Burton , Cabrera And Frank Research Articles

Revisiting the Meandering Instability During Step-Flow Epitaxy

This paper starts with a generalized Burton, Cabrera and Frank (BCF) model by considering the energetic contribution of the adjacent terraces to the step chemical potential. We use the linear stability analysis of the quasistatic free-boundary problem for a two-dimensional step separated by broad terraces to study the step-meandering instabilities. The results show that the equilibrium adatom coverage has influence on the morphological instabilities.

Step density waves on growing vicinal crystal surfaces – Theory and experiment

The Burton, Cabrera and Frank (BCF) theory plays a key conceptual role in understanding and modeling the crystal growth of vicinal surfaces. In BCF theory the adatom concentration on a vicinal surface obeys to a diffusion equation, generally solved within quasi-static approximation where the adatom concentration at a given distance x from a step has a steady state value n(x). Recently, we show that going beyond this approximation (Ranguelov and Stoyanov, 2007) [6], for fast surface diffusion and slow attachment/detachment kinetics of adatoms at the steps, a train of fast-moving steps is unstable against the formation of steps density waves. More precisely, the step density waves are generated if the step velocity exceeds a critical value related to the strength of the step–step repulsion. This theoretical treatment corresponds to the case when the time to reach a steady state concentration of adatoms on a given terrace is comparable to the time for a non-negligible change of the step configuration leading to a terrace adatom concentration n(x,t) that depends not only on the terrace width, but also on its “past width”. This formation of step density waves originates from the high velocity of step motion and has nothing to do with usual kinetic instabilities of step bunching induced by Ehrlich–Schwoebel effect, surface electromigration and/or the impact of impurities on the step rate. The so-predicted formation of step density waves is illustrated by numerical integration of the equations for step motion. In order to complete our previous theoretical treatment of the non-stationary BCF problem, we perform an in-situ reflection electron microscopy experiment at specific temperature interval and direction of the heating current, in which, for the first time, the step density waves instability is evidenced on Si(111) surface during highest possible Si adatoms deposition rates.

Utilization of Anodized Aluminum Oxide Substrate for the Growth of ZnO Microcrystals on Polygonized Spirals

Anodized Aluminum Oxide (AAO) has been utilized as a substrate for the screw dislocation assisted growth of polygonize spirals (PS) of ZnO with diameter of the order of ~230 μm by Chemical Vapour Deposition (CVD) process. Stoichiometric ZnO microcrystals nucleated on the terraces and tops of these polygonized spirals. Stress inherent in the ZnO polygonized spiral morphology (~ 3.57 GPa) was deciphered from the values of the magnitude of shift in observed 2θ values of Glancing Incidence angle XRD (GIXRD) peaks from the standard values (JCPDS 36-1451) for hexagonal Zincite. The growth mechanism of these PS was explained albeit to a limited extent on the basis of the Burton, Cabrera and Frank (BCF) theory and its later modification, wherein data obtained from exsitu SEM measurements concomitant with numerical analysis was utilized to decipher values of the critical radius and supersaturation ratios. Nucleation of ZnO microcrystals on the PS was explained on the basis of the supersaturation ratio and the plausible values of diffusion lengths, existent on the summits of these PS. Retardation of the step rotation of the PS, due to elastic stress around the dislocation source and the Gibbs–Thomson effect, was explained on the basis of numerical coefficient ω0, the dimensionless frequency of spiral rotation. Role of stress in inhibition of ZnO nucleation on PS of smaller heights and with larger supersaturation ratio, has been discussed albeit qualitatively. The optical characteristics of a single ZnO microcrystal has been analyzed by room temperature CL measurements in the wavelength range 350 nm to 650 nm, revealing a single high intensity peak at 382 nm corresponding to a excitonic bandgap of 3.25 eV.

From crystal steps to continuum laws: Behavior near large facets in one dimension

The passage from discrete schemes for surface line defects (steps) to nonlinear macroscopic laws for crystals is studied via formal asymptotics in one space dimension. Our goal is to illustrate by explicit computations the emergence from step motion laws of continuum-scale power series expansions for the slope near the edges of large, flat surface regions (facets). We consider surface diffusion kinetics via the Burton, Cabrera and Frank (BCF) model by which adsorbed atoms diffuse on terraces and attach–detach at steps. Nearest-neighbor step interactions are included. The setting is a monotone train of N steps separating two semi-infinite facets at fixed heights. We show how boundary conditions for the continuum slope and flux, and expansions in the height variable near facets, may emerge from the algebraic structure of discrete schemes as N → ∞ . Our technique relies on the use of self-similar discrete slopes, conversion of discrete schemes to sum equations, and their reduction to nonlinear integral equations for the continuum-scale slope. Approximate solutions to the continuum equations near facet edges are constructed formally by direct iterations. For elastic-dipole and multipole step interactions, the continuum slope is found in agreement with a previous hypothesis of ‘local equilibrium’.

