Step Bunching Instability Research Articles

Step bunching instability and its effects in electrocatalysis on platinum surfaces

Step bunching instability and its effects in electrocatalysis on platinum surfaces

Achieving atomically flat copper surface: Formation of mono-atomic steps and associated strain energy mechanisms

Recent studies have demonstrated that achieving an atomically flat surface for metals can significantly enhance their oxidation resistance and further advance their electronic-optical applications. However, traditional heat treatments for polycrystalline metals tend to create numerous low-surface-energy steps and facets due to surface energy minimization. Moreover, surface diffusion is further limited by three-dimensional Ehrlich-Schwoebel barriers due to the formation of steps and facets. Therefore, achieving atomically flat surfaces is energetically unfavorable and kinetically unstable. Here, we covered graphene (Gr) on Copper (Cu) surface and performed systematic and statistical analysis of microstructures on three types of graphene-Cu (Gr/Cu) interfaces: annealed Cu, transferred and high-temperature deposited Gr/Cu interfaces. We found that mono-atomic steps formed at high-temperature deposited Gr/Cu interface, in comparison with multi-atomic steps at annealed Cu and transferred Gr/Cu interfaces. Molecular statics/dynamics simulations and thermodynamic analysis suggest that formation of mono-atomic steps can be ascribed to the minimization of Gr strain energy and high-temperature assisted surface diffusion. When a step height (h) is smaller than five atomic planes (h < 5), strain energy minimization of Gr will prevent step bunching (a non-uniform surface morphology), accelerate the formation of atomically flat surface. When h ≥ 5, Gr strain energy minimization will trigger step-bunching instability, decompose large steps and thus facilitate the surface diffusion to develop atomically flat surface. The present results may suggest a potential strategy to achieve atomically flat surface of bulk metals by high-temperature treatment.

Step-bunching Instability of Growing Interfaces between Ice and Supercooled Water

We review our recent study on step-bunching instability (SBI) of ice melt growth, one of the interfacial instabilities driven by self-organization of elementary step flow associated with crystal growth dynamics. With the aid of an advanced optical microscope and its combination with a two-beam interferometer, we realized quantitative in situ observations of the spatiotemporal dynamics of the SBI. This enables us to examine the origin of the SBI at the level of the step–step interaction. Our finding offers a significant clue to understanding the general mechanism of melt growth beyond ice-crystal growth, inseparably involving several broad research fields, including cryobiological, geophysical, and material branches.

Step-bunching instability of growing interfaces between ice and supercooled water

SignificanceStep-bunching instability (SBI) is one of the interfacial instabilities driven by self-organization of elementary step flow associated with crystal-growth dynamics, which has been observed in diverse crystalline materials. However, despite theoretical suggestions of its presence, no direct observations of SBI for simple melt growth have been achieved so far. Here, with the aid of a type of optical microscope and its combination with a two-beam interferometer, we realized quantitative in situ observations of the spatiotemporal dynamics of the SBI. This enables us to examine the origin of the SBI at the level of the step-step interaction. We also found that the SBI spontaneously induces a highly stable spiral growth mode, governing the late stage of the growth process.

Revealing electromigration on dielectrics and metals through the step-bunching instability

Electromigration, due to its technological and scientific significance, has been a subject of extensive studies for many years. We present evidence of electromigration in dielectric materials, namely $C$-plane sapphire, obtained from direct experimental observation of an atomic step-bunching instability driven by electromigration. We further expand upon our previously reported findings of electromigration induced step-bunching transformation of a metal surface. The only system where electromigration driven step bunching has been observed and comprehensively investigated is the low index surfaces of silicon. In this study we show that electromigration driven SB can be induced on a variety of crystallographic surfaces, including metals and insulating oxides, and may be more prevalent than previously thought. Electric fields were applied at high temperature to W(110) and $\mathrm{A}{\mathrm{l}}_{2}{\mathrm{O}}_{3}(0001)$ crystals whereupon their surface reordered to a morphology closely resembling that of Si(111) with atomic steps bunched by electromigration. This suggests that the mechanism of step bunching on the W(110), $\mathrm{A}{\mathrm{l}}_{2}{\mathrm{O}}_{3}(0001)$, and Si(111) can be fundamentally the same. Annealing W(110) offcut in the [001] direction with an up-step current produced a morphology with the bunch edges composed of zigzag segments meeting at a right angle.

Stability of Vicinal Surfaces: Beyond the Quasistatic Approximation.

We revisit the step bunching instability without recourse to the quasistatic approximation and show that the stability diagrams are significantly altered, even in the low-deposition regime where it was thought sufficient. In particular, steps are unstable against bunching for attachment-detachment limited growth. By accounting for the dynamics and chemical effects, we can explain the onset of step bunching in Si(111)-(7×7) and GaAs(001) without resort to the inverse Schwoebel barrier or step-edge diffusion. Further, the size-scaling analysis of step-bunch growth, as induced by these two combined effects, agrees with the bunching regime observed at 750 °C in Si(111)-(7×7).

