# Multiscale Kinetic Monte Carlo Simulation of Self-Organized Growth of GaN/AlN Quantum Dots

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## Abstract

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## 1. Introduction

## 2. Description of the Modeling Strategy

#### 2.1. KMC Model and Implementation

- (i)
- During the growth simulation, the adatoms are randomly deposited onto the surface, with a deposition rate ${r}^{\mathrm{dep}}=F{N}_{l}$, where F represent the deposition flux (in ML/s).
- (ii)
- The adatoms sitting on the surface are allowed to randomly hop to one of its six nearest neighbor sites, see Figure 1b, with a jump rate defined by an Arrhenius-type expression:$${r}_{\mathrm{dif}}={\nu}_{0}\phantom{\rule{0.166667em}{0ex}}\mathrm{exp}\left(-\frac{{E}_{\mathrm{dif}}}{{k}_{B}T}\right)\phantom{\rule{0.222222em}{0ex}},$$
- (iii)
- The adatom desorption process is analogously described by the rate:$${r}_{\mathrm{des}}={\nu}_{0}\phantom{\rule{0.166667em}{0ex}}\mathrm{exp}\left(-\frac{{E}_{\mathrm{des}}}{{k}_{B}T}\right)\phantom{\rule{0.222222em}{0ex}},$$

- At every time step, the systematic search across the entire surface for the sites attainable by the process selected by the BKL algorithm has a cost in CPU time scaling with the surface size. To avoid this problem, we have used an inverted-list algorithm [42]. Since the updating of the inverted lists is a local procedure, the adatom search process becomes now independent of the surface size. This approach is specially efficient in the case of having a not too extensive catalog of possible events. In our case, in the absence of Schwöbel and strain-induced barriers, the system would have a very simple rate structure: There are only seven transitions for the diffusion and desorption processes, corresponding to the possible bonds to nearest neighbors ($n=0,\dots ,6$). To include the (Ehrlich-)Schwöbel effect (in cases where the jump occurs across a step) and the inhomogeneous strain-induced barrier ${E}_{\mathrm{str}}$, without having to modify the structure of inverted lists mentioned above, we use an acceptance-rejection algorithm [42].
- In growth regimes where surface diffusion plays a dominant role, most of the computation time is spent in calculating the adatom diffusion random walks. However, the associated jump rates for isolated and bonded adatoms can differ by orders of magnitude, due to the relative factor $\mathrm{exp}\left(-\frac{n{E}_{b}}{{k}_{B}T}\right)$. We use the multiscale kinetic Monte Carlo method proposed in Ref. [32] that takes advantage of this disparity in the adatom dynamics by allowing the isolated adatoms to make longer jumps than the bonded ones. This reduces drastically the number of Monte Carlo steps required to complete the simulation. For example, for a surface with ${N}_{l}=500\times 500$ lattice sites, a tenfold reduction in computation time can be easily achieved. The long jumps in the multiscale model are an effective way of simulating by a single event a chain of multiple short jumps. We mentioned earlier that the presence of a nonuniform surface elastic energy is expected to lead, for sufficiently long times, to a preferential diffusion towards more relaxed regions. Therefore, it is natural to introduce, for long jumps (with associated long elapsed times), a bias in the jump direction that gives preference to the arrival sites with the lowest values of ${E}_{\mathrm{str}}$. In this work, we have implemented that idea as follows: Once an isolated adatom ($n=0$) has been selected, we scan its neighborhod in search of obstacles (steps to upper terraces, other isolated adatoms or clusters). This search sets the number of sites L (>1) for the long jump, and the jump direction $\alpha $ ($\alpha =1,\cdots ,6$) is chosen according to the following probability distribution:$${p}_{\alpha}={w}_{\alpha}/{\displaystyle \sum _{\beta =1}^{6}{w}_{\beta}}\phantom{\rule{0.222222em}{0ex}},$$$${w}_{\alpha}=\mathrm{exp}\left(-\frac{{\eta}_{\alpha}{E}_{\mathrm{step}}+k\left(L-1\right)\Delta {E}_{\mathrm{str}}^{\left(\alpha \right)}}{{k}_{B}T}\right)\phantom{\rule{0.222222em}{0ex}},$$

#### 2.2. Calculation of the Strain-Induced Energy Barrier ${E}_{str}$

## 3. Simulation Results

#### 3.1. Calibration of the Activation Energies by Homoepitaxial Growth Simulations

#### 3.2. Onset of the Stranski–Krastanov Growth Mode and Formation of Quantum Dots

#### 3.3. QD Growth as a Function of ${\varphi}_{\mathrm{Ga}}/{\varphi}_{\mathrm{N}}$ Ratio

#### 3.4. QD Growth as a Function of Substrate Temperature

#### 3.5. Growth of Stacks of GaN/AlN QDs: Correlation Effects

## 4. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

KMC | Kinetic Monte Carlo |

SK | Stranski–Krastanov |

QD | Quantum dot |

PA-MBE | Plasma-assisted molecular beam epitaxy |

ML | Monolayer |

BKL | Bortz–Kalos–Lebowitz |

## Appendix A. Model for the Effective Flux ${\varphi}_{\mathrm{Ga}}^{\mathrm{eff}}$

**Figure A1.**Growth rate of GaN as a function of the impinging Ga flux, for two substrate temperatures and two impinging N fluxes. The dots of (

**a**,

**b**) are experimental data taken from Refs. [20,53], respectively. The lines represent fits calculated within the framework of the model described in the text.

