Burton-Cabrera-Frank Research Articles

Influencing the morphological stability of MOVPE-grown β-Ga2O3 films by O2/Ga ratio

The morphology evolution of (100) β-Ga2O3 films grown by metalorganic vapor phase epitaxy (MOVPE) is studied by atomic force microscopy (AFM). In particular, when grown under a high O2/Ga ratio (O2/Ga = 1250) and above a thickness of 350 nm, these films exhibit striking morphological instabilities, including step meandering and bunching, which contribute significantly to surface roughening. Transmission electronic microscopy (TEM) measurements reveal a morphological transition on the growing surface with a coexistence of step-flow and step-bunching growth modes resulting from these instabilities. In contrast, growth conditions with low O2/Ga ratios (O2/Ga = 350) effectively suppress meandering and bunching instabilities, resulting in films with excellent electrical properties. These observations are rationalized through the Burton-Cabrera-Frank (BCF) theory and Bales-Zangwill (BZ) instability (Gibbs-Thompson effect and Mullins-Sekerka instability) , thus bridging theoretical models and the experimental results.

Estimation of supersaturation at steps during chemical vapor deposition of 4H-SiC () from reported growth rate and cross-sectional profile of spiral hillock

Based on the Burton–Cabrera–Frank (BCF) and Cabrera–Levine theories, supersaturation at steps (σ step) during chemical vapor deposition of 4H-SiC (0001̄) at 1570 °C under the condition of C/Si ratio of unity was estimated from the reported growth rate and cross-sectional profile of a spiral hillock, as a function of surface diffusion length of C2H2 molecules (λC2H2). The previously assumed zero σ step in the BCF theory was found to be a possible source of error in the case λC2H2 > 0.63 μm because of the estimated ratio of σ step to the surface supersaturation exceeding 0.1.

Reflections on the effect of an external flux in surface physics

Sublimating surfaces are out of equilibrium. It has been proposed that sublimation can be compensated by an impinging atomic flux to obtain equilibrium. This work concerns the effect of such an impinging flux on the stability of surfaces in various situations. For this purpose we combine Kinetic Monte Carlo Simulations with analytical developments based on the Burton-Cabrera-Frank (BCF) classical theory. We show that a perfect compensation of the sublimation is possible for vicinal surfaces but not when 2D islands are present on a surface. We thus study the effect of an impinging flux on the dynamic of a 2D island on a surface. We show that the 2D island area generally varies with time t as −tα. In absence of any impinging flux the value of the exponent α enables to identify the main mechanism at work (diffusion limited or attachment-detachment limited). On the contrary, in presence of an impinging flux the value of the exponent α is not enough to identify the main mechanism limiting the area change.

Burton-Cabrera-Frank theory for surfaces with alternating step types

Burton-Cabrera-Frank (BCF) theory has proven to be a versatile framework to relate surface morphology and dynamics during crystal growth to the underlying mechanisms of adatom diffusion and attachment at steps. For an important class of crystal surfaces, including the basal planes of hexagonal close-packed and related systems, the steps in a sequence on a vicinal surface can exhibit properties that alternate from step to step. Here we develop BCF theory for such surfaces, relating observables such as alternating terrace widths as a function of growth conditions to the kinetic coefficients for adatom attachment at steps. We include the effects of step transparency and step-step repulsion. A general solution is obtained for the dynamics of the terrace widths, assuming quasi-steady-state adatom distributions on the terraces. An explicit simplified analytical solution is obtained under widely applicable approximations. From this we obtain expressions for the full-steady-state terrace fraction as a function of growth rate. Fits of the theoretical predictions to recent experimental determinations of the steady state and dynamics of terrace fractions on GaN (0001) surfaces during organometallic vapor phase epitaxy give values of the kinetic coefficients for this system. In Appendixes, we also connect a model for diffusion between kinks on steps to the model for diffusion between steps on terraces, which quantitatively relates step transparency to the kinetics of atom attachment at kinks, and consider limiting cases of diffusion-limited, attachment-limited, and mixed kinetics.

