Numerical Simulation of a Space-Fractional Molecular Beam Epitaxy Model without Slope Selection

Abstract

In this paper, we introduce a space-fractional version of the molecular beam epitaxy (MBE) model without slope selection to describe super-diffusion in the model. Compared to the classical MBE equation, the spatial discretization is an important issue in the space-fractional MBE equation because of the nonlocal nature of the fractional operator. To approximate the fractional operator, we employ the Fourier spectral method, which gives a full diagonal representation of the fractional operator and achieves spectral convergence regardless of the fractional power. And, to combine with the Fourier spectral method directly, we present a linear, energy stable, and second-order method. Then, it is possible to simulate the dynamics of the space-fractional MBE equation efficiently and accurately. By using the numerical method, we investigate the effect of the fractional power in the space-fractional MBE equation.

1. Introduction

The molecular beam epitaxy (MBE) is a versatile technique for growing thin epitaxial structures made of semiconductors, metals or insulators [1]. One of models for simulating the MBE growth is the gradient flow for the following energy functional [2]:

$$\begin{array}{c}\hfill \mathcal{E}\left(\varphi \right):={\int }_{\mathsf{\Omega }}\left(-\frac{1}{2}{ln(1+|\nabla \varphi |}^2)+\frac{\delta }{2}{|\Delta \varphi |}^2\right)d\mathbf{x},\end{array}$$

(1)

where $\varphi$ is a scaled height function of a thin film in a co-moving frame and $\delta >0$ is a constant. The first and second terms in $\mathcal{E}\left(\varphi \right)$ model the Ehrlich–Schwoebel effect [3,4,5] and surface diffusion, respectively. Because of the first term in $\mathcal{E}\left(\varphi \right)$, this model is characterized by the absence of a preferred slope, i.e., there is no slope selection [6]. The $L^2$-gradient flow for $\mathcal{E}\left(\varphi \right)$ takes the form [6]

$$\begin{array}{c}\hfill \frac{\partial \varphi }{\partial t}=-\frac{\delta \mathcal{E}}{\delta \varphi }=-\left(\nabla ·\left(\frac{\nabla \varphi }{1+{|\nabla \varphi |}^2}\right)+\delta {\Delta }^2\varphi \right),\end{array}$$

(2)

where $\frac{\delta }{\delta \varphi }$ denotes the variational derivative and the boundary condition for $\varphi$ is considered periodic in all spatial directions. It has been shown analytically and numerically that the MBE Equation (2) predicts $\mathcal{E}\left(t\right)\sim O(-ln(t\left)\right)$ and $w\left(t\right)\sim O\left(t^{1/2}\right)$ as $t\to \infty$ [6,7,8,9,10,11,12,13,14,15], where $w\left(t\right)$ is the roughness (the standard deviation of the height profile) defined as [7,16]

$$\begin{array}{c}\hfill w\left(t\right)=\sqrt{\frac{1}{|\Omega |}{\int }_{\Omega }{\left(\varphi (\mathbf{x},t)-\overline{\varphi }\left(t\right)\right)}^{2}d\mathbf{x}},\end{array}$$

(3)

where $\overline{\varphi }\left(t\right)=\frac{1}{|\Omega |}{\int }_{\Omega }\varphi (\mathbf{x},t)d\mathbf{x}$. One might wonder whether anomalous MBE affects the scaling rates. A straightforward idea is to utilize fractional derivatives, i.e., formulate fractional MBE equations. Fractional partial differential equations have been proved to be valuable tools for modeling diffusive processes associated with anomalous diffusion [17,18,19,20,21,22,23,24,25,26,27,28,29,30,31]. In recent years, the time-fractional version of the MBE equation has been introduced and solved numerically to describe sub-diffusion in the MBE Equation [32,33,34,35,36,37]. However, to the best of our knowledge, there is no study that investigates the effect of super-diffusion on the MBE growth.

