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Submitted on 1 Jan 1976
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POINT DEFECTS, GENERAL.THE CALCULATION
OF LATTICE DEFECTS AT SURFACES FOR CUBIC
IONIC CRYSTALS
R. Stewart, W. Mackrodt
To cite this version:
POINT DEFECTS, GENERAL.
THE
CALCULATION OF LATTICE DEFECTS
AT SURFACES FOR CUBIC IONIC CRYSTALS
R. F. STEWART and W. C. MACKRODTICI Ltd, Corporate Laboratory, PO Box No 11, The Heath, Runcorn, Cheshire WA7 4QE, U. K.
R6sum6. - On prksente un calcul des proprietb des dkfauts de surface pour un certain nombre d'oxydes cubiques, dans le cadre de la thBorie de Mott-Littleton. Pour MgO et MnO les Bnergies des defauts ponctuels dans le bulk et sur les faces (001) et (011) sont donnies. On examine Bgalement l'effet du dopage en impuretks cationiques pour tous les substrats considerCs dans ce travail. On Btudie en particulier l'effet du remplacement de l'ion Mn2+ par des ions a charge plus BlevBe tels que U4+ dans MnO, aussi bien dans le bulk qu'en surface. L'effet du remplacement en surface de Mgz+ par CO4+ dans MgO est egalement examine B la lumikre de resultats experimentaux rBcents.
Abstract. - Calculations carried out within a Mott-Littleton framework are presented for the surface defect properties of a number of cubic oxides. For MgO and MnO the energies of the point defects in the bulk and at the (001) and (011) faces are given, while for the full range of substrates considered here, the doping by foreign cations is examined. Particular attention is paid to the bulk and surface substitution in MnO by a highly-charged species such as U4+, and to the surface substitution of Mg2+ by Co2+ in MgO in relation to recent experimental work.
1. Introduction.
-
In recent years there has been an accumulation of evidence to support the utility of theoretical methods in the investigation of the defect structure of solids. In particular, developments in certain computational techniques (e.g. Norgett and Fletcher 1970, Lidiard and Norgett 1972) have led to a position wherein the properties of the point defects of the simpler ionic crystals can often be described with an accuracy comparable to the avai- lable experimental data. The basic approach to the calculation of lattice defects derives from that of Mott and Littleton (1938) and is currently imple- mented in the HADES computer package (Lidiard and Norgett 1972, Norgett 1972, 1974) among others. Now in situations such as those encountered in corrosion or catalysis, for example, surface properties, as distinct from those of the bulk, are likely to exert a controlling influence on the fundamental atomic processes. A quantitative knowledge of the surface defect structure, therefore, would seem to be essential for an accurate description of the reactive behaviour of solids, particularly for those involved in the cir- cumstances cited above. In a previous report (Mac- krodt and Stewart 1977) the intrinsic defect structures of the (001) and (011) surfaces of NaCl and MgO have been given, together with an account of the doping of NaF by Li', K + and Ca2+, in the bulk and at the two lowest index faces. The calculated energies of formation and migration of the basic lattice defects suggested that there were significant differences between surfaces and the interior, and thatthese differences arose as a result of changes in both the adiabatic and elastic contributions to the total defect energy. In the paper presented here, the same approach is extended to cover a range of surface and bulk properties of the simple cubic transition metal oxides, and includes calculations of surface energies, point defects and their interactions and aspects of iso- and aliovalent doping. The substitution of Mg2' by Co2+ in MgO is of special interest in view of the recent experimental work by Hagen, Arean and Stone (1976) on the possible four-fold surface co-ordination of the extrinsic ion.
