# A Fokker-Planck reaction model for the epitaxial growth and shape transition of quantum dots

## TL;DR

This paper develops a Fokker-Planck reaction model to simulate the coupled epitaxial growth and shape transition dynamics of quantum dot arrays, revealing how external parameters influence shape distributions and transition kinetics.

## Contribution

The paper introduces a mean-field Fokker-Planck reaction model that captures asymmetric shape transitions and their size dependence during quantum dot growth.

## Key findings

- Shape distribution modes depend on deposition flux and temperature.
- Shape transition rate influences the fraction of asymmetric islands.
- Model predicts evolution of shape distributions during growth.

## Abstract

We construct a Fokker-Planck reaction model to investigate the dynamics of the coupled epitaxial growth and shape transition process of an array of quantum dots. The Fokker-Planck reaction model is based on a coupled system of Fokker-Planck equations wherein the distribution of each island type is governed by its own Fokker-Planck equation for growth, with reaction terms describing the shape transitions between islands of different types including asymmetric shapes. The reaction terms for the shape transitions depend on the island size and are determined from explicit calculations of the lowest-barrier pathway for each shape transition. This mean-field model enables us to consider the kinetics of asymmetric shape transitions and study the evolution of island shape distributions during the coupled growth and transition process. Through numerical simulations over a range of growth parameters, we find multimodal and unimodal evolution modes of the shape distribution of island arrays, which depend on the external deposition flux rate and temperature rather than the shape transition rate. However, the shape transition rate governs the kinetics of shape transitions and determines the fraction of islands that form via asymmetric states, which has implications for the development of asymmetric composition profiles within alloy islands.