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Nous avons utilisk des techniques fiables de simulation sur ordinateur et de prkcis ... dopant cations for NaCI, KC1 and KBr containing Mg2+ and for the NaCl  ...
Dopant aggregation and precipitation in alkali halides doped with divalent ions J. Corish, J. Quigley, C. Catlow, P. Jacobs

To cite this version: J. Corish, J. Quigley, C. Catlow, P. Jacobs. Dopant aggregation and precipitation in alkali halides doped with divalent ions. Journal de Physique Colloques, 1980, 41 (C6), pp.C6-68-C671. .

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JOURNAL DE PHYSIQUE

Colloyue C6, supplkment au no 7 , Tome 41, Juillet 1980, page C6-68

Dopant aggregation and precipitation in alkali halides doped with divalent ions J. Corish (*), J. M. Quigley (*), C. R. A. Catlow (**) and P. W. M. Jacobs (***) Department of Chemistry, University College, Dublin 4, Ireland

Rhumb. - Nous avons utilisk des techniques fiables de simulation sur ordinateur et de prkcis potentiels d'interaction pour calculer les Cnergies de liaison pour une sQie de structures possibles pour des agregats mettant en jeu deux ou plusieurs impuretks cationiques dans NaCI, KC1 et KBr dopks par Mg2+ ainsi que dans le systeme NaCl : BaZ+.Nous avons considkrk des mecanismes d'agrkgation impliquant a la fois des dip6les de type nn et nnn. L'importance relative des ktapes transitoires a Btk examinke. Nous dkrivons egalement des resultats obtenus avec les techniques d'knergie de rkseau du modele de Born sur les Cnergies de precipitation de la phase mktastable de Suzuki dans ces systemes. Abstract. - We have used reliable computer simulation techniques and accurate lattice and defect interionic potentials to calculate the binding energies of a range of possible structures for clusters involving two or more dopant cations for NaCI, KC1 and KBr containing Mg2+ and for the NaCl : BaZt system. Aggregation mechanisms involving both nn and nnn dipoles have been considered and the relative importance of alternative pathways examined. We also report results obtained using Born-Model lattice energy techniques on the energetics of metastable Suzuki-phase precipitation in these systems.

1. Introduction. - We present results of a theoretical survey of the aggregation of impurity-vacancy dipoles in alkali-halides doped with divalent cations. This has been undertaken to help resolve difficulties in the analyses of conductivity data [l-31 and thermal depolarization experiments [4, 51 and because of the inherent interest in the kinetics of decay of these dipoles [6-101 and in precipitation in these systems [I 1141. Our study includes calculations of the energy of the metastable Suzuki phase precipitates. A theoretical survey of this type is necessary in view of the complexity of these aggregation processes and because previous theoretical studies in this field [8, 15-19] used methods and lattice potentials which have since been shown to be inadequate. 2. Method and potential. - The energies of the defects were calculated using the HADES program [20] which employs a generalized and automated Mott-Littleton procedure [21]. The effect of changing the number of ions in region I, which immediately surrounds the defect, was investigated and it was found to be adequate, even for the largest aggregate studied, if 171 sites were treMed explicitly. The lattice (*) Department of Chemistry, University College, Dublin 4, Ireland. (**) Department of Chemistry, University College, London,

U.K. (***) Department of Chemistry, University of Western Ontario,

London, Ontario, Canada.