Kinetics of gypsum crystal growth from high ionic strength solutions: A case study of Dead Sea – seawater mixtures

Gypsum precipitation kinetics were examined from a wide range of chemical compositions ( 11 < Ca 2 + / SO 4 2 - < 115 ) , ionic strengths (4.75–10 m) and saturation state with respect to gypsum (1.16–1.74) in seeded batch experiments of mixtures of Ca 2+-rich Dead Sea brine and SO 4 2 - -rich seawater. Despite the variability in the experimental solutions, a single general rate law was formulated to describe the heterogeneous precipitation rate of gypsum from these mixtures: Rate het = k 1 · ( Ω 0.5 - 1 ) 10 + k 2 · ( Ω 0.5 - 1 ) 2 mol m - 2 s - 1 , where k 1 and k 2 are heterogeneous rate coefficients (mol s −1 m −2) that vary as a function of the solution compositions, and is the saturation state with respect to gypsum. It is suggested that two parallel mechanisms control the heterogeneous precipitation rate. Under closer-to-equilibrium conditions, the reaction is dominated by a mechanism best described as a 2nd order reaction with respect to Ω 0.5 − 1, which fits to the predictions of both the Burton Cabrera and Frank (BCF) crystal growth theory ( Burton et al., 1951) and other layer-by-layer growth mechanisms ( Goto and Ridge, 1967; Van Rosmalen et al., 1981; Bosbach and Rammensee, 1994). Under further-away-from-equilibrium conditions, the reaction is dominated by an apparent 10th order reaction. A conceptual model for gypsum growth kinetics is presented. The model is based on the 2nd order kinetic coefficients determined in the present study and data from the literature and is valid under a wide range of ionic strengths and Ca 2 + / SO 4 2 - ratios. According to this model, the integration of SO 4 2 - to kinks on the surface of the growing crystals is the rate-limiting step in the precipitation reaction. At ionic strengths above 8.5 m the precipitation rate of gypsum is enhanced, possibly due to the formation of CaSO 4 ° ion pairs and/or a decrease in hydration frequencies.

Anisotropic diffusion in continuum relaxation of stepped crystal surfaces

We study the continuum limit in 2+1 dimensions of nanoscale anisotropic diffusion processes on crystal surfaces relaxing to become flat below roughening. Our main result is a continuum law for the surface flux in terms of a new continuum-scale tensor mobility. The starting point is the Burton, Cabrera and Frank (BCF) theory, which offers a discrete scheme for atomic steps whose motion drives surface evolution. Our derivation is based on the separation of local space variables into fast and slow. The model includes: (i) anisotropic diffusion of adsorbed atoms (adatoms) on terraces separating steps; (ii) diffusion of atoms along step edges; and (iii) attachment–detachment of atoms at step edges. We derive a parabolic fourth-order nonlinear partial differential equation (PDE) for the continuum surface height profile. An ingredient of this PDE is the surface mobility for the adatom flux, which is a nontrivial extension of the tensor mobility for isotropic terrace diffusion derived previously by Margetis and Kohn (2006 Multisci. Model. Simul. 5 729–58). Approximate, separable solutions of the PDE are discussed.

Arrays of widely spaced atomic steps on Si(1 1 1) mesas due to sublimation

Steps with spacings of microns form on top of mesas fabricated on Si(1 1 1) that is annealed at temperatures where sublimation becomes important. Upon annealing, mesas first develop ridges along their edges, effectively creating craters which then become step-free by a step flow process described in the literature [S. Tanaka, C.C. Umbach, J.M. Blakely, R.M. Tromp, M. Mankos, Appl. Phys. Lett. 69 (9) (1996) 1235; Y. Homma, N. Aizawa, T. Ogino, Jpn. J. Appl. Phys. 35 (2B) (1996) L241]. Due to the miscut of the average surface from (1 1 1), ridge breakdown occurs on one edge of each mesa as sublimation proceeds. The breakdown point then acts as a source of steps which spread out over the mesa surface. The distribution of steps in the resulting step train depends on the sublimation rate, direct step-step interaction and the diffusive exchange of atoms among the steps. Insight into the role of these processes on the self-organization of the wide terrace distributions is provided by computer simulations using BCF (Burton, Cabrera and Frank) theory. This shows that step spacing can be controlled by varying the annealing temperature and the deposition flux. Comparison of the experimental and predicted step distributions suggest that the dynamics of the widely spaced steps are sublimation limited.

