Ehrlich-Schwoebel Barrier Research Articles

Effects of three-dimensional Ehrlich-Schwoebel barrier on texture selection during Cu nanorod growth

This work explores the effectiveness of the three-dimensional Ehrlich-Schwoebel (3D ES) barrier in the selection of textures during nanorod growth. The authors use molecular dynamics simulations to show that the 3D ES barrier, together with geometrical shadowing and the thermal dynamics of surface faceting, defines the texture of nanorods. The results offer insights into when and how the 3D ES barrier will be effective in texture selection, and may be applied to texture design during growth of various nanorods.

Linear stability analysis for step meandering instabilities with elastic interactions and Ehrlich-Schwoebel barriers

Vicinal surfaces are known to exhibit morphological instabilities during step-flow growth. Through a linear stability analysis of step meandering instabilities, we investigate two effects that are important in many heteroepitaxial systems: elastic monopole-monopole interactions arising from bulk stress and the Ehrlich-Schwoebel (ES) barriers due to the asymmetric adatom incorporation rates. The analysis shows that the effects of the ES barriers increase as the average terrace width increases, whereas the effects of elastic monopole-monopole interactions decrease. The ES barriers favor an in-phase step pattern with a zero phase shift between consecutive steps, while elastic stress favors an out-of-phase pattern with a phase shift of pi. However, our analysis shows that the instability growth rate becomes nearly independent of the phase shift when either the ES-barrier effect or the stress effect is large. In particular, for ES-barrier-driven instability, the in-phase step pattern develops only within an intermediate range of terrace widths when bulk stress exists. Similarly, for the elastic-interaction-driven instability, an out-of-phase pattern only forms within a certain range of monopole strength; if the strength is too small, the ES barrier effect dominates, and if it is too large, the peak in the instability growth rate becomes delocalized in the phase shift and no patterns form. This transition between patterned and random step morphologies depends on the monopole strength, but is independent of the terrace width. A phase diagram that describes the regions of the ES-barrier-dominant instability and the elastic-interaction-dominant instability is established, along with the morphological phase diagrams that predict the step configurations as a function of the controlling parameters for the two types of instabilities.

The Calculation of Adatom Diffusion Process on Si and Ge Surfaces from First-Principles

We study Si adatom diffusion process near Si/Ge stepped surfaces from first-principles. Dependence of surface growth morphology on growing conditions such as temperature is not clearly understood. We calculate Ehrlich-Schwoebel(ES) barrier of stepped surfaces from first-principles and analyze adatom diffusion process to understand growth mechanism of the surfaces. The configuration of the surfaces with ES barrier would play a key role in the diffusion process because it controls hopping rates in the presence of step edges. Our results are likely to help correctly access the adatom diffusion process on the surfaces.

Growth Mode Diagram for the Epitaxial Growth on the Vicinal Surface of Strained Si (001)

We present the linear stability analysis for the epitaxial thin film growth on the vicinal surface of strained Si and the growth mode diagrams of the epitaxial growth under various operation conditions. Competition between step-step elastic interactions and the asymmetry of incorporation of adatoms from the terraces to step edge is considered. Force monopoles at steps and their interaction lead to it on the vicinal surface while kinetic asymmetry of the adatom incorporation at steps due to Ehrlich-Schwoebel barrier prevents the step bunching instability. Growth mode on the vicinal surface is determined by the competition between elastic step-step interactions and Ehrlich-Schwoebel barrier.

Influence of two different adatom mobilities on the initial heteroepitaxial growth kinetics

In the heteroepitaxy deposited adatoms in the first layer migrate on a substrate with a heterodiffusion coefficient, while those in all the next layers – with a self-diffusion coefficient. In this paper the effect of two different adatom surface mobilities on the layer heteroepitaxial growth in the presence of an Ehrlich–Schwoebel barrier is studied quantitatively using kinetic model based on rate equations for adatom and 2D island density and coverage in successive layers combined with a feeding zone taking into account interlayer mass transport. It is shown that this effect is essential in the first few monolayers and strongly depends on a ratio hetero-to-self-diffusion coefficient and Ehrlich–Schwoebel barrier. A faster heterodiffusion leads to a more rough initial growth whereas a slower heterodiffusion – to a smoother initial growth. In a transient (homoepitaxial) growth regime from smooth layer-by-layer – to rough growth a slower heterodiffusion retards this transition, while a faster heterodiffusion promotes it, especially at higher Ehrlich–Schwoebel barrier. It is shown that a critical coverage for the next layer nucleation is more appropriate parameter for characterization growth mode transition than a widely used critical island radius because it does not depend on island density and uniquely determines growth mode transition and its dependence on hetero- and self-diffusion coefficients and a critical nucleus size is obtained. The results suggest an opportunity for improving ultrathin (a few monolayers) film surface morphology by manipulating growth mode during deposition by initial (first layer) deposition either at low substrate temperature (low diffusivity) or with high deposition rate and consecutive one either at higher temperature (high diffusivity) or with a lower deposition rate.