energy of the Suzuki phase was calculated using the PLUTO [22] program and the unit cell was given the same lattice constant as that of the host crystal. The potentials have been listed previously [23]. 3. Stability of aggregates. - We shall compare the energies of the aggregates with those of both isolated impurity ions and vacancies and also with those of the various combinations of dipoles from which the aggregates may form. Thus in table I we give the calculated defect energies for these possible component species : U," is the energy to remove a cation from the perfect crystal to infinity; U(M2+) is the energy of a dopant ion on a substitutional cation site ; Ual is the defect energy of a (I 10) nn vacancy-impurity ion complex and U,, that of a (200) nnn complex with AUal and AUa2being the energy changes accompanying the formation of these dipoles from their isolated constituents. The trend evident here i.e. that the nn complex is more strongly bound for the larger impurity ions in a given host while the nnn complex predominates for the smaller substitutional ion, has been found to extend to the full range of systems for which calculations have been made [23]. The structures for which defect energies are reported here are illustrated in figure 1 and the calculated energies U(x) are listed in table 11. The estimation of the energy changes AU(x) on the formation ofa cluster x is made by comparing this defect energy with that of the isolated dopant ions and vacancies involved in its

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980618

DOPANT AGGREGATION AND PRECIPITATION IN ALKALI HALIDES DOPED WITH DIVALENT IONS

C6-69

Table I. - Calculated defect energies and dipole binding energies. All energies are in units of eV and the symbols are defined in the text. System u? U(M2+) ua 1 ua 2 Aua~ Aua~ -

-

-

NaCl : Mg2+ NaC1 : Ba2+ KC1 : Mg2+ KBr : Mg2+

4.929 4.929 4.866 4.573

13.210 - 8.613 - 14.134 - 13.249

-

8.837 - 4.365 - 9.865 - 9.241

-

-

-

8.844 4.145 9.932 9.299

-

-

- 0.556 - 0.681 - 0.597 - 0.565

-

0.563 0.461 0.663 0.623

Table 11. - Calculated defect energies U(x)for the aggregates a-j illustrated injigure 1. All energies are in units of eV. a

b - 17.562 - 18.171 - 9.015 - 8.556 - 19.227 - 20.463 - 18.045 - 19.161 -

System -

NaCl : Mg2+ NaC1 : Ba2+ KC1 : MgZ+ KBr : MgZ+

U(x) d e f h g 17.717 - 17.801 - 27.025 - 27.397 - 26.158 - 27.308 8.871 - 8.934 - 13.871 - 12.800 - 13.562 - 12.761 19.692 - 19.746 - 29.992 - 30.842 - 29.014 - 30.778 -28.889 -27.150 -28.823 18.460 - 18.514 -28.116 c

j

1

-

-

- 25.940

- 36.962 - 17.285 - 41.741 - 28.786 -27.046 - 39.088

(*)

(*) .Not minimized correctly due to excessive displacement of the Na+ ion which is in nn position to the three vacancies.

Table 111. - Calculated association energies for the aggregates shown in jigure 1. All the energies AU/n are in units of eV. They are expressedper vacancy-impurity ion pair and are relative to the same number of isolated vacancies and impurities. System

a

b

c

d

-

-

-

-

-

NaCl : Mg2+ NaCl : BaZ+ KC1 : Mg2+ KBr : Mg2+

0.500 -0.824 - 0.345 - 0.347

- 0.805 -0.594 - 0.963 - 0.905

-

- 0.577 . - 0.616 -0.804 -0.751 - 0.577 - 0.616 - 0.554 - 0.590

A U(x)ln, e f - 0.727 - 0.851 -0.940 -0.583 - 0.729 - 1.012 - 0.696 - 0.954

g - 0.438 -0.837 - 0.403 - 0.374

h

i -

- 0.821 -0.570 - 0.991 - 0.932

-

0.366

( ) - 0.327 - 0.339

j - 0.960 -0.637 - 1.167 - 1.096

(*) Value for U(h) for this system not available, see table 11.

1

Fig. 1. - Structures of the dipole aggregates for which defect energies are given in table 11. The ions of the host crystal are not a cation represents a divalent impurity ion and shown ; vacancy.

formation. If n, is the number of vacancies or impurity ions in the cluster then AU(x)

=

U ( x ) - n, { U y

+ U ( M 2 + )j .

