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THE ENERGETICS OF SHEAR PLANE
FORMATION IN REDUCED TiO2
R. James, C. Catlow
To cite this version:
JOURNAL DE PHYSIQUE Colloque C7, supplement au n° 12, Tome 38, dtcembre 1977, page C7-32
THE ENERGETICS OF SHEAR PLANE FORMATION IN REDUCED Ti0
2R. JAMES and C. R. A. CATLOW
Department of Chemistry, University College London, 20 Gordon Street, London WC1H OAJ, U.K.
Résume. — Des calculs exacts d'énergie de réseau sont utilisés dans une étude théorique des
énergies de formation de plan de cisaillement dans T i 02. Les énergies calculées sont en bon accord
avec les valeurs déduites de données expérimentales. De plus, à partir de ces résultats, on met en évidence des facteurs apparemment critiques pour la stabilisation des plans de cisaillement. On montre que la condition la plus importante est celle d'une relaxation étendue au voisinage des plans de cisaillement. La valence du cation réduit peut aussi jouer un rôle important. On examine l'effet de dopants, et on montre que, outre l'effet de valence, l'interaction dopant-plan de cisaillement peut provenir de la variation de la taille de l'ion et de sa polarisabilité.
Abstract. — We use exact lattice energy calculations in a theoretical study of the energetics of shear plane formation in T i 02. Our calculated energies agree well with values deduced from
experi-mental data. Moreover from our results we identify a number of factors that are apparently critical in stabilising shear planes. We show that the most important requirement is for extensive relaxation in the region of the shear plane. The valence of the reduced cation may also play an important role. We consider the effect of dopants, and find that in addition to the valence effect, dopant-shear plane interactions may arise from variations in ion size and polarisability.
1. Introduction. — Electron microscopy studies on
a number of transition metal oxides, e.g. T i 02_x [1]
have shown that the defect structure of non-stoichio-metric phases may be based on extended rather than point defects. In these defects, planes of oxygen ions are eliminated from the crystal with an accompanying crystallographic shear. Considerable detail is now available on the structure of shear planes in T i 02_x
and other oxides [1]. Moreover, extensive studies are reported on the reduced oxide at larger deviations from stoichiometry. An important feature of this work is the discovery of an homologous series of reduced . oxides Ti„02 n_i, whose structures contain ordered
arrays of shear planes in which the interplane separa-tion increases with n.
There remain, however, a number of fundamental questions concerning the energetics of shear plane formation and the relationship between point and extended defect structures. The nature of shear plane interactions — clearly of considerable importance in view of the observation of shear plane ordering, as discussed above — is also not fully understood.
We have therefore undertaken a detailed study of these problems, of which preliminary results are reported in the present paper. Our discussion will use theoretical estimates of shear plane energetics based on exact atomistic calculations. Our primary aim is to identify the factors controlling shear plane formation, and to explain why extended defects are formed in such a restricted range of compounds. We also consider briefly the question of the point defect structure of
crystals containing shear planes. Finally we discuss the properties of doped crystals, in particular the effect of dopants on the stability and structure of extended defects.
We describe our calculations in the following section ; they are at present restricted to T i 02, but they
lead to general conclusions which will be of value in discussing the defect structure of related crystals.
2. Calculations. — Our study is based on lattice energy calculations on T i407 and T i509 for which
accurate structural data are available [2,3]. As remarked, these structures contain regularly spaced shear planes in a rutile host; the structures differ principally by the inter-plane spacing. The lattice energy calculations used the PLUTO program deve-loped recently by Catlow and Norgett [4]. This is an efficient general module in which both electrostatic and short range contributions to the lattice energy are summed to high accuracy ; in the former summation, convergence is improved by the use of the Ewald transformation. The high efficiency of the computa-tional methods is essential for the feasibility of the calculation on the large, low symmetry unit cells for which we report results in this paper.
The reliability of our lattice energies is determined entirely by that of the lattice potentials employed by the calculations. We use the fully ionic model for T i 02,
with a shell model treatment of ionic polarisation and short range interionic forces described by simple analytical forms of the Born-Mayer or Buckingham
ENERGETICS OF SHEAR PLANE FORMATION IN REDUCED TiO, C7-33
forms. The variable parameters in the model are obtained by fitting to empirical data, comprising elastic, dielectric and structural properties of the solid. Details of the potential will be published shortly [5]. The use of this type of lattice model in calculations of defect energetics is justified by the success of similar studies on other oxides [6, 71.
