arXiv subjects

T. L. Einstein

Publications and source records attributed to T. L. Einstein.

31 records · Page 2Linked to original sources

arXiv2005

Step Position Distributions and the Pairwise Einstein Model for Steps on Crystal Surfaces

The Pairwise Einstein Model (PEM) of steps not only justifies the use of the Generalized Wigner Distribution (GWD) for Terrace Width Distributions (TWDs), it also predicts a specific form for the Step Position Distribution (SPD), i.e., the probability density function for the fluctuations of a step about its average position. The predicted form of the SPD is well approximated by a Gaussian with a finite variance. However, the variance of the SPD measured from either real surfaces or Monte Carlo simulations depends on $Δy$, the length of step over which it is calculated, with the measured variance diverging in the limit $Δy \to \infty$. As a result, a length scale $L_{\rm W}$ can be defined as the value of $Δy$ at which the measured and theoretical SPDs agree. Monte Carlo simulations of the terrace-step-kink model indicate that $L_{\rm W} \approx 14.2 ξ_Q$, where $ξ_Q$ is the correlation length in the direction parallel to the steps, independent of the strength of the step-step repulsion. $L_{\rm W}$ can also be understood as the length over which a {\em single} terrace must be sampled for the TWD to bear a "reasonable" resemblence to the GWD.

cond-mat.stat-mech↗

arXiv2005

Beyond the Wigner Distribution: Schr{ö}dinger Equations and Terrace Width Distributions

The so-called generalized Wigner distribution has earlier been shown to be an excellent approximation for the terrace width distribution (TWD) of vicinal surfaces characterized by step-step interactions that are perpendicular to the average step direction and fall off as the inverse square of the step spacing. In this paper we show that the generalized Wigner distribution corresponds to a mean-field treatment of a single step, discuss generalizations to more general step-step interactions, and show that the mean-field predictions are in good agreement with TWDs derived from numerical transfer-matrix calculations and Monte Carlo simulations. We also show that this mean-field treatment allows the step-step interaction to be extracted from experimental TWDs.

cond-mat.stat-mech↗

arXiv2004

Low-Temperature Orientation Dependence of Step Stiffness on {111} Surfaces

For hexagonal nets, descriptive of {111} fcc surfaces, we derive from combinatoric arguments a simple, low-temperature formula for the orientation dependence of the surface step line tension and stiffness, as well as the leading correction, based on the Ising model with nearest-neighbor (NN) interactions. Our formula agrees well with experimental data for both Ag and Cu{111} surfaces, indicating that NN-interactions alone can account for the data in these cases (in contrast to results for Cu{001}). Experimentally significant corollaries of the low-temperature derivation show that the step line tension cannot be extracted from the stiffness and that with plausible assumptions the low-temperature stiffness should have 6-fold symmetry, in contrast to the 3-fold symmetry of the crystal shape. We examine Zia's exact implicit solution in detail, using numerical methods for general orientations and deriving many analytic results including explicit solutions in the two high-symmetry directions. From these exact results we rederive our simple result and explore subtle behavior near close-packed directions. To account for the 3-fold symmetry in a lattice gas model, we invoke a novel orientation-dependent trio interaction and examine its consequences.

cond-mat.mtrl-sci↗

arXiv2004

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.

cond-mat.mtrl-sci↗

arXiv2004

A facet is not an island: step-step interactions and the fluctuations of the boundary of a crystal facet

In a recent paper [Ferrari et al., Phys. Rev. E 69, 035102(R) (2004)], the scaling law of the fluctuations of the step limiting a crystal facet has been computed as a function of the facet size. Ferrari et al. use rigorous, but physically rather obscure, arguments. Approaching the problem from a different perspective, we rederive more transparently the scaling behavior of facet edge fluctuations as a function of time. Such behavior can be scrutinized with STM experiments and with numerical simulations.

cond-mat.stat-mech↗

arXiv2004

Interactions Mediated by Surface States: From Pairs and Trios to Adchains and Ordered Overlayers

Since metallic surface states on (111) noble metals are free-electron like, their propagators can be evaluated analytically. Since they are well-screened, one can use simple tight-binding formalism to study their effects. The needed phase shifts can be extracted from experiment. Hence, one can now make quantitative predictions of these slowly-decaying, oscillatory indirect interactions. For the (isotropic!) pair interactions (which decay as the inverse square of adatom-adatom separation), remarkable agreement has been obtained with experiments by two groups. We have extended the formalism to consider the full indirect ("triple") interaction of 3 adsorbates, which is the sum of the 3 constituent pair interactions plus the non-pairwise "trio" contribution, which tends to decay with the 5/2 power of perimeter. Here, we concentrate on interactions due to ordered overlayers and to linear defects, relating the latter to the interactions of (nx1) ordered overlayers and to the constituent pair interactions. We compare with experimental studies of interactions of adatoms with adchains and of consequent 1D motion of adatoms trapped between two such parallel chains. We discuss implications for step-step interactions (on vicinal surfaces), with attention to the modification of the surface state itself for small terrace widths.

cond-mat.mtrl-sci↗

arXiv2004

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).

cond-mat.mtrl-sci↗

arXiv2004

Distinguishing step relaxation mechanisms via pair correlation functions

Theoretical predictions of coupled step motion are tested by direct STM measurement of the fluctuations of near-neighbor pairs of steps on Si(111)-root3 x root3 R30 - Al at 970K. The average magnitude of the pair-correlation function is within one standard deviation of zero, consistent with uncorrelated near-neighbor step fluctuations. The time dependence of the pair-correlation function shows no statistically significant agreement with the predicted t^1/2 growth of pair correlations via rate-limiting atomic diffusion between adjacent steps. The physical considerations governing uncorrelated step fluctuations occurring via random attachment/detachment events at the step edge are discussed.

