Interstep Distance Research Articles

Scaling laws for step bunching on vicinal surfaces: Role of the dynamical and chemical effects.

We study the evolution of step bunches on vicinal surfaces using a thermodynamically consistent step-flow model. By accounting for the dynamics of adatom diffusion on terraces and attachment-detachment at steps (referred to collectively as the dynamical effect), this model circumvents the quasistatic approximation that prevails in the literature. Furthermore, it generalizes the expression of the step chemical potential by incorporating the necessary coupling between the diffusion fields on adjacent terraces (referred to as the chemical effect). Having previously shown that these dynamical and chemical effects can explain the onset of step bunching without recourse to the inverse Ehrlich-Schwoebel (iES) barrier or other extraneous mechanisms, we are here interested in the evolution of step bunches beyond the linear-stability regime. In particular, the numerical resolution of the step-flow free boundary problem yields a robust power-law coarsening of the surface profile, with the bunch height growing in time as H∼t^{1/2} and the minimal interstep distance as a function of the number of steps in the bunch cell obeying ℓ_{min}∼N^{-2/3}. Although these exponents have previously been reported, the novelty of the present approach is that these scaling laws are obtained in the absence of an iES barrier or adatom electromigration. In order to validate our simulations, we take the continuum limit of the discrete step-flow system via Taylor expansions with respect to the terrace size, leading to a novel nonlinear evolution equation for the surface height. We investigate the existence of self-similar solutions of this equation and confirm the 1/2 coarsening exponent obtained numerically for H. We highlight the influence of the combined dynamical-chemical effect and show that it can be interpreted as an effective iES barrier in the setting of the standard Burton-Cabrera-Frank theory. Finally, we use a Padé approximant to derive an analytical expression for the velocity of steadily moving step bunches and compare it to numerical simulations.

Effect of long-term confinement on metabolic and physiological parameters in mice

Long-term and mild confinement or isolation in an enclosed environment can occur in situations such as disasters, specific political, economic or social events, nuclear shelters, seabed exploration, polar expeditions, and space travel. To investigate the effects of stress caused by long-term confinement in an enclosed environment in mammals, we divided 8-week-old C57BL/6J mice into four groups that were housed in a closed environment with a narrow metabolic cage (stress group), normal metabolic cage (control group), conventional cage (conventional group) or conventional cage with wire mesh floor (wire mesh group). The phenotypes of the mice were examined for four weeks, followed by behavioral tests. Weight gain suppression was observed in the stress group. Continuous analysis of these mice every two minutes for four weeks using an implanted measuring device showed a significantly decreased amount of spontaneous activity and subcutaneous temperature in the stress group. After housing in each environment for four weeks, the behavioral tests of mice in the stress group also revealed a shorter latency to fall off in the rotarod test and shorter stride length and interstep distance in the footprint test. Interestingly, the lower spontaneous activity of mice in the stress group was rescued by housing in conventional cages. These results suggest a temporary effect of long-term confinement in an enclosed environment as a chronic and mild stress on homeostasis in mammals.

Spiral growth of a crystal due to chemical reaction

The article is devoted to the study of crystal growth due to chemical reactions in multicomponent media via terrace-kink-step mechanism. The theory of Burton, Cabrera and Frank is developed and applied to multicomponent case. Equations are obtained for speed of movement of single step, group of steps at certain interstep distance. Growth spirals caused by screw dislocations are considered. Growth rate of the crystal is found. Impact of partial pressures of reaction products on diffusion length of adatoms and growth rate is analyzed.

Experimental determination of dependence of vertical growth rate on surface supersaturation in GaAs(001) microchannel epitaxy and growth optimization

Surface supersaturation in microchannel epitaxy (MCE) by liquid-phase epitaxy is a key parameter to control the width-to-thickness ratio (aspect ratio), which determines the dimensions of the dislocation-free area. The interstep distance of a spiral step at a screw-dislocation and curve-fitting of screw-like steps at stacking faults (SFs) are employed to estimate the surface supersaturation during MCE. The dependence of the vertical growth rate on the surface supersaturation was experimentally studied in GaAs(001) MCE. As a result, 2D nucleation at SFs was determined to act as a strong step source, which is stronger than a typical spiral step at a screw-dislocation over a wide range of surface supersaturation. Precise control of the surface supersaturation is important to optimize the growth conditions for MCE. An MCE layer with a large aspect ratio can thus be reproducibly obtained using appropriately controlled surface supersaturation.