Population-balance study of protein crystal growth from solution using a hyperbolic rate law

Kinetic data for crystal size, recently reported for bulk protein crystal growth from solution, is shown to obey a hyperbolic rate law beginning at some critical time after the commencement of the growth run. Given this rate law, the linear growth rate is established and mass-balance methods are then used to derive analytic time dependent expressions for the nucleation rate, the supersaturation and the distribution function for crystal sizes. Our population-balance approach involves a truncated Fokker–Planck equation along with an integro-differential equation for mass conservation. This model is used to describe three cases of protein crystal growth from solution for which kinetic data has been reported: lysozyme, β–lactoglobulin and insulin. Additionally, we show that this empirical rate law for crystal size is approximately related to the Burton-Cabrera-Frank (BCF) and Kossel-Stranski (KS) growth rates. Further, insight into the so called two-step, process is obtained. For two of the growth cases studied here, two-step activity was experimentally confirmed. The hyperbolic tangent rate law most accurately describes growth beginning at some critical time when two-step activity begins to wane and the bulk crystal phase becomes the more dominate form in the solution. Therefore, there seems to be a transition between growth modes at the critical time. We propose here that two-step activity creates a diffusion limited growth situation in the solution for times less than the critical time. The kinetics of crystal size are estimated, during the early diffusion limited time period, using a known model for diffusion limited growth leading to the crystal radius being directly proportional to the square root of time. At the critical time, the two models meet and the kinetics of the crystal radius can thereafter be described using the hyperbolic rate law. Additionally, features of the hyperbolic growth rate model are used to develop a lever rule that can be used to predict the critical time when the transition between growth modes occurs.

Formulation of partition coefficient of impurities in a growing crystal based on the Burton-Cabrera-Frank model

This paper presents an extension of a classical model of the crystal growth, the Burton–Cabrera–Frank (BCF) model, to a system that includes impurity molecules and discusses the formulation of the amount of the impurity uptake into the crystal as a partition coefficient. The BCF model considers the following surface processes of host molecules: adsorption from a solution to a terrace, desorption from the terrace to the solution, surface diffusion on the terrace, incorporation at a step edge, and release from the step edge to the terrace. The above-mentioned processes have been formulated not only for the host molecules but also for the impurity molecules (Miura, 2020). In this paper, the author analytically solves the previous model and derives an analytic formula for the partition coefficient as a ratio of the uptake of the impurity molecule to that of the host molecule into the crystal. The dependence of the partition coefficient on supersaturation is systematically investigated. The analytic formula is applied to the data obtained in the experiments of protein crystal growth to discuss the behavior of impurity molecules on the growing crystal surface. This model provides a theoretical framework for eliciting quantitative data from the crystal growth experiments on impurity partitioning.

Phase-field modelling of 2D island growth morphology in chemical vapor deposition.

The rich island morphology of two-dimensional (2D) materials during chemical vapor deposition (CVD) growth process is studied using a computational model based on a Burton-Cabrera-Frank (BCF) type crystal growth theory. A previously formulated phase-field (PF) model for the BCF crystal growth process is employed to investigate the effect of various growth conditions, such as the concentration of absorbed atoms on the substrate and the growth temperature, that have been experimentally known to significantly impact the island morphology. It is shown that, within this simple model, the rich morphology of 2D islands in CVD growth can be well reproduced. With increasing substrate temperature, the 2D island changes from dendritic to compact shape. When considering the energy difference between the zigzag and the armchair edges of the 2D island, most commonly known morphologies, from quasi-sixfold compact islands to spiky triangular and compact triangular shapes, are observed in the model. Growth mechanisms associated with different island shapes and potential model improvements are also discussed.