Table I11 gives the values of AU(x)/n, i.e. the energy

of formation of the cluster per vacancy-impurity ion pair which it contains. The most striking feature of these results is that the preferences evident in the formation of nn as against nnn dipoles, as seen in table I and to which we have already referred, are more pronounced in the formation of the aggregates. The changes in energy which occur as the dipoles aggregate are given in table IV where the defect energies of the aggregates (a)-(j) are compared with those of their precursors. Dimers are taken to form from the relevant dipoles but for the higher aggregates the energy changes are given for formation both from isolated dipoles and from possible intermediates where these energies have been calculated. We may now use the results of tables I11 and IV to predict the likely fate of either nn or nnn dipoles in these systems but it should be emphasized that the actual aggregation behaviour in any particular case will depend on the relative concentrations of these dipoles initially present. If we consider magnesium as dopant then the nn dipoles are most likely to aggregate first to the dimer (d) and hence to the hexagonal trimer on the (1 11) plane (e) [15].

J. CORISH, J. M. QUIGLEY, C. R. A. CATLOW AND P. W. M. JACOBS

For the ~ g ion~ aggregates + studied here the existence of this aggregation pathway for the nn dipoles is vital since no other cluster (with the possible exception of the linear dimer (c) which is marginal) was found to be more stable than the isolated dipoles. For the NaCl : Ba2+ system, where nn dipoles are expected to predominate (cf. Table I), the trimer is even more stable but now the formation of the nn dimer (a) oriented in the (100) plane becomes a competing process and would ultimately result in the (100) trimer (g) which is now also found to be stable. The nnn dipoles in the Mg2+ doped systems were found to show a progressive stabilization as a result of aggregation through the nnn dimer (b) to the tetramer (j). The addition of a third dipole to this dimer results in either the planar trimer (f) or the outof-plane trimer (h) which are of comparable energy. The sequential addition of further nnn dipoles would lead to the formation of a Suzuki phase the stability of which we will consider in the next section. It is noteworthy that in the NaCl : Ba2+ system the same sequence of stabilities is evident for any nnn dipoles present although magnitudes of the energy changes are markedly less favourable than in the Mg2+ systems. 4. Energies of Suzuki phases. - The calculated values for the lattice energies per M' X2 : 6MX unit of the Suzuki phases, ULs, and those of the host crystals, U,, are given in table V. The former were

Table V. - Calculated lattice energies for Suzuki phases and host crystals and energy changes which occur when the Suzuki phase forms from isolated defects. All energies are in eV and the symbols are defined in the text. The Set I potentials of Catlow et al. [17] were used in these calculations. System

-

l i C(

0

5 I s 3.d

E

C)

c

B

*

2

2 r

9

NaCI : MgZ+ NaC1 : Ba2+ KC1 : Mg2+ KBr : M g 2 +

ULS

- 72.171 - 67.113 - 68.911 - 64.663

UL -

- 7.932 - 7.932 - 7.277 - 6.864

A~SI

*

AUs2

-

-

- 1.392 - 0.764 - 1.623 - 1.441

- 0.847 - 0.320 - 1.017 - 0.908

calculated using the PLUTO program to which a Newton-Raphson minimization routine had been added. From a thermodynamic analysis of the solution process and as has been, shown previously [16] the quantity which must be compared with the energy of the isolated defects to obtain the relative stability of the Suzuki phase is ULs - 8 U,. Thus

k

-$8 -r 8

represents the stabilization energy per impurity ion of the Suzuki phase compared with isolated vacancies and impurities.