Quartz precipitation kinetics at 180°C in NaCl solutions—Implications for the usability of the principle of detailed balancing

The kinetics of heterogeneous quartz precipitation were determined at 180°C and a pH near 4 as a function of the degree of super saturation and dissolved NaCl concentrations. Stirred batch reactors were used and the changes in H 4SiO 4 concentrations over time were used to model the reaction rates. As the change in concentration in the batch experiment is not linear with time during all sampling intervals, the precipitation rates were retrieved by fitting the experimental observations to a forward model, i.e., a model that uses a prescribed rate law and compares the results with the experiments. Quartz precipitation occurs via an overall reaction, and therefore one can not presuppose the applicability of the predictions of Transition State Theory (TST). Therefore, different functions describing the dependence of the rate on deviation from equilibrium were examined in the forward model. The obtained precipitation rates were within error the same, regardless of the function that was used. By substituting the prediction of TST into the forward model, the change of quartz precipitation rate (at 180°C, pH 4 and ionic strength of 0.1M) with deviation from equilibrium was found to be r a t e = 1.9 ± 0.4 ⋅ 10 − 8 [ 1 − exp ⁡ ( Δ G r R T ) ] mol m − 2 s − 1 Within error, this observed rate agreed with predictions of quartz precipitation rates, which were calculated using the principle of detailed balancing, far from equilibrium quartz dissolution rates of Dove (1994) and equilibrium constant from Kharaka et al. (1988). Similar agreement between observations and predictions was obtained using the BCF (Burton Cabrera and Frank) crystal growth theory (Burton et al., 1951). In contrast to TST and BCF, a more general form of the rate dependence of deviation from equilibrium failed to predict the observed precipitation rates. Based on the observation that the principle of detailed balancing was verified for quartz dissolution and precipitation at 180°C and pH 4, we suggest that the principle of detailed balancing could be fruitfully applied to the huge database of quartz dissolution rates, together with appropriate thermodynamic data, for modeling quartz precipitation under natural conditions.

Accelerated convergence in numerical simulations of surface supersaturation for crystal growth in solution under steady-state conditions

This is an investigative paper which reports the results of comparisons of two numerical techniques for the solution of the Burton Cabrera and Frank (BCF) equation for the growth on crystal surfaces under steady state conditions. A successive over-relaxation (SOR) scheme for the equivalent finite difference equation gives rapid convergence to the static solution. It is known that a suitable choice of scattering parameters in a transmission line matrix (TLM) network analogue of the Laplace equation yields ultra-fast convergence. The results of numerical experiments which are reported here suggests that a similar situation also applies to the solution of the Poisson equation with shunt losses (the BCF equation), although the choice of optimum conditions appears to be different for different spatial positions within the solution space. Copyright © 2005 John Wiley & Sons, Ltd.

Spontaneous Formation of Ridges on Patterned Mesas and Their Role in the Evolution of

ABSTRACTMesa structures fabricated on Si(111) surfaces have been found experimentally to develop step arrays with large spacing of the order of a micron or more after annealing at temperatures where sublimation becomes important. Ridges around the edges initially develop during annealing and form barriers to step motion before eventually breaking down. This produces an array of steps of the same sign with a few wide terraces. Computer simulations using one dimensional Burton, Cabrera and Frank (BCF) theory including attachment-detachment rates and step-step repulsion for this configuration show that the terraces evolve under different dynamics depending on the terrace widths. For large terrace widths, sublimation dominates the step dynamics and the Ehrlich-Schwoebel effect is negligible. Sinusoidal terrace width distributions result in this case. The experimentally measured step distribution has such a sinusoidal shape suggesting that the step dynamics is sublimation dominated on the mesas after ridge breakdown.