To describe super-diffusion in the MBE equation, we consider a space-fractional version of the MBE equation

$$\begin{array}{c}\hfill \frac{\partial \varphi }{\partial t}=-\frac{\delta {\mathcal{E}}^s}{\delta \varphi }=-\left({\nabla }^{\frac{s}{2}}·\left(\frac{{\nabla }^{\frac{s}{2}}\varphi }{1+|{\nabla }^{\frac{s}{2}}{\varphi |}^2}\right)+\delta {(-\Delta )}^s\varphi \right),\end{array}$$

(4)

where

$$\begin{array}{c}\hfill {\mathcal{E}}^s\left(\varphi \right):={\int }_{\Omega }\left(-\frac{1}{2}ln(1+|{\nabla }^{\frac{s}{2}}{\varphi |}^2)+\frac{\delta }{2}{\left|{(-\Delta )}^{\frac{s}{2}}\varphi \right|}^2\right)d\mathbf{x},\end{array}$$

(5)

${(-\Delta )}^{\frac{s}{2}}$ is the fractional Laplacian ($1<s\le 2$), and ${\nabla }^{\frac{s}{2}}$ is the fractional gradient defined as ${\nabla }^{\frac{s}{2}}:=\nabla {(-\Delta )}^{\frac{s}{2}-1}$ [17,22,29]. Note that the space-fractional MBE Equation (4) reduces to the MBE equation when $s=2$. The space-fractional MBE equation satisfies the energy dissipation and mass conservation properties:

$$\begin{array}{c}\hfill \frac{d{\mathcal{E}}^s}{dt}={\int }_{\Omega }\frac{\delta {\mathcal{E}}^s}{\delta \varphi }\frac{\partial \varphi }{\partial t}d\mathbf{x}=-{\int }_{\Omega }{\left(\frac{\partial \varphi }{\partial t}\right)}^{2}d\mathbf{x}\le 0\end{array}$$

(6)

and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{d}{dt}{\int }_{\Omega }\varphi d\mathbf{x}={\int }_{\Omega }\frac{\partial \varphi }{\partial t}d\mathbf{x}& =-{\int }_{\partial \Omega }\left(\frac{{\nabla }^{\frac{s}{2}}\varphi }{1+|{\nabla }^{\frac{s}{2}}{\varphi |}^2}-\delta \nabla {(-\Delta )}^{\frac{s}{2}}\varphi \right)·\mathbf{n}ds\hfill \\ & +{\int }_{\Omega }\left(\frac{{\nabla }^{\frac{s}{2}}\varphi }{1+|{\nabla }^{\frac{s}{2}}{\varphi |}^2}-\delta \nabla {(-\Delta )}^{\frac{s}{2}}\varphi \right)·{\nabla }^{\frac{s}{2}}\mathbf{1}d\mathbf{x}=0,\hfill \end{array}\end{array}$$

(7)

where $\mathbf{n}$ is a unit normal vector to $\partial \Omega$. Since the space-fractional MBE equation cannot generally be solved analytically, efficient and accurate numerical methods are essential.

The aim of this paper is to simulate the dynamics of the space-fractional MBE equation by using an efficient and accurate method. Compared to the classical MBE equation, the spatial discretization is an important issue in the space-fractional MBE equation because of the nonlocal nature of the fractional operator, which leads to large, dense matrices. Finite difference [18,19,20], finite element [21,22,23], and finite volume [24,25,26] methods have been investigated to solve space-fractional models. In this paper, we employ the Fourier spectral method [27,28,29,30,31,38,39,40,41] for the spatial discretization, which gives a full diagonal representation of the fractional operator and achieves spectral convergence regardless of the fractional power. And we present a linear, energy stable, and second-order method that can be combined with the Fourier spectral method directly. Then, it is possible to produce more efficient and accurate long-time simulation results.

The organization of this paper is as follows. In Section 2, we develop the numerical method for solving the space-fractional MBE equation. In Section 3, we provide numerical examples including a long-time simulation for the coarsening process. Finally, conclusions are given in Section 4.

2. Numerical Method

In this section, we describe an efficient and accurate method for the space-fractional MBE Equation (4). Firstly, we introduce the method at a discrete-time, continuous-space level. Then, we use the Fourier spectral method to discretize the space.