Roughening of GaN homoepitaxial surfaces due to step meandering and bunching instabilities and their suppression in hydride vapor phase epitaxy

In this letter, we have shown that well-known surface instabilities in crystal growth, i.e., in-phase step meandering and step bunching, are the main causes of surface roughening of GaN homoepitaxial layers on vicinal +c-oriented GaN substrates. Both instabilities were effectively suppressed in hydride vapor phase epitaxy (HVPE) under appropriate conditions, which produced highly smooth as-grown surfaces suitable for reproducible device fabrication. Wavy surface morphologies, often observed in homoepitaxial GaN layers grown not only by HVPE but also by metal-organic chemical vapor deposition (MOCVD), were found to be covered by an array of meandered bunched steps consisting of m- and a-oriented sections. Because the bunched steps meandered in an in-phase fashion, facets consisting of m- and a-oriented bunched steps formed narrow macro-steps and wide macro-terraces, respectively, leading to the formation of ridges and grooves in-between the facets. Although the use of a substrate with large off-angles (≥0.4°) effectively suppressed the step meandering, i.e., wavy surface morphology, it induced a strong tendency for step bunching. Only growth on surfaces having relatively small off-angles (0.25°), under conditions providing large degrees of adatom desorption, i.e., at high temperatures or low V/III-ratios, suppressed both the meandering and bunching instabilities simultaneously and produced highly smooth as-grown surfaces. The use of the HVPE method seems better than the use of the MOCVD method in this strategy to increase the surface flatness of GaN homoepitaxial layers because it can maintain a sufficiently high growth rate even with a large degree of adatom desorption.

Energy Scaling and Asymptotic Properties of Step Bunching in Epitaxial Growth with Elasticity Effects

In epitaxial growth on vicinal surfaces, elasticity effects give rise to step bunching instability and some self-organization phenomena, which are widely believed to be important in the fabrication of nanostructures. It is challenging to model and analyze these phenomena due to the nonlocal effects and interactions between different length scales. In this paper, we study a discrete model for epitaxial growth with elasticity. We rigorously identify the minimum energy scaling law and prove the formation and appearance of one bunch structure. We also provide sharp bounds for the bunch size and the slope of the optimal step bunch profile. Both periodic and Neumann boundary conditions are considered.

Diffusion-controlled growth rate of stepped interfaces.

For many materials, the structure of crystalline surfaces or solid-solid interphase boundaries is characterized by an array of mobile steps separated by immobile terraces. Despite the prevalence of step-terraced interfaces a theoretical description of the growth rate has not been completely solved. In this work the boundary element method (BEM) has been utilized to numerically compute the concentration profile in a fluid phase in contact with an infinite array of equally spaced surface steps and, under the assumption that step motion is controlled by diffusion through the fluid phase, the growth rate is computed. It is also assumed that a boundary layer exists between the growing surface and a point in the liquid where complete convective mixing occurs. The BEM results are presented for varying step spacing, supersaturation, and boundary layer width. BEM calculations were also used to study the phenomenon of step bunching during crystal growth, and it is found that, in the absence of elastic strain energy, a sufficiently large perturbation in the position of a step from its regular spacing will lead to a step bunching instability. Finally, an approximate analytic solution using a matched asymptotic expansion technique is presented for the case of a stagnant liquid or equivalently a solid-solid stepped interface.

Interface kinetics in phase-field models: Isothermal transformations in binary alloys and step dynamics in molecular-beam epitaxy

We present a unified description of interface kinetic effects in phase-field models for isothermal transformations in binary alloys and steps dynamics in molecular-beam-epitaxy. The phase-field equations of motion incorporate a kinetic cross-coupling between the phase field and the concentration field. This cross-coupling generalizes the phenomenology of kinetic effects and was omitted until recently in classical phase-field models. We derive general expressions (independent of the details of the phase-field model) for the kinetic coefficients within the corresponding macroscopic approach using a physically motivated reduction procedure. The latter is equivalent to the so-called thin-interface limit but is technically simpler. It involves the calculation of the effective dissipation that can be ascribed to the interface in the phase-field model. We discuss in detail the possibility of a nonpositive definite matrix of kinetic coefficients, i.e., a negative effective interface dissipation, although being in the range of stability of the underlying phase-field model. Numerically we study the step-bunching instability in molecular-beam-epitaxy due to the Ehrlich-Schwoebel effect, present in our model due to the cross-coupling. Using the reduction procedure we compare the results of the phase-field simulations with the analytical predictions of the macroscopic approach.