cond-mat.stat-mech↗

arXiv2003

Applications of Ideas from Random Matrix Theory to Step Distributions on "Misoriented" Surfaces

Arising as a fluctuation phenomenon, the equilibrium distribution of meandering steps with mean separation $<\ell>$ on a "tilted" surface can be fruitfully analyzed using results from RMT. The set of step configurations in 2D can be mapped onto the world lines of spinless fermions in 1+1D using the Calogero-Sutherland model. The strength of the ("instantaneous", inverse-square) elastic repulsion between steps, in dimensionless form, is $β(β-2)/4$. The distribution of spacings $s< \ell>$ between neighboring steps (analogous to the normalized spacings of energy levels) is well described by a {\it "generalized" Wigner surmise}: $p_β(0,s) \approx a s^β\exp(-b s^2)$. The value of $β$ is taken to best fit the data; typically $2 \le β\le 10$. The procedure is superior to conventional Gaussian and mean-field approaches, and progress is being made on formal justification. Furthermore, the theoretically simpler step-step distribution function can be measured and analyzed based on exact results. Formal results and applications to experiments on metals and semiconductors are summarized, along with open questions. (conference abstract)

cond-mat.stat-mech↗

arXiv2003

Orientation Dependence of Step Stiffness: Failure of SOS and Ising Models to Describe Experimental Data

We have investigated the step stiffness on Cu(001) surfaces as a function of step orientation by two independent methods at several temperatures near 300 K. Both sets of data agree well and show a substantial dependence of the stiffness on the angle of orientation. With the exception of steps oriented along $<110>$, the experimental stiffness is significantly larger than the stiffness calculated within the solid-on-solid (SOS) model and the Ising-model, even if next nearest-neighbor interactions are taken into account. Our results have considerable consequences for the understanding and for the theoretical modeling of equilibrium and growth phenomena, such as step meandering instabilities.

cond-mat.mtrl-sci↗

arXiv2000

Terrace-Width Distributions and Step-Step Repulsions on Vicinal Surfaces: Symmetries, Scaling, Simplifications, Subtleties, and Schrödinger

For more than three decades, measurement of terrace width distributions (TWDs) of vicinal crystal surfaces have been recognized as arguably the best way to determine the dimensionless strength $\tilde{A}$ of the elastic repulsion between steps. For sufficiently strong repulsions, the TWD is expected to be Gaussian, with $\tilde{A}$ varying inversely with the squared variance. However, there has been a controversy over the proportionality constant. From another perspective the TWD can be described as a continuous generalized Wigner distribution (CGWD) essentially no more complicated than a Gaussian but a much better approximation at the few calibration points where exact solutions exist. This paper combines concisely the experimentally most useful results from several earlier papers on this subject and describes some advancements that are in progress regarding numerical tests and in using Schrödinger-equation formalism to give greater understanding of the origin of the CGWD and to give hope of extensions to more general interaction potentials between steps. There are many implications for future experiments.

cond-mat.mtrl-sci↗

arXiv2000

Electromigration of Single-Layer Clusters

Single-layer atom or vacancy clusters in the presence of electromigration are studied theoretically assuming an isotropic medium. A variety of distinctive behaviors distinguish the response in the three standard limiting cases of periphery diffusion (PD), terrace diffusion (TD), and evaporation-condensation (EC). A general model provides power laws describing the size dependence of the drift velocity in these limits, consistent with established results in the case of PD. The validity of the widely used quasistatic limit is calculated. Atom and vacancy clusters drift in opposite directions in the PD limit but in the same direction otherwise. In absence of PD, linear stability analysis reveals a new type of morphological instability, not leading to island break-down. For strong electromigration, Monte Carlo simulations show that clusters then destabilize into slits, in contrast to splitting in the PD limit. Electromigration affects the diffusion coefficient of the cluster and morphological fluctuations, the latter diverging at the instability threshold. An instrinsic attachment-detachment bias displays the same scaling signature as PD in the drift velocity.

cond-mat.mtrl-sci↗

arXiv2000

Extraction of Step-Repulsion Strengths from Terrace Width Distributions: Statistical and Analytic Considerations

Recently it has been recognized that the so-called generalized Wigner distribution may provide at least as good a description of terrace width distributions (TWDs) on vicinal surfaces as the standard Gaussian fit and is particularly applicable for weak repulsions between steps, where the latter fails. Subsequent applications to vicinal copper surfaces at various temperatures confirmed the serviceability of the new analysis procedure but raised some theoretical questions. Here we address these issues using analytical, numerical, and statistical methods. We propose an extension of the generalized Wigner distribution to a two-parameter fit that allows the terrace widths to be scaled by an optimal effective mean width. We discuss quantitatively the approach of a Wigner distribution to a Gaussian form for strong repulsions, how errors in normalization or mean affect the deduced interaction, and how optimally to extract the interaction from the variance and mean of the TWD. We show that correlations reduce by two orders of magnitude the number of {\em independent} measurements in a typical STM image. We also discuss the effect of the discreteness ("quantization") of terrace widths, finding that for high misorientation (small mean width) the standard continuum analysis gives faulty estimates of step interactions.

cond-mat.mtrl-sci↗