2. Theory.
-
The essential feature of the Mott- Littleton method is the partition of the crystal sur- rounding the defect into an explicit inner region in which the co-ordinates of the distorted lattice are relaxed to equilibrium, and an outer region, which is treated as a polarisable dielectric continuum. Suitable boundary conditions describe the interface of the two. Ionic interactions are assumed to be exclusively two-body, while electronic polarisation of the ions is included as some variant of the shell model originally proposed by Dick and Overhau- ser (1958). For surfaces, the necessary modifications to this approach have been fully described elsewhere (Mackrodt and Stewart 1977), so that only the salient points will be included here.It has previously been shown that the Ewald method for lattice summation is particularly suited to bulk defect calculations (Norgett 1972, 1974)
C7-248 R. F. STEWART AND I W. C. MACKRODT
and similar procedures can equally be applied to surfaces (Mackrodt and Stewart 1977). The principal difference between the two relates to the evaluation of the long range part of the electrostatic potential due to the compensating charge distribution. For the bulk, the (assumed) infinite lattice ensures the periodicity of the superimposed density which can be expanded as a Fourier series in reciprocal lattice vectors (Ewald, 1932), in all cases. For surfaces, on the other hand, the semi-infinite compensating lattice is periodic (up to the boundary) only for a restricted class of distributions. In the calculations reported here, cut-off Gaussians, defined by
$(r-R) = Nexp[-q2
I
r-R12],
for ( r - R 1G0.5 (2. la)= 0, otherwise (2. lb)
were used, the unit of length being the lattice spacing parallel to the surface. The corresponding total charge density is clearly periodic in the range
in which the Z-axis is perpendicular to the boundary, and so has a well defined Fourier expansion in reci- procal lattice space. From a computational point of view, the use of truncated Gaussians necessitates an increased number of vectors in the Fourier expansion for a given accuracy, but is otherwise identical to the normal bulk procedure.
Two body potentials were used throughout and were assumed to be of the usual form, viz.
in which the constants Aij etc., together with the cor-
responding shell parameters, Yi. and Ki, were taken from Catlow, Faux and Norgett (1976) for MgO and from Catlow, Mackrodt, Norgett and Stone- ham (1977) for the transition metal oxides.
The essential problem in defect calculations is the minimisation of the total lattice energy, appropriate numerical procedures for which have previously been described (Norgett and Fletcher 1970, Lidiard and Norgett 1972). In contrast to the bulk, sur- face calculations need to account for a distortion from crystal symmetry perpendicular to the surface plane, even in the absence of a defect. The simplest approach to this and one which was used throughout was to allow the ions in the inner region to relax to equilibrium in a preliminary calculation prior to the introduction of a defect, and to use this configuration and zero energy as the starting point for the defect calculation. Defect energies obtained in this way were found to converge rapidly as the size of the inner region was increased (though less so than in the bulk), so that the procedure would seem to be satis- factory, at least for the crystals in question. For highly charged defects it was found to be more
expedient to extrapolate the results for a series of inner regions of moderate size than to relax a very large number of ions. In every case, however, the increment due to extrapolation was less than 3
%
of the total relaxation energy.3. Results and discussion. - The methods outlined
in section 2 were used to investigate a range of bulk and surface properties of MgO, MnO, FeO, COO and NiO : the results are collected in tables I to VI. Surface energies, including relaxation, are listed in table I. Without exception the (001) face is calculated
Surface energies (in ergs/cm2) for some cubic oxides
System and
Crystal Face This Work Other
-
- - MnO 001 0.64 x 103 FeO 001 0.98 x 103 COO 001 1.03 x 103 NiO 001 1.14 x 103 MgO 001 1.17 x 103 1.45 x 103 (a) 1.04 x 103, 1.2 x 103, 1.15 x 103(b) MnO 011 2.09 x 103 FeO 011 2.59 x 103 COO 011 2.72 x 103 NiO 011 2.97 x 103 MgO 011 2.98 x 103 3.89 x 103(a)( a ) and ( b ) Theoretical values and experimental estimates res-
pectively reviewed by Tosi (1964).