Nucleus and spiral growth mechanisms of nitride semiconductors in metalorganic vapor phase epitaxy

Nucleus and spiral growth mechanisms of GaN and InN are investigated using the selective-area metalorganic vapor phase epitaxy (SA-MOVPE) technique on GaN bulk substrates. Nucleus growth of GaN occurs within selective areas having no screw-type dislocations, while spiral growth occurs within selective areas having screw-type dislocations. These growth modes are simultaneously observed on a single substrate in a single growth run. The nucleus and spiral growths of GaN result in the formation of step-free surfaces and growth spirals, respectively, wherein the interstep distance of growth spirals enables us to estimate the degree of surface supersaturation (σ). The σ dependences of nucleus and spiral growth rates of GaN are experimentally investigated. We found that these dependences are well explained by the classical crystal growth theories advocated by Burton, Cabrera, and Frank. We also investigate nucleation of InN using step-free GaN surfaces as an ideal platform.

Formulas for the force dipole interaction of surface line defects in homoepitaxy

Using asymptotics, we derive explicit, simplified formulas for integrals representing the force dipole interaction energy per unit length between line defects (steps) of the same sign that form perturbations of circles in homoepitaxy. Our starting point is continuum linear elasticity in accordance with the classic model by Marchenko and Parshin (1981 Sov. Phys.—JETP 52 129). In the case with concentric circular steps, we define a small geometric parameter, δ2, which expresses the smallness of interstep distance relative to the circle radii. We invoke the Mellin transform with respect to δ2 and derive systematically an approximation for the requisite integral. This technique offers an alternative to an exact evaluation in terms of elliptic integrals. We then demonstrate the use of the Mellin transform when calculating the force dipole interaction energy between smoothly, slowly varying steps that form perturbations of circles. We discuss the implications of our results for small-amplitude modulations of circular step profiles.

Scaling and universality in models of step bunching: the “C+–C-” model

We recently introduced a novel model of step flow crystal growth – the so-called “C+–C-” model [B. Ranguelov et al., C.R. Acad. Bulgare Sci. 60, 389 (2007)]. In this paper we aim to develop a complete picture of the model’s behaviour in the framework of the notion of universality classes. The basic assumption of the model is that the reference (“equilibrium”) densities used to compute the supersaturation might be different on either side of a step, so CL/CR ≠ 1 (L/R stands for left/right in a step train descending from left to right), and that this will eventually cause destabilization of the regular step train. Linear stability analysis considering perturbation of the whole step train shows that the vicinal is always unstable when the condition CL /CR >1 is fulfilled. Numerical integration of the equations of step motion combined with an original monitoring scheme(s) results in obtaining the exact size- and time- scaling of the step bunches in the limit of long times (including the numerical prefactors). Over a broad range of parameters the surface morphology is characterized by the appearance of the minimal interstep distance at the beginning of the bunches (at the trailing edge of the bunch) and may be described by a single universality class, different from those already generated by continuum theories [A. Pimpinelli et al., Phys. Rev. Lett. 88, 206103 (2002), J. Krug et al., Phys. Rev. B 71, 045412 (2005)]. In particular, the scaling of the minimal interstep distance lmin in the new universality class is shown to be lmin = (Sn /N)1/(n+1), where N is the number of steps in the bunch, n is the exponent in the step-step repulsion law U ~ 1/d0 n for two steps placed a distance d0 apart and Sn is a combination of the model parameters. It is also shown that N scales with time with universal exponent 1/2 independent of n. For the regime of slow diffusion it is obtained for the first time that the time scaling depends only on the destabilization parameter CL/CR. The bunching outside the parameter region where the above scaling exists cannot be assigned to a specific universality class and thus should be considered non-universal.

