Double-exponential scaling function for island-size distributions.

Area of Science:

• Surface science
• Materials science
• Physical chemistry

Background:

• Homogeneous nucleation and growth are fundamental processes in thin-film formation.
• Island-size distributions are crucial for understanding surface morphology and properties.
• Rate equations and capture numbers are key theoretical tools for modeling these phenomena.

Purpose of the Study:

• To investigate the impact of capture number (σ(s)) functional forms on island-size distributions during submonolayer deposition.
• To analyze the validity and limitations of existing scaling functions (Family-Vicsek) under different growth conditions.
• To develop a new, singularity-free analytic scaling function for improved theoretical modeling.

Main Methods:

• Solving the continuum rate equation for island nucleation and growth.
• Analyzing power-law size dependencies of capture numbers (σ(s)∼s^{α}).
• Revisiting the Bartelt-Evans theory and proposing a novel analytic form for scaled capture numbers C(x).

Main Results:

• Existing Family-Vicsek scaling functions exhibit singularities for 0≤α<1/2, which worsen with increasing α.
• No normalizable scaling solutions exist for 1/2<α<1.
• An analytic scaling function with a double-exponential shape was derived, resolving singularities and satisfying sum rules.

Conclusions:

• The singularities in scaling functions arise from zero growth rates at maximum island sizes.
• The proposed analytic scaling function provides a more robust theoretical description of island-size distributions.
• The new function closely matches results from kinetic Monte Carlo simulations, validating its accuracy.