to be significantly more stable than the (OlI), while the uniform trend for the transition metal oxides is similar to that for other properties (Catlow, Mac- krodt, Norgett and Stoneham 1977). The extent to which the defect structure can be modified by the surface of a crystal is illustrated in tables I1 and I11
THE CALCULATION OF SURFACE LATTICE DEFECTS
Defect energies for the bulk, 001 and 01 1 surfaces of MgO
All energies, here and throughout this work, are in eV unless otherwise stated
Defect Type Bulk (eV) 001 Face
- - -
Cation vacancy 23.8 (40.6) (") 24.6 (40.0) ( b )
Anion vacancy 24.7 (41.2) 25.4 (40.4)
Neutral divacancy 45.9 (55.6) 44.0 (54.2)
Cation-anion vacancy interaction energy
-
2.6 (- 26.2) - 6.0 (- 26.2)Cation-cation divacancy 49.0 (100.5) 55.0 (99.3)
Cation-cation vacancy interaction energy 1.4 (19.3) 5.8 (19.3)
Anion-anisn divacancy 50.8 (101.6) 56.5 (100.0)
Anion-anion divacancy interaction energy 1.4 (19.3) 5.7 (19.3) Square planar neutral vacancy quartet 88.7 (97.4) 85.5 (94.7) Pair-pair interaction energy in above
-
3.1 (- 13.7) - 2.5 (- 13.7) Total vacancy-vacancy interaction in above - 8.3 (- 66.0)-
14.5 (- 66.0) Cation migration activation energy by vacan-cy mechanism 2.1 0.7
Anion migration activation energy by vacan-
cy mechanism 2.1 0.7 01 1 Face - 22.2 (37.3) 23.0 (37.6) 39.5 (48.8) - 5.7 (- 26.2) -
(a) Where appropriate, unrelaxed energies are given in parentheses. For the surfaces the unrelaxed defect energy is with respect to
the undistorted surface.
( 0 ) The error in the defect energy, owing to the effect of the finite size of the relaxed region, is very unlikely to exceed 0.2 eV for most of the defect types considered. The exceptions are the charged divacancies, for which the relaxation energy is very large and for which the error may be proportionately larger.)
Defect energies for MnO. Where appropriate, unrelaxed defect energies are again given in parentheses
Defect Type Bulk
- -
Cation vacancy 20.9 (37.2)
Anion vacancy 21.3 (37.5)
Neutral divacancy 40.7 (50.0)
Formation energy of above - 1.6 (- 24.6) Cation migration activation energy by vacxncy
mechanism 1.6
Anion migration activation energy 1.5
Defect Site 01 1 Face 01 1 Face
-
21.9 (36.9) 19.6 (34.6) 22.2 (37.1)-
39.8 (49.4) - - 4.3 (- 24.6)between these two contributions that distinguishes the surface from the bulk. With charged defects, for example, the interaction with the induced dipoles in the lattice leads to an appreciable binding energy, the long range contribution of which is approximately halved at the surface. A decrease in the Madelung potential, therefore, provided it is not too large, can be compensated by an even greater decrease in the polarization energy of the outer region, so that the net vacancy formation energy a t a surface can be increased. Apart from charge polarisation effects, however, surface relaxation is also influenced to a large extent by the defect configuration. For example, the energy of the symmetric transition state for migra- tion by a vacancy mechanism is substantially reduced by the relaxation of the migrating ion perpendicular to the surface, in contrast with the bulk process.
It would seem reasonable to expect that the surface defect energies of the other oxides viz. NiO, Co and FeO lie uniformly between those of MgO and MnO. This is certainly so for the surface energies as shown in table I and would seem to follow from figures 1 and 2 in which bulk vacancy formation and interaction energies are plotted as a function of the static die- lectric constant, E ~ . Correlations of this type appear
to be remarkably good and would seem to merit further attention, particularly so with respect to the formation of extended defects.
