We consider a lattice gas model which in addition to the canonical nearest neighbor pair interato... more We consider a lattice gas model which in addition to the canonical nearest neighbor pair interatomic interaction accounts for a many-body interaction inside atomic trios. Interactions of this kind arise in the coherent strained epitaxy and were recently used by us to describe some surface phenomena. With the use of the Monte Carlo simulation we show that in two dimensions the model at low temperature exhibits glassy behaviour, in particular, undergoes a gelation transition. We argue that this model may belong to the universality class of non-equilibrium critical phenomena which may comprise also some off-lattice structural glass transitions, provided such a class exists.
The harmonic Frenkel-Kontorova model is used to illustrate with an exactly solvable example a gen... more The harmonic Frenkel-Kontorova model is used to illustrate with an exactly solvable example a general technique of mapping a coherently strained epitaxial system with continuous atomic displacements onto a lattice gas model (LGM) with only discrete variables. The misfit strain of the original model is transformed into cluster interatomic interactions of the LGM. The clusters are contiguous atomic chains of all lengths but the interaction strength for long chains is exponentially small. This makes possible the application of efficient Monte Carlo techniques developed for discrete variables both in kinetic and equilibrium simulations. The formalism developed can be applied to 1D as well as to 2D systems. As an illustrative example we consider the problem of self-organization of 1D size calibrated clusters on the steps of the vicinal surfaces.
ABSTRACT The scaling approach to the irreversible epitaxial growth gained wide recognition due to... more ABSTRACT The scaling approach to the irreversible epitaxial growth gained wide recognition due to its ability to describe with the use of a universal function the island size distributions (ISDs) corresponding to a broad range of experimental conditions. The approach, however, is operative only in the case of large average island sizes sav and large diffusion to deposition rates ratios R. Physically this corresponds to long deposition times and/or high temperatures. We argue that the ISDs exhibit yet another universality property which holds for much broader range of growth conditions, in particular, for low temperatures (small R) and small sav (short deposition times). We show that the normalized ISDs corresponding to the same sav are accurately described by the same universal distribution.
Journal of Statistical Mechanics: Theory and Experiment, 2010
The Wako-Saitô-Muñoz-Eaton (WSME) model, initially introduced in the theory of protein folding, h... more The Wako-Saitô-Muñoz-Eaton (WSME) model, initially introduced in the theory of protein folding, has also been used in modeling the RNA folding and some epitaxial phenomena. The advantage of this model is that it admits exact solution in the general inhomogeneous case (Bruscolini and Pelizzola, 2002) which facilitates the study of realistic systems. However, a shortcoming of the model is that it accounts only for interactions within continuous stretches of native bonds or atomic chains while neglecting interstretch (interchain) interactions. But due to the biopolymer (atomic chain) flexibility, the monomers (atoms) separated by several non-native bonds along the sequence can become closely spaced. This produces their strong interaction. The inclusion of non-WSME interactions into the model makes the model more realistic and improves its performance. In this study we add arbitrary interactions of finite range and solve the new model by means of the transfer matrix technique. We can therefore exactly account for the interactions which in proteomics are classified as medium-and moderately long-range ones.
We consider a one-dimensional lattice gas model of strained epitaxy with the elastic strain accou... more We consider a one-dimensional lattice gas model of strained epitaxy with the elastic strain accounted for through a finite number of cluster interactions comprising contiguous atomic chains. Interactions of this type arise in the models of strained epitaxy based on the Frenkel-Kontorova model. Furthermore, the deposited atoms interact with the substrate via an arbitrary periodic potential of period L. This model is solved exactly with the use of an appropriately adopted technique developed recently in the theory of protein folding. The advantage of the proposed approach over the standard transfer-matrix method is that it reduces the problem to finding the largest eigenvalue of a matrix of size L instead of 2 L−1 , which is vital in the case of nanostructures where L may measure in hundreds of interatomic distances. Our major conclusion is that the substrate modulation always facilitates the size calibration of self-assembled nanoparticles in one-and two-dimensional systems.
