Silicon diffusion in molybdenum disilicide: correlation effects

Correlation factors for silicon diffusion by a vacancy mechanism in the silicon sublattice of the tetragonal MoSi₂ structure have been calculated by combining an analytical and a Monte Carlo approach. The ratio of the silicon diffusivity perpendicular to the tetragonal axis to that parallel to the tetragonal axis is also deduced. An effect of forward correlation of tracer atom jumps in the silicon sublattice with the corresponding partial correlation factor of 1.5 appears at small frequencies of silicon atom jumps along the tetragonal axis with respect to the jump frequencies in the silicon layer perpendicular to the tetragonal axis of the MoSi₂ structure. The anisotropy of silicon diffusion in MoSi₂ measured by Salamon et al. is explained in terms of correlation effects of silicon diffusion on its own sublattice.     

Diffusion in ordered Fe-Si alloys

The measurement of the diffusional Mössbauer line broadening in single crystalline samples at high temperatures provides microscopic information about atomic jumps. We can separate jumps of iron atoms between the various sublattices of Fe-Si intermetallic alloys (D0₃ structure) and measure their frequencies. The diffusion of iron in Fe-Si samples with Fe concentrations between 75 and 82 at% shows a drastic composition dependence: the jump frequency and the proportion between jumps on Fe sublattices and into antistructure (Si) sublattice positions change greatly. Close to Fe₃Si stoichiometry iron diffusion is extremely fast and jumps are performed exclusively between the three Fe sublattices. The change in the diffusion process when changing the alloy composition from stoichiometric Fe₃Si to the iron-rich side is discussed.     

Interstitial solute diffusion in metals

The diffusion of interstitial solute atoms, which were subject to nearest neighbour solute-solute interactions, was simulated by means of a modified Monte Carlo method. Solute correlation factors, nearest neighbour vacancy availability factors and effective jump frequency factors were calculated as functions of composition and temperature. Compared with non-interacting atoms attraction between solute atoms always retarded diffusion whereas repulsion always enhanced diffusion except within the ordered region.     

A dynamical mean field theory for the study of surface diffusion constants

We present a combined analytical and numerical approach based on the Mori projection operator formalism and Monte Carlo simulations to study surface diffusion within the lattice-gas model. In the present theory, the average jump rate and the susceptibility factor appearing are evaluated through Monte Carlo simulations, while the memory functions are approximated by the known results for a Langmuir gas model. This leads to a dynamical mean field theory (DMF) for collective diffusion, while approximate correlation effects beyond DMF are included for tracer diffusion. We apply our formalism to three very different strongly interacting systems and compare the results of the new approach with those of usual Monte Carlo simulations. We find that the combined approach works very well for collective diffusion, whereas for tracer diffusion the influence of interactions on the memory effects is more prominent.     

Tracer diffusion in concentrated lattice gas models. Rectangular lattices with anisotropic jump rates

An approximate theory is developed for tracer diffusion in rectangular lattice gas models with anisotropic jump rates to neighboring unoccupied sites in different directions. Comparison with Monte Carlo simulations on quadratic lattices with several ratios for the jump rates in orthogonal directions shows a satisfactory agreement in all cases investigated.     

Crystalline silicon films sputtered on molybdenum: A study of the silicon–molybdenum interface

Polycrystalline silicon films were grown on molybdenum (Mo)-coated substrates at high deposition rate using the pulsed magnetron sputtering technique. Our study investigates the silicon–molybdenum interface of these films to elucidate stimulating mechanisms for an ordered crystalline silicon thin film growth. Both Auger electron spectroscopy and Rutherford backscattering reveal that at a substrate temperature as low as T_S=450 °C during the deposition process intermixing of Si and Mo at the Si–Mo interface takes place leading to a compositional ratio Mo:Si of about 1:2. By Raman spectroscopy hexagonal β-MoSi₂ could be identified as the dominant phase in this intermixed region. The dependence of the resulting thickness of the reacted interface layer on the deposition conditions is not fully understood yet.     

PAC Probes as Diffusion Tracers in Solids

Perturbed angular correlation (PAC) probe atoms have been used as tracers to study diffusion in solids. The method works for diffusion on a sublattice for which the point symmetry is noncubic and the electric field gradient (EFG) at the probe nucleus reorients in each jump. Such motion leads to relaxation of the nuclear quadrupole interaction. Precise values of the tracer jump frequency have been obtained from fits of measured PAC perturbation functions. Results obtained to date are reviewed for Cd tracer atoms in rare-earth indides such as LaIn3 that have the L1₂ crystal structure, for which each jump on the In-sublattice reorients the EFG by 90°. New results are presented for LaSn3 and prospects for future studies are outlined.     

