Impurity-vacancy interactions in the alkali halides

Volume 39, number 3

CHEMICAL PHYSICS LEITERS

MWRITY-VACANCY

INTERACTIONS

I May 1976

IN THE ALK.Al!J HALIDES

C.R.A. CATLOW* Department of TheoretictzlChemistry, Oxford OXI 3QR, UK Received X2 December 1975

The interaction energies of vacancies with divalent impurities are. calculated using recently developed efficient computer simulation methods. The results show that nearest neighbour and next nearest neighbour complexes are roughly equally bound, thus emphasising the importance of including both types of defect pair in anaIyses of spin resonance and other experimental data on doped alkali halides. We also find that a simple Coulomb expression for defect interactions gives energies close to those obtained by the full mlculation, for all but the nearest neighbour complex. This result encourages the use of Debye-Hiickel treatments of defect activities in alkali halide crystals.

1. Introduction

(a) nn (1101 OOPANT-

Divalent cations, when doped in the alkali halide crystals occupy lattice cation sites. The electroneutrality of the crystal is &ten maintained by the creation of cation vacancies; and the defect properties of the crystal are largely controlled by the interaction between these vacancies and the impurity cations. The nature of such interactions has been studied principally by the spin resonance of magnetic divalent ions e.g. Mn2+ (see e.g. Watkins [l] )_ Other techniques used have included dielectric relaxation, as well as conductivity and diffusion studies. These, however, give less detailed information than the spin resonance studies; a review is given by Corish end Jacobs f2]. The results of these experimental studies have not presented a clear picture. There are evidently two important impurity-vacancy clusters: the nearest neighbour (110) and next nearest neigbbour (200) pairs, shown in figs. la and lb. But the question of their relative concentrations has not been settled. Earlier work of Watkins [1] demonstrated the presence of both clusters, and sugested roughly equal binding energies for the two complexes. Similar conclusions were reached in studies of Fong and Wong [3] and Fong [4] _SeverZJstudies have, however, ignored the. existence of-two * On attachsient Herwell.

to Thexeticd

IMPURITY

PAIR LATTICE

CATION

IHPURITI

CATION

x

LATTICE

ANION

q

CATION

VACANCY

(b)

(Cl

I2101

DOPANT

u4pumT~

PAIR

Physics Division, A.E.R.E.,

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‘CHEMICAL

PHYSICS

-&es of defect; and the evidence regarding their re!ative binding energes remains inconclusive. A second question cbncerns the reliability of the simple Coulomb ‘expression for the defect interaction energy. Here the binding energy is written as: F-&j

= Crlcr&f,

(1)

where q1 ana q2 are the effective charges of vacancy and impurity, and ~0 is the macroscopic dielectric COIL stant of the material. The validity of simple DebyeHiickel treatments of ionic interactions is determined by this, amongst other factors. Fong [4] suggested the use of the Coulomb expression for all but the nearest neighbour pairs; but no justification for this assumption has been obtained. In the present paper, we report a theoretical study of vacancy-impurity interactions in an attempt to resolve the problems outlined above. We will calculate i$e binding energies of the three clusters shown in figs. la-lc, using methods which have been shown to be reliable. Our study will concern only KC1 for which there is the greatest experimental data. This work follows earlier calculations of Ramdas et al. [S] _ Our results have some resemblance to these earlier calculations, but there are important differences which we can specifically attribute to improvements in the lattice models used in our calculations.

2. Calculations and lattice potentials Our calculations are performed using standard methods for the estimation of defect energies; these meth$ds are now programmed as a general package -- HADES - applicable to defects in cubic ionic crystals. The methods used by the program are described by Norgett and Lidiard [6] and by Norgett [7] _They are based ORatomistic simulation of the defect and surrounding regions of the lattice, using specified interionic pair potentials. The minimum energy configuration of this region is then calculated by relaxation of every ion to zero force using a Newton-aphson procedure employing efficient fast matrix techniques discussed by Norgett and Fletcher [8]. The explicitly described inner region is surrounded by an outer regiotl which is .* HMvell

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Automatic Defect Evduarion System.