DOPANT AGGREGATION A N D PRECIPITATION IN ALKALl HALIDES DOPED WITH DIVALENT IONS

gives the analogous energy relative to nnn dipoles. AUsl is directly comparable with the energies given in table I11 and whereas it is evident that the Suzuki phase is more stable than any of the aggregates this effect is much more pronounced in the Mg2+ doped systems. This is in agreement with the available experimental evidence (summarized in reference [16]). Thus Suzuki phases are expected to precipitate when the impurity ion is small relative to the host cation. The same aggregation sequence is available for nnn dipoles in the NaC1 : Ba2+ system although with much less favourable energetics. In addition our calculated

C6-71

dipole binding and aggregation energies suggest that most of the dipoles will be of the nn type which will then form the (1 11) trimer (e). Since rearrangement of this trimer or of the linear dimer (c) to nnn type clusters is energetically unfavourable we may conclude that formation of a Suzuki phase in this system is not likely. Acknowledgment. - J. M. Q. and J. C. thank Amdahl Ireland Ltd for the use of the 470/V6 computers and the Computer Centre U. C. Dublin for facilities and the arrangements with Amdahl Ireland.

DISCUSSION Question. - F . GRANZER.

In calculating the Suzuki phase, a random distribution of dipoles ?

you start with

Reply. - J . CORISH.

No, we start with the superlattice cell and allow the ions to relax to their equilibrium positions during the calculation.

Question. - L. W . HOBBS. I presume you have not included the elastic strain energy term which is present for nucleation of Sumki phase. This presents an energy barrier which must be overcome so long as the matrix-Sumki phase interface remains coherent. In fact, nucleation is observed to take place heterogeneously, along dislocations, etc., and the interface eventually loses coherency by shear.

- J. COR1sH. Your presumption is quite correct.

References [l] BROWN,N. and JACOBS,P. W. M., J. Physique Colloq. 34 (1973) C9-437. [2] CHAPMAN, J. A. and LILLN, E., J. Physique Colloq. 34 (1973) C9-455. [3] GUERRERO, A. L., JAIN,S. C. and PRATT,P. L., Phys. Status Solidi (a) 49 (1978) 353. P. W. M. and MOODIE,K. S., Phys. [4] HOR, A. M., JACOBS, Status Solidi (a) 38 (1976) 293. [5] KIRK,D. L. and INNES,R. M., J. Phys. C . 11 (1978) 1105. [6] COOK,J. S. and DRYDEN, J. S., Proc. Phys. Soc. 80 (1962) 479. J. S. and HARVEY, G. G., J. Phys. C 2 (1969) 603. [7] DRYDEN, [8] CRAWFORD, J. H. Jr., J. Phys. Chem. Solids 31 (1970) 399. [9] UNGER,S. and PERLMAN, H. M., Phys. Rev. B 10 (1974) 3692. 101 DIENES,G. J., Semicond. and Insul. 4 (1978) 159. 111 SUZUKI,K. and MIYAKE, S., J. Phys. Soc. Japan 9 (1954) 702. 121 SUZUKI, K., J. Phys. Soc. Japan 10 (1955) 794. 131 SUZUKI,K., J. Phys. Soc. Japan 16 (1961) 67.

[14] SORS,A. I. and LILLN, E., Phys. Status Solidi (a) 32 (1975) 533. [I51 STRUTT, J. E. and LILLEY, E., Phys. Status Solidi (a) 33 (1976) 229. [16] BOSWARVA, I. M., Phys. Status Solidi (a) 37 (1976) 65. I171 BERG,G., FROHLICH, F. and SIEBENHUNER, S., Phys. Status Solidi (a) 31 (1975) 385. [I81 BERG, G., FROHLICH,F. and SIEBENHUNER, S.; Krist. und Technik 10 (1975) 1091. [19] BERG, G., FROHLICH,F. and SCHNEIDER, D., Phys. Status Solidi (a) 42 (1977) 73. [20] NORGETT, M. J., AERE Harwell Rep. ~76'50(1974). [21] MOTT,N. F. and LITTLETON, M. I., Trans. Faraday Soc. 34 (1938) 485. 1221 CATLOW,C. R. A. and JAMES,R., Nature 272 (1978) 603. [23] CATLOW,C. R. A., CORISH,J., QUIGLEY,J. M. and JACOBS,P. W. M., J. Phys. Chem. Solids 41 (1980) 231.