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**Figure 1.**(

**a**) Cross section of the system under study. The actual domain of simulation is the indicated GaN film, modeled as a set of adatoms located at predefined lattice sites, and represented here as spheres in hexagonal unit cells of parameters ${a}^{*}$ and ${c}^{*}$. The in-plane sites are labeled by integers I, and the corresponding local heights are denoted by ${h}_{I}$ (see text). Also sketched are the possible processes that a given adatom can experience: (i) deposition, (ii) diffusion, and (iii) desorption. (

**b**) Top view of the GaN film, with indication of the 2D hexagonal lattice, and illustration of the possible diffusion jumps that an adatom at ${\mathit{x}}_{I}$ can make.

**Figure 2.**Schematic illustration of an epitaxial GaN layer on a semi-infinite AlN substrate. Both the original discrete description $\left\{{h}_{I}\right\}$ (

**a**) and the smoothed version $h\left({\mathit{x}}_{\perp}\right)$ (

**b**) of the surface morphology are shown. The topography in part (

**b**) is overlaid by a color map representing the strain energy ${E}_{\mathrm{str}}\left({\mathit{x}}_{\perp}\right)$, as calculated within continuum elasticity theory (see text).

**Figure 3.**Comparison between the RHEED traces measured during GaN homoepitaxial growth at $T=730{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C under different ratios ${\varphi}_{\mathrm{Ga}}/{\varphi}_{\mathrm{N}}$ in Ref. [20] with the step density $1-{S}_{d}$ curves obtained from simulations with the fitted activation energies. In all cases, ${\varphi}_{\mathrm{N}}=0.28$ ML/s.

**Figure 4.**Simulation of the process of growth of 5 ML of GaN on AlN. The various panels show the GaN surface morphology at different instants (corresponding to the specified values of the GaN coverage $\mathrm{\Theta}$). The color scale indicating the discrete height is independently chosen in each panel, but the corresponding maximum height is specified in every case.

**Figure 5.**(

**a**) Evolution of the step density $1-{S}_{d}$ as a function of $\mathrm{\Theta}$. The vertical line at ≈2.25 ML indicates the critical thickness at which the 2D-3D SK transition occurs. (

**b**) QD density as a function of $\mathrm{\Theta}$. The results obtained from the KMC simulations are compared with the experimental data of Ref. [21].

**Figure 6.**Surface morphology of a GaN epitaxial layer grown on an AlN substrate for simulations with different nominal Ga fluxes ${\varphi}_{\mathrm{Ga}}$ (specified in each panel of the figure). The N flux ${\varphi}_{\mathrm{N}}$ has been set at $0.28$ ML/s. The color scale indicating the discrete height is the same for all panels. A reference height is provided in the panel for ${\varphi}_{\mathrm{Ga}}=0.28$ ML/s.

**Figure 7.**Surface morphology of a GaN epitaxial layer grown on an AlN substrate as obtained by simulations performed at different temperatures (specified in each panel of the figure). The color scale indicating the discrete height is the same for all panels. A reference height is provided in the panel for $T=760{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C.

**Figure 8.**QD density obtained from the KMC simulations as a function of temperature (blue circles). For comparison, we also display two sets of experimental densities (red circles and green squares) measured in Ref. [21] (see text).

**Figure 9.**Vertical cross sections through the simulated stacks consisting of five QD layers separated by different spacer thicknesses: $d=10.2$ nm (stack

**A**), $d=5.2$ nm (

**B**), $d=4.2$ nm (

**C**), and $d=3.2$ nm (

**D**). For each stack, the distribution of the elastic energy density $U\left(\mathit{x}\right)$ is shown by means of a pixelated color map. The color scale has been truncated at 0.1 eV/nm${}^{3}$, so that most the GaN volume, which is strained beyond that value, appears as black colored.

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**MDPI and ACS Style**

Budagosky, J.A.; García-Cristóbal, A.
Multiscale Kinetic Monte Carlo Simulation of Self-Organized Growth of GaN/AlN Quantum Dots. *Nanomaterials* **2022**, *12*, 3052.
https://doi.org/10.3390/nano12173052

**AMA Style**

Budagosky JA, García-Cristóbal A.
Multiscale Kinetic Monte Carlo Simulation of Self-Organized Growth of GaN/AlN Quantum Dots. *Nanomaterials*. 2022; 12(17):3052.
https://doi.org/10.3390/nano12173052

**Chicago/Turabian Style**

Budagosky, Jorge A., and Alberto García-Cristóbal.
2022. "Multiscale Kinetic Monte Carlo Simulation of Self-Organized Growth of GaN/AlN Quantum Dots" *Nanomaterials* 12, no. 17: 3052.
https://doi.org/10.3390/nano12173052