A morphological study of vicinal surface growth based on a hybrid algorithm

A hybrid algorithm that combines a phase-field model and a lattice gas model evolving according to a kinetic Monte-Carlo (KMC) simulation scheme is used to investigate the dynamics of vicinal surface growth during vapor phase epitaxy. The algorithm is computationally far more efficient than pure KMC schemes, and this gain in efficiency does not correspond to a loss in information on the kinetics of individual atoms. We present numerical studies on the temperature dependence of macroscopic properties of the growing surface, evaluating the relevant stochastic processes (attachment, detachment, diffusion and island dynamics) as a function of their rates. We show that the temperature at which step flow is replaced by island nucleation depends on incoming flux, diffusion parameters and interstep distance. Moreover, we validate these finding by comparison to experiments and by analytical investigations.

Domain ordering of strained 5 ML SrTiO3 films on Si(001)

High resolution x-ray diffraction data indicate ordered square shaped coherent domains, ∼1200Å in length, coexisting with longer, ∼9500Å correlated regions in highly strained 5 ML SrTiO3 films grown on Si(001). These long range film structures are due to the Si substrate terraces defined by the surface step morphology. The silicon surface “step pattern” is comprised of an “intrinsic” terrace length from strain relaxation and a longer “extrinsic” interstep distance due to the surface miscut.

Formation mechanism of rotational twins in beam-induced lateral epitaxy on (1 1 1)B GaAs substrate

The formation mechanism of rotational twins was systematically studied in beam-induced lateral epitaxy (BILE) on (1 1 1)B GaAs substrates using in situ scanning electron microscope (SEM) and ex situ atomic force microscope (AFM). As a result, it was found that rotational twins were produced from two-dimensional nuclei on the surface with 2×2 reconstruction by introducing stacking faults beneath them. As-trimers on 2×2 reconstruction probably induced the change. This was confirmed by the result that the formation of rotational twins was suppressed by the use of substrates with offset angles and a high growth temperature. The former is characterized by the high density of surface steps, which inhibits two-dimensional nucleation by the narrow inter-step distance. The latter brings 19 × 19 reconstruction, which includes no As-trimer, and leads to the normal stacking of growth with a smooth surface even with two-dimensional nucleation.

Step-Induced Anisotropic Growth of Pentacene Thin Film Crystals on a Hydrogen-Terminated Si(111) Surface

We investigate thin film crystals of an organic semiconductor, pentacene (Pn), grown on hydrogen-terminated Si(111) (H−Si(111)) surfaces with various interstep distances to elucidate the effects of vicinal steps of H−Si(111) on the growth of the Pn layers. By observing the morphology of the thin films with atomic force microscopy, we conclude that the vicinal steps induce significant anisotropy in the growth of the first layers of Pn: dendritic branches evolve in a lower-terrace side (the Si[112̄] direction), but a compact shape appears in an upper-terrace side (the Si[1̄1̄2] direction), although the first layers grow in an isotropic shape on a flat H−Si(111) surface. Furthermore, the growth of the first layers is much faster in the lower-terrace-side direction than in the upper-terrace-side direction. The anisotropy of the growth increases with decreasing interstep distances of H−Si(111), in particular ≤10 nm. Since such anisotropic growth was observed in a similar way irrespective of the directions of an incident Pn molecular beam, we conclude that the cause of the anisotropic growth is not the anisotropy of the surface diffusion and admolecule distribution of Pn molecules. Under the substrate temperature of 30−90 °C, the degree of the anisotropy remained constant within experimental error.

Fabrication of homoepitaxial ZnO films by low-temperature liquid-phase epitaxy

Highly crystalline undoped ZnO films were grown by liquid-phase epitaxy on (0 0 0 1) ZnO substrates with a misorientation of 0.5° toward 〈 1 0 1 ¯ 0 〉 . ZnO was continuously formed by the chemical reaction of ZnCl 2 and K 2CO 3. Films of thickness ⩽4 μm were grown in the temperature range 630–640 °C, resulting in step-flow growth as observed by differential interference microscopy and scanning electron microscopy. An interstep distance of 2±0.5 μm and step height 0.5 nm corresponding to one monolayer was observed by atomic force microscopy. Measurement of the (0 0 0 2) reflection of the X-ray rocking curve revealed the high quality of the ZnO films with a full-width half-maximum of 31 arcsec. The photoluminescence spectrum obtained at 4 K is dominated by emission from the band edge at 3.36 eV with a sharp splitting of the free exciton emission. A damaged surface layer in the ZnO substrate due to machining was revealed by radioluminescence at 80 K.