Journal of Statistical Mechanics: Theory and Experiment, 2015
Irreversible growth of strained epitaxial nanoislands has been studied with the use of the kineti... more 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.
The kinetic Monte Carlo technique has been used to simulate irreversible growth of epitaxial isla... more The kinetic Monte Carlo technique has been used to simulate irreversible growth of epitaxial islands in two consecutive steps. At the first step a small quantity of adatoms was simultaneously deposited at the surface at random positions and allowed us Q4 to freely diffuse until nucleating new islands or until being caught by the earlier nucleated ones. It was found that the distribution of the Voronoi cell (VC) areas around the island centers could be accurately described by the Gaussian distribution (GD) which was narrower than the GD describing the VCs of randomly distributed nucleation centers. Thus, our simulations provide an alternative explanation of the narrowing that was observed experimentally and attributed to elastic forces. At the second step the surface was exposed to an atomic deposition flux that was chosen to be small enough for the nucleation of new islands was strongly suppressed and the growth was dominated by the aggregation of deposited atoms into existing islands. At this step the island size distributions (ISDs) obtained could be also well described by the GD only more peaked than the corresponding VC area distributions. The narrowing has been explained in the framework of an effective medium theory. In several cases the simulated VC area distributions and ISDs semi-quantitatively agreed with those observed experimentally. Furthermore, the two-step growth made the island diameter distributions much more symmetric than those obtained under the conventional irreversible growth setup. It is suggested that this technique may provide a method of controlled growth of the island ensembles with narrow and symmetric size distributions in practically any system: homo-or heteroepitaxial. 30
The van Hove autocorrelation function of the impurity is obtained which is exact to the first ord... more The van Hove autocorrelation function of the impurity is obtained which is exact to the first order in the vacancy concentration. It is found that in the case of strong vacancy-impurity binding a singularity in the van Hove function corresponding to a resonant bound state develops on the unphysical sheet in the complex frequency plane close to the real axis. It is argued that this bound state corresponds to the defect-impurity pairs widely used in models of diffusion in semiconductors.
A technique allowing for a perturbative treatment of nonlocal corrections to the single-site dyna... more A technique allowing for a perturbative treatment of nonlocal corrections to the single-site dynamical mean-field theory (DMFT) in finite dimensions is developed. It is based on the observation that in the case of strong electron correlation the one-electron Green's function is strongly spatially damped so that its intersite matrix elements may be considered as small perturbations. Because the non-local corrections are at least quadratic in these matrix elements, DMFT in such cases may be a very accurate approximation in dimensions d = 1-3. This observation provides a rigorous justification for the application of DMFT to physical systems. Furthermore, the technique allows for a systematic evaluation of the nonlocal corrections. This is illustrated with the calculation of the magnetic short range order parameter for nearest neighbor spins in the half filled Hubbard model on the square lattice in its insulating phase which exhibits an excellent agreement with the results of a rece...
A formalism for solving field-theoretic lattice models by means of seri.es expansions in powers o... more A formalism for solving field-theoretic lattice models by means of seri.es expansions in powers of a parameter), = exp(-1/~), where ~ is the correlation length, is developed. Applications of the method to several lattice models and its connection with some of the known approximations are briefly discussed.
ABSTRACT We consider an atomistic model of a coherent epitaxial submonolayer which is strained du... more ABSTRACT We consider an atomistic model of a coherent epitaxial submonolayer which is strained due to the lattice mismatch with the substrate. We argue that the same interactions which lead to the size calibration of two-dimensional islands according to the mechanism proposed by Priester and Lannoo may lead to a considerable slowing down of atomistic kinetics similar to that observed in glassy materials. To show this, we develop a formalism that is suitable for large-scale numerical simulations. Assuming pair-wise interatomic interactions and treating the lattice-mismatch- induced atomic displacements in the harmonic approximation, we map the system onto a lattice-gas model with multi-atom cluster interactions. In the simplest case, the cluster interactions reduce to a repulsion between linear atomic triplets. In the presence of competing attractive pair interactions of comparable magnitude this leads to frustration and kinetic glassy phenomena such as aging.
