Exact Renormalization-Group Study of Aperiodic Ising Quantum Chains and Directed Walks

We consider the Ising model and the directed walk on two-dimensional layered lattices and show that the two problems are inherently related: The zero-field thermodynamical properties of the Ising model are contained in the spectrum of the transfer matrix of the directed walk. The critical properties of the two models are connected to the scaling behavior of the eigenvalue spectrum of the transfer matrix which is studied exactly through renormalization for different self-similar distributions of the couplings. The models show very rich bulk and surface critical behaviors with nonuniversal critical exponents, coupling-dependent anisotropic scaling, first-order surface transition, and stretched exponential critical correlations. It is shown that all the nonuniversal critical exponents obtained for the aperiodic Ising models satisfy scaling relations and can be expressed as functions of varying surface magnetic exponents.Comment: 22 pages, 8 eps-figures, uses RevTex and epsf, minor correction

Fluctuations of steps on crystal surfaces

Fluctuations of isolated and pairs of ascending steps of monoatomic height are studied in the framework of SOS models, using mainly Monte Carlo techniques. Below the roughening transistion of the surface, the profiles of long steps show the same scaling features for terrace and surface diffusion. For a pair of short steps, their separation distance is found to grow as $t^{1/3}$ at late stages. Above roughening, simulational data on surface diffusion agree well with the classical continuum theory of Mullins.Comment: 4 pages, 2 eps figure

Scaling Analysis of the Site-Diluted Ising Model in Two Dimensions

A combination of recent numerical and theoretical advances are applied to analyze the scaling behaviour of the site-diluted Ising model in two dimensions, paying special attention to the implications for multiplicative logarithmic corrections. The analysis focuses primarily on the odd sector of the model (i.e., that associated with magnetic exponents), and in particular on its Lee-Yang zeros, which are determined to high accuracy. Scaling relations are used to connect to the even (thermal) sector, and a first analysis of the density of zeros yields information on the specific heat and its corrections. The analysis is fully supportive of the strong scaling hypothesis and of the scaling relations for logarithmic corrections.Comment: 15 pages, 3 figures. Published versio

The Effects of Next-Nearest-Neighbor Interactions on the Orientation Dependence of Step Stiffness: Reconciling Theory with Experiment for Cu(001)

Within the solid-on-solid (SOS) approximation, we carry out a calculation of the orientational dependence of the step stiffness on a square lattice with nearest and next-nearest neighbor interactions. At low temperature our result reduces to a simple, transparent expression. The effect of the strongest trio (three-site, non pairwise) interaction can easily be incorporated by modifying the interpretation of the two pairwise energies. The work is motivated by a calculation based on nearest neighbors that underestimates the stiffness by a factor of 4 in directions away from close-packed directions, and a subsequent estimate of the stiffness in the two high-symmetry directions alone that suggested that inclusion of next-nearest-neighbor attractions could fully explain the discrepancy. As in these earlier papers, the discussion focuses on Cu(001).Comment: 8 pages, 3 figures, submitted to Phys. Rev.

Surface critical behavior of two-dimensional dilute Ising models

Ising models with nearest-neighbor ferromagnetic random couplings on a square lattice with a (1,1) surface are studied, using Monte Carlo techniques and star-tiangle transformation method. In particular, the critical exponent of the surface magnetization is found to be close to that of the perfect model, beta_s=1/2. The crossover from surface to bulk critical properties is discussed.Comment: 6 pages in RevTex, 3 ps figures, to appear in Journal of Stat. Phy

Correlations in nano-scale step fluctuations: comparison of simulation and experiments

We analyze correlations in step-edge fluctuations using the Bortz-Kalos-Lebowitz kinetic Monte Carlo algorithm, with a 2-parameter expression for energy barriers, and compare with our VT-STM line-scan experiments on spiral steps on Pb(111). The scaling of the correlation times gives a dynamic exponent confirming the expected step-edge-diffusion rate-limiting kinetics both in the MC and in the experiments. We both calculate and measure the temperature dependence of (mass) transport properties via the characteristic hopping times and deduce therefrom the notoriously-elusive effective energy barrier for the edge fluctuations. With a careful analysis we point out the necessity of a more complex model to mimic the kinetics of a Pb(111) surface for certain parameter ranges.Comment: 10 pages, 9 figures, submitted to Physical Review

Dynamics of surface steps

In the framework of SOS models, the dynamics of isolated and pairs of surface steps of monoatomic height is studied, for step--edge diffusion and for evaporation kinetics, using Monte Carlo techniques. In particular, various interesting crossover phenomena are identified. Simulational results are compared, especially, to those of continuum theories and random walk descriptions.Comment: 13 pages in elsart style, 4 eps figures, submitted to Physica

Boundary critical behaviour of two-dimensional random Ising models

Using Monte Carlo techniques and a star-triangle transformation, Ising models with random, 'strong' and 'weak', nearest-neighbour ferromagnetic couplings on a square lattice with a (1,1) surface are studied near the phase transition. Both surface and bulk critical properties are investigated. In particular, the critical exponents of the surface magnetization, 'beta_1', of the correlation length, 'nu', and of the critical surface correlations, 'eta_{\parallel}', are analysed.Comment: 16 pages in ioplppt style, 7 ps figures, submitted to J. Phys.

Fluctuations, line tensions, and correlation times of nanoscale islands on surfaces

We analyze in detail the fluctuations and correlations of the (spatial) Fourier modes of nano-scale single-layer islands on (111) fcc crystal surfaces. We analytically show that the Fourier modes of the fluctuations couple due to the anisotropy of the crystal, changing the power spectrum of the fluctuations, and that the actual eigenmodes of the fluctuations are the appropriate linear combinations of the Fourier modes. Using kinetic Monte Carlo simulations with bond-counting parameters that best match realistic energy barriers for hopping rates, we deduce absolute line tensions as a function of azimuthal orientation from the analyses of the fluctuation of each individual mode. The autocorrelation functions of these modes give the scaling of the correlation times with wavelength, providing us with the rate-limiting kinetics driving the fluctuations, here step-edge diffusion. The results for the energetic parameters are in reasonable agreement with available experimental data for Pb(111) surfaces, and we compare the correlation times of island-edge fluctuations to relaxation times of quenched Pb crystallites.Comment: 11 pages, 8 figures; to appear in PRB 70, xxx (15 Dec 2004), changes in MC and its implication

Nanoscale Equilibrium Crystal Shapes

The finite size and interface effects on equilibrium crystal shape (ECS) have been investigated for the case of a surface free energy density including step stiffness and inverse-square step-step interactions. Explicitly including the curvature of a crystallite leads to an extra boundary condition in the solution of the crystal shape, yielding a family of crystal shapes, governed by a shape parameter c. The total crystallite free energy, including interface energy, is minimized for c=0, yielding in all cases the traditional PT shape (z x3/2). Solutions of the crystal shape for c≠0 are presented and discussed in the context of meta-stable states due to the energy barrier for nucleation. Explicit scaled relationships for the ECS and meta-stable states in terms of the measurable step parameters and the interfacial energy are presented.Comment: 35 page