3. Result and discussion.
-
3 . 1 POINT DEFECT PROPERTIES.-
Before presenting our calculated shear plane energies, we consider briefly the point defect structure of reduced rutile.The calculation of reliable point defect energies in TiO; is difficult, as the high dielectric constant of the niaterial results in very extensive lattice relaxation around the defect. However, preliminary results have been obtained, using the HADES program [8], by James [9]. These calculations clearly show that the anion vacancy would be the most energetically favourable point defect to compensate the Ti3+ cation created in reduced rutile. We suggest therefore that these defects may form in very near stoichio- metric Ti0,-, ; at higher values of x vacancies will aggregate to form shear planes - possibly by 'the mechanism discussed by Anderson and Hyde [lo].
3.2 SHEAR PLANE CALCULATION.
-
Our calculatedlattice energies for the Ti,O, and Ti,O, structures are given in table Ia. We have assumed in both structures that the Ti3+ ions are located at the sites adjacent-to the shear plane (see section 3.3.2). From the results we obtain the lattice energy term associated with the elimination of 0'- and creation of Ti3+ in TiO, with accompanying formation of the shear plane. We express the results as the lattice energy change per eliminated oxygen atoin ; values are given in table Ib.
Shear plane formation energies a ) Lattice energies of observed structures
Lattice energy (eV)
b) Shear plane energy
Shear plane energy per eliminated 0' - (eV) Ti4.07
' Ti,O, Experimental (*)
(*) PICARD, C. and GERDANIAN, P. Ill].
In order. to compare our calculated values with experiment, we deduce an experimental shear plane lattice energy, via a Born-Haber cycle, from the thermodynamic data of Picard and Gerdanian [ l l ]
(for greater details of this procedure see James [9]). From the measured partial molar heat of reduction of Ti0,-, at x = 0.01 - a value of x at which shear plane rather than point defect formation is probable -
we obtain an experimental shear plane energy of 100.0 eV. This is close to our calculated values given in table Ib. The results are not strictly comparable, as the shear plane interactions in Ti407 and TiiO, will be considerably larger than in TiO,.,, where the experi- mental enthalpy is measured. However, the shear plane interaction energies are low (of the order of 1 eV, as discussed below) relative to the formation energies. The close agreement between theory and experiment thus confirms the reliability of our calculations ,for studying shear plane energetics.
Our estimate of
-
1 eV for shear plane interaction energies is obtained from the difference between our calculated shear plane energies for Ti407 and Ti,O,. Although this energy is small compared with the formation energy, it confirms, nevertheless, that appreciable interactions between shear planes exist over spacings several times the lattice parameter. As commented in the Introduction, it is these long range interactions that are responsible for the ordering between shear planes observed in the homologous series of reduced oxides of which Ti40, and Ti,O, are members.3 . 3 CRITICAL FACTORS IN SHEAR PLANE STABILISA- TION. - In this section we present the results of calculations which have helped us to identify the following factors which play an important role in stabilising shear planes.
3.3.1 Ion relaxation.
-
In the Ti407 and Ti,O, structures discussed above, the titanium ions neigh- bouring the shear plane,are displaced by-
0.3A
from the centres of their oxygen octahedra. This large relaxation is caused by the strong repulsive electro- static forces acting between titanium ions across. the plane. To investigate the magnitude of the stabilisation energy arising from these relaxations we have calculat- ed the energy of the ideal structure of Ti,O,, in which titanium ions remain at the sites they occupy in the perfect rutile structure. Shear plane formation is less favourable by 10.1 eV. A change in energy of this magnitude would almost certainly destabilise the shear plane relative to a point defect structure based on anion vacancies or simple aggregates of these defects. It appears therefore that metal ion relaxationis essential to shear plane formation
-
a point which is further emphasised by calculations reported in table 11. We find that if cation displacements are permitted within an otherwise ideal Ti,O, structure, the relaxa- tion energy approaches that for full relaxation from the ideal to the observed structure ; the energy asso- ciated with oxygen displacement is therefore a relati- vely small proportion of the total relaxation energy.R. JAMES AND C. R. A. CATLOW
TABLE I1
Effect of lattice relaxations on shear planevenergies
Ti407 (298 K) Ti509
Shear plane Shear plane energy (eV) energy (eV)
- -
Observedstructure 94.82 (97.88)- 96.20 (100.69)
Ideal structure (**) - 106.31 (1 10.58)
Ideal structure af- ter allowing the Ti3+ ions on the shear plane to re-
lax. - 97.82 (1 10.58)
Figures in btackets are the shear plane energies before core-shell relaxation (i.e. in an unpolarisable lattice).