A new method of eliminating short-wavelength degrees of freedom from hamiltonians of the Landau-G... more A new method of eliminating short-wavelength degrees of freedom from hamiltonians of the Landau-Ginzburg-Wilson model is proposed. For a class of models with hamiltonians of special form the problems of calculating the free energies and
SL (4, C) symmetry of the Hubbard interaction 1-lamiltonian is established. It is shown that most... more SL (4, C) symmetry of the Hubbard interaction 1-lamiltonian is established. It is shown that most of the symmetry generators cannot be realized as operators in the canonical formalism due to the Fermi statistics of electrons. A functional integer derivation ofnontrivial Ward-Takahashi identities for all the generators of the SL(4, C) group is presented.
If a stochastic system during some periods of its evolution can be divided into non-interacting p... more If a stochastic system during some periods of its evolution can be divided into non-interacting parts, the kinetics of each part can be simulated independently. We show that this can be used in the development of efficient Monte Carlo algorithms. As an illustrative example the simulation of irreversible growth of extended one dimensional islands is considered. The new approach allowed to simulate the systems characterized by parameters superior to those used in previous simulations.
ABSTRACT The length distribution of the one-dimensional atomic rows self-assembled on a crystal s... more ABSTRACT The length distribution of the one-dimensional atomic rows self-assembled on a crystal surface in thermal equilibrium at low coverage is calculated both in the framework of an analytical theory and with the use of the Monte Carlo simulations. It has been shown that in the case of interatomic interactions restricted to nearest-neighbor atoms the distribution exhibits the scaling properties formally similar to those found in the theories of irreversible growth. The difference consisted in the behavior of the scaling functions which in the equilibrium case was monotonously decreasing while in the case of irreversible growth exhibits a monomodal character. We found that our scaled distribution described without any fitting parameters the monotonous distributions recently observed in the growth of Ga rows on Si(001) by Albao et al. Phys. Rev. B 72 035426 (2005). The implications of the scaling for the experimental definition of interatomic interaction parameters is briefly discussed.
We consider a lattice gas model which in addition to the canonical nearest neighbor pair interato... more We consider a lattice gas model which in addition to the canonical nearest neighbor pair interatomic interaction accounts for a many-body interaction inside atomic trios. Interactions of this kind arise in the coherent strained epitaxy and were recently used by us to describe some surface phenomena. With the use of the Monte Carlo simulation we show that in two dimensions the model at low temperature exhibits glassy behaviour, in particular, undergoes a gelation transition. We argue that this model may belong to the universality class of non-equilibrium critical phenomena which may comprise also some off-lattice structural glass transitions, provided such a class exists.
The harmonic Frenkel-Kontorova model is used to illustrate with an exactly solvable example a gen... more The harmonic Frenkel-Kontorova model is used to illustrate with an exactly solvable example a general technique of mapping a coherently strained epitaxial system with continuous atomic displacements onto a lattice gas model (LGM) with only discrete variables. The misfit strain of the original model is transformed into cluster interatomic interactions of the LGM. The clusters are contiguous atomic chains of all lengths but the interaction strength for long chains is exponentially small. This makes possible the application of efficient Monte Carlo techniques developed for discrete variables both in kinetic and equilibrium simulations. The formalism developed can be applied to 1D as well as to 2D systems. As an illustrative example we consider the problem of self-organization of 1D size calibrated clusters on the steps of the vicinal surfaces.
ABSTRACT The scaling approach to the irreversible epitaxial growth gained wide recognition due to... more ABSTRACT The scaling approach to the irreversible epitaxial growth gained wide recognition due to its ability to describe with the use of a universal function the island size distributions (ISDs) corresponding to a broad range of experimental conditions. The approach, however, is operative only in the case of large average island sizes sav and large diffusion to deposition rates ratios R. Physically this corresponds to long deposition times and/or high temperatures. We argue that the ISDs exhibit yet another universality property which holds for much broader range of growth conditions, in particular, for low temperatures (small R) and small sav (short deposition times). We show that the normalized ISDs corresponding to the same sav are accurately described by the same universal distribution.