Spontaneous Formation of Ridges on Patterned Mesas and Their Role in the Evolution of Step Arrays

AbstractMesa structures fabricated on Si(111) surfaces have been found experimentally to develop step arrays with large spacing of the order of a micron or more after annealing at temperatures where sublimation becomes important. Ridges around the edges initially develop during annealing and form barriers to step motion before eventually breaking down. This produces an array of steps of the same sign with a few wide terraces. Computer simulations using one dimensional Burton, Cabrera and Frank (BCF) theory including attachment-detachment rates and step-step repulsion for this configuration show that the terraces evolve under different dynamics depending on the terrace widths. For large terrace widths, sublimation dominates the step dynamics and the Ehrlich-Schwoebel effect is negligible. Sinusoidal terrace width distributions result in this case. The experimentally measured step distribution has such a sinusoidal shape suggesting that the step dynamics is sublimation dominated on the mesas after ridge breakdown.

High temperature Si(0 0 1) surface defect evolution during extended annealing: experimental results and modelling

Square surface defects evolving in a triangular shape were observed on Si(0 0 1) as the consequence of long annealing step at very high temperature and in slightly oxidizing conditions. The evolution of the size and shape of these defects were studied as a function of time, of surface misorientation and of oxygen/neutral gas ratio. The association of the Burton Cabrera and Frank (BCF) model developed for ultra high vacuum conditions with the Deal and Groves (DG) law for oxidation allows to explain the influence of the studied parameters on the defect evolution. This shows that the oxidation step proceeds with the help of diffusion along the interface rather than with a localized reaction, at least in the conditions of the experiment.

Constant composition kinetics study of carbonated apatite dissolution

The carbonated apatites (CAP) may be more suitable models for biominerals such as bone and dental hard tissues than is pure hydroxyapatite (HAP) since they have similar chemical compositions. Although they contain only a relatively small amount of carbonate, the solubility and dissolution properties are different. The solubility product of the CAP particles used in this dissolution study, 2.88×10 −112 mol 18 l −18, was significantly greater than that of HAP, 5.52×10 −118 mol 18 l −18. The kinetics of dissolution of CAP has been studied using the constant composition (CC) method. At low undersaturations, the dissolution reaction appeared to be controlled mainly by surface diffusion with an effective reaction order of 1.9±0.1 with respect to the relative undersaturation. These results together with those obtained by scanning electron microscopy (SEM) suggest a dissolution model. Based on the surface diffusion theory of Burton, Cabrera and Frank (BCF). The interfacial tension between CAP and the aqueous phase calculated from this dissolution model, 9.0 m J m −2, was consistent with its relatively low solubility. An abnormal but interesting dissolution behavior is that the CAP dissolution rate was relatively insensitive to changes in calcium and phosphate concentrations at higher undersaturations, suggesting the importance of the carbonate component under these conditions.

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.

The study of behaviour of surface supersaturation caused by variations in bulk supersaturation

Variations in the surface supersaturation σ s, caused by an abrupt (stepped) change in bulk supersaturation σ and subsequent exponential decrease or increase in σ, are studied with the use of the surface diffusion theory of Burton, Cabrera and Frank (BCF). The study is based on the analytical solution [M. Rak, Surf. Sci. 442 (1999) 149] of the BCF time-dependent equation. It is shown that the solution can be approximated by a simple expression convenient for the analysis. It is found that at the time t=0, the instantaneous rate of a change in the surface supersaturation σ s is determined only by the exchange of growth units between the crystal surface and the solution bulk, and is not affected by the surface diffusion. If an abrupt increase in bulk supersaturation σ at t=0 is followed by an exponential decrease in σ, then at t M>0 the function σ s of t has a maximum. In the opposite case, i.e. if an abrupt decrease in σ at t=0 is followed by an exponential increase in σ, then σ s has a minimum at t M>0. The time t M is estimated and an effect of variations in σ on value of t M is analysed.

Transient behaviour of surface supersaturation caused by variation in bulk supersaturation

The time-dependent equation of Burton, Cabrera and Frank (BCF) is solved analytically for time-dependent bulk supersaturation tending exponentially to a constant value. As a result, the expression for the transient surface supersaturation is found. Special attention is focused on the particular case of the solution presented, corresponding to an abrupt change in bulk supersaturation from σ o to another constant value σ 1. It is shown that, in this particular case, the solution of the time-dependent BCF equation can be approximated by a simple expression which is accurate for large distances from the step edge. The study of the transient behaviour of surface supersaturation reveals that the time necessary to attain the steady-state value of the surface supersaturation does not depend on the difference between the values σ o and σ 1 of the bulk supersaturation. This time is proportional to the relaxation time for leaving the surface adsorption layer, and increases with increasing interstep distance.