Because the space-fractional MBE equation is of a gradient type, it has the energy decay property. To preserve energy stability without compromising efficiency, we consider the following linear convex splitting:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{E}}^s\left(\varphi \right)& ={\mathcal{E}}_c^s\left(\varphi \right)-{\mathcal{E}}_e^s\left(\varphi \right)\hfill \\ & ={\int }_{\Omega }\left(\frac{1}{2}|{\nabla }^{\frac{s}{2}}{\varphi |}^2+\frac{\delta }{2}{\left|{(-\Delta )}^{\frac{s}{2}}\varphi \right|}^2\right)d\mathbf{x}-{\int }_{\Omega }\frac{1}{2}\left(|{\nabla }^{\frac{s}{2}}{\varphi |}^2+ln(1+|{\nabla }^{\frac{s}{2}}\varphi {|}^2)\right)d\mathbf{x}.\hfill \end{array}\end{array}$$

(8)

The convexity of ${\mathcal{E}}_c^s\left(\varphi \right)$ is obvious, and the convexity of ${\mathcal{E}}_e^s\left(\varphi \right)$ comes from the convexity of the following function:

$$\begin{array}{c}\hfill F\left(\mathbf{y}\right)=\frac{1}{2}\left({\left|\mathbf{y}\right|}^2{+ln(1+|\mathbf{y}|}^2)\right).\end{array}$$

(9)

To have second-order time accuracy, we combine the linear convex splitting (8) with the second-order strong-stability-preserving implicit–explicit Runge–Kutta method [42]:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\varphi }^{\left(1\right)}& ={\varphi }^n-\Delta t\left(\frac{\delta {\mathcal{E}}_c^s\left({\varphi }^{\left(1\right)}\right)}{\delta \varphi }-\frac{\delta {\mathcal{E}}_e^s\left({\varphi }^n\right)}{\delta \varphi }\right),\hfill \\ \hfill {\varphi }^{\left(2\right)}& =\frac{3}{2}{\varphi }^{\left(1\right)}-\frac{1}{2}{\varphi }^n-\frac{\Delta t}{2}\left(\frac{\delta {\mathcal{E}}_c^s\left({\varphi }^{\left(2\right)}\right)}{\delta \varphi }-\frac{\delta {\mathcal{E}}_e^s\left({\varphi }^{\left(1\right)}\right)}{\delta \varphi }\right),\hfill \\ \hfill {\varphi }^{n+1}& =-{\varphi }^{\left(2\right)}+\frac{5}{2}{\varphi }^{\left(1\right)}-\frac{1}{2}{\varphi }^n-\frac{\Delta t}{2}\left(\frac{\delta {\mathcal{E}}_c^s\left({\varphi }^{n+1}\right)}{\delta \varphi }-\frac{\delta {\mathcal{E}}_e^s\left({\varphi }^{\left(2\right)}\right)}{\delta \varphi }\right),\hfill \end{array}\end{array}$$

(10)

where $\frac{\delta {\mathcal{E}}_c^s\left(\varphi \right)}{\delta \varphi }={(-\Delta )}^{s-1}\varphi +\delta {(-\Delta )}^s\varphi$ and $\frac{\delta {\mathcal{E}}_e^s\left(\varphi \right)}{\delta \varphi }={(-\Delta )}^{s-1}\varphi -{\nabla }^{\frac{s}{2}}·\left(\frac{{\nabla }^{\frac{s}{2}}\varphi }{1+|{\nabla }^{\frac{s}{2}}{\varphi |}^2}\right)$.

Theorem 1.

The method (10) satisfies the following energy dissipation and mass conservation laws:

$$\begin{array}{c}\hfill {\mathcal{E}}^s\left({\varphi }^{n+1}\right)\le {\mathcal{E}}^s\left({\varphi }^n\right)\mathrm{and}{\int }_{\Omega }{\varphi }^{n+1}d\mathbf{x}={\int }_{\Omega }{\varphi }^{n}d\mathbf{x}.\end{array}$$

(11)

Proof. 

