Reconstruction of silicon surfaces: A stochastic optimization problem

Abstract

Over the last two decades, scanning tunnelling 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 highindex 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 article, 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 example, with two main results: (a) the correct single-step rebonded structure [e.g., Fujikawa et al., Phys. Rev. Lett. 88, 176101 (2002)] is obtained even when starting from the paired-dimer model [Mo et al., 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.

Finding the reconstructions of semiconductor surfaces via a genetic algorithm

“…Our choice for developing this genetic algorithm (GA) was motivated by its successful application for the structural optimization of atomic clusters [11,12]. Like the previous study [10], the present algorithm also circumvents the intuitive process when proposing candidate models for a given high-index surface. Except for the periodic vectors of the surface unit cell [which can be determined from scanning tunnelling microscopy (STM) or from low-energy electron diffraction (LEED) measurements], no other experimental input is necessary.…”

“…However, the publication of numerous studies that report different structures for a given high-index silicon surface (see, e.g., [1,2,3,4,5]) indicates a need to develop methodologies capable of actually searching for the atomic structure in a way that does not predominantly rely on the heuristic reasoning associated with interpreting STM data. Recently, it was shown that parallel-tempering Monte Carlo (PTMC) simulations combined with an exponential cooling schedule can successfully address the problem of finding the reconstructions of high-index silicon surfaces [10]. The PTMC simulations, however, have a broader scope, as they are used to perform a thorough thermodynamic sampling of the surface systems under study.…”

“…Here we test the genetic procedure on the (105) surface, which, at least in conditions of compressive strain, is known to have a single-height rebonded step structure SR [13,14,15,16,17]. The PTMC study [10] indicates that the SR structure is the lowest surface energy even in the absence of strain, although there are several other reconstructions with very similar surface energies. It is interesting to note that the number of reported reconstructions for the (105) orientation has increased very rapidly from two (models SU and SR [13,14,15]), to a total of fourteen reported in Refs.…”

“…The PTMC simulations, however, have a broader scope, as they are used to perform a thorough thermodynamic sampling of the surface systems under study. Given their scope, such calculations [10] are very demanding, usually requiring several tens of processors that run canonical simulations at different temperatures and exchange configurations in order to drive the low-temperature replicas into the ground state. If we focus only on finding the reconstructions at zero Kelvin (which can be representative for crystal surfaces in the low-temperature regimes achieved in laboratory conditions), it is then justified to explore alternative methods for finding the structure of high-index surfaces.…”

“…In the present work, we test the algorithm for the case of Si(105); the surface slab is made of four bulk unit cells of dimensions 1 a √ 6.5 × a × a √ 6.5 (a = 5.431Å is the bulk lattice constant of Si), stacked two by two along the [010] and [105] directions. In terms of atomic interactions, we have used the highly-optimized empirical model developed by Lenosky et al [19], which was found to have superior transferability to the diverse bonding environments present on high-index silicon surfaces [10].…”