Atom transport in random two sublattice structures: analogue of the random alloy sum rule

A simple and often used model of atom transport by the vacancy mechanism on two physically distinct interpenetrating sublattices assumes that each atom–vacancy exchange frequency depends only on the species of the atom and the sublattice from which it jumps. In the kinetic theory of this model, the phenomenological coefficients can be expressed as sums of partial correlation functions, each labelled by the sublattices associated with the atoms making the first and last jumps in the sequence of correlated jumps which it represents. A sum rule, a set of exact relations among these partial correlation functions, is derived for the model, assuming arbitrary vacancy content and any number of chemical species. It reduces to a widely used sum rule for the random lattice gas [L.K. Moleko and A.R. Allnatt, Phil. Mag. A 58 677 (1988)] in the limit that atom jump frequencies are made independent of sublattice. For the two sublattice model at very low vacancy contents, a more powerful sum rule is also derived; it is essentially the same as that of Belova and Murch [Defect Diffus. Forum 194/199 547 (2001)]. The efficiencies of the three sum rules are briefly compared. The low vacancy concentration sum rule is illustrated by numerical simulations for a binary two sublattice system.     

Modeling band spectra of compound crystals in the basis of states of their sublattices

A technique for simulation of band spectra of compound crystals in the basis of states of their sublattices is developed. The Hamiltonian of the crystal is written as a sum of sublattice Hamiltonians and a perturbation operator resulting from hybridization of the sublattice states. The simplest models are proposed to account for the sublattice hybridization. The methods for calculating the average crystal potential are discussed to represent the band spectra of sublattices in a common energy scale when calculating band structures by the pseudopotential method. The band spectra of MgO and MgS crystals are calculated to show the application of the methods under consideration. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 81–84, July, 2006.     

Atomistic Modeling of Point Defects and Diffusion in Copper Grain Boundaries

The atomic structure of several symmetrical tilt grain boundaries (GBs) in Cu and their interaction with vacancies and interstitials as well as self-diffusion are studied by molecular statics, molecular dynamics, kinetic Monte Carlo (KMC), and other atomistic simulation methods. Point defect formation energy in the GBs is on average lower than in the lattice but variations from site to site within the GB core are very significant. The formation energies of vacancies and interstitials are close to one another, which makes the defects equally important for GB diffusion. Vacancies show interesting effects such as delocalization and instability at certain GB sites. They move in GBs by simple vacancy-atom exchanges or by long jumps involving several atoms. Interstitial atoms can occupy relatively open positions between atoms, form split dumbbell configurations, or form highly delocalized displacement zones. They diffuse by direct jumps or by the indirect mechanism involving a collective displacement of several atoms. Diffusion coefficients in the GBs have been calculated by KMC simulations using defect jump rates determined within the transition state theory. GB diffusion can be dominated by vacancies or interstitials, depending on the GB structure. The diffusion anisotropy also depends on the GB structure, with diffusion along the tilt axis being either faster or slower than diffusion normal to the tilt axis. In agreement with Borisov's correlation, the activation energy of GB diffusion tends to decrease with the GB energy.     

Influence of the Si(111)-7×7 surface reconstruction on the diffusion of strontium atoms

The diffusion of strontium atoms on the Si(111) surface at room temperature has been investigated using scanning tunnel microscopy and simulation carried out in terms of the density functional theory and the Monte Carlo method. It has been found that the reconstruction of a clean silicon surface with a 7 × 7 structure has a profound effect on the diffusion process. The average velocity of motion of a strontium atom in a unit cell of the 7 × 7 structure has been calculated. The main diffusion paths of a strontium atom and the corresponding activation energies have been determined. It has been demonstrated that the formation of scanning tunnel microscope images of the Si(111)-7 × 7 surface with adsorbed strontium atoms is significantly affected by the shift of the electron density from the strontium atom to the nearest neighbor silicon adatoms in the 7 × 7 structure.     

Diffusion of particles over triangular inhomogeneous lattice with two non-equivalent sites

We investigate the diffusion of particles adsorbed on a triangular lattice with deep and shallow sites. It is shown that the character of the particle migration depends substantially on the relative jump rates from the deep and shallow sites. The site inhomogeneity imposes specific correlation betweeen successive jumps: particles perform pairs of slow and fast jumps. General analytical expressions have been derived for the chemical and jump diffusion coefficients. We have calculated coverage dependencies of the diffusion coefficients and some thermodynamic quantities for different lateral interactions between the particles. The analytical data have been compared with the numerical data obtained by kinetic Monte Carlo simulations. The agreement between the results obtained by these quite different approaches is found to be very satisfactory.     