The convexity of ${\mathcal{E}}_c^s\left(\varphi \right)$ and ${\mathcal{E}}_e^s\left(\varphi \right)$ gives the following inequality:

$$\begin{array}{c}\hfill {\mathcal{E}}^s\left(\varphi \right)-{\mathcal{E}}^s\left(\psi \right)\le {\int }_{\Omega }\left(\frac{\delta {\mathcal{E}}_c^s\left(\varphi \right)}{\delta \varphi }-\frac{\delta {\mathcal{E}}_e^s\left(\psi \right)}{\delta \varphi }\right)(\varphi -\psi )d\mathbf{x}.\end{array}$$

(12)

Using the inequality, we have

$$\begin{array}{c}\hfill \begin{array}{cc}& {\mathcal{E}}^s\left({\varphi }^{n+1}\right)-{\mathcal{E}}^s\left({\varphi }^n\right)\hfill \\ & =\left({\mathcal{E}}^s\left({\varphi }^{n+1}\right)-{\mathcal{E}}^s\left({\varphi }^{\left(2\right)}\right)\right)+\left({\mathcal{E}}^s\left({\varphi }^{\left(2\right)}\right)-{\mathcal{E}}^s\left({\varphi }^{\left(1\right)}\right)\right)+\left({\mathcal{E}}^s\left({\varphi }^{\left(1\right)}\right)-{\mathcal{E}}^s\left({\varphi }^n\right)\right)\hfill \\ & \le -\frac{1}{\Delta t}\left({\int }_{\Omega }(2{\varphi }^{n+1}+2{\varphi }^{\left(2\right)}-5{\varphi }^{\left(1\right)}+{\varphi }^n)({\varphi }^{n+1}-{\varphi }^{\left(2\right)})d\mathbf{x}\right.\hfill \\ & \left.+{\int }_{\Omega }(2{\varphi }^{\left(2\right)}-3{\varphi }^{\left(1\right)}+{\varphi }^n)({\varphi }^{\left(2\right)}-{\varphi }^{\left(1\right)})d\mathbf{x}+{\int }_{\Omega }{({\varphi }^{\left(1\right)}-{\varphi }^n)}^{2}d\mathbf{x}\right)\hfill \\ & =-\frac{1}{\Delta t}{\int }_{\Omega }\left(\frac{1}{4}{({\varphi }^{n+1}-3{\varphi }^{\left(1\right)}+2{\varphi }^n)}^2+\frac{7}{4}{({\varphi }^{n+1}-{\varphi }^{\left(1\right)})}^2\right)d\mathbf{x}\le 0.\hfill \end{array}\end{array}$$

(13)

And integrating each stage of (10) over $\Omega$ and using the fact that ${\int }_{\Omega }\frac{\delta {\mathcal{E}}_c^s\left(\varphi \right)}{\delta \varphi }d\mathbf{x}=0$ and ${\int }_{\Omega }\frac{\delta {\mathcal{E}}_e^s\left(\varphi \right)}{\delta \varphi }d\mathbf{x}=0$, we obtain

$$\begin{array}{c}\hfill {\int }_{\Omega }{\varphi }^{n+1}d\mathbf{x}={\int }_{\Omega }{\varphi }^{\left(2\right)}d\mathbf{x}={\int }_{\Omega }{\varphi }^{\left(1\right)}d\mathbf{x}={\int }_{\Omega }{\varphi }^{n}d\mathbf{x}.\end{array}$$

(14)

□

Next, we will present the fully discrete method based on the Fourier spectral method. Let $N_1$ and $N_2$ be positive integers and $h=L_1/N_1=L_2/N_2$ be the space step. For the periodic boundary condition, we employ the discrete Fourier transform [14]:

$$\begin{array}{c}\hfill {\widehat{\varphi }}_{k_1{k}_2}=\sum _{l_1=0}^{N_1-1}\sum _{l_2=0}^{N_2-1}{\varphi }_{l_1{l}_2}{e}^{-i(x_{l_1}{\xi }_{k_1}+y_{l_2}{\xi }_{k_2})}\end{array}$$