A cellular automata algorithm for step dynamics in continuum modeling of epitaxial growth

We adapt a cellular automata algorithm in order to track the steps in a continuum model of epitaxial growth and couple it to adatom diffusion at the terraces. The model is capable to treat the Ehrlich–Schwoebel barrier at steps, which leads to instabilities during epitaxial growth. In order to examine the mound formation a nucleation model beyond the mean field approximation is included in our simulations. We show that the meander instability as well as the onset of mound formation can be treated by this model.

Island coalescence and diffusion along kinked steps on Cu(0 0 1): Evidence for a large kink Ehrlich–Schwoebel barrier

Using temperature-variable scanning tunneling microscopy, we studied the coalescence of vacancy islands on Cu(0 0 1) in ultra-high vacuum. From the temperature dependence of the relaxation of merged vacancy islands to the equilibrium shape we obtain an activation energy of the island coalescence process of 0.76 eV. From that value we deduce an activation energy for the atomic hopping coefficient of E Γ h = 0.89 eV. Comparing our result with previous STM data on step fluctuations with dominant diffusion along straight step segments ( E Γ h = 0.68 eV; [M. Giesen, S. Dieluweit, J. Mol. Catal. A: Chem. 216 (2004) 263]) and step fluctuations with kink crossing ( E Γ h = 0.9 eV; [M. Giesen-Seibert, F. Schmitz, R. Jentjens, H. Ibach, Surf. Sci. 329 (1995) 47]), we conclude that there is a large extra barrier for diffusion of atoms across kinks on Cu(0 0 1) of the order of 0.23 eV. This is the first direct experimental evidence for the existence of a large kink Ehrlich–Schwoebel barrier on Cu(0 0 1).

Computation of the Ehrlich-Schwoebel barrier to adatom diffusion in heteroepitaxial systems

The Ehrlich-Schwoebel (ES) barriers to the diffusion of adatoms (Pd,Ag and Cu) on different bench terraces of fcc Cu are compared by molecular dynamics computation. The computation reveals that: 1) For both heteroepitaxial and homoepitaxial systems, the ES barriers increase similarly with the increase of bench terrace height until a certain height is reached. 2) The ES barrier of adatom Pd is the biggest, the second one is that of adatom Cu, and the smallest one is that of adatom Ag for the hopping diffusion mechanism, while the ES barrier of adatom Ag is the biggest, the second one is that of adatom Cu, and the smallest one is that of adatom Pd for the exchange diffusion mechanism. 3)The ES barrier of exchange is lower than that of hopping mechanism in most cases of bench terraces.

Interplay between elastic interactions and kinetic processes in stepped Si (001) homoepitaxy

A vicinal Si (001) surface may form stripes of terraces, separated by monatomic-layer-high steps of two kinds, ${S}_{\mathrm{A}}$ and ${S}_{\mathrm{B}}$. As adatoms diffuse on the terraces and attach to or detach from the steps, the steps move. In equilibrium, the steps are equally spaced due to elastic interaction. During deposition, however, ${S}_{\mathrm{A}}$ is less mobile than ${S}_{\mathrm{B}}$. We model the interplay between the elastic and kinetic effects that drives step motion, and show that during homoepitaxy all the steps may move in a steady state, such that alternating terraces have time-independent, but unequal, widths. The ratio between the widths of neighboring terraces is tunable by the deposition flux and substrate temperature. We study the stability of the steady-state mode of growth using both linear perturbation analysis and numerical simulations. We elucidate the delicate roles played by the standard Ehrlich-Schwoebel (ES) barriers and inverse ES barriers in influencing growth stability in the complex system containing $({S}_{\mathrm{A}}+{S}_{\mathrm{B}})$ step pairs.

Determination of the Ehrlich-Schwoebel barrier in epitaxial growth of thin films

We demonstrate an approach for determining the ``effective'' Ehrlich-Schwoebel (ES) step-edge barrier, an important kinetic constant to control the interlayer mass transport in epitaxial growth of thin films. The approach exploits the rate difference between the growth and/or decay of an adatom and a vacancy two-dimensional island, which allows the ``effective'' ES barrier to be determined uniquely by fitting with a single parameter. Application to growth of Pb islands produces an effective ES barrier of $\ensuremath{\sim}83\ifmmode\pm\else\textpm\fi{}10\phantom{\rule{0.3em}{0ex}}\mathrm{meV}$ on Pb(111) surface at room temperature.