Kink Distance and Binding Energy of Colloidal Crystals

Particle interaction is a critical parameter for growth kinetics of colloidal crystals. The equilibrium concentration and step free energy are strongly dependent on the particle interaction. The growth mechanism of the attractive system of colloidal crystals is similar to that of vapor or solution growth, where crystals grow by incorporating diffusing ad-particles on the terrace into kinks of steps (Terrace-Step-Kink model). Here, we have applied the theory of crystal growth to experimentally determine the binding energy that originates from the particle interaction. We focus on the relationship between kink distance and bond energy, which is described by BCF (Burton-Cabrera-Frank) theory. The value of the step free energy determined by the kink distance measurement agrees well with the values obtained from other measurements, including nucleation rate and critical radius. The obtained value also accounts well for the change in step velocity of two-dimensional islands, which is due to the Gibbs–Thomson ef...

STERIC HINDRANCE OF CRYSTAL GROWTH: NONLINEAR STEP FLOW IN 1+1 DIMENSIONS.

By linking atomistic and mesoscopic scales, we formally show how a local steric effect can hinder crystal growth and lead to a buildup of adsorbed atoms (adatoms) on a supersaturated, (1+1)-dimensional surface. Starting from a many-adatom master equation of a kinetic restricted solid-on-solid (KRSOS) model with external material deposition, we heuristically extract a coarse-grained, mesoscale description that defines the motion of a line defect (i.e., a step) in terms of statistical averages over KRSOS microstates. Near thermodynamic equilibrium, we use error estimates to show that this mesoscale picture can deviate from the standard Burton-Cabrera-Frank (BCF) step flow model in which the adatom flux at step edges is linear in the adatom supersaturation. This deviation is caused by the accumulation of adatoms near the step, which block one another from being incorporated into the crystal lattice. In the mesoscale picture, this deviation manifests as a significant contribution from many-adatom microstates to the corresponding statistical averages. We carry out kinetic Monte Carlo simulations to numerically demonstrate how certain parameters control the aforementioned deviation. From these results, we discuss empirical corrections to the BCF model that amount to a nonlinear relation for the adatom flux at the step. We also discuss how this work could be used to understand the kinetic interplay between accumulation of adatoms and step motion in recent experiments of ice surfaces.

Continuum Limit of a Mesoscopic Model with Elasticity of Step Motion on Vicinal Surfaces

This work considers the rigorous derivation of continuum models of step motion starting from a mesoscopic Burton-Cabrera-Frank (BCF) type model following the work [Xiang, SIAM J. Appl. Math. 2002]. We prove that as the lattice parameter goes to zero, for a finite time interval, a modified discrete model converges to the strong solution of the limiting PDE with first order convergence rate.

Morphology and Growth of Methyl Stearate as a Function of Crystallization Environment

In situ studies of methyl stearate growing from supersaturated n-dodecane, kerosene, and toluene solutions reveal strong evidence that solvent choice influences the crystal morphology and crystal growth kinetics. Crystals with similar habit are observed in all solvents, with the exception of lower supersaturations in kerosene, where a less symmetric morphology was observed. BFDH analysis based on the monoclinic C2 crystal structure of methyl stearate yielded the morphological indexation to be (110), (1–10), (−110), and (−1–10) for the dominant observed habit and (110) (1–10) (−1–10) (−240) (−3–10) for the less symmetric habit observed in kerosene solvent. Measurements of the growth rate for the (110) and (1–10) faces are similar for all solutions ranging from 0.02 to 1.13 μm/s, for significantly lower values of supersaturation in the case of toluene. The tendency of the growth rate dependence on σ was consistent with the Burton-Cabrera-Frank (BCF) growth mechanism in n-dodecane, the Birth and Spread (B&S)...