Reactive diffusion in the Ti–Si system and the significance of the parabolic growth constant

Solid diffusion couple experiments are conducted to analyse the growth mechanism of the phases and the diffusion mechanism of the components in the Ti–Si system. The calculation of the parabolic growth constants and the integrated diffusion coefficients substantiates that the analysis is intrinsically prone to erroneous conclusions if it is based on just the parabolic growth constants determined for a multiphase interdiffusion zone. The location of the marker plane is detected based on the uniform grain morphology in the TiSi₂ phase, which indicates that this phase grows mainly because of Si diffusion. The growth mechanism of the phases and morphological evolution in the interdiffusion zone are explained with the help of imaginary diffusion couples. The activation enthalpies for the integrated diffusion coefficient of TiSi₂ and the Si tracer diffusion are calculated as 190 ± 9 and 197 ± 8?kJ/mol, respectively. The crystal structure, details on the nearest neighbours of the components, and their relative mobilities indicate that the vacancies are mainly present on the Si sublattice.     

The distribution of the valence electron density in predominantly ionic crystals with different Bravais sublattices

Based on the theory of the local-density functional, self-consistent valence electron densities are calculated for CaO with a rock-salt lattice, CaF₂ with a fluorite lattice, K₂O with an antifluorite lattice, and for their constituent sublattices. It is shown that in the crystals with different Bravais sublattices, the anionic sublattice is a framework with covalent bonds containing a metal sublattice inside of them. The coupling between the sublattices is characterized by the density difference, which is deflned as the difference between the total electron density and the densities of the individual sublattices. The density difference is found to be an order of magnitude smaller than the crystal and sublattice densities, which is evidence of weak hybridization of the sublattices and of the predominately ionic character of the bonding between them.     

Static correlations of higher order and diffusion in an interacting lattice gas

For a lattice gas with extended hard core interaction on a square lattice the static correlation functions of higher order, which determine the average jump rate in the diffusion process, are calculated both by the Monte Carlo method and by analytic approximations. It is found that the superposition approximation is very inaccurate for the correlation functions of third and fourth order, but gives better results for the average jump rate. Up to concentrations ofc = 0.3 better consistency with the Monte Carlo data for both quantities is obtained by treating the site occupation numbers as Gaussian random variables and accordingly expressing the correlation functions of higher order by products of averages of two particle correlations. For concentrationsc > 0.3, however, a Bethe-Peierls cluster approximation is superior to the superposition approximation.The results of this paper were presented at the I.L.L, workshop Beyond Radial Distribution, Grenoble, July 15–16, 1985.     

Random alloy diffusion kinetics for the application to multicomponent alloy systems

In this paper, extensive Monte Carlo simulation results are reported on tracer and collective diffusion correlation effects in the random ternary alloy, as an example of a multicomponent alloy system. The problem of analytically describing both collective and tracer diffusion kinetics is also addressed for the random multicomponent alloy by application of a combination of the Manning theory and Holdsworth and Elliott theory. It is found that the overall results from the combined theory agree reasonably well with Monte Carlo results. This combined approach is much more accurate than Manning’s approach itself and much more manageable than the almost exact, but unfortunately difficult to use, self-consistent theory of Moleko, Allnatt and Allnatt. Some relations between the Onsager phenomenological coefficients and tracer diffusion coefficients are derived and are tested with our Monte Carlo data. Good agreement is found.     

Influence of a uniform driving force on tracer diffusion in a one-dimensional hard-core lattice gas

The influence of a uniform driving force on tracer diffusion is investigated for a one-dimensional lattice gas where particles jump stochastically to unoccupied neighboring sites. A new, simple calculation is presented for the diffusion coefficient of a tracer particle with respect to its average drift, obtained recently by rigorous methods by De Masi and Ferrari. A theoretical expression describing the tracer particle mean square displacement approximately for all times is derived and found to be in excellent agreement with the results of Monte Carlo simulations.     

Radiation-stimulated diffusion in Al–Si alloys

A di-vacancy low-temperature diffusion is proposed to explain diffusion-controlled processes in Al–Si alloys responsible for neutron-induced silicon precipitation. Ab initio calculations of potential barriers for Si atom hopping in aluminium lattice showed that in the case of di-vacancy diffusion, they are small compared with that of mono-vacancy diffusion. The low temperature diffusivity of mono-vacancies is too small to account for the measured Si diffusivities in aluminium. The dependencies of radiation-stimulated diffusion on the neutron flux and on the temperature are obtained and can be used for the experimental verification of the developed model.