Transient analysis of surface supersaturation after crystal face submersion using the analytical and transmission-line matrix (TLM) methods

The time-dependent equation of Burton, Cabrera and Frank (BCF) is analytically solved for transient-state growth conditions occurring immediately after submersion of crystal face in supersaturated solution. The time-dependent BCF model is also solved using the transmission-line matrix (TLM) numerical method. We find that the time necessary to attain the steady-state value of surface supersaturation does not depend on bulk supersaturation but increases with increasing interstep distance. Calculated results obtained by the two methods are shown to be in excellent agreement, indicating a promising potential of the TLM method for solving analytically intractable problems of interest.

Step Structure Transformation of Si(001) Surface Induced by Current

The domain conversion mechanism on a Si(001) surface induced by DC current is investigated, and the origin of the critical step spacing is clarified. The conversion between 2×1 and 1×2 domains on the Si(001) surface proceeds with the movement of each step, caused by the transport of adatoms across the terrace. The uniform driving force on adatoms is assumed to be induced by the supply of DC current. The step kinetics is treated by developing the BCF (Burton, Cabrera and Frank) theory to take account of the three effects: contribution of the drift flux of adatoms, anisotropy of a diffusion constant, and repulsive interaction between steps. The domain conversion is induced by the coupling of the first and second effects. The balance condition between the drift flux and the backward diffusion flux caused by the repulsive step-step interaction determines the critical spacing. The time evolution of step configuration is calculated numerically, and it explains well the observed behaviour.

Monte Carlo simulation study on the dissolution of a train of infinitely straight steps and of an infinitely straight crystal edge

The dissolution of a train of infinitely straight steps and of an infinitely straight crystal edge was studied over a wide range of undersaturations by Monte Carlo simulation using the solid-on-solid (SOS) Kossel model for the SC(100) solid/fluid interface. The simulation was carried out below the roughening temperature to ensure that step movement is involved. Surface diffusion was not included in order to minimise computing time. Both the movement of the infinite steps and of the steps produced from layer opening by means of nucleation were apparent in the case of step trains at high undersaturations. For this reason, dissolution involving only pure step movement was also studied with nucleation excluded. A linear dependence of rate on undersaturation in terms of concentration ( C/ C eq) was found for dissolution involving solely step train movement and for the dissolving crystal edge. In the dissolution of the step train, the rate was also found to be proportional to the step density. This rate equation is in agreement with theoretical predictions with combined volume and surface diffusion. The involvement of nucleation at large undersaturations in the step train movement case resulted in enhancement of the rate, and deviation of the linearity of the rate equation. However, the extent of nucleation and hence its influence on the dissolution rate were found to decrease with increasing step density. Nucleation was not apparent in the dissolution of the infinite edge at all undersaturations. At the steady state, both the separation between steps in the step trans fluctuate slightly. In the case of high step density, steps were found to infrequently catch up with preceding ones which results in bunching. Similarly, instead of producing steps with equidistant spacing as assumed in a previous theoretical studies, the dissolution of the crystal edge resulted in a persistent bunching of steps with the exception of the region near the boundary of which the leading step and with its image and annihilated. Such roughness attributable to the dissolving edge, however, is not consistent with the kinetic roughening proposed earlier because nucleation was not involved in the dissolution. The mean density of kinks along the steps on the entire surface and for leading steps were almost constant at all undersaturations. Both measurements of kink density, though unequal when the step densities gets larger, indicated agreement with the assumption of constant density of kinks made by Burton, Cabrera and Frank (BCF) and not the proportion to ( C/ C eq) 1 2 , as suggested recently by Nielsen.

A magnetoresistance probe for magnetic fields at helium temperatures using n-type indium antimonide

Sixty-five years have passed since Burton, Cabrera and Frank (BCF) published the seminal paper (W. K. Burton, N. Cabrera and F. C. Frank, Phil. Trans. Royal Soc. London, 243 (1951), 299). Since then, the paper provided us the basic scheme of growth of crystals. In this lecture, the BCF theory is introduced for beginners as the basis of modern crystal growth study. The BCF theory explained the growth of facets with the help of screw dislocations. It introduced the concept of the roughening transition, which distinguishes the crucial difference of lateral growth of facets and normal growth of round surfaces.