Kinetic Monte Carlo model for homoepitaxial growth of ${\mathrm{Ga}}_2{\mathrm{O}}_3$

We developed a kinetic Monte Carlo (KMC) model for the homoepitaxy of $\beta \text{−}{\mathrm{Ga}}_2{\mathrm{O}}_3$. It comprises adsorption, diffusion, and desorption and reflects the structure of $\beta \text{−}{\mathrm{Ga}}_2{\mathrm{O}}_3$ with its two kinds of atoms: Ga and O. The knowledge gained from metal organic vapour phase experiments (MOVPE) experiments combined with AFM and TEM characterisation was used for the setup of rules and activation energies for the various surface processes. We performed a set of runs for the growth on flat and vicinal (100) surfaces. The nucleation on the flat surface requires a minimum ratio of the impingement rate of ${\mathrm{O}}_2$ and Ga. The behavior at different substrate temperatures was similar in experiment and in simulation. At high temperatures, we observe the formation of large islands whereas at low temperatures small islands are formed. The growth rate is increasing with decreasing temperature. On a vicinal surface $\left(6^{\circ }\right)$ different growth modes have been observed when using different desorption energies. Low desorption energy (high desorption rate) leads to step bunching, intermediate to step growth, and high energy (low desorption rate) to nucleation on terraces with a final configuration similar to step bunching.

Figure 1

Numbering of cells in the $x\text{−}y$ plane. The numbering is given for the first layer ($l=1$). For other layers, one has to add $l*Nx*Ny$ to the given numbers. Periodic boundary conditions are applied in $x$ and $y$ directions. The neighboring cells are numbered in blue color. In $x$ direction, we allow to have a step of a unit cell. In this case, the neighboring cells at the left and right boundary are given by the outer expression in the black box.

Figure 2

Atom labeling in the Unit cell. (a) $c\text{−}{a}^{*}$ plane. Please note that the crystallographic $c$ axis is in the minus $x$ direction . (b) $b\text{−}{a}^{*}$ plane. For a better viewing, we also show one neighboring cell.

Figure 3

Surface planes as observed in the STEM analysis of grown layers by step flow mode [17]. Only the relevant atom labeling is given. Please note that in contrary to Fig. 2 the positive $c$ direction is going to the right.

Figure 4

Adsorption of an oxygen molecule on a Ga atom (here a5). This is just a schematic view to describe the rules for KMC. Probably in reality hydrogen is involved at least in process (b) and one of the oxygen atoms will form an OH bond.

Figure 5

Configurations for diffusion of clusters. For every configuration the activation energy $E_A$ is given. $E_A$ might be increased by additional bondings in the layer of the cluster. The possibilities are shown for the Ga atom a5. The same possibilities are set for a7.

Figure 6

Number of Ga atoms per unit cell area ($bc$) as a function of time for different runs on a flat surface. At the initial time, the number is defined as 0. Bluish lines represent runs with the domain size of $20\times 32$ cells. Red lines represent runs with the domain sizes $64\times 64$ and $40\times 100$ for A2c and A2b, respectively. For case A2b and domain size $40\times 100$, four runs with different initialization for the random generator have been performed resulting in quite different times for the onset of nucleation. For comparison, we also show the dynamics for the step growth of case A2b (A2b step). For an increased desorption energy growth on a flat surface is also observed for the case ${\mathcal{R}}_{{\mathrm{O}}_2}^{\mathrm{ads}}=50{\mathrm{s}}^{-1}$ (A3a).

Figure 7

MOVPE growth on a flat (100) plane for three different substrate temperatures. AFM images of the results of four runs stopped after 10 s [(a), (e), and (i)], 15 s [(b), (f), and (j)], 20 s [(c), (g), and (k)], and 25 s [(d), (h), and (l)]. $b$ and $c$ axes are vertical and horizontal, respectively. The size of the domain is $500\times 500{\mathrm{nm}}^2$. For comparison in the bottom left image the size of the computational domain ($40\times 100$) is shown as a white box. The residual mean square (RMS) value of the roughness is defined as $\sqrt{\frac{1}{n}{\sum }_{ı=1}^n{y}_i^2}$, where $y_i$ means the vertical distance from the $i^{\mathrm{th}}$ data point to the mean line of the AFM tip trace.

Figure 8

Number of Ga atoms per unit cell area ($bc$) as a function of time for runs of A2b (${\mathcal{R}}_{{\mathrm{O}}_2/\mathrm{Ga}}^{\mathrm{ads}}=20$) at different temperatures. Blue and red lines represent results from KMC runs with different number of cells as denoted in the upper left corner. Black lines are from coverage measurements of experimental results [24]. These results are scaled by a factor of 1/10 in time to be comparable with the numerical results. For On top the morphologies are shown at 2 s for all temperatures and 4.5 s for $T=850{}^{\circ }\mathrm{C}$ (visualisation made by ovito [25]). $b$ and $c$ axes are vertical and horizontal, respectively. The size of the domain is $30.4\times 20.3{\mathrm{nm}}^2$ ($40\times 100$). The Ga atoms a8 (top of B1 plane) are in light red, the Ga atoms a6 (top of B2-plane) are in blue, all other Ga atoms are in grey.

Figure 9

Number of Ga atoms per unit cell area ($bc$) as a function of time for runs of case B at different temperatures. Blue and red lines represent runs with the domain size of $20\times 32$ cells and $40\times 100$, respectively. For $T=750{}^{\circ }\mathrm{C}$ and $850{}^{\circ }\mathrm{C},$ runs with different initialization of the random generator have been performed. For $T=850{}^{\circ }\mathrm{C},$ in total four runs have been performed. In one case, no nucleation have been observed until 30 s. For all temperatures the morphologies are shown on top for the same coverage of 0.45 Ga adatoms per unit cell area: (a) $T=750{}^{\circ }\mathrm{C}$, (b) $800{}^{\circ }\mathrm{C}$, and (c) $850{}^{\circ }\mathrm{C}$.

Figure 10

Number of Ga atoms per unit cell area ($bc$) as a function of time. At the initial time the number is defined as 0. The results for three different $E_{\mathrm{des}}^{\mathrm{Ga}}$ are shown. Bluish lines represent runs with the domain size of $20\times 32$ cells. Red lines represent runs with the domain size $20\times 64$. For all cases we performed runs with different initialisations of the random number generator and show the average curves including the RMS errors. For comparison also the case A2b is shown, where ${\mathcal{R}}_{{\mathrm{O}}_2}^{\mathrm{ads}}$ is twice as for A2a. The lower dotted black line represents the growth rate for the minimal sticking coefficient for Ga atoms, the upper dotted black line represents the growth rate for $\mathcal{S}=1$.

Figure 11

Surface morphologies for different runs at 10 s (left) and 20 s (right). The desorption rate of ${\mathrm{Ga}}_2\mathrm{O}$ is increasing from top to bottom. The Ga atoms a8 (top of B1 plane) are in light red, the Ga atoms a6 (top of B2 plane) are in blue, all other Ga atoms are in grey. Oxygen atoms have smaller radii and are in red. Visualisation made by ovito [25].