Singular response to a dopant of an evaporating crystal surface

Moving crystal surfaces can undergo step-bunching instabilities, when subject to an electric current. We show analytically that an infinitesimal quantity of a dopant may invert the stability, whatever the sign of the current. Our study is relevant for experimental results [S. S. Kosolobov et al., JETP Lett. 81, 117 (2005)] on an evaporating Si(111) surface, which show a singular response to Au doping, whose density distribution is related to inhomogeneous Si diffusion.

Surface organization of homoepitaxial InP films grown by metalorganic vapor-phase epitaxy

We present a systematic study of the morphology of homoepitaxial InP films grown by metalorganic vapor-phase epitaxy which are imaged with ex situ atomic force microscopy. These films show a dramatic range of different surface morphologies as a function of the growth conditions and substrate (growth temperature, V/III ratio, and miscut angle $&lt;0.{6}^{\ensuremath{\circ}}$ and orientation toward A or B sites), ranging from stable step flow to previously unreported strong step bunching, over 10 nm in height. These observations suggest a window of growth parameters for optimal quality epitaxial layers. We also present a theoretical model for these growth modes that takes account of deposition, diffusion, and dissociation of molecular precursors, and the diffusion and step incorporation of atoms released by the precursors. The experimental conditions for step flow and step bunching are reproduced by this model, with the step bunching instability caused by the difference in molecular dissociation from above and below step edges, as was discussed previously for GaAs (001).

Simultaneous step meandering and bunching instabilities controlled by Ehrlich-Schwoebel barrier and elastic interaction

Through phase-field simulations, we investigate simultaneous step meandering and bunching instabilities with the presence of Ehrlich-Schwoebel barrier and elastic interaction. The meandering instability induced by the Ehrlich-Schwoebel barrier is found to be dependent on the elastic interaction at low adatom deposition rate. The ordered step meandering-bunching structure is designed by using the predefined magnitude distribution of the force monopoles on vicinal surfaces based on interplay between the Ehrlich-Schwoebel barrier and the elastic interaction.

Driving Ge Island Ordering on Nanostructured Si surfaces

Semiconductor epitaxial nanostructures have been recently proposed as the key building blocks of many innovative applications in materials science and technology. To bring their tremendous potential to fruition, a fine control of nanostructure size and placement is necessary. We present a detailed investigation of the self-ordering process in the prototype case of Ge/Si heteroepitaxy. Starting from a bottom-up strategy (step-bunching instabilities), our analysis moves to lithographic techniques (scanning tunneling lithography, nanomechanical stamping, focused ion beam patterning) with the aim of developing a hybrid approach in which the exogenous intervention is specifically designed to suit and harness the natural self-organization dynamics of the system.

Critical field behavior and antiband instability under controlled surface electromigration on Si(111)

In this study we investigate step bunching and antiband surface instabilities on Si(111). We experimentally study the effects of a controlled electromigration field on the onset of antibands. We analyze the initial stage of antiband formation on step bunched surfaces under conditions of constant temperature of 1270 \ifmmode^\circ\else\textdegree\fi{}C, while systematically varying the applied electromigration field. The relationship between the electromigration field and minimum terrace width required to initiate the antiband formation has been established. Also, we systematically measured values of the critical electromigration field, which is required to initiate the step-bunching process on Si(111) at 1130 \ifmmode^\circ\else\textdegree\fi{}C (regime II) and 1270 \ifmmode^\circ\else\textdegree\fi{}C (regime III). The dependence of the critical field on the mean atomic terrace width has been investigated and discussed.

Step velocity tuning of SrRuO3step flow growth on SrTiO3

Taking advantage of vicinal (001) SrTiO3 substrates with different mean terrace widths, the heteroepitaxial growth of SrRuO3 in the step flow mode has been mapped as a function of mean step velocity. Transition between stable and unstable step flow is shown to occur at a well-defined critical step velocity, with a step bunching instability observed below this threshold. The ability to pass from unstable to stable step flow during growth upon increasing the mean step velocity is demonstrated. This result is discussed in terms of a stress-based driving force for step bunching in competition with an effective step-up adatom current.

Experimental quantitative study into the effects of electromigration field moderation on step bunching instability development on Si(111)

