The energy and elastic dipole tensor of defects in ionic crystals calculated by the supercell method

Methods which have been developed and employed for the calculation of point defects in ionic crystals are described. These methods involve a detailed Hartree-Fock-like treatment by way of angular-momentum-dependent exchange potentials of the ions within an inner region ${\ensuremath{\tau}}_{1}$ centered about the defect. In the outer region ${\ensuremath{\tau}}_{2}$, composed of the rest of the crystal, the ions are treated in two different approximations, the most successful of which involves an effective-mass formalism. Polarization effects are included by expressions of the form resulting from the work of Toyozawa, Haken, and Schottky. Although the main purpose of the paper is to describe the methods, applications are made to the calculation of the electronic energy levels in the optical absorption of the $F$ center in KCl and NaCl. Good agreement with experiment is found only if polarization effects, which are important even in the ground state, are included. Additional applications will be given in subsequent papers.

The electrostatic potential which arises from a lattice array of point ions is computed in terms of a Taylor's series expansion for small distances from a lattice site. This expansion gives the change in electrostatic energy when an ion moves in the background of a perfect point ion lattice potential. The Taylor's series coefficients for terms up to fourth order in the ion displacement are evaluated for the NaCl and CaF2 lattice structures.

The subsurface electric potential in oriented single crystals of silver chloride has been directly mapped out by a technique based on the equilibrium distribution of a very small concentration of a charged radiotracer. The potential was found to vary with depth according to the Gouy-Chapman solution of Poisson's equation, with screening lengths in the range 10\char21{}20 nm. The surfaces were all negative, relative to the deep interior, by 0.10\char21{}0.30 V, depending on temperature and surface orientation. The present results on the subsurface potential difference can be employed to resolve the Frenkel-pair formation enthalpy and entropy, as determined from transport measurements, into the formation enthalpies and entropies of the individual point defects comprising the pair. For the silver chloride (110) surface, for example, ${G}_{v}$=0.74 eV and ${G}_{i}$=0.37 eV at 150 \ifmmode^\circ\else\textdegree\fi{}C; the best values obtained to date for ${H}_{v}$, ${H}_{i}$, ${S}_{v}$, and ${S}_{i}$, are 0.77 eV, 0.70 eV, 0.79k, and 9.0k, respectively.

Various mechanisms for the formation of point defects in ionic crystals are considered. Their probable importance as a function of the energy of the ionizing radiation and the temperature at which the irradiation takes place is discussed. Complex centers formed from aggregates of simple point defects are reviewed and their occurrence is considered in terms of the expected spatial distribution of the point defects. The effect of high-energy X-rays and protons on the ionic conductivity is dis- cussed in terms of a new interpretation of the conductivity curve as a function of temperature, and it is suggested that radiation-enhanced diffusion may be important in explaining the experimental data.

Any deviation from a perfect periodic lattice or structure in a crystal is an imperfection. There are three main types of imperfections, point, line and plane imperfections1, of which we shall deal only with the point defects in this chapter. The point imperfections encountered in ionic solids are: (i) chemical impurities (ii) vacant lattice sites and (iii) interstitial ions. Real crystals are always imperfect in some respect. In thermal equilibrium in an otherwise perfect crystal a certain number of lattice vacancies or interstitials are always present, because the entropy is increased by the presence of disorder in the structure and at a finite temperature, the equilibrium condition of a crystal happens to be a state of minimum free energy.

The Stark effect of localized electronic centres is reviewed by using a few selected examples of impurity centres or simple defects. Important information on the symmetry of centres and on the position of energy levels is obtained through the electric-field-induced absorption changes. Typical results are summarized for the Stark effect on the zerophonon lines and on the broad absorption band of point defects in ionic crystals and of impurities in semiconductors.

The authors present and describe a new embedded-cluster approach for calculating both the electronic and the spatial structure of point defects in ionic crystals with self-consistent incorporation of the polarisation effects into the eigenvalue spectrum and the total energy of a crystal. For the first time both the electronic structure and the ionic displacements within a large cluster containing a defect are obtained in the framework of the same calculating scheme. It permits one to describe correctly back-coupling between redistribution of the electron density in a defective region and polarisation of the rest of the crystal outside the cluster, which can play a vital role for the study of small polaron effects. Making use of a 45-atom cluster with 175 electrons and the semi-empirical INDO method, the authors have simulated the impurity-induced trapping of a hole in the (Li)0 centre in an MgO crystal. The calculations do not use a priori assumptions and do confirm that one-centre localisation of a hole in the ground state is energetically favourable. They show that polarisation energies can differ considerably for the ground and charge-transfer excited states of the defect. It is also demonstrated that the generally accepted model of the O- hole polaron is oversimplified: a considerable charge redistribution between Li+ and another O- along the (100) direction takes place.