Nucleation and growth mechanism of GaAs epitaxial growth

The kinetic Monte Carlo simulations were used to study the nucleation and growth mechanism of GaAs. The Ehrlich–Schwoebel (ES) barrier has an important effect on the growth mechanism. It was shown that the island density mainly depends on the growth temperature: low island density for high temperature, and high island density for low temperature. The islands in the second layer appear at small surface coverage for the high ES barrier or low temperature, while the islands appear at large surface coverage for the low ES barrier or high temperature. In GaAs epitaxial growth the critical size has its original meaning only for the low temperature and the high ES barrier. The single value of the critical size has to be replaced by two sizes in the two different directions because of the anisotropic surface migration.

Effects of short-range attraction in unstable metal (110) epitaxial growth: Molecular dynamics simulations for Cu, Ag, and Al

The results of molecular dynamics simulations of the process of adatom deposition near steps for Cu(110), Ag(110), and Al(110) surfaces are presented in order to determine the effects of short-range (SR) attraction on the surface current and selected mound angle during unstable epitaxial growth. Our results indicate that, in qualitative agreement with previous work on metal (100) and (111) surfaces, for [001] steps the short-range attraction of depositing atoms to step edges can significantly increase the selected mound angle for typical energies used in epitaxial growth. In contrast, for the case of deposition near $[1\overline{1}0]$ steps, due to geometric effects the effects of short-range attraction are significantly weaker. As a result the overall ``uphill funneling'' probability ${P}_{\mathrm{av}}^{1\overline{1}0}$ is essentially the same as in the absence of SR attraction. Predictions for the selected mound angles along [001] and $[1\overline{1}0]$ steps are also presented for the case of irreversible growth with a large Ehrlich-Schwoebel barrier. Our results are also compared with recent experimental results in which anisotropic hut clusters with (111) and (100) facets have been observed. While our results are in qualitative agreement with experimental observations that the selected mound slopes are typically larger for [001] steps than for $[1\overline{1}0]$ steps, they also indicate that the effects of short-range attraction are not sufficient to explain the large-angle facets observed in $\mathrm{Al}∕\mathrm{Al}(110)$ growth at $400\phantom{\rule{0.3em}{0ex}}\mathrm{K}$, thus confirming that in this case other effects such as uphill diffusion at step edges must play an important role.

Morphology transition in a heteroepitaxial system: Co/Cu(111)

The initial stages of multilayer Co thin film grown on Cu(111) surface were simulated by means of kinetic Monte Carlo (KMC) method, where the realistic growth model and physical parameters were presented. The effects of edge diffusion along the islands and mass transport between interlayers were included in the simulation model. Emphasis was placed on revealing the transition of growth morphology in heteroepitaxial Co/Cu(111) system with the changing of surface temperature. The simulation results show that the dendritic islands form at low temperature ( T = 210 K), while compact islands grow at room temperature (RT). The Volmer-Webber (three-dimensional, 3D) growth mode is presented due to the relative higher Ehrlich-Schwoebel (ES) barrier. Our simulation results are in good agreement with the real scanning tunneling microscopy (STM) experiments.

High-order surface relaxation versus the Ehrlich–Schwoebel effect

We consider a class of continuum models of epitaxial growth of thin films with two competing mechanisms: (1) the surface relaxation described by high-order gradients of the surface profile and (2) the Ehrlich–Schwoebel (ES) effect which is the asymmetry in the adatom attachment and detachment to and from atomic steps. Mathematically, these models are gradient-flows of some effective free-energy functionals for which large slopes are preferred for surfaces with low energy.We characterize the large-system asymptotics of the minimum energy and the magnitude of gradients of energy-minimizing surfaces. We also show that, in the large-system limit, the renormalized energy with an infinite ES barrier is the Γ-limit of those with a finite one, indicating the enhancement of the ES effect in a large system. Introducing λ-minimizers as energy minimizers among all candidates that are spatially λ-periodical, we show the existence of a sequence of such λ-minimizers that are in fact equilibriums. For the case of a finite ES effect, we prove the well-posedness of the initial-boundary-value problem of the continuum model and obtain bounds for the scaling laws of interface width, surface slope and energy, all of which characterize the surface coarsening during the film growth. We conclude with a discussion on the implications of our rigorous analysis.

