Burton Cabrera Frank Theory Research Articles

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.

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.

Growth Kinetics of the (110) Faces of Complex Potassium Cobalt–Nickel Sulphate K2CoxNi1−x(SO4)2·6H2O Crystals

The normal growth rate, the steepness of polygonized growth hillocks and the velocity of step movement on the (110) faces of potassium cobalt–nickel sulphate crystals in aqueous solutions with cobalt to nickel ratios of 1:1 and 1:2 were investigated as a function of supersaturation by the geometry of growth hillocks using laser interferometry. It was found that the morphologies of growth hillocks on the (110) faces of the crystals grown from 1:1 and 1:2 solutions are similar and that the growth hillocks are formed by multiple screw dislocation sources. The experimental data on the growth kinetics of the (110) faces of the crystals were analyzed by using the Burton–Cabrera–Frank theory. It was found that (1) there is a critical supersaturation for the growth of the (110) faces, and the value of this supersaturation in the 1:2 solution is higher than that in the 1:1 solution, and (2) the kinetic coefficient of the step movement in the sectors of growth hillocks is highly anisotropic, and the values of this coefficient are larger in 1:2 solution than in 1:1 solution. These results are discussed in the presented work.

Development of Burton–Cabrera–Frank Theory for the Growth of a Non-Kossel Crystal via Chemical Reaction

The growth of an arbitrary multicomponent crystal via the Burton–Cabrera–Frank mechanism due to chemical reaction is studied. The rates of advancement of a single step, a group of equidistant steps...

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...

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.

Influence of (7×7)–“1×1” phase transition on step-free area formation in molecular beam epitaxial growth of Si on Si (111)

The step-flow growth condition of Si on Si (111) near the (7×7)–“1×1” surface phase transition temperature TC is analysed within the framework of Burton–Cabrera–Frank theory. In particular, coexistence of both surface phases well below TC and their specific influence on the step-flow growth behaviour is considered. We presume that under dynamical condition of growth, the surface initially covered by only the (7×7) phase separates into domains surrounded by “1×1” areas. On such a surface, the overall supersaturation should be reduced drastically compared to a surface with only (7×7), resulting in much larger critical terrace width for nucleation.

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 ...

Atomic‐scale study of vapour growth morphology of crystalline urea

AbstractThe role of surface relaxation on habit controlling energetics and growth morphology are investigated within the framework of Burton‐Cabrera‐Frank (BCF) and Hartman‐Perdok (HP) models. The habit controlling energetics has been calculated using first principles method. The growth morphology obtained using BCF theory shows that the structural relaxation has considerable effect on growth morphology. The relaxed growth morphology obtained using BCF model match with the experimental result from vapour phase. On the other hand the shape obtained using HP model does not correspond with the experimental shape. Observed polar growth morphology of urea crystal has been discussed particularly in the context of different atomic environments of (111) and (111) faces. (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Kinetics of mass crystallization of calcium carbonate at 25, 30 and 37 °C

Calcium carbonate crystal growth kinetics was studied by simple mass crystallization, recording pH as function of time. The phase crystallizing was vaterite at 25°C, calcite at 30°C and aragonite at 37°C. Growth kinetics at the initial stage followed that of the polynuclear surface nucleation mechanism, and from the slopes of linearized graphs edge free energies of the three phases could be determined. The values were: calcite 35.4pJ/m at 30°C, aragonite 32.3pJ/m at 37°C and vaterite 24.2pJ/m at 25°C. These values agree reasonably well with literature values determined by other methods, and they are useful in testing the validity of the relation between step distance of a growth spiral and size of a critical surface nucleus as assumed in the Burton–Cabrera–Frank theory.

Finding polymorphic structures during vicinal surface growth

A hybrid scheme is developed to describe vicinal surface growth during epitaxy on two different time and length scales. For this purpose this algorithm combines two modules based on a continuum and an atomistic approach. The continuum module is realized by a phase-field-model which traces back to the Burton–Cabrera–Frank theory, the atomistic module is based on the anisotropic Ising model which is mapped onto a lattice-gas model. The latter provides thermal density fluctuations resulting in adatom clustering. With increasing temperature the probability for island nucleation on the terraces decreases according to 1-p where p is an Arrhenius-type activation probability which prevents clusters from becoming islands. Within this framework it is possible to find the transition from a rough surface at low temperatures to an evenly stepped surface at high temperatures where slight step meandering is observed. Furthermore two competing mechanisms of step bunching are investigated within this scale bridging algorithm: alternating anisotropic diffusion and different Ehrlich–Schwoebel barriers at the step edges. It is shown that a simulation of step bunching displaying the full variety of phenomena observed in experiments can only be achieved by the consideration of different time and length scales.

