Elastic interaction of point defects in cubic and hexagonal crystals

The elastic interaction of two point defects in cubic and hexagonal structures has been considered. On the basis of the exact expression for the tensor Green’s function of the elastic field obtained by the Lifschitz–Rozentsveig for a hexagonal medium, an exact formula for the interaction energy of two point defects has been obtained. The solution is represented as a function of the angle of their relative position on the example of semiconductors such as III-nitrides and α-SiC. For the cubic medium, the solution is found on the basis of the Lifschitz–Rozentsveig Green’s tensors corrected by Ostapchuk, in the weak-anisotropy approximation. It is proven that the calculation of the interaction energy by the original Lifschitz–Rozentsveig Green’s tensor leads to the opposite sign of the energy. On the example of the silicon crystal, the approximate solution is compared with the numerical solution, which is represented as an approximation by a series of spherical harmonics. The range of applicability of the continual approach is estimated by the quantum mechanical calculation of the lattice Green’s function.     

Elastic dipole interaction of point defects in a sphere of elastically anisotropic cubic material

The elastic field of centres of dilatation at arbitrary positions in a spherical specimen with cubic elastic constants is studied. The influence of the (free) boundary, which is of central importance due to the long-range nature of the elastic interaction, is included. For materials with small elastic anisotropy expansions in terms of vector spherical harmonics are obtained for the displacements due to one defect, as well as for the interaction energy of two defects. Infinite material expressions and surface contributions are given separately. The general treatment is exemplified by presenting explicit results for the case of hydrogen atoms dissolved in vanadium.     

Screening of the electric field in covalent crystals containing point defects

An expression is derived within the framework of the Debye-Hückel approximation for the screening length of a field in a p-type semiconductor taking into account the energy spread of immobile acceptor levels and the density of states tail of the valence band. It is shown that the screening length depends additively on the product of the carrier density and their diffusion coefficient ratio (for free holes and holes hopping via acceptors).Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 41–43, November, 1984.     

Elastic theory of defects in toroidal crystals

We report a comprehensive analysis of the ground-state properties of axisymmetric toroidal crystals based on the elastic theory of defects on curved substrates. The ground state is analyzed as a function of the aspect ratio of the torus, which provides a non-local measure of the underlying Gaussian curvature, and the ratio of the defect core energy to the Young modulus. Several structural features are discussed, including a spectacular example of curvature-driven amorphization in the limit of the aspect ratio approaching one. The outcome of the elastic theory is then compared with the results of a numerical study of a system of point-like particles constrained on the surface of a torus and interacting via a short-range potential. Electronic supplementary material  Supplementary material in the form of a pdf file available from the journal web page at and are accessible for authorised users.     

Energetics of native point defects in cubic silicon carbide

In this work we present a detailed investigation of native point defects energetics in cubic SiC, using state-of-the-art first principles computational method. We find that, the carbon vacancy is the dominant defect in p-type SiC, regardless the growth conditions. Silicon and carbon antisites are the most common defects in n-type material in Si-rich and C-rich conditions respectively. Interstitial defects and silicon vacancy are less favorite from the energetic point of view. The silicon vacancy tends to transform into a carbon vacancy-antisite complex and the carbon interstitial atom prefers to pair to a carbon antisite. The dumbbell structure is the lowest-energy configuration for the isolated carbon interstitial defect, and the tetrahedral interstitial silicon is a stable structure in p-type and intrinsic conditions, while in n-type material the dumbbell configuration is the stable one. Our results suggest that, in samples grown in Si-rich stoichiometric conditions, native defects are a source of n-doping and of compositional unbalance of nominally intrinsic SiC, in accord with experimental evidence.Received: 9 January 2004, Published online: 28 May 2004PACS: 61.72.Ji Point defects (vacancies, interstitials, color centers, etc.) and defect clusters - 68.55.Ln Defects and impurities: doping, implantation, distribution, concentration, etc. - 74.62.Dh Effects of crystal defects, doping and substitution     

Localized vibrations of point defects in body-centred cubic lattices

The paper gives an exact calculation of the localized frequencies of substitutional defects in a body-centred cubic lattice by the method of Green's functions and compares it with the approximate calculation carried out after [14]. The exact calculation is based on newly computed Green's functions of a b.c.c. lattice [18]. It is shown how by means of group theory the symmetry of the system can be used both in an approximate and in the exact calculation. Some symmetry relations between Green's functions, which limit the number of functions necessary for numerical calculations, are derived.     