LETTERS

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1 May 1976

treated as a-dielectric continuum, following the method-of Mott and Littleton [9]. Otir potential for the interaction of lattice ions is taken from the recent work of Catlow et al. [lo], who obtained a comprehensive set of potentials for the alkali halide crystals by fitting to bulk elastic and dielectric data. An important feature of their potentials is the use of the shell mm&l in describ@ ionic polarisation. This model, proposed by Dick and Overhauser [I I] describes the ionic dipole as a shell displaced from a core to which it is connected by an isotropic harmonic spring constant. LJrJike the simpler point dipole model, used in the earlier calculations of Ramdas et al. [5], the shell model permits the correct simulation of both elastic and dielectric properties of a crystal. The latter is particularly important for calculations on defects in ionic crystals, as the lattice relaxation around a defect is principally due to the dielectric response to the electrostatic perturbation of the lattice provided by the effective charge of the defect. The importance of using the shell model in defect calculations has clearly been shown by work of Faux and Lidiard [12] and Catlow and Norgett [13]. The short range potential for the impurity-lattice interactions are taken as identical to those for the interactions involving lattice cations; similarly, for the polarisation parameters we take the values derived by Catlow et al. [lo] for K”. The~impurity is given a charge of i2 and is therefore treated entirely as an electrostatic perturbation of the lattice; the elastic perturbations due to differences between the ionic radii of the impurity ion and K+ are thus ignored. The latter will unquestionably be much smaller than the electrostatic term; they may, however, be sufficient to cause small differences in the binding energies ef vacancies to different divalent dopants, which cannot be reflected in OUTresults. Our calculated energies are reported in tables 1 and 2; in the-former we give the energies of the isolated impurity and vacancy, and in the latter the total energies of the three defect pairs shown in figs. la-lc, together with their binding energies with respect to the isolated components. The calculations are all performed with al00 ions in the explicitly simulated region; the work of Catlow et al. [lo] has shown that the calculated defect energy is no longer sensitive to the size of the region when this number of ions has been included. We should note that such insensitivity of the calculated

Volume 39, number 3

CHEMICAL PHYSICS LETTERS

Table 1 Isolated defect energies Defect

Energy (eVJ

divalent subsritutional ion’) cation vacancyb)

-11.66 4.92

a) Calculnted e~~ergyis that required to remove a lattice cation to infinity and replace it by a diva!ent ion, also introduced

from infinity.

b) Calculated energy is that required to remove a to infinity.

Tab!e 2 Energies of defect

lattice

cation

clusters

Cluster

(110) nearest neighbour pair (fig. la) (200) next nearest ncighbour pair (fig. lb) (210) pair (fig. ICI

Absolute enerm (eV)

Binding energy (eV)3)

-7.35

-O-61(-0.73)

-7.32

-0.58(-0.51)

-7.11

-O-37(-0.41)

a) Given with respect to isoIated defect energies reported in table 1. Value quoted in brackets is obtained from the simple Coulomb formula.

energy to expansion of the inner region is a feature cf shell model calculations which is not found when point dipole models are used; the reliability of the results using point dipole potentials is thus seriously reduced. 3. Discussion Our results show that there is little difference between the binding energies of vacancies in the nearest neighbour and next nearest neighbour sites with respect to a substitutional impurity. This is in line with the apparent detection of both centres in spin resonance [ 11 and other studies. It may also explain the confusion that has arisen-from experimental studies over the relative concentrations of these defects. As if their binding energies are roughly equal, then given different entropies of formation, the relative concentration of the defects may be changed by a detectable amount within a small temperature range. Different experimental conditions may therefore produce quite different results.