The differences between surface and bulk charac- teristics of doping are illustrated in tables IV to VI which list substitution energies and enthalpies of solution for combinations of MgO and the transition metal oxides, and for U4+ in MnO. In the absence
R. F. STEWART AND W. C. MACKRODT
- 4 0 . 0 ' 6 I I
10.0 15.0 2 0 - 0
€ 0
FIG. 1. - The relaxation energy, a$ a percentage of the for- mation energy of the unrelaxed defect, versus static dielectric constant, EO, for a series of cubic oxides. Results for cation vacancy formation are shown as crosses whilst those for the
anion vacancy are displayed as circles.
mined almost solely by the nearest neighbour (metal- oxygen) interaction and co-ordination number. The former, therefore, should lead to a systematic variation in the series Mn, Fe, Co and Ni and the latter to an approximate ratio of 6 : 5 : 4 for the bulk to (001) to (001) surface energies. This is indeed the case as the unrelaxed values indicate. The inclusion of lattice relaxation does not modify this simple picture radi- cally except for ~ g doping MnO for which, with ~ + the occurrence of considerable reorganization of the lattice about the relatively small foreign cation, there is an inversion of the order of energies for the
FIG. 2. - Divacancy formation energies (from isolated defects)
versus E O for some cubic oxides. Values for anion-anion diva- cancy formation are given in the upper curve whilst those for the
neutral pair are in the lower.
two surfaces. Overall, as may be observed from figure 3, for doping both of the transition metal oxides and of MgO the ease of substitution can be correlated surprisingly well with the difference in host and dopant ion size even for the most sensitive cases for which (a,,
-
aMg0) is very small. The enthalpies of solution, X, defined by :in which E and Ware substitution and lattice cohesive
energies respectively, are presented in table V for solutions of MnO-NiO in MgO at infinite dilution.
Substitution energies for the doping, by M ~ ' , of cubic oxides. MO Unrelaxed substitution energies (in the case of substitution at the surface with respect to the undistorted surface) are given in parentheses
System Mg2 +/M~o Mg2+/Fe0 Mg2 + / c o o Mg2 + /NiO Mn2 + /MgO Fe2+/Mg0 Co2 + /MgO Ni2+/Mg0 Bulk
-
- 3.37 (- 3.19)-
1.31 (- 1.30)-
0.84 (- 0.83) 0.20 (0.21) 3.83 (3.99) 1.39 (1.40) 0.86 (0.86) - 0.19 (- 0.18) Defect Site 001 Face - 2.77 (- 2.57) - 1.07 (- 1.04)-
0.67 (- 0.66) 0.17 (0.19) 2.98 (3.17) 1.11 (1.12) 0.68 (0.68) - 0.18 (- 0.16) 01 1 Face --
2.93 (- 2.06)-
0.97 (- 0.83) -0.57 (-0.53) 0.22 (*) (0.15) 2.56 (2.53) 0.95 (0.89) 0.57 (0.55)-
0.25( (*) (- 0.13)THE CALCULATION OF SURFACE LATTICE DEFECTS C7-251
-
6 . 0 ~FIG. 3.
-
Energies for M2+ substitution in MgO and Mg2+ substitution in MO versus the dil7erence in lattice parameter( U M O - U M ~ O ) .
Enthalpies of solution of the transition metal cubic oxides in MgO at injinite dilution. Quantities, as before, are in eV a (*) b c - - - MnO 0.38 - 0.47 - 0.89 FeO 0.02 - 0.26 - 0.42 COO
-
0.02 - 0.20 - 0.31 NiO-
0.05-
0.04 - 0.1 1but nevertheless there would seem to be no doubt of the marked preference for substitution by MnO- COO at the surface rather than in the bulk of MgO, in harmony with earlier results on the alkali halides (Mackrodt and Stewart 1977).
Recently Hagan, Arean and Stone (1976) have examined the UV reflectance spectra of Co-doped MgO and interpreted the results in terms of 4-coor- dinated Co2+ ions at the surface.
In view of the low probability for (011) faces, it was surmised that the Co2' ions might be associated either with (001) surface complexes involving an anion vacancy and an octahedral Mg2+ ion, or with edges or kinks. Calculations of the type reported here sug- gest that the (001) complex is energetically highly unfavourable (> 5eV) but that the substitution energy at a (001)/(010) edge (which can be con- sidered as a line through a (011) plane) is very close to the surface value of 0.57 eV listed in table IV.