Self-organization of step bunching instability on vicinal substrate

The authors investigate quantitatively the self-organization of step bunching instability during epitaxy of Si on vicinal Si(001). They show that growth instability evolution can be fitted by power laws L∼tα and A∼tβ (where L is the correlation length and A is the instability amplitude) with critical exponents α∼0.3 and β∼0.5 in good agreement with previous studies and well reproduced by kinetic Monte Carlo simulation. They demonstrate that the main phenomenon controlling step bunching is the anisotropy of surface diffusion. The microscopic origin of the instability is attributed to an easier adatom detachment from SA step, which can be interpreted as a pseudoinverse Ehrlich-Schwoebel barrier [J. Appl. Phys. 37, 3682 (1967); J. Chem. Phys. 44, 1039 (1966)].

Influence of epitaxial strain on the terrace and inter-layer diffusions in metal epitaxy

The influence of epitaxial strain on the surface and inter-layer diffusions are investigated using molecular statics and transition state theory with several types of embedded atom method potentials for Ag and Ni. Quantitatively to analyze the competition of the surface and inter-layer diffusions in the instability to the multi-layer growth, Ehrlich–Schwoebel barrier and the attempt frequencies of the surface and inter-layer diffusions by both hopping and exchange mechanisms are considered simultaneously. The attempt frequencies of exchange mechanism are larger by order one than those of hopping mechanism and especially, the difference becomes more severe for the inter-layer diffusion. Considering both the attempt frequency and activation energy barrier shows that the layer-by-layer growth is enhanced by compressive strain for Ag(0 0 1), which is confirmed by the existing linear stability theory and kinetic Monte Carlo simulations.

Adsorption and diffusion of an alkali-metal adatom on transition-metal dichalcogenides

Employing total-energy density-functional theory calculations the adsorption energies, the diffusion pathways, and the barriers for a Li adatom on stepped ${\mathrm{TiSe}}_{2}$ surfaces with full relaxation of the atomic environment were studied. The minimum-energy paths for Li diffusion on a ${\mathrm{TiSe}}_{2}$ (0001) surface and across the $(10\overline{1}0)$ and $(\overline{1}2\overline{1}1)$ steps have been calculated. At the $(\overline{1}2\overline{1}1)$ step edge an Ehrlich-Schwoebel barrier of $0.31\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$ was obtained, which is additional to the diffusion barrier of $0.25\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$ on the flat surface. The energies for subsequent diffusion within the van der Waals gap have been calculated, too.

Growth dynamics of pentacene thin films: Real-time synchrotron x-ray scattering study

Real-time synchrotron x-ray scattering in the anti-Bragg configuration was used to monitor the dynamics of pentacene film growth on inert substrates. A distributed-growth model, according to which pentacene molecules adsorbed on the nth layer can either nucleate and contribute to the growth of the (n+1)th layer or transfer downward and contribute to the growth of the nth layer, gave a good description of the data. For molecules adsorbed on the first and second layers, the probability of downward transfer was found to be dependent on the substrate, and independent of temperature within the range from 25 to 60 deg. C. For films grown on SiO{sub 2}, an Ehrlich-Schwoebel barrier of the order of 70 meV dominated downward transfer of pentacene molecules in layers away from the substrate. For films grown on an alkylated self-assembled monolayer, significant desorption of pentacene molecules from the substrate at elevated temperatures forced the growth mode toward the three-dimensional limit.

Simulation of multilayer homoepitaxial growth on Cu (100) surface

The processes of multilayer thin Cu films grown on Cu (100) surfaces at elevated temperature (250–400 K) are simulated by mean of kinetic Monte Carlo (KMC) method, where the realistic growth model and physical parameters are used. The effects of small island (dimer and trimer) diffusion, edge diffusion along the islands, exchange of the adatom with an atom in the existing island, as well as mass transport between interlayers are included in the simulation model. Emphasis is placed on revealing the influence of the Ehrlich–Schwoebel (ES) barrier on growth mode and morphology during multilayer thin film growth. We present numerical evidence that the ES barrier does exist for the Cu/Cu(100) system and an ES barrier EB>0.125 eV is estimated from a comparison of the KMC simulation with the realistic experimental images. The transitions of growth modes with growth conditions and the influence of exchange barrier on growth mode are also investigated.

Effect of step-step separation on surface diffusion processes

We present results of calculations of activation energy barriers for several processes involving a diffusing atom around a step edge on the (111), (100), and (110) surfaces of Cu using the nudged elastic band method with interaction potentials based on the embedded atom method. In the presence of a neighboring step, the Ehrlich-Schwoebel barrier for an adatom to undergo jump or exchange at a step edge is found to be independent of the separation between the steps unless they are at two-atom width apart, in which case the barrier is larger than that for an isolated step. The influence of the local step geometry on diffusion energetics is also discussed.