Rigorous derivation of the rate equations for epitaxial growth

Tokar, V. I.

doi:10.1088/1742-5468/2013/06/p06001

Cited by 6 publications

(12 citation statements)

References 46 publications

(222 reference statements)

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“…The formulation of rate equations for particular systems is based typically on phenomenological considerations, though with input from experiment and first-principles calculations for complex systems. Although direct connections between rate equations and KMC simulations can be made [9][10][11], and derivations of rate equations from the master equation are available [12], the rates of processes that are not directly accessible from experiment are typically assigned Arrhenius rates in which the frequency prefactor and the energy barrier are regarded as fitting parameters. This effectively assumes the validity of transition-state theory.…”

Section: Introductionmentioning

confidence: 99%

Optimization algorithm for rate equations with an application to epitaxial graphene

Boer

Ford

Kantorovich

et al. 2014

J. Phys.: Condens. Matter

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We describe an algorithm that searches the parameter space of rate theories to optimize the associated rate coefficients based on a fit to experimental (or any other) data. Beginning with an initial set of parameters, which may be estimated, partially calculated, or indeed random, the algorithm follows a path, calculating the error at each point, until a minimum error is reached. We illustrate our method by correcting a previously proposed rate theory for the nucleation and growth of graphene on Ru(0 0 0 1) and Ir(1 1 1) to account for the temperature dependence of the graphene island density. This quantity shows an exponential decrease as the temperature is raised, in contrast to the power law decrease predicted by conventional nucleation theory, which indicates that a qualitatively different mechanism is operative for graphene island formation. Other applications of our method are also discussed.

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Section: Introductionmentioning

confidence: 99%

Optimization algorithm for rate equations with an application to epitaxial graphene

Boer

Ford

Kantorovich

et al. 2014

J. Phys.: Condens. Matter

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“…In this section we introduce the 1D point-island model of epitaxial growth in the framework of the SQ formalism that will be used in our calculations below. A more detailed explanation of the model can be found in [1,2,4] and of the SQ formalism in [10,[15][16][17][18][19] where further references to the literature on the subject can be found.…”

Section: The Model and The Second Quantization Formalismmentioning

confidence: 99%

“…In the present paper we aim to suggest ways to alleviate these difficulties. To this end, in section 2 we extend the rigorous second quantization (SQ) approach developed in [10] for the island densities of extended islands to the densities of the interisland gaps during the growth in 1D PIM. The latter is widely used in theoretical studies of epitaxial growth due to its simplicity [4,6,8,9,11,12].…”

Section: Introductionmentioning

confidence: 99%

“…This description implies that we are going to discuss only the so-called i = 1 case when two monomers are sufficient to nucleate a stable island [1,2]. The SQ formalism, however, is very flexible so that with minor modifications other values of i as well as extended islands can also be described (see [10,[15][16][17][18][19] and references therein). In section 5 we suggest two ways of improving some deficiencies of the MF solution.…”

Section: Introductionmentioning

confidence: 99%

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Rigorous approach to fragmentation equation for irreversible epitaxial growth in the one-dimensional point island model

Tokar¹

2017

J. Phys. A: Math. Theor.

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Rigorous second quantization formalism has been used to derive a fragmentation equation (FE) for the one-dimensional (1D) epitaxial growth. For simplicity, the point island model (PIM) and the case have been studied. In the continuum approximation and a mean-field decoupling the exact FE reduces to that suggested in the literature but this time with rigorous formal definitions of its terms. Furthermore, a correction factor accounting for the discrete nature of the monomers that was missing in known FEs for the 1D epitaxy has been included in the formalism. A novel technique for a numerical solution of the FEs has been suggested and illustrated in the problem of the growth following the instant deposition of the monomers. Good agreement with our earlier Monte Carlo simulations of the growth was found. In the calculations a heuristic ansatz for the fragmentation kernel appropriate to the instant deposition case has been used but the technique is also suitable for the solution of the FEs corresponding to the more conventional case of continuous monomer deposition.

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“…Equation ( 6) allows us to understand this. According to [35] the island capture numbers in the rate equations that define the rate at which the islands capture the mobile adatoms is proportional to the density of the adatoms at the sites situated one hopping step away from the island. They may be called the island nearest neighbors (NN).…”

Section: The Dipole-dipole Interactionmentioning

confidence: 99%

Size calibration of strained epitaxial islands due to dipole–monopole interaction

Tokar¹

2015

J. Stat. Mech.

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Abstract. Irreversible growth of strained epitaxial nanoislands has been studied with the use of the kinetic Monte Carlo (KMC) technique. It has been shown that the strain-inducing size misfit between the substrate and the overlayer produces long range dipole-monopole (d-m) interaction between the mobile adatoms and the islands. To simplify the account of the long range interactions in the KMC simulations, use has been made of a modified square island model. Analytic formula for the interaction between the point surface monopole and the dipole forces has been derived and used to obtain a simple expression for the interaction between the mobile adatom and the rectangular island. The d-m interaction was found to be longer ranged than the conventional dipole-dipole potential. The narrowing of the island size distributions (ISDs) observed in the simulations was shown to be a consequence of a weaker repulsion of adatoms from small islands than from large ones which led to the preferential growth of the former. Furthermore, similarly to the unstrained case, the power-law behavior of the average island size and of the island density on the coverage has been found. In contrast to the unstrained case, the value of the scaling exponent was not universal but strongly dependent on the strength of the long range interactions. Qualitative agreement of the simulation results with some previously unexplained behaviors of experimental ISDs in the growth of semiconductor quantum dots was observed.

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Rigorous derivation of the rate equations for epitaxial growth

Cited by 6 publications

References 46 publications

Optimization algorithm for rate equations with an application to epitaxial graphene

Optimization algorithm for rate equations with an application to epitaxial graphene

Rigorous approach to fragmentation equation for irreversible epitaxial growth in the one-dimensional point island model

Size calibration of strained epitaxial islands due to dipole–monopole interaction

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