Showing papers by "Joachim Krug published in 2002"

Four lectures on the physics of crystal growth

Abstract: Several aspects of the theory of epitaxial crystal growth from atomic or molecular beams are developed from the perspective of statistical physics. Lectures are devoted to the rate equation theory of two-dimensional nucleation and its limitations; the growth of multilayer wedding cakes in the presence of strong step edge barriers; the continuum theory of mound coarsening; and growth-induced step meandering on vicinal surfaces.

Pattern dynamics of vortex ripples in sand: nonlinear modeling and experimental validation.

Abstract: Vortex ripples in sand are studied experimentally in a one-dimensional setup with periodic boundary conditions. The nonlinear evolution, far from the onset of instability, is analyzed in the framework of a simple model developed for homogeneous patterns. The interaction function describing the mass transport between neighboring ripples is extracted from experimental runs using a recently proposed method for data analysis, and the predictions of the model are compared to the experiment. An analytic explanation of the wavelength selection mechanism in the model is provided, and the width of the stable band of ripples is measured.

Competing mechanisms for step meandering in unstable growth

Abstract: The meander instability of a vicinal surface growing under step flow conditions is studied within a solid-on-solid model. In the absence of edge diffusion the selected meander wavelength agrees quantitatively with the continuum linear stability analysis of Bales and Zangwill [Phys. Rev. B $41,$ 4400 (1990)]. In the presence of edge diffusion a local instability mechanism related to kink rounding barriers dominates, and the meander wavelength is set by one-dimensional nucleation. The long-time behavior of the meander amplitude differs in the two cases, and disagrees with the predictions of a nonlinear step evolution equation [O. Pierre-Louis et al., Phys. Rev. Lett. $80,$ 4221 (1998)]. The variation of the meander wavelength with the deposition flux and with the activation barriers for step adatom detachment and step crossing (the Ehrlich-Schwoebel barrier) is studied in detail. The interpretation of recent experiments on surfaces vicinal to Cu(100) [T. Maroutian et al., Phys. Rev. B $64,$ 165401 (2001)] in the light of our results yields an estimate for the kink rounding barrier at the close-packed steps.

New mechanism for impurity-induced step bunching

Abstract: Codeposition of impurities during the growth of a vicinal surface leads to an impurity concentration gradient on the terraces, which induces corresponding gradients in the mobility and the chemical potential of the adatoms Here it is shown that the two types of gradients have opposing effects on the stability of the surface: Step bunching can be caused by impurities which either lower the adatom mobility, or increase the adatom chemical potential In particular, impurities acting as random barriers (without affecting the adatom binding) cause step bunching, while for impurities acting as random traps the combination of the two effects reduces to a modification of the attachment boundary conditions at the steps In this case, attachment to descending steps, and thus step bunching, is favored if the impurities bind adatoms more weakly than the substrate

Coarsening of sand ripples in mass transfer models.

Abstract: Coarsening of sand ripples is studied in a one-dimensional stochastic model, where neighboring ripples exchange mass with algebraic rates, $\ensuremath{\Gamma}(m)\ensuremath{\sim}{m}^{\ensuremath{\gamma}},$ and ripples of zero mass are removed from the system. For $\ensuremath{\gamma}l0,$ ripples vanish through rare fluctuations and the average ripple mass grows as $〈m〉(t)\ensuremath{\sim}\ensuremath{-}{\ensuremath{\gamma}}^{\ensuremath{-}1}\mathrm{ln}(t).$ Temporal correlations decay as ${t}^{\ensuremath{-}1/2}$ or ${t}^{\ensuremath{-}2/3}$ depending on the symmetry of the mass transfer, and asymptotically the system is characterized by a product measure. The stationary ripple mass distribution is obtained exactly. For $\ensuremath{\gamma}g0,$ ripple evolution is linearly unstable, and the noise in the dynamics is irrelevant. For $\ensuremath{\gamma}=1,$ the problem is solved on the mean-field level, but the mean-field theory does not adequately describe the full behavior of the coarsening. In particular, it fails to account for the numerically observed universality with respect to the initial ripple size distribution. The results are not restricted to sand ripple evolution since the model can be mapped to zero range processes, urn models, exclusion processes, and cluster-cluster aggregation.