1 May

1976

The magnitude of our calculated binding energies may be compared with the experimental data for a variety of dopants (see Corish and Jacobs [2]). The experimental binding energies are generally close to 0.5 eV, but ranging from 0.43 eV-0.58 eV. It is not clear whether these variations represent experimental error, or genuine differences between the binding energies of different ions. Our ca!culated energies, however, fall within the range of the experimental values; the agreement is therefore satisfactory. On comparing our calculated energies with those predicted from eq. (1) (see table 2), which we recall assumes solely a simple Coulomb term for the defect interaction energy, we find good agreement for all but the nearest neighbour pair for which the detailed calcu!ation gives a smaller binding energy than the approximate expression_ This is an important result, as it justifies the application of the Debye-Hiickel theory to defect solutions (when suitably modified for the discrete nature of lattice, as by Allnatt and Rowley [14] ); the assumption of a simple Coulomb interaction between defects, on which such theories are based, only fails for nearest neighbour pairs which may be included separately in the theory as an ion pair. Finally, our calculations provide a good illustration of the detailed differences that may arise between defect calculations using the shell model and those empioymg point dipole potentials. Our results are broadly similar to those of Ramdas et al. [S] - But tfre use of the point dipole models, we find, has distorted the result of this latter study. The calculated binding energies are smaller than given by our calculations. This reflects the overestimation of polarisation energies by the point dipole models (which predict static dielectric

that are considerably in excess of the experimental values). The energies of electroneutral clusters are thus underestimated compared with those of their charged components. The ‘same factor has led the point dipole calculations to give quite appreciably larger binding energies for the next nearest neighbour than for the nearest neighbour pairs, as the greater dipole moment of the former is favoured by a model which exaggerates the dielectric response of the lattice. The use of reliable lattice models is therefore required not only for ensur;ag correct quantitative agreement with experiment, but for obtaining a reliable qualitative guide to interpretation of the data. constants

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CiTEMICAL PHYSICS LMTERS

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1 May 1976

References

-

-The work reported in the paper_ has led to two irn-

’ porta@ qualitati$ conch&m: first, that nearest neigh. hour andnext nearest neighbcur defect pairs have ioughly equal b&i& energies and consequently both &ay be present in appreciable concentrations in doped alkali h&ides crystal; second, that the simple Coulomb ternt-gives an adequate representation of defect interaction energies for all but the nearest neigbbour pair:. the apphcation of Debye-Hiickel and related treatments is thus encourage& However, we have also shown how stlch results depend upon the use of accurate lattice models. This, we believe, applies in general to de.fect studies.

111 G.D. Watkins,phys. Rev. 113 (1959) 79. 12] J. Corish and P.W.M. Jacobs, Chemical Society Specialist [3]

[4] [5]

PI [71 rg1 [91 [lOI [ill

&knowIedgement

(121

I am grateful to Professor C.N.R. Rae for sugsting this study.

I131

I141

Periodical Reports, Vol. 11, Surface and Defect Roperties of Solids (2973) p. 162. F.K. Fongand E.Y. Wang, P’nys.Rev. 162 (1967) 348. F.K. Fang, Phys. Rev:187 (1969) 1099. S. Ramdas, A-K. Shukla and C-N-R. Rao. Chem. phvs. Letters 16 (1972) 14. A-B. Lidiid and M.J. Norgett, in: Computational solid state physics, eds. F. Herman, N.W. Dalton and T.R. Koehler (Plenum Press, New York, 1972} p. 385. M.J. Norgett, A.E.R.E. Report AERE-R 76.50. M.J. Norgett and R. Fletcher, J. Phys. C 3 (1970) L190. N.F. Mott and M.I. Lit&ton, Trans. Faraday Sot. 34 (1938) 48.5. C.R.A. Catlow, X.hf. DiJJer and h1.J. Norgett, J. Phys. C, to be published. B.G. Dick and A.W. Gverhauser, Phys. Rev. 133A (1958) 419. I.D. Faux and A.B. Lidiard, 2. Naturforsch. 26a (197i) 62. C.R.A. Catlow and M.J. Norgett, J. Phys. C 5 (1972) L237. A.R. Allnatt and L.A. Rowley, J. Chem. Phys. 53 (1970) 3217.