Excitation of Plasmons and Interband Transitions by Electrons. By H. Raether. (Berlin, Heidelberg, New York: Springer-Verlag, 1980.) [Pp. 192+subject index.] Price DM 59, – –/US $32.50 hardback. Physics of Superionic Conductors. Edited by M. B. Salamon. (Berlin, Heidelberg, New York: Springer-Verlag, 1979.) [Pp. xii+248+subject index.] Price DM 59, – –/US $32.50 hardback. Electronic and Vibrational Properties of Point Defects in Ionic Crystals. By Y. Farces and M. P. Fontana. (Amsterdam: North Holland, 1979.) [Pp. xix+247+author and subject index.] Price Dfl. 100.00/US $48.75 hardback. Defects and Diffusion in Solids. An Introduction. By S. Mrowec. (Amsterdam, Oxford, New York: Elsevier, 1980.) [Pp. 464+subject index.] Price Dfl. 170.00/US $83.00 hardback. Fluctuation Phenomena. Edited by E. W. Montroll and J. L. Lebowitz. (Amsterdam: North-Holland, 1979.) [Pp. 340+author and subject index.] Price Dfl. 120.00/US $58.50 hardback. Crystal Growth: A Tutorial Approach. Edited by W. Bardsley, D. T. J. Hurle and J. B. Mullin. (Amsterdam: North-Holland, 1979.) [Pp. vi+401+subject index.] Price Dfl. 110.00/US $53.75 hardback. Liquid Crystals. By S. Chandrasekhar. (Cambridge University Press, 1980.) [Pp. x+336+subject index.] Price £8.95 paperback. Light Scattering in Solids. Edited by J. L. Birman, H. Z. Cummins and K. K. Rebane. (New York: Plenum Press, 1979.) [Pp. xxi+523+author and subject index.] Price US $55.00 hardback (camera-ready typescript). Electrostatics 1979. Edited by J. Lowell. (London, Bristol: The Institute of Physics, 1979.) [Pp. 349+author index.] Price £32.00 hardback. Auger Electron Spectroscopy Reference Manual. By G. E. Mcguire. (New York: Plenum Press, 1979.) [Pp. ix+129.] Price US $39.50 paperback. Laser Handbook, Volume 3. Edited by M. L. Stitch. (Amsterdam: North-Holland, 1979.) [Pp. ix+846+author and subject index.] Price Dfl. 250.00/ US $122.00 hardback.

Investigation of the electronic structure of point defects in ionic crystals by the cluster scattered wave method with the self-consistent calculation of the lattice distortion and long-range polarization

A mechanism of interaction of dislocations with point defects is suggestedon the basis of a study of deformation-stimulated luminescence (DL)from irradiated KCl. DL is considered to be a result of destruction ofhole colour centres by dislocations. The centres destroyed by dislocationsare responsible for irradiation-induced strengthening of alkali halides.

We investigate the influence of elastic properties of point defects on dislocation climb under stress and irradiation. For this purpose, elastic dipole tensors and diaelastic polarizabilities are evaluated in aluminum for vacancies and self-interstitial atoms in their stable and saddle configurations, using density functional theory calculations. These parameters are introduced in an object kinetic Monte-Carlo code and a continuous diffusion model to estimate the stress dependence of dislocation climb, using a dipole of straight dislocations. We show that both parameters have an influence on absorption of point defects under stress, in agreement with previous analytical models. However, the effect of dipole tensor is found only 5 times larger than polarizability, whereas models predict a factor up to 30. In addition, including polarizability reverses the stress angular dependence when a uniaxial stress is applied orthogonal to the dislocation line, so in general polarizability cannot be ignored for simulations under applied stress. Further comparison with analytical models shows that they give a good description of angular dependence, provided saddle point configuration of point defects is not too anisotropic. For vacancies, which are strongly anisotropic in their saddle configuration, models fail to reproduce quantitatively lattice effects on stress angular dependence observed in simulations. Calculations show that dislocation climb velocity under irradiation is expected to be the highest if the stress is approximately orthogonal to the dislocation line, especially along the Burgers vector, and the lowest if the stress is applied close to the 〈100〉 direction with the largest projection on the dislocation line.

Stress-biased loop nucleation in irradiated type 316 stainless steel

The paper briefly reviews the development of the atomistic method of calculating the properties of defects in ionic crystals, as it derives from the early work of Mott and Littleton (1938). The physical principles and assumptions of the method are described, although the main emphasis is on the development of the technique. It is pointed out that appeal to the quasi-harmonic approximation has made it possible to extend the technique so that the characteristics of defects can now be successfully obtained as functions of temperature. The insights into atomic transport phenomena which these calculations provide are illustrated by three examples.

Methods described in a previous paper are applied to the calculation of the electronic structure of the $F$ center in KCl, KBr, KI, and NaCl. Structure of the ions is explicitly taken into account in a region close to the defect site, and an effective-mass method is used in the more distant regions. Dielectric polarization effects are included on the basis of the formulation due to Toyozawa, Haken, and Schottky, but results in best agreement with experiment are obtained by a procedure which deviates somewhat from that formulation, as might be expected also on purely theoretical grounds. It does not appear to be possible to obtain agreement with experiment even for absorption energies if polarization effects are neglected. The calculated energy differences of the $1s$, $2p$, $3p$, and $2s$ states in absorption agree very well with presently available experimental data. Agreement with experiment is fairly good for the $1s$, $2s$, and $2p$ energies in emission. The calculated optical transition matrix elements are very sensitive to the details of the model; those obtained from our best model and wave functions are compatible with experimental results. The results for the spin density at the nearest-neighbor ions are reasonably close to the experimental values.

CALANIE: Anisotropic elastic correction to the total energy, to mitigate the effect of periodic boundary conditions