Specific heat due to substitutional defects in cubic crystals

An expression is obtained for the specific heat of an isotopically disordered cubic crystal considering both mass and force constant changes using the method of double-time thermal Green's functions. It is shown that in the low concentration limit, the defect contribution to the specific heat depends only on the force constant change.     

Elastic interaction of point defects with dislocations

Elastic interaction between dislocation and point defects with spherical symmetry is investigated theoretically by means of the sphere-in-hole model. The interaction energy due to the different elastic moduli of the point defect and matrix is computed and then added to the interaction energy due to the size effect. The force on and the probable position of the point defect is determined. The numerical results for some selected solute atoms are computed by means of available experimental data.The results are compared with other recent papers and some experimental applications are briefly discussed.The author is much indebted to F. Kroupa, CSc., for fruitful discussions and valuable comments.     

Magnetic field induced double refraction in cubic crystals

Magnetic field induced spatial dispersion of the dielectric permeability is discussed for a crystal possessing a T_d symmetry. Bilinear terms of the ground state exciton hamiltonian are obtained using the method of invariants. Relevant material constants are expressed in terms of the band parameters.     

Universal symmetry property of point defects in crystals

Point defects (vacancies, solute atoms, and disorder) are ubiquitous in crystalline solids. With in situ transmission electron microscopy we find clear evidence for the existence of a universal symmetry property of point defects; i.e., the symmetry of short-range order of point defects follows the crystal symmetry when in equilibrium. We further show that this symmetry-conforming property can lead to various interesting effects including "aging-induced microstructure memory" and the associated "time-dependent two-way shape memory."     

Elastic interaction of dislocation loops and point defects

The elastic interaction energy of a circular dislocation loop with interstitial atoms and vacancies characterized as dilatation or relaxation centres is calculated. Further, the forces which the dislocation loops exert on point defects through elastic interaction are discussed.     

Elastic interaction of point defects on biological membranes

The interaction between inclusions mediated by biological membranes undulations is analyzed. The interaction law is interpreted in terms of symmetries associated with inclusions. We show, in particular, that for a C3 and C3v symmetries the interaction law falls as 1/r3. We show that for completely isotropic inclusions, the dominant interaction vanishes to all orders, and the first contribution stems from the induced interaction which behaves as 1/r4. The same law holds for any other symmetry which is higher than C4. We introduce a straightforward method to compute these interactions. We point out important differences with results in the literature and explain the discrepancies.     

Anelasticity and stress-induced ordering of point defects in crystals

Point defects may exist in a crystal on a set of crystallographically equivalent orientations or sites. The response of a crystal containing point defects to an externally applied stress takes the form of ‘stress-induced ordering’, or preferential alignment of the defects. Recent experiments have used optical and spin resonance techniques to observe this stress-induced ordering directly. This same process also gives rise to anelasticity, e.g. to an internal friction peak. In describing such behaviour, the concept of a point defect as an ‘elastic dipole’, introduced by Kröner and others, is very useful. The elastic dipole is here defined as a second-rank tensor, the ‘λ-tensor’, equal to the average strain per mole fraction of defects all aligned in a particular orientation. This definition is then used to develop a thermodynamic theory of stress-induced ordering, which includes the interaction among defects and the possibility of a ‘reaction’ or interconversion between two species of defects. Applications of the theory are made to defects of various point symmetries in cubic crystals, and a number of specific examples are discussed to illustrate each type of defect symmetry. The usefulness of the theory is that it enables one to calculate the values of the components of the λ-tensor from experimental information and the defect symmetry. Typical values of these λ-parameters range from 10^-2 unity.     