Refined BCF-type boundary conditions for mesoscale surface step dynamics

Deposition on a vicinal surface with alternating rough and smooth steps is described by a solid-on-solid model with anisotropic interactions. Kinetic Monte Carlo (KMC) simulations of the model reveal step pairing in the absence of any additional step attachment barriers. We explore the description of this behavior within an analytic Burton-Cabrera-Frank (BCF)-type step dynamics treatment. Without attachment barriers, conventional kinetic coefficients for the rough and smooth steps are identical, as are the predicted step velocities for a vicinal surface with equal terrace widths. However, we determine refined kinetic coefficients from a two-dimensional discrete deposition-diffusion equation formalism which accounts for step structure. These coefficients are generally higher for rough steps than for smooth steps, reflecting a higher propensity for capture of diffusing terrace adatoms due to a higher kink density. Such refined coefficients also depend on the local environment of the step and can even become negative (corresponding to net detachment despite an excess adatom density) for a smooth step in close proximity to a rough step. Our key observation is that incorporation of these refined kinetic coefficients into a BCF-type step dynamics treatment recovers quantitatively the mesoscale step-pairing behavior observed in the KMC simulations.

The Blue LED Nobel Prize: Historical context, current scientific understanding, human benefit

The Blue LED Nobel Prize: Historical context, current scientific understanding, human benefit

Emergence of step flow from an atomistic scheme of epitaxial growth in 1+1 dimensions.

The Burton-Cabrera-Frank (BCF) model for the flow of line defects (steps) on crystal surfaces has offered useful insights into nanostructure evolution. This model has rested on phenomenological grounds. Our goal is to show via scaling arguments the emergence of the BCF theory for noninteracting steps from a stochastic atomistic scheme of a kinetic restricted solid-on-solid model in one spatial dimension. Our main assumptions are: adsorbed atoms (adatoms) form a dilute system, and elastic effects of the crystal lattice are absent. The step edge is treated as a front that propagates via probabilistic rules for atom attachment and detachment at the step. We formally derive a quasistatic step flow description by averaging out the stochastic scheme when terrace diffusion, adatom desorption, and deposition from above are present.

Kinetic modeling of multi‐component crystallization of industrial‐grade oils and fats

Transient crystallization kinetics is investigated for complex, industrial‐grade vegetable oils consisting of more than ten triacylglycerols (TAG). The classical nucleation model has been used to describe primary nucleation, while secondary nucleation has been described by a semi‐empirical approach. Growth is modeled using a modified Burton‐Cabrera‐Frank (BCF) model. Surface tensions and growth constants have been determined using focused‐beam‐reflectance measurements (FBRM). The required adjustable parameters in the model have been fitted to overall crystallization curves obtained by solid‐fat content (SFC) measurements for a given oil at different cooling rates and degrees of dilution. The developed model can accommodate more polymorphs simultaneously and performs well with respect to predicting crystallization onset, rate of crystallization and final SFC value. It can also qualitatively describe how higher cooling rates lead to formation of more meta‐stable crystals and smaller mean‐crystal sizes.Practical applications: The model provides a good starting point for developing more realistic, transient models for TAG crystallization with the ability to accommodate processing conditions and complex chemical compositions. Such a predictive model may provide a powerful tool to screen and optimize oil formulations in industrial processes and allow product developers to evaluate recrystallization events.An advanced transient, mathematical model describing the course of crystallization in natural, multi‐component, multi‐phase vegetable oils and fats has been developed. The model is based on well described nucleation and growth kinetics and accommodates the presence of more polymorphic crystal forms. The result is an industry‐relevant model aiding understanding and predictability of oil and fat crystallization with respect to processing conditions and chemical compositions.

Step Growth and Meandering in a Precursor-Mediated Epitaxy with Anisotropic Attachment Kinetics and Terrace Diffusion

Step meandering instability in a Burton-Cabrera-Frank (BCF)-type model for the growth of an isolated, atomically high step on a crystal surface is analyzed. It is assumed that the growth is sustained by the molecular precursors deposition on a terrace and their decomposition into atomic constituents; both processes are explicitly modeled. A strongly nonlinear evolution PDE for the shape of the step is derived in the long-wave limit and without assuming smallness of the amplitude; this equation may be transformed into a convective Cahn-Hilliard-type PDE for the step slope. Meandering is studied as a function of the precursors diffusivity and of the desorption rates of the precursors and adatoms. Several important features are identified, such as: the interrupted coarsening, “facet” bunching, and the lateral drift of the step perturbations (a traveling wave) when the terrace diffusion is anisotropic. The nonlinear drift introduces a disorder into the evolution of a step meander, which results in a pronounced oscillation of the step velocity, meander amplitude and lateral length scale in the steady-state that emerged after the coarsening was interrupted. The mean values of these characteristics are also strongly affected by the drift.

