathrm{\Gamma }\left(r\right)/\mathrm{\Gamma }\left(0\right)$ for both considered growth conditions in the KMC and the LM model, respectively.