A Simple Model for the Formation of Step-Free Surfaces

ABSTRACTArrays of step-free regions on the surface of silicon have been created either by evaporating atoms from craters[1] or by depositing atoms on mesas[2]. In most cases the maximum extent of the step-free regions is limited by the occurrence of circular pits or islands in the crater or mesa structures. We model the process of step clearing and nucleation of these pits and islands by approximating the initial surface by an array of circular steps whose movement is mediated by adatoms. BCF (Burton-Cabrera-Frank) theory[3] is used to incorporate the effects of surface diffusion, evaporation and the deposition of atoms on the surface. We include the effects of step curvature and step interactions. If the step spacing is large enough, we find that the innermost step moves outwards to create a step-free region; otherwise it moves inward and leads to large scale smoothening of the surface. Pit or island nucleation in the center of the craters or mesas is also included in the model by using classical nucleation theory. We investigate the effect of deposition flux and temperature on the formation of step-free surfaces and compare the results to reported experiments on silicon and to some of our recent work on sapphire.

Misorientation effect on the monolayer terrace topography of (100) InP substrates annealed under a PH3/H2 ambient and homoepitaxial layers grown by metalorganic chemical vapor deposition

The detailed monolayer terracelike topography of nominal on-axis and misoriented vicinal (100) InP surfaces was systematically investigated by ‘‘ex situ’’ atomic force microscopy. Vicinal (100) InP reconstructs into steps around 550 °C in either argon or phosphine/hydrogen. Widths of observed monoatomic terraces are measured to be in close agreement with misorientation angles. Homoepitaxial growth by metalorganic chemical vapor deposition preserves the terracelike structure. Growth acts through a monoatomic step flow mode following the classical Burton–Cabrera–Frank theory. Surface diffusion length is estimated up to 350 nm in our standard growth conditions at 630 °C and 2.5 μm/h. Spiral growth is also observed and takes its origin from a screw dislocation emerging on the surface of the substrate. The critical misorientation angle below which the spiral growth mechanism occurs and competes with the vicinal steps is in the range of 0.05°–0.1°.

Spiral growth of GaAs by molecular beam epitaxy

Steps of monolayer height (0.28 nm) were observed by atomic force microscopy on a GaAs surface grown by molecular beam epitaxy. The monolayer terrace width between steps was found to be as large as 1000 nm in some areas. Surface reconstruction affects the surface diffusion process during growth and the shape of the step edges. Growth spirals were observed. Spirals originate from screw dislocations. The growth mechanism is according to the Burton–Cabrera–Frank theory.

Spiral growth of InP by metalorganic vapor phase epitaxy

Spirals were observed on InP grown by metalorganic vapor phase epitaxy. Atomic force microscopy is the technique used. The growth took place on a vicinal surface and the growth mechanism is according to the classical Burton–Cabrera–Frank theory. Spirals originate from screw dislocations. Successive turns of steps are sent out by the dislocations. These steps are generally of monolayer height (0.29 nm) except close to the dislocation emergence points where they are submonolayers. It is predicted that spiral growth will become the dominant mechanism if the vicinal steps are eliminated.

Spiral growth of GaAs by metalorganic vapor phase epitaxy

Growth spirals are observed on metalorganic vapor phase epitaxy grown GaAs surfaces by atomic force microscopy. The growth mechanism is according to the classical Burton–Cabrera–Frank theory. Spirals originate from screw dislocations. Successive turns of the steps are sent out by the dislocations. These steps are of monolayer height (0.28 nm) and the interstep distance is around 150 nm.

Spiral growth of GaAs by metalorganic vapor phase epitaxy

Growth spirals are observed on metalorganic vapor phase epitaxy grown GaAs surfaces by atomic force microscopy. The growth mechanism is according to the classical Burton–Cabrera–Frank theory. Spirals originate from screw dislocations. Successive turns of steps are sent out by the dislocations. These steps are of monolayer height (0.28 nm) and the interstep distance is around 150 nm. The spiral steps are well developed around the screw dislocations, while the adjacent vicinal steps lack the regularity of the spiral steps. Two-dimensional nucleation islands are also observed on the vicinal steps.

Surface morphology of metalorganic vapor phase epitaxy grown GaAs observed by atomic force microscopy

Steps of monolayer height (0.28 nm) are observed by atomic force microscope on a metalorganic vapor phase epitaxy grown GaAs surface. Monolayer terrace width was found to be as large as 430 nm, the same as the vicinal substrate surface. The growth mechanism is according to the classical Burton–Cabrera–Frank theory. We may have a larger (≳500 nm) terrace width and surface diffusion length if an exactly oriented [100] substrate is used.

The stability of rapidly growing or evaporating crystals

Solutions for the growth rate of perturbations in the locations of moving steps on a growing or evaporating crystal are presented. They are obtained by solving an equation derived by R. Ghez, H. G. Cohen, and J. B. Keller [J. Appl. Phys. 73, 3685 (1993)] based upon the Burton–Cabrera–Frank theory of crystal growth. They agree with the results derived via the adiabatic approximation when the dimensionless growth rate is small, which shows that those results are correct. However, when the growth rate is large the present exact results differ from those of the adiabatic approximation, as might be expected.