| Summary: | The temperature dependent density of Al and Ga droplets deposited on AlGaAs with molecular beam epitaxy is studied theoretically. Such droplets are important for applications in quantum information technology and can be functionalized e.g., by droplet epitaxy or droplet etching for the self-assembled generation of quantum emitters. After an estimation based on a scaling analysis, the droplet densities are simulated using first a mean-field rate model and second a kinetic Monte Carlo (KMC) simulation basing on an atomistic representation of the mobile adatoms. The modeling of droplet nucleation with a very high surface activity of the adatoms and ultra-low droplet densities down to 5 × 10<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow></mrow><mn>6</mn></msup></semantics></math></inline-formula> cm<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup></semantics></math></inline-formula> is highly demanding in particular for the KMC simulation. Both models consider two material related model parameters, the energy barrier <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mi>S</mi></msub></semantics></math></inline-formula> for surface diffusion of free adatoms and the energy barrier <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mi>E</mi></msub></semantics></math></inline-formula> for escape of atoms from droplets. The rate model quantitatively reproduces the droplet densities with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mi>S</mi></msub></semantics></math></inline-formula> = 0.19 eV, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mi>E</mi></msub></semantics></math></inline-formula> = 1.71 eV for Al droplets and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mi>S</mi></msub></semantics></math></inline-formula> = 0.115 eV for Ga droplets. For Ga, the values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mi>E</mi></msub></semantics></math></inline-formula> are temperature dependent indicating the relevance of additional processes. Interestingly, the critical nucleus size depends on deposition time, which conflicts with the assumptions of the scaling model. Using a multiscale KMC algorithm to substantially shorten the computation times, Al droplets up to 460 ℃ on a 7500 × 7500 simulation field and Ga droplets up to 550 ℃ are simulated. The results show a very good agreement with the experiments using <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mi>S</mi></msub></semantics></math></inline-formula> = 0.19 eV, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mi>E</mi></msub></semantics></math></inline-formula> = 1.44 eV for Al, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mi>S</mi></msub></semantics></math></inline-formula> = 0.115 eV, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mi>E</mi></msub></semantics></math></inline-formula> = 1.24 eV (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mo>≤</mo></mrow></semantics></math></inline-formula> 300 ℃) or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mi>E</mi></msub></semantics></math></inline-formula> = 1.24 + 0.06 (<i>T</i>[℃]-300)/100 eV (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mo>></mo><mn>300</mn></mrow></semantics></math></inline-formula> ℃) for Ga. The deviating <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mi>E</mi></msub></semantics></math></inline-formula> is attributed to a re-nucleation effect that is not considered in the mean-field assumption of the rate model.
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