(15)

for $k_1=0,1,\dots ,N_1-1$ and $k_2=0,1,\dots ,N_2-1$, where $x_{l_1}=l_{1}h$, $y_{l_2}=l_{2}h$, ${\xi }_{k_1}=2\pi k_1/L_1$, and ${\xi }_{k_2}=2\pi k_2/L_2$. By applying the Fourier transform to (10), we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\widehat{\varphi }}_{k_1{k}_2}^{\left(1\right)}& ={\widehat{\varphi }}_{k_1{k}_2}^n-\Delta t\left(A_{k_1{k}_2}{\widehat{\varphi }}_{k_1{k}_2}^{\left(1\right)}-{\widehat{\frac{\delta {\mathcal{E}}_e^s\left({\varphi }^n\right)}{\delta \varphi }}}_{k_1{k}_2}\right),\hfill \\ \hfill {\widehat{\varphi }}_{k_1{k}_2}^{\left(2\right)}& =\frac{3}{2}{\widehat{\varphi }}_{k_1{k}_2}^{\left(1\right)}-\frac{1}{2}{\widehat{\varphi }}_{k_1{k}_2}^n-\frac{\Delta t}{2}\left(A_{k_1{k}_2}{\widehat{\varphi }}_{k_1{k}_2}^{\left(2\right)}-{\widehat{\frac{\delta {\mathcal{E}}_e^s\left({\varphi }^{\left(1\right)}\right)}{\delta \varphi }}}_{k_1{k}_2}\right),\hfill \\ \hfill {\widehat{\varphi }}_{k_1{k}_2}^{n+1}& =-{\widehat{\varphi }}_{k_1{k}_2}^{\left(2\right)}+\frac{5}{2}{\widehat{\varphi }}_{k_1{k}_2}^{\left(1\right)}-\frac{1}{2}{\widehat{\varphi }}_{k_1{k}_2}^n-\frac{\Delta t}{2}\left(A_{k_1{k}_2}{\widehat{\varphi }}_{k_1{k}_2}^{n+1}-{\widehat{\frac{\delta {\mathcal{E}}_e^s\left({\varphi }^{\left(2\right)}\right)}{\delta \varphi }}}_{k_1{k}_2}\right),\hfill \end{array}\end{array}$$

(16)

where $A_{k_1{k}_2}={({\xi }_{k_1}^2+{\xi }_{k_2}^2)}^{s-1}+\delta {({\xi }_{k_1}^2+{\xi }_{k_2}^2)}^s$.

It could easily be shown that the fully discrete method also preserves the energy dissipation and mass conservation properties. Since the proofs are similar, we omit the details for simplicity.

3. Numerical Experiments

3.1. Accuracy, Efficiency, and Energy Stability Tests

We evolve the solution $\varphi (x,y,t)$ of the space-fractional MBE equation from the initial condition [7]

$$\varphi (x,y,0)=0.1(\sin 3x\sin 2y+\sin 5x\sin 5y)$$

(17)

on $\Omega =[0,2\pi ]\times [0,2\pi ]$. We set $\delta =0.1$, $h=2\pi /64$ (which provides enough spatial accuracy), and $\Delta t=0.01\times 2^{-9}$ (which is a sufficiently small time step). For $s=2$, 1.5, and 1.05, Figure 1, Figure 2 and Figure 3 show the evolution of $\varphi (x,y,t)$ for $0<t\le 30$, respectively. And Figure 4 shows the evolution of energy ${\mathcal{E}}^s\left(t\right)$ and roughness $w\left(t\right)$. When $s=2$, the initial modes $(3,2)$ and $(5,5)$ disappear at $t=2.5$. After a buffering time to $t=4$, a new mode $(1,2)$ appears at $t=5.5$. But this unstable mode disappears and only one mode $(1,1)$ is observed in a steady state. This rough–smooth–rough pattern also occurs when $s=1.5$ and 1.05. However, it takes a longer buffering time for a new mode to appear as s decreases.

