Martin Ahr

Kinetic Monte Carlo simulations of heteroepitaxial growth

We introduce an algorithm for off-lattice kinetic Monte Carlo simulations of heteroepitaxial crystal growth. In heteroepitaxy a mismatch of the lattice constants in adsorbate and substrate can lead to a variety of phenomena already within the first monolayers of growth. This includes the appearance of misfit dislocations or the formation of self-assembled islands. In order to gain general insight into the relevant mechanisms we study, as a first example, a Lennard-Jones system in 1q1 dimensions. The critical layer thickness for the occurrence of dislocations is determined as a function of the misfit. Furthermore, we investigate the 2D-3D transition which marks the emergence of mounds from initially flat islands on the surface. 2002 Elsevier Science B.V. All rights reserved.

A Kinetic Monte Carlo method for the simulation of heteroepitaxial growth

Abstract We introduce a simulation algorithm which allows the off-lattice simulation of various phenomena observed in heteroepitaxial growth (see e.g. [Politi et al., Phys. Rep. 324 (2000) 271–404]) like a critical layer thickness for the appearance of misfit dislocations, or self-assembled island formation in 1+1 dimensions. The only parameters of the model are deposition flux, simulation temperature and an interaction potential between the particles of the system.

A lattice gas model of II-VI(001) semiconductor surfaces

We introduce an anisotropic two-dimensional lattice gas model of metal-termi nated II-VI(001) semiconductor surfaces. Important properties of this class of materials are represented by effective NN and NNN interactions, which results in the competition of two vacancy structures on the surface. We demonstrate that the experimentally observed transition from the dominant c(2 × 2) ordering of the CdTe(001) surface to a local (2 × 1) arrangement of Cd atoms can be explained as a phase transition in thermal equilibrium. The model is studied by means of transfer-matrix and Monte Carlo techniques. The analysis shows that the small energy difference of the competing reconstructions determines to a large extent the nature of the different phases. Possible implications for further experimental research are discussed.

The influence of the crystal lattice on coarsening in unstable epitaxial growth

Abstract We report the results of computer simulations of epitaxial growth in the presence of a large Schwoebel barrier on different crystal surfaces: simple cubic(001), bcc(001), simple hexagonal(001) and hcp(001). We find that mounds coarsen by a step-edge diffusion-driven process, if adatoms can diffuse relatively far along step edges without being hindered by kink-edge diffusion barriers. This yields the scaling exponents α=1, β= 1 3 . These exponents are independent of the symmetry of the crystal surface. The crystal lattice, however, has strong effects on the morphology of the mounds, which are by no means restricted to trivial symmetry effects. Whereas we observe pyramidal shapes on the simple lattices, on bcc and hcp there are two fundamentally different classes of mound, which are accompanied by characteristic diffusion currents: a metastable one with rounded corners, and an actively coarsening configuration, which breaks the symmetry given by the crystal surface.

Phase transitions in soft-committee machines

Equilibrium statistical physics is applied to the off-line training of layered neural networks with differentiable activation functions. A first analysis of soft-committee machines with an arbitrary number (K) of hidden units and continuous weights learning a perfectly matching rule is performed. Our results are exact in the limit of high training temperatures (β → 0). For K = 2 we find a second-order phase transition from unspecialized to specialized student configurations at a critical size P of the training set, whereas for K ≥ 3 the transition is first order. The limit K → ∞ can be performed analytically, the transition occurs after presenting on the order of NK/β examples. However, an unspecialized metastable state persists up to P NK2/β.

Flat (001) surfaces of II–VI semiconductors: a lattice gas model

Abstract We present a two-dimensional lattice gas with anisotropic interactions which model the known properties of the surface reconstructions of CdTe and ZnSe. In contrast to an earlier publication [M. Biehl, M. Ahr, W. Kinzel, M. Sokolowski, T. Volkmann, Europhys. Lett. 53 (2001) 169] the formation of anion dimers is considered. This alters the behaviour of the model considerably. We determine the phase diagram of this model by means of transfer matrix calculations and Monte Carlo simulations. We find qualitative agreement with the results of various experimental investigations.

Kinetic model of II-VI(001) semiconductor surfaces: Growth rates in atomic layer epitaxy

We present a zinc-blende lattice gas model of II-VI(001) surfaces, which is investigated by means of kinetic Monte Carlo simulations. Anisotropic effective interactions between surface metal atoms allow for the description of, e.g., the sublimation of CdTe(001), including the reconstruction of Cd-terminated surfaces and its dependence on the substrate temperature T. Our model also includes Te dimerization and the potential presence of excess Te in a reservoir of weakly bound atoms at the surface. We study the self-regulation of atomic layer epitaxy (ALE) and demonstrate how the interplay of the reservoir occupation with the surface kinetics results in two different regimes: at high T the growth rate is limited to 0.5 layers per ALE cycle, whereas at low enough T each cycle adds a complete layer of CdTe. The transition between the two regimes occurs at a characteristic temperature and its dependence on external parameters is studied. Comparing the temperature dependence of the ALE growth rate in our model with experimental results for CdTe we find qualitative agreement.

Statistical physics and practical training of soft-committee machines

Abstract:Equilibrium states of large layered neural networks with differentiable activation function and a single, linear output unit are investigated using the replica formalism. The quenched free energy of a student network with a very large number of hidden units learning a rule of perfectly matching complexity is calculated analytically. The system undergoes a first order phase transition from unspecialized to specialized student configurations at a critical size of the training set. Computer simulations of learning by stochastic gradient descent from a fixed training set demonstrate that the equilibrium results describe quantitatively the plateau states which occur in practical training procedures at sufficiently small but finite learning rates.

Modeling (001) surfaces of II-VI semiconductors

Abstract First, we present a two-dimensional lattice gas model with anisotropic interactions which explains the experimentally observed transition from a dominant c (2×2) ordering of the CdTe(001) surface to a local (2×1) arrangement of the Cd atoms as an equilibrium phase transition. Its analysis by means of transfer-matrix and Monte Carlo techniques shows that the small energy difference of the competing reconstructions determines to a large extent the nature of the different phases. Then, this lattice gas is extended to a model of a three-dimensional crystal which qualitatively reproduces many of the characteristic features of CdTe which have been observed during sublimation and atomic layer epitaxy.

Singularity spectra of rough growing surfaces from wavelet analysis

We apply the wavelet transform modulus maxima method [A. Arneodo, N. Decoster, and S. G. Roux, Phys. Rev. Lett. 83, 1255 (1999)] to the analysis of simulated surfaces grown by molecular-beam epitaxy. In contrast to the structure function approach commonly used in the literature, this method permits an investigation of the complete singularity spectrum. We focus on a kinetic Monte Carlo model with Arrhenius dynamics, which in particular takes into consideration the process of thermally activated desorption of particles. We find a wide spectrum of Holder exponents, which reflects the multiaffine surface morphology. Although our choice of parameters yields small desorption rates (<3%), we observe a dramatic change in the singularity spectrum, which is shifted toward smaller Holder exponents. Our results offer a mathematical foundation of anomalous scaling: We identify the global exponent alpha(g) with the Holder exponent that maximizes the singularity spectrum.