'(**) After BURSILL, L. A. and HYDE, B. G. [I].
placement polarisabilities in shear.plane stabilisation. In these calculations we allowed relaxation of shells but not of cores in the ideal structure. For the shell model potentials used in all calculations (see section 2) this corresponds to electr.onic without nuclear relaxa- tion. The relaxation energy is small compared with the total value of 10 eV. Large displacement polari- sation is therefore essential to shear plane stability. A measure of the extent to which a crystal will permit ion displacement isgiven by the static dielectric constant, go. The large value of E, (- 100) in TiO, is therefore consistent with the large measured relaxa- tion of cations neighbouring the shear plane. Moreover for SnO, where reduction does not apparently lead to shear plane formation [12], a much smaller static dielectric constant is measured. We suggest therefore that in the rutile structure, shear planes will form onIy
in those crystals with high static dielectric constants.
The importance of a high value of E, has been suggested previously [12, 131. Our calculations have provided a specific atomistic explanation of this,.correlation. The high dielectric constant reflects the ease with which large ion displacements may occur in the material ;
such displacements are essential for shear plane sta- bility.
3.3.2 Shear plane-electron binding.
-
We have investigated the binding to the shear plane of Ti3+ ions - that is the electrons produced on reduction of TiO, - by calculating the energies of the observed Ti407 and Ti,Og structures in which, however; the trivalent cations instead of occupying the sites adja- cent to the shear plane, are displaced first to nearest neighbour (n.n.) and then to next nearest neighbour,' cation (n.n.n.) sites with respect to the shear plane. (Displacement to more distant sites is not possible in the Ti407 and Ti509 structures.) From the results we deduce a relatively small decrease (- 0.1 eV) in the binding energy for displacements to n.n. sites. Displa-cements to n.n.n. sites, however, result in a bigger decrease ( N 0.6 eV).
These binding energies are sufficiently large to have pronounced effects on electrical conductivity. They are, however, small compared with the relaxation effects discussed above. Electron-shear plane interac- tions do not therefore appear to play a major role in stabilising the extended defects. Greater binding energies would, of course, follow from calculations in which larger displacement of electrons from the shear plane were possible. It is unlikely, however, that this would affect our qualitative conclusions.
3.3.3 Valence. - In the above calculations we have assumed that the reduction of TiO, leads to the formation of trivalent metal cations. The calculations discussed in this section suggest, however, that if divalent ions were formed a big increase in shear plane stabilisation would result. We performed calculations on both Ti,07 and Ti,Og structures, with divalent rather than trivalent ions bound to the shear plane. We find that shear plane formation is stabilised by 11 eV (per eliminated 02-). This effect may be attri- buted mainly to the decrease in Coulomb repulsions across the plane which follows from the replacement of trivalent by divalent ion compensation. Polarisation energies are, however, important. Thus if the Ti2' ions are placed on the n.n. sites rather than on those adja- cent to the shear plane, a further small increase (0.3 eV) in the stabilisation of the shear plane is obtained. The lo& of Coulomb binding between the shear plane and the divalent ion is compensated by the increased polarisation energy which accompanies the charge separation.
The stabilisation of the shear plane by divalent ions is nevertheless not sufficient to result in Ti2+ formation in Ti0,-,. From measured values of the 3rd and 4th Ionisation Potentials of titanium we deduce that an extra stabilisation energy of
-
16 eV would be required - a value greater than the lattice term calculated above. The results, however, suggest that the introduction of dopants which form more stable divalent ions may have a strong stabilising effect on shear planes, and such dopants may indeed give shear planes in oxides in which these structures would otherwise not form. The results might also suggest that shear planes would be more stable in an oxide such as SnO,, in which we would expect reduction to lead to the formation of divalent ions. As the converse behaviour is observed [12], we conclude that the favourable valence factor is insufficient to compensate the loss of stabilisation from the lower cation relaxa- tion which we predict for this material, as discussed above.3.4 EFFECTS OF DOPANTS.