Journal of Statistical Mechanics: Theory and Experiment, 2010
The Wako-Saitô-Muñoz-Eaton (WSME) model, initially introduced in the theory of protein folding, h... more The Wako-Saitô-Muñoz-Eaton (WSME) model, initially introduced in the theory of protein folding, has also been used in modeling the RNA folding and some epitaxial phenomena. The advantage of this model is that it admits exact solution in the general inhomogeneous case (Bruscolini and Pelizzola, 2002) which facilitates the study of realistic systems. However, a shortcoming of the model is that it accounts only for interactions within continuous stretches of native bonds or atomic chains while neglecting interstretch (interchain) interactions. But due to the biopolymer (atomic chain) flexibility, the monomers (atoms) separated by several non-native bonds along the sequence can become closely spaced. This produces their strong interaction. The inclusion of non-WSME interactions into the model makes the model more realistic and improves its performance. In this study we add arbitrary interactions of finite range and solve the new model by means of the transfer matrix technique. We can therefore exactly account for the interactions which in proteomics are classified as medium-and moderately long-range ones.
We consider a one-dimensional lattice gas model of strained epitaxy with the elastic strain accou... more We consider a one-dimensional lattice gas model of strained epitaxy with the elastic strain accounted for through a finite number of cluster interactions comprising contiguous atomic chains. Interactions of this type arise in the models of strained epitaxy based on the Frenkel-Kontorova model. Furthermore, the deposited atoms interact with the substrate via an arbitrary periodic potential of period L. This model is solved exactly with the use of an appropriately adopted technique developed recently in the theory of protein folding. The advantage of the proposed approach over the standard transfer-matrix method is that it reduces the problem to finding the largest eigenvalue of a matrix of size L instead of 2 L−1 , which is vital in the case of nanostructures where L may measure in hundreds of interatomic distances. Our major conclusion is that the substrate modulation always facilitates the size calibration of self-assembled nanoparticles in one-and two-dimensional systems.
Journal of Statistical Mechanics: Theory and Experiment, 2015
Irreversible growth of strained epitaxial nanoislands has been studied with the use of the kineti... more 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.
The kinetic Monte Carlo technique has been used to simulate irreversible growth of epitaxial isla... more The kinetic Monte Carlo technique has been used to simulate irreversible growth of epitaxial islands in two consecutive steps. At the first step a small quantity of adatoms was simultaneously deposited at the surface at random positions and allowed us Q4 to freely diffuse until nucleating new islands or until being caught by the earlier nucleated ones. It was found that the distribution of the Voronoi cell (VC) areas around the island centers could be accurately described by the Gaussian distribution (GD) which was narrower than the GD describing the VCs of randomly distributed nucleation centers. Thus, our simulations provide an alternative explanation of the narrowing that was observed experimentally and attributed to elastic forces. At the second step the surface was exposed to an atomic deposition flux that was chosen to be small enough for the nucleation of new islands was strongly suppressed and the growth was dominated by the aggregation of deposited atoms into existing islands. At this step the island size distributions (ISDs) obtained could be also well described by the GD only more peaked than the corresponding VC area distributions. The narrowing has been explained in the framework of an effective medium theory. In several cases the simulated VC area distributions and ISDs semi-quantitatively agreed with those observed experimentally. Furthermore, the two-step growth made the island diameter distributions much more symmetric than those obtained under the conventional irreversible growth setup. It is suggested that this technique may provide a method of controlled growth of the island ensembles with narrow and symmetric size distributions in practically any system: homo-or heteroepitaxial. 30
The van Hove autocorrelation function of the impurity is obtained which is exact to the first ord... more The van Hove autocorrelation function of the impurity is obtained which is exact to the first order in the vacancy concentration. It is found that in the case of strong vacancy-impurity binding a singularity in the van Hove function corresponding to a resonant bound state develops on the unphysical sheet in the complex frequency plane close to the real axis. It is argued that this bound state corresponds to the defect-impurity pairs widely used in models of diffusion in semiconductors.
