Reconstruction of silicon surfaces: A stochastic optimization problem

Over the last two decades, scanning tunneling microscopy (STM) has become one of the most important ways to investigate the structure of crystal surfaces. STM has helped achieve remarkable successes in surface science such as finding the atomic structure of Si(111) and Si(001). For high-index Si surfaces the information about the local density of states obtained by scanning does not translate directly into knowledge about the positions of atoms at the surface. A commonly accepted strategy for identifying the atomic structure is to propose several possible models and analyze their corresponding simulated STM images for a match with the experimental ones. However, the number of good candidates for the lowest-energy structure is very large for high-index surfaces, and heuristic approaches are not likely to cover all the relevant structural models. In this paper, we take the view that finding the atomic structure of a surface is a problem of stochastic optimization, and we address it as such. We design a general technique for predicting the reconstruction of silicon surfaces with arbitrary orientation, which is based on parallel-tempering Monte Carlo simulations combined with an exponential cooling. The advantages of the method are illustrated using the Si(105) surface as an example, with two main results: (a) the correct single-step rebonded structure [e.g., Fujikawa, Akiyama, Nagao, Sakurai, Lagally, Hashimoto, Morikawa, and Terakura, Phys. Rev. Lett. 88, 176101 (2002)] is obtained even when starting from the paired-dimer model [Mo, Savage, Swartzentruber, and Lagally, Phys. Rev. Lett. 65, 1020 (1990)] that was assumed to be correct for many years, and (b) we have found several double-step reconstructions that have lower surface energies than any previously proposed double-step models.