The Energetics of Hut-Cluster Self-Assembly in Ge/Si(001) from Linear-Scaling DFT Calculations

Abstract

Density functional theory (DFT) in a linear-scaling implementation is used to study the energetics of three-dimensional (3D) Ge islands (hut clusters) grown on Si(001) surface. DFT calculations on the fully relaxed energies of a series of hut clusters of increasing size are reported, finding a 2D to 3D cross-over near three monolayers; the number of atoms in the largest simulated system is over 20,000. A variety of technical issues which are important in addressing the accuracy and validity of the calculations are described and assessed. The results suggest that energetics alone is responsible for the initial transition from 2D to 3D growth. Self-assembly is one of the major routes to the formation of surface nanostructures (quantum dots, nanowires, etc.), but our understanding of the underlying processes is limited. In the initial stages of growth, structures which are metastable and whose formation is dominated by kinetic effects can often form before the energetically stable structure; 1) this is particularly true on semiconductor surfaces where the interplay between strain and electronic energies compete to give a rich variety of one-dimensional 2) and two-dimensional (2D) structures.3) Atomistic modeling based on electronic structure theory has a key role to play in this area, but progress depends on addressing a number of issues, including accuracy, the scaling of computer effort with system size, the control of computational artefacts, and the sampling of energetically competing geometries. The much studied system of self-assembled Ge hut clusters on Si (001) 4,5) serves as a paradigm where modeling encounters all these issues. We describe here how we have used OðNÞ (linear-scaling) density-functional theory (DFT) to tackle these questions.Ge hut clusters, which have a pyramid-like shape, are formed when Ge is deposited epitaxially on Si(001): the first few monolayers (ML) grow two-dimensionally, forming a 2 Â N missing-dimer reconstruction which relieves the strain from the 4.2% lattice mismatch between Ge and Si. Subsequently, three-dimensional (3D) islands form, 6) their edges lying along the elastically soft (100) and (010) directions. The coverage at which huts appear depends on substrate temperature, deposition rate and the presence or absence of hydrogen, because these affect the diffusion rates on the surface as well as the amount of Ge/Si alloying. The lowest coverage at which huts appear is $3 ML. STM measurements show that the hut clusters have four (105) facets; details of the Ge(105) surface reconstruction have been elucidated by combining STM measurements and DFT calculations. 7,8) However, the mechanisms underlying hut formation are not understood; in particular, the energetic balance between strain, surface energy and the energies of edges between facets is subtle, and may be subject to kinetic effects. Such balances are important even for simple surface reconstructions, and will be crucial for small sizes of hut cluster.Attempts have been made to model semiconductor nanostructures using e...

\mathcal{O}(N) methods in electronic structure calculations

“…From Ref. [471] The stability of the 3D structures grown on surfaces is usually governed by the competition between the release of the strain energy from the formation of a 3D structure and the energy increase due to the larger surface area of the facets on the surface. So far, theoretical approach on the Stranski-Krastanov growth mode has usually used continuum elasticity theory for the first part, and employed DFT calculations for the latter.…”

Linear-scaling first-principles molecular dynamics of complex biological systems with the Conquest code

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Computing the Optical Properties of Large Systems

“…Figures 6.3, 6.5 and 6.8, as well as table 6.2 have already appeared in an earlier publication [99] and are reproduced here with permission of AIP Publishing. (10,0) carbon nanotube with different numbers of atoms in the system for fully dense density matrices and for truncated density matrices with a truncation radius of 60 a 0 . The blue line represents a linear fit while the red line is a cubic fit to the data points.…”

“…While the development of linear-scaling techniques has opened up a wide field of new applications for DFT (see, for example [8][9][10][11]), it also comes with a new set of challenges. Since calculations of ground state properties of systems containing thousands of atoms are now possible, the problem shifts to extracting information from these calculations that can be compared directly to experimental measurements.…”

“…This defines the maximum kinetic energy the Kohn-Sham electrons are allowed to have and thus provides a physically motivated, systematically improvable accuracy of the basis set representation. While plane-waves form a very effective basis in an infinite solid in periodic boundary conditions, they are not a suitable choice for isolated molecules 10 . In this scenario, the most efficient basis sets come in the form of localised, atom-centered functions, that are generally non-orthogonal.…”