A technique allowing for a perturbative treatment of nonlocal corrections to the single-site dyna... more A technique allowing for a perturbative treatment of nonlocal corrections to the single-site dynamical mean-field theory (DMFT) in finite dimensions is developed. It is based on the observation that in the case of strong electron correlation the one-electron Green's function is strongly spatially damped so that its intersite matrix elements may be considered as small perturbations. Because the non-local corrections are at least quadratic in these matrix elements, DMFT in such cases may be a very accurate approximation in dimensions d = 1-3. This observation provides a rigorous justification for the application of DMFT to physical systems. Furthermore, the technique allows for a systematic evaluation of the nonlocal corrections. This is illustrated with the calculation of the magnetic short range order parameter for nearest neighbor spins in the half filled Hubbard model on the square lattice in its insulating phase which exhibits an excellent agreement with the results of a rece...
A formalism for solving field-theoretic lattice models by means of seri.es expansions in powers o... more A formalism for solving field-theoretic lattice models by means of seri.es expansions in powers of a parameter), = exp(-1/~), where ~ is the correlation length, is developed. Applications of the method to several lattice models and its connection with some of the known approximations are briefly discussed.
ABSTRACT We consider an atomistic model of a coherent epitaxial submonolayer which is strained du... more ABSTRACT We consider an atomistic model of a coherent epitaxial submonolayer which is strained due to the lattice mismatch with the substrate. We argue that the same interactions which lead to the size calibration of two-dimensional islands according to the mechanism proposed by Priester and Lannoo may lead to a considerable slowing down of atomistic kinetics similar to that observed in glassy materials. To show this, we develop a formalism that is suitable for large-scale numerical simulations. Assuming pair-wise interatomic interactions and treating the lattice-mismatch- induced atomic displacements in the harmonic approximation, we map the system onto a lattice-gas model with multi-atom cluster interactions. In the simplest case, the cluster interactions reduce to a repulsion between linear atomic triplets. In the presence of competing attractive pair interactions of comparable magnitude this leads to frustration and kinetic glassy phenomena such as aging.
A new method of eliminating short-wavelength degrees of freedom from hamiltonians of the Landau-G... more A new method of eliminating short-wavelength degrees of freedom from hamiltonians of the Landau-Ginzburg-Wilson model is proposed. For a class of models with hamiltonians of special form the problems of calculating the free energies and
SL (4, C) symmetry of the Hubbard interaction 1-lamiltonian is established. It is shown that most... more SL (4, C) symmetry of the Hubbard interaction 1-lamiltonian is established. It is shown that most of the symmetry generators cannot be realized as operators in the canonical formalism due to the Fermi statistics of electrons. A functional integer derivation ofnontrivial Ward-Takahashi identities for all the generators of the SL(4, C) group is presented.
If a stochastic system during some periods of its evolution can be divided into non-interacting p... more If a stochastic system during some periods of its evolution can be divided into non-interacting parts, the kinetics of each part can be simulated independently. We show that this can be used in the development of efficient Monte Carlo algorithms. As an illustrative example the simulation of irreversible growth of extended one dimensional islands is considered. The new approach allowed to simulate the systems characterized by parameters superior to those used in previous simulations.
ABSTRACT The length distribution of the one-dimensional atomic rows self-assembled on a crystal s... more ABSTRACT The length distribution of the one-dimensional atomic rows self-assembled on a crystal surface in thermal equilibrium at low coverage is calculated both in the framework of an analytical theory and with the use of the Monte Carlo simulations. It has been shown that in the case of interatomic interactions restricted to nearest-neighbor atoms the distribution exhibits the scaling properties formally similar to those found in the theories of irreversible growth. The difference consisted in the behavior of the scaling functions which in the equilibrium case was monotonously decreasing while in the case of irreversible growth exhibits a monomodal character. We found that our scaled distribution described without any fitting parameters the monotonous distributions recently observed in the growth of Ga rows on Si(001) by Albao et al. Phys. Rev. B 72 035426 (2005). The implications of the scaling for the experimental definition of interatomic interaction parameters is briefly discussed.
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Papers by Vasyl Tokar