-
Our calculations show that interactions between shear planes and dopants may arise from three factors, namely, valence, ionicENERGETICS O F SHEAR PLANE FORMATION IN REDUCED TiO, C7-35
shear planes. If therefore such ions can be introduced an ion whose polarisability was half that of Ti3+, but into the TiO, structure, pronounced effects should which otherwise was identical to the lattice cation. We follow on shear plane formation energies. found a smaller increase in the cation binding energy The effects of the other factors are illustrated by to the shear plane of
-
0.03 eV relative to that results on the dopants Fe3+ and C r 3 + . Our calcula- obtained for Ti3+. We conclude therefore that a tions here used potential parameters derived from a reduction in ionic radius and electronic polarisability separate study on Cr,03 and F e 2 0 3 [15]. We found will enhance dopant-shear plane interactions. that the binding energy of both ions to the cation sitesadjacent to the shear plane is 0.3 eV greater than that for Ti3+. We predict therefore that the two dopants will accumulate in the region of the shear plane. This increased binding of Cr3+ and Fe3+ is attributable largely to an ion size effect. In the octahedra adjacent to the shear plane smaller metal-oxygen distances are observed ; Fe3+ and Cr3+ are smaller than Ti3+ and are therefore stabilised at these sites.
Changes in polarisability may, however, contribute to the change in binding energy. To study effects due to polarisability and radius separately we performed a set of model calculations. First we obtained the magnitude of the binding energy to the shear plane of an ion whose radius was 0.1
A
less than that of Ti4+. (The variation in the Born-Mayer parameters for this change in ionic radius, was estimated using the Huggins Mayer scheme - see e.g. Tosi and Fumi [16].) The binding energy of the dopant relative to that of ~ i ~ +was enhanced by
-
0.3 eV. Secondly we considered4. Conclusions. - The work in this paper has shown first that our lattice energy techniques may simulate reliably shear plane energetics in TiO,. But our calculations have also led to important conclu- sions as to the fundamental factors controlling the formation of extended defects. Most important are the extent of cation relaxations close to the shear plane, and the valence of the reduced cation. Dopant- shear plane interactions may, however, play an impor- tant role; they are affected by ion-size and polarisa- bility in addition to valence factors.
Several extensions of the calculations are clearly required for a complete understanding of the problems we have discussed. The present study, however, demonstrates the power of our theoretical techniques in investigating fundamental problems in this area.
Acknowledgments. - We are grateful to B. E. F. Fen- der, M. J. Norgett and A. M. Stoneham for useful
discussions.
References
[I] BURSILL, L. A. and HYDE, B. G., Progress in Solid-Stale
Chemistry 7 (1972) 177.
[2] MAREZIO, M. el al., J . Solid Stufe Chem. 6 (1973) 213. [3] ANDERSON, S., Acta Chemica Scand. 14 (1960) 1161.
[4] CATLOW, C. R. A. and N o R c m , M. J., UKAEA report, in press.
[5] CATLOW, C. R. A. and JAMES, R., to be published.
[6] CATLOW, C. R. A. and FENDER, B. E. F., J. Phys. C 8 (1975) 3267.
[7] CATLOW, C. R. A,, Proc. R. Soc. A 353 (1977) 533.
[8] NORGE~T, M. J., UKAEA report, AERE, R7650 (1974).
[9] JAMES, R., UKAEA report, AERE, TP 666 (1976). [lo] ANDERSON, J. S. and HYDE, B. G., J. Phys. & Chem. Solids
28 (1967) 1393.
[ I l l PICARD, C. and GERDANIAN, P., J. Solid Stare Chem. 14 (1975) 66.
[I21 ARMYTAGE, D. and FENDER, B. E. F., unpublished work. [13] HYDE, B. G., Nature 250 (1974) 411.
[14] IGUCHI, E. and TILLEY, R. J. D., J. Solid Slate Chem. 21
(1977) 49.
[IS] JAMES, R. and CATLOW, C. R. A., to be published. [I61 Tosl, M. P. and FUMI, F. G., J. Phys. & Chem. Solidr 23
