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We study the role of the surface diffusion anisotropy in the fabrication of nanostructures by laser-focused (“cold”) atom lithography, and on the stability of nanostructures using two-dimensional kinetic Monte Carlo simulations. The simulations were performed using one- and two-directional standing-wave fields (using two different wave profiles) and two different diffusion models (limited and total diffusion model) on anisotropic surfaces, such as (110) surfaces of face-centered-cubic crystals, or reconstructed surfaces such as Si(100). The simulations were made with and without Schwoebel barriers. Our diffusion-deposition models reproduce qualitatively experimental results.