Dissolution on Cholesterol Monohydrate Single-Crystal Surfaces Monitored by in Situ Atomic Force Microscopy

The mechanism(s) and dissolution rates on the (001) surface of cholesterol monohydrate (ChM) in aqueous solutions containing 10−50% ethanol were monitored using in situ atomic force microscopy (AFM). Dissolution was found to proceed mechanistically via the layer-by-layer retreat of 34 Å bilayer or multilayer steps and/or the creation and expansion of bilayer and multilayer etch pits. In general, the dissolution rate is very strongly dependent on the local surface topography, which is highly variable on (001) ChM. Since the overall dissolution on the surface is by necessity a function of the relative density of different types of surface features, dissolution on both typical “smooth” and “rough” micron-sized areas was examined. For areas exhibiting low topographical relief, the rate of solute loss from isolated surface features was monitored by tracing the position of particular step fronts in sequential images over time. Although the absolute local dissolution rates are variable, the loss of solute molecules from isolated bilayer islands and/or pits occurred on the order of ∼105 molecules/min. The critical interstep distance for nondissolving features (observed in multiple experiments) appeared to decrease with increasing ethanol concentration, suggesting that surface diffusion effects likely influence this process. Dissolution in regions exhibiting much greater topographical relief tended to occur most readily by the rapid expansion of multilayer pits rather than by bilayer step retreat. In such areas, the alternative method of roughness analysis provided a more reliable means to track surface changes over time. The relatively high frequency with which such pit features were observed presumably speaks to the significantly large defect density in conventionally grown ChM crystals.

22.18 Methods of constructing stimulus-response curves for motor evoked potentials induced by transcranial magnetic stimulation

Surface supersaturation in microchannel epitaxy (MCE) by liquid-phase epitaxy is a key parameter to control the width-to-thickness ratio (aspect ratio), which determines the dimensions of the dislocation-free area. The interstep distance of a spiral step at a screw-dislocation and curve-fitting of screw-like steps at stacking faults (SFs) are employed to estimate the surface supersaturation during MCE. The dependence of the vertical growth rate on the surface supersaturation was experimentally studied in GaAs(001) MCE. As a result, 2D nucleation at SFs was determined to act as a strong step source, which is stronger than a typical spiral step at a screw-dislocation over a wide range of surface supersaturation. Precise control of the surface supersaturation is important to optimize the growth conditions for MCE. An MCE layer with a large aspect ratio can thus be reproducibly obtained using appropriately controlled surface supersaturation.

Scaling properties of step bunches induced by sublimation and related mechanisms

This work provides a ground for a quantitative interpretation of experiments on step bunching during sublimation of crystals with a pronounced Ehrlich-Schwoebel (ES) barrier in the regime of weak desorption. A strong step bunching instability takes place when the kinetic length is larger than the average distance between the steps on the vicinal surface. In the opposite limit the instability is weak and step bunching can occur only when the magnitude of step-step repulsion is small. The central result are power law relations of the between the width, the height, and the minimum interstep distance of a bunch. These relations are obtained from a continuum evolution equation for the surface profile, which is derived from the discrete step dynamical equations for. The analysis of the continuum equation reveals the existence of two types of stationary bunch profiles with different scaling properties. Through a mathematical equivalence on the level of the discrete step equations as well as on the continuum level, our results carry over to the problems of step bunching induced by growth with a strong inverse ES effect, and by electromigration in the attachment/detachment limited regime. Thus our work provides support for the existence of universality classes of step bunching instabilities [A. Pimpinelli et al., Phys. Rev. Lett. 88, 206103 (2002)], but some aspects of the universality scenario need to be revised.