Crystal Morphology Prediction of Hexahydro-1,3,5-trinitro-1,3,5-triazine by the Spiral Growth Model

The crystal morphology of hexahydro-1,3,5-trinitro-1,3,5-triazine (RDX) was predicted by the advanced Burton–Cabrera–Frank (BCF) model with consideration of non-centrosymmetric growth units. The present modeling showed that the advanced BCF model provides reliable results and understanding of the growth habit of RDX crystals grown from acetone and γ-butyrolactone, in which the {210} and {111} faces are dominantly developed. In the present work, the kink rate, which is the net flux input and output from kink sites along the edge, is found to be a critical factor for the crystal growth of RDX, where the energy required for detaching the growth unit from a kink site determines the existence probability of a growth unit. The {210} and {111} faces were found to show slow kink rates originated from the kink sites with a low existence probability and consequently became morphologically important faces. Moreover, the effect of solvent on the crystal growth habit of RDX, the difference in morphology of RDX grown f...

From atoms to steps: The microscopic origins of crystal evolution

The Burton–Cabrera–Frank (BCF) theory of crystal growth has been successful in describing a wide range of phenomena in surface physics. Typical crystal surfaces are slightly misoriented with respect to a facet plane; thus, the BCF theory views such systems as composed of staircase-like structures of steps separating terraces. Adsorbed atoms (adatoms), which are represented by a continuous density, diffuse on terraces, and steps move by absorbing or emitting these adatoms. Here we shed light on the microscopic origins of the BCF theory by deriving a simple, one-dimensional (1D) version of the theory from an atomistic, kinetic restricted solid-on-solid (KRSOS) model without external material deposition. We define the time-dependent adatom density and step position as appropriate ensemble averages in the KRSOS model, thereby exposing the non-equilibrium statistical mechanics origins of the BCF theory. Our analysis reveals that the BCF theory is valid in a low adatom-density regime, much in the same way that an ideal gas approximation applies to dilute gasses. We find conditions under which the surface remains in a low-density regime and discuss the microscopic origin of corrections to the BCF model.

Connection of Kinetic Monte Carlo Model for Surfaces to One-Step Flow Theory in 1+1 Dimensions

The Burton--Cabrera--Frank (BCF) theory of step flow has been recognized as a valuable tool for describing nanoscale evolution of crystal surfaces. We formally derive a single-step BCF-type model from an atomistic, kinetic Monte Carlo (KMC) model of a crystal surface in 1+1 dimensions without external material deposition. Through an averaging procedure, consistent with Boltzmann statistics, we introduce definitions of the continuous quantities of the BCF theory, i.e., the step position and density of adsorbed atoms (adatoms), as expectation values taken over the discrete probabilities of the KMC model. The equations of our BCF-type model describe the time evolution of these expectation values. A central idea of our approach is to exploit the observation that the number of adatoms on a surface is typically small at experimentally relevant temperatures. Accordingly, we (i) show that our BCF-type theory arises from a KMC model in which only one adatom is allowed to move, and (ii) characterize corrections to ...

Phase-field model for reconstructed stepped surface

We formulate a phase-field, or diffuse-interface, model for the evolution of stepped surfaces under surface diffusion in the presence of distinct material parameters across nanoscale terraces. In the sharp-interface limit, our model reduces to a Burton-Cabrera-Frank (BCF)-type theory for the motion of noninteracting steps separating inhomogeneous terraces. This setting aims to capture features of reconstructed semiconductor, e.g., Si surfaces below the roughening transition. Our work forms an extension of the phase-field construction by Hu et al. [Physica D 241, 77 (2012)].