For aliovalent doping, the situation is slightly more complicated in that long range polarization effects are introduced and charge compensation must be considered. The isolated substitution of Mn2+ by
u4+
in MnO is largely dominated by elec- trostatic forces which favour the bulk by virtue of both the enhanced Madelung potential and the binding to the chargeinduced dipoles in the lattice. Compensation by either cation vacancies or anion excess, however, alters the balance in favour of the surface, as shown in table VI. For vacancies the diffe-rence between the bulk and (001) surface binding is very nearly the same as for the neutral vacancy pair, i.e.
-
2.6 eV (table 11), and for precisely the same reasons. The calculated enthalpies of solution suggest that this is the preferred mode of compensation with preference for the surface.(*) The figures in the first column employ equation (3.1)
with the substitution energy (from Table IV) appropriate to the - We like thank bulk. Forb and c the substitution energies determined for the 001
-
Drs A. M. Stoneham and M. J. Norgett (Harwell),and 01 1 faces of MgO are used. and C. R. A. Catlow (Oxford) for many helpful
discussions. We are also indebted to MJN and CRAC For small absolute values of
x
the error in this quan- for communication, prior to publication, of some of tity may be high due to the cancellation in (3. I), the potentials used in this report.Energies of substitution of
u4+
for ~ n in MnO (eV). Potentials involving ~ +u4+
are taken from Catlow (1974) Site of DefectSystem Bulk 001 Face 01 1 Face
- - -
-
U4+ substitution with no counterbalancing effect
-
42.5-
42.3-
41.5 U4+ substitution with nn vacancy - 22.4 <-
23.9 < - 25.0U4+ substitution with isolated vacancy - 21.5 - 20.4
-
21.9Interaction energy of U4+ and nn vacancy
-
0.9 <-
3.5 < - 3.1 U4 + substitution with adjacent 0' --
56.7-
60.8 < - 60.3x
for UO, solvation in MnO at infinite diIution with va-cancy formation 5.9
<
4.3<
3.2As above but with excess 0'- ions rather than vacancy
R. F. STEWART AND W. C. MACKRODT
References
NORGETT, M. J. and FLETCHER, R., J. Phys. C : Solid State
Phys. 3 (1970) L190.
LIDIARD, A. B. and NORGETT, M. J., Computational Solid
State Physics, Ed. F . Herman, N. W. Dalton and
T. R. Koehler (New York : Pkenum), 1972, p. 385.
NORGETT, M. J., AEREReport, A ERE,R7015 (1972).
MOTT, N. F. and LITTLETON, M. J., Trans. Faraday Soc. 34
(1938) 485.
NORGETT, M. J., AERE Report, AERE, R 7650 (1974).
MACKRODT, W. C. and STEWART, R. F., J. Phys. C : Solid
State Phys. (1977). To be published.
HAGAN, A. P., AXEAN, C. 0. and STONE, F. S., Proceedings
of the 8th International Conference on the Reactivity of Solids, June 14-19. Gothenburg, Sweden (1976) 22.
DICK, B. G. and OVERHAUSER, A. W., Phys. Rev. 112 (1958) 90.
EWALD, P. P., Phys. Rev. 39 (1932) 675.
CATLOW, C. R. A., FAUX, I. D. and NORGETT, M. J., J. Phys. C :
Solid State Phys. 9 (1976) 419.
CATLOW, C. R. A., MACKRODT, W. C., NORGETT, M. J. and
STONEHAM, A. M., Phil. Mag. (1977). To be published.
Tos~, M. P., Sol. Stat. Phys. 16 (1964) 1.
CATLOW, C. R. A., D. Phil. Thesis, The University of Oxford
(1974).
DISCUSSION
P. W. M. JACOBS. - What is the topography of your surface ? Is it a plane surface or have you been able to consider the effect of surface steps ?