We experimentally studied the effects of a moderated electromigration field on the dynamics of the step bunching process on the Si(111) surface at 1130 \ifmmode^\circ\else\textdegree\fi{}C (regime II) and 1270 \ifmmode^\circ\else\textdegree\fi{}C (regime III). The surfaces with step bunch morphologies were created by annealing vicinal Si(111) at fixed temperatures while the applied electric field $E$ was adjusted for every experiment. Scaling relations, ${y}_{m}~{h}^{\ensuremath{\alpha}}{E}^{q}$, between the slope of a step bunch ${y}_{m}$, step bunch height $h$, and electromigration field $E$ were experimentally probed. Scaling exponents $\ensuremath{\alpha}$ \ensuremath{\approx} 2/3 and $q$ \ensuremath{\approx} 1/3 were extracted from the step bunch morphologies created by annealing Si(111) in the regime III (1270 \ifmmode^\circ\else\textdegree\fi{}C), which are in good agreement with the predictions of the generalized BCF theory. Scaling exponents $\ensuremath{\alpha}$ \ensuremath{\approx} 3/5 and $q$ \ensuremath{\approx} 1/3 were extracted from the morphologies created by annealing in regime II (1130 \ifmmode^\circ\else\textdegree\fi{}C). This result was compared to the scaling relations derived within the frame of the transparent step model, which correctly predicts the formation of the step bunching instability by step-up adatom electromigration. The scaling relation obtained by experiment was found to differ from the model predictions. We measured values of critical electric field (${E}_{\mathrm{cr}}$), i.e., minimum electric field required for the step bunching to take place. A relatively weak field of $E$ &gt; 0.5 V/cm was found to be sufficient to initiate the step bunching process in regime II. This contrasts with regime III, where ${E}_{\mathrm{cr}}=1.0$ and 2.0 V/cm were measured for Si miscut from the (111) plane by 1.1\ifmmode^\circ\else\textdegree\fi{} and 2.5\ifmmode^\circ\else\textdegree\fi{}, respectively. The increased values of ${E}_{\mathrm{cr}}$ were attributed to the enhanced step-step repulsion in regime III. The theoretically predicted formation of compressed step density waves was observed upon annealing in both regimes with $E&lt;{E}_{\mathrm{cr}}$.

Hydride vapor phase epitaxy of high structural perfection thick AlN layers on off-axis 6H-SiC

The employment of more than 10 μm thick AlN epilayers on SiC substrates for AlGaN/GaN high-electron-mobility transistors (HEMTs) substantially raises their performance in high-power energy-efficient amplifiers for 4G wireless mobile stations. In this paper, structural properties and surface morphology of thick AlN epilayers deposited by hydride vapor phase epitaxy (HVPE) on off-axis conductive 6H-SiC substrates are reported. The epilayers were examined in detail by high-resolution X-ray diffraction (XRD), atomic force microscopy (AFM), Nomarski differential interference contrast (DIC), scanning electron microscopy (SEM), and selective wet chemical etching. At optimal substrate preparation and growth conditions, a full width at half-maximum (FWHM) of the XRD rocking curve (RC) for the symmetric (00.2) reflex was very close to that of the substrate (less than 40 arcsec) suggesting low screw dislocation density in the epilayer (∼10 6 cm −2) and small in-plane tilt misorientation. Reciprocal space mapping around asymmetric reflexes and measured lattice parameters indicated a fully relaxed state of the epilayers. The unit-cell-high stepped areas of the epilayers with 0.5 nm root mean square (RMS) roughness over 1×1 μm 2 scan were alternated with step-bunching instabilities up to 350 nm in height. Low warp of the substrates makes them suitable for precise epitaxy of HEMT structures.

Anticoarsening and complex dynamics of step bunches on vicinal surfaces during sublimation

A sublimating vicinal crystal surface can undergo a step bunching instability when the attachment-detachment kinetics is asymmetric, in the sense of a normal Ehrlich-Schwoebel effect. Here we investigate this instability in a model that takes into account the subtle interplay between sublimation and step-step interactions, which breaks the volume-conserving character of the dynamics assumed in previous work. On the basis of a systematically derived continuum equation for the surface profile, we argue that the nonconservative terms pose a limitation on the size of emerging step bunches. This conclusion is supported by extensive simulations of the discrete step dynamics, which show breakup of large bunches into smaller ones as well as arrested coarsening and periodic oscillations between states with different numbers of bunches.

Continuum model for the long-range elastic interaction on stepped epitaxial surfaces in2+1dimensions

In heteroepitaxy, the mismatch of lattice constants between the crystal film and the substrate causes misfit strain and stress in the bulk of the film, driving the surface of the film to self-organize into various nanostructures. Below the roughening transition temperature, an epitaxial surface consists of facets and steps and changes its morphology by lateral motion of steps. In this paper, we present a $2+1$-dimensional continuum model for the long-range elastic interaction on stepped surface of a strained film. The continuum model is derived rigorously from the discrete model for the interaction between steps; thus it incorporates the discrete features of the stepped surfaces. Examples show that our continuum model is much more accurate as an approximation to the discrete model than the traditional continuum approximation. Moreover, in the linear instability of a planar surface, our continuum model gives the transition from step bunching instability to step undulation instability as the distance between adjacent steps increases, which agrees with the experimental observations and the results of discrete models and is missing using the traditional continuum approximation. Numerical simulations of the surface evolution using our model in the nonlinear regime show several different surface morphologies, including the morphology of step bunching which cannot be obtained using the traditional continuum approximation.