In situ observation of elementary growth steps on the surface of protein crystals by laser confocal microscopy

Surface micro-topography of the {1 1 0} faces of tetragonal lysozyme crystals was observed in situ by laser confocal microscopy (LCM) combined with differential interference contrast microscopy (DIM). We could observe two-dimensional (2D) nucleation and subsequent growth of the 2D islands in real time. When steps of neighboring 2D islands coalesced, the contrast of the steps disappeared completely and the 2D islands merged smoothly. This result proved that the height of the steps observed by LCM combined with DIM was always the same and these steps were elementary ones (5.6 nm high). The combination of LCM and DIM was necessary to observe elementary steps with sufficient contrast. The elementary steps on a spiral growth hillock could also be observed using the LCM–DIM system, although their contrast was much smaller than that of the 2D islands because of much smaller inter-step distance.

Fluctuations in the step velocity and the generation of a dislocation spiral on the (101) face of monoclinic lysozyme crystals

The generation of a dislocation spiral and fluctuations in the step velocity on the (101) face of monoclinic lysozyme crystals have been studied by in situ atomic force microscopy (AFM). It is shown that the (101) face grows by the dislocation mechanism and that the steps move via the formation of one-dimensional nuclei. The velocity of a part of the step fluctuates, with the fluctuations increasing proportionally to the fourth-order root of time. In the process of spiral generation, a segment of the step attains a certain critical length and then moves with a constant velocity. Even under constant supersaturation, the fluctuations can give rise to changes in the segment length. The interstep distance in the step echelon also varies.

Energetics of stepped and kinked surfaces of Rh, Pd and Cu from electronic structure calculations

We apply the spd tight-binding models, that have been recently developed, to the calculations of the surface energies of several vicinal surfaces of Rh, Pd and Cu with (1 1 1), (1 0 0) and (1 1 0) terraces of increasing widths. From these results we extract the isolated step energies and the step–step electronic interactions. These interactions are most often decaying oscillatory functions of the interstep distance which are strongly dependent of the step geometry. Kink energies are also computed. Our results are in complete agreement with the existing experimental data, in particular on the equilibrium shapes of adislands, which are only available in the literature for Cu. Finally the electronic structure of the vicinal surfaces is also discussed.

Experimental evidence for an Ehrlich-Schwoebel effect on Si(111)

We have used reflection electron microscopy to study (1) the drift velocity of positive monolayer islands on Si(111) under near equilibrium conditions and direct current (dc) heating and (2) the difference between the velocity of monoatomic steps heated by direct current in the step-up and step-down direction under free sublimation. Our results obtained at around $1200\ifmmode^\circ\else\textdegree\fi{}\mathrm{C}$ show that the islands move in the opposite direction of the dc and the step velocity for ascending steps with respect to the dc direction is smaller than the velocity for opposite steps with the same interstep distance. These two experimental results are a clear qualitative prove of the existence of an Ehrlich-Schwoebel effect on Si(111). We analyze the experimental results for the step dynamics within a modified Burton, Cabrera, and Frank model. We find that a quantitative agreement between the model and the experimental results needs an enhanced effect of the electric field on the diffusion species.

Transient behaviour of surface supersaturation caused by an abrupt change in bulk supersaturation

The transient behaviour of the surface supersaturation, caused by an abrupt (stepped) change in bulk supersaturation from one constant value σ 0 to another σ 1, is studied with the use of the surface diffusion theory of Burton, Cabrera and Frank (BCF). The presented analytical solution of the time-dependent BCF equation takes into account a phenomenological coefficient Λ s describing the kinetics of growth unit exchange between the step edge and the adsorbed layer. It is shown that the solution can be approximated by a simple expression, which enables an easy estimate of the time t s required to attain the steady-state value of the surface supersaturation. The time t s is independent of the difference between values σ 0 and σ 1 but it increases with increasing coefficient Λ s or increasing interstep distance. On the other hand, the average rate of change in the surface supersaturation during the transient state is determined neither by the coefficient Λ s nor by the interstep distance but it is proportional to the difference between values of σ 0 and σ 1 of bulk supersaturation.