Elastic scattering of acoustic phonons on point mass defects

The relaxation of homogeneous states of long-wave acoustic phonon gas scattered by point mass defects in transversely—isotropic media is studied. The spectrum of the suitable collision operator of the Boltzmann-Peierls equation is investigated. It consists of a continuous part and several discrete eigenvalues. Both continuous and discrete part of the spectrum depend on the values of components of the elastic constant tensor. For some values of elastic constants the continuous part splits up into two separate intervals and some of the discrete eigenvalues appear in the gap. The number of discrete eigenvalues and their arrangement are also affected by elastic properties of medium.     

Elastic scattering of acoustic phonons on point mass defects

The problem of eigenvalues of the collision operator for a gas of acoustic phonons scattered by point mass defects of small concentration embedded in transversely isotropic media is considered. For this purpose the properties of solution of the Boltzmann-Peierls kinetic equation for spatially homogeneous states are studied. An analytic formula for the Laplace transform of the distribution function is obtained. The singularities of this Laplace transform and the initial distribution function determine the dependence of this distribution function on time. For several hexagonal materials characteristics of the singularity set are calculated. Usually the singularity set consists of a continuous part and four discrete points. However, there exist elastic hexagonal materials (⁴He, Cd, Ta, Zn) for which some of discrete points are absent. For some materials (e.g. Zr) the continuous part is very narrow.     

A Debye model for point defects in aluminosilicate crystals

Summary In aluminosilicate lattices, exchanges between silicon and aluminum ions are expected to originate charged point defects of opposite signs. Thermodynamic arguments, based on a model analogous to the Debye electrolyte model, allow evaluation of the equilibrium density of defects as a function of temperature. The electric field of point defects is found to be screened by charge distribution even at distances small enough to affect the ground-state energy of electrons trapped by the defect positive charges. Calculations show that large fluctuations of trapped electron energy are expected. These results agree with data from thermoluminescence experiments on different aluminosilicates, which showed broad distributions of electron traps.     

Elastic wave surfaces for the (111) plane of cubic crystals

The nature of inverse velocity surfaces as well as energy surfaces for elastic wave propagation in the (111) plane have been studied for a number of cubic crystals. The sections of inverse velocity surfaces by the (111) plane exhibit six-fold symmetry in all cases. Cuspidal edges are exhibited with a six-fold symmetry by both the slow transverse and fast transverse shear modes in the (111) plane, unlike the case of the (100) and (110) planes for which only the slow transverse shear mode exhibits cuspidal edges. The slow transverse mode energy surface exhibits cuspidal edges along direction or an equivalent symmetry direction. The inverse velocity surfaces of the A-15 compounds exhibit unusually large inflexions for the slow transverse mode, whereas their energy surfaces have large cuspidal edges which intersect each other resulting in common regions of cusps.     

Interactions of point defects with dislocations in silver chloride crystals

Internal friction techniques at 35 kHz were used to study the interactions between point defects and dislocations in crystals of silver chloride. The Granato-Lücke plots of the amplitude-dependent decrement were not linear, but were concave upward. Various analyses of these data all gave pin-dislocation binding energies of approximately 0·2 eV. An extensive study was made of the decay of the amplitude-independent decrement after “excitation” by high-amplitude oscillation; the Granato-Lücke L⁴ law was found valid, but the arrival of pins proceeded as t^1/3, rather than the usual t^2/3 found in deformed specimens. A first-order model of the response of a slightly perturbed Maxwell atmosphere about the dislocation is presented. The model quantitatively describes the experimental results. The activation energy of the rate constant was found to be 0·42 eV, consistent with data on solute diffusion and strain-aging.