To evaluate the accuracy and efficiency of the proposed method, we perform simulations by varying $s=2,1.5,1.05$ and $\Delta t=0.01·2^{-7},0.01·2^{-6},\dots ,0.01$ on Intel Core i5-7500 CPU at 3.40 GHz with 8 GB RAM. Figure 5a shows the relative $l_2$-errors of $\varphi (x,y,t)$ at $t=2.5$ and $5.5$ for various fractional powers and time steps. Here, the errors are calculated by comparison with the solutions given in Figure 1, Figure 2 and Figure 3. It is observed that the method is second-order accurate in time. And Figure 5b presents the CPU times taken until the last iteration. The results indicate that the CPU time is nearly linear in the number of iterations.

To demonstrate the energy stability and mass conservation of the proposed method, we take much larger time steps. Figure 6 shows the evolution of ${\mathcal{E}}^s\left(t\right)$ and ${\int }_{\Omega }(\varphi (x,y,t)-\varphi (x,y,0))dxdy$ for different fractional powers and time steps. As predicted by Theorem 1, the energy dissipation and mass conservation properties are satisfied even for sufficiently large time steps.

3.2. Coarsening Dynamics

It is known that the classical MBE Equation (2) predicts $\mathcal{E}\left(t\right)\sim O(-ln(t\left)\right)$ and $w\left(t\right)\sim O\left(t^{1/2}\right)$ as $t\to \infty$ [6,7,8,9,10,11,12,13,14,15]. One might wonder whether the fractional power affects the scaling rates. To answer this question numerically, we take an initial condition as $\varphi (x,y,0)=\mathrm{rand}(x,y)$ on $\Omega =[0,12.8]\times [0,12.8]$, where $\mathrm{rand}(x,y)$ is a random number between $-0.001$ and $0.001$ at the grid points, and use $\delta =0.{03}^2$, $h=12.8/512$, and $\Delta t=0.01$. For $s=2$, 1.5, and 1.05, Figure 7, Figure 8 and Figure 9 show the evolution of $\varphi$ and its fractional Laplacian ${(-\Delta )}^{\frac{s}{2}}\varphi$ for $0<t\le 5000$. For all cases, coarsening dynamics with shapes of valleys and hills in the system is obvious. However, the coarsening dynamics goes slower as s decreases at the beginning, but it is much faster after reaching a time point. Figure 10 shows the evolution of ${\mathcal{E}}^s\left(t\right)$ and $w\left(t\right)$. The energy decays like $-ln\left(t\right)$ and the roughness grows like $t^{1/2}$, with different coefficients dependent upon s. Figure 11 shows the evolution of ${\mathcal{E}}^s\left(t\right)$ for different fractional powers and time steps. The unconditional energy stability of the proposed method is also demonstrated in a long-time simulation.

We also simulate coarsening dynamics in 3D. We take an initial condition as $\varphi (x,y,z,0)=\mathrm{rand}(x,y,z)$ on $\Omega =[0,3.2]\times [0,3.2]\times [0,3.2]$, where $\mathrm{rand}(x,y,z)$ is a random number between $-0.001$ and $0.001$ at the grid points, and use $\delta =0.{03}^2$, $h=3.2/128$, and $\Delta t=0.01$. Figure 12 shows the evolution of isosurface of $\varphi =0$ for various fractional powers. The phenomenon captured in 2D is also observed in 3D.

4. Conclusions

In recent years, the time-fractional MBE model without slope selection has been introduced and solved numerically [32,33,34,35,36,37]. However, it is an open question what a space-fractional generalization of the MBE model without a slope selection is and whether the space-fractional generalization satisfies the energy dissipation and mass conservation properties. This paper was a first attempt to introduce a space-fractional version of the MBE model without slope selection. The numerical idea was not entirely new, but its application to the new energy dissipative system was particularly appropriate since it preserves the original (not modified) energy dissipation law without any assumptions and stabilizers. Our numerical simulations indicated that the energy decays like $-ln\left(t\right)$ and the roughness grows like $t^{1/2}$, with different coefficients dependent upon the fractional power.

In future work, we have a plan to carry out rigorous analyses for the well-posedness of the considered problem and for the convergence rate of the proposed method.