Tempo and mode in quasispecies evolution

Abstract: Evolutionary dynamics in an uncorrelated rugged fitness landscape is studied in the framework of Eigen’s molecular quasispecies model. We consider the case of strong selection, which is analogous to the zero temperature limit in the equivalent problem of directed polymers in random media. In this limit the population is always localized at a single temporary master sequence σ*(t), and we study the statistical properties of the evolutionary trajectory which σ*(t) traces out in sequence space. Numerical results for binary sequences of length N = 10 and exponential and uniform fitness distributions are presented. Evolution proceeds by intermittent jumps between local fitness maxima, where high lying maxima are visited more frequently by the trajectories. The probability distribution for the total time T required to reach the global maximum shows a T -2-tail, which is argued to be universal and to derive from near-degenerate fitness maxima. The total number of jumps along any given trajectory is always small, much smaller than predicted by the statistics of records for random long-ranged evolutionary jumps.

Second Layer Nucleation and the Shape of Wedding Cakes

Abstract: The rate of second layer nucleation—the formation of a stable nucleus on top of a two-dimensional island—determines both the conditions for layer-by-layer growth, and the size of the top terrace of multilayer mounds in three-dimensional homoepitaxial growth. It was recently shown that conventional mean-field nucleation theory overestimates the rate of second layer nucleation by a factor that is proportional to the number of times a given site is visited by an adatom during its residence time on the island. In the presence of strong step-edge barriers this factor can be large, leading to a substantial error in previous attempts to experimentally determine barrier energies from the onset of second layer nucleation. In the first part of the paper simple analytic estimates of second layer nucleation rates based on a comparison of the relevant time scales will be reviewed. In the main part the theory of second layer nucleation is applied to the growth of multilayer mounds in the presence of strong but finite step-edge barriers. The shape of the mounds is obtained by numerical integration of the deterministic evolution of island boundaries, supplemented by a rule for nucleation in the top layer. For thick films the shape converges to a simple scaling solution. The scaling function is parametrized by the coverage θc of the top layer, and takes the form of an inverse error function cut off at θc. The surface width of a film of thickness d is. Finally, we show that the scaling solution can be derived also from a continuum growth equation.

Four Lectures on the Physics of Crystal Growth

Abstract: Several aspects of the theory of epitaxial crystal growth from atomic or molecular beams are developed from the perspective of statistical physics. Lectures are devoted to the rate equation theory of two-dimensional nucleation and its limitations; the growth of multilayer wedding cakes in the presence of strong step edge barriers; the continuum theory of mound coarsening; and growth-induced step meandering on vicinal surfaces.

Effects of Atomic Interactions in Two-Component Submonolayer Growth

Abstract: We discuss the effects of different chemical bondings on the island morphology and on the island density scaling in two-component sub- monolayer growth. Different regimes, depending on the strength of the mutual interactions and on the relative mobility of species, are described and studied with kinetic Monte Carlo simulations using a two-component solid- on-solid growth model. Results for the temperature and flux dependence of the island density as well as examples of the surface morphologies are presented.

Effect of kink-rounding barriers on step edge fluctuations

Abstract: The effect that an additional energy barrier E_{kr} for step adatoms moving around kinks has on equilibrium step edge fluctuations is explored using scaling arguments and kinetic Monte Carlo simulations. When mass transport is through step edge diffusion, the time correlation function of the step fluctuations behaves as C(t) = A(T) t^{1/4}. At low temperatures the prefactor A(T) shows Arrhenius behavior with an activation energy (E_{det} + 3 epsilon)/4 if E_{kr} epsilon, where epsilon is the kink energy and E_{det} is the barrier for detachment of a step adatom from a kink. We point out that the assumption of an Einstein relation for step edge diffusion has lead to an incorrect interpretation of step fluctuation experiments, and explain why such a relation does not hold. The theory is applied to experimental results on Pt(111) and Cu(100).