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The calculated defect structure of thoria
Journal
50
THE CALCULATED
E.A. COLBOURN
DEFECT
STRUCTURE
of Nuclear Materials 118 (1983) 50-59 North-Holland Publishing Company
OF THORIA
AND W.C. MACKRODT
ICI PLC, New Science Group, PO Box 11, The Heath, Runcorn, Cheshire WA7 4QE, UK Received
3 March
1983; accepted
11 April
1983
The present paper is concerned with calculated energies of lattice and electronic disorder in thoria. From these we deduce the fundamental defect structure, electronic energy levels and oxidation-reduction potentials. This enables us to re-assess the reported diffusion and electrical conductivity data and to suggest alternative defect mechanisms for these processes.
1. Introduction Thoria is an important nuclear material, the properties of which are of interest in other areas of application such as solid state electrolytes [ 11. It is a wide band-gap (5.75 eV) [2], high melting point (33OO“C) [3] oxide, whose defect structure has been inferred largely from conductivity [4- 1 l] and diffusion [ 13- 181 measurements. In common with other fluorite-structured materials, Frenkel defects are thought to predominate the intrinsic disorder [ 17,181, while at high temperatures (- 1OOO’C) and oxygen pressures down to - IO-l4 atm, the observed electronic conductivity is p-type ]5,9,111. Self-diffusion coefficients of oxygen in the temperature range 800-16OO“C were determined by Edwards et al. [ 131, and, more recently, by Ando et al. [ 17,181, who found a break in the temperature dependence at approximately 1100°C. The results have been interpreted in terms of intrinsic and extrinsic mechanisms. Arrhenius energies of 2.85 eV [13] and 2.16 eV [17,18] (from 1 lOO-1600°C) have both been attributed to intrinsic diffusion, the latter being in broad agreement with activation energies of electrical conduction determined by Lee (1.91 eV) [8] and by Choudhury and Patterson (1.80 eV and 1.92) [9] at the same temperatures. Lasker and Rapp [6], on the other hand, have reported a somewhat smaller value of 1.34 eV for electrical conduction. Between 800- 1100°C Ando et al. [ 17,181 derived an Arrhenius energy of 0.78 eV, which they ascribed to extrinsic diffusion, possibly dislocation-enhanced, while Bransky and Tallan [7] obtained a similar value (0.88 eV) for ionic conduction. Self-diffusion coefficients of Th4’ have also been measured. Hawkins and Alcock
0022-3115/83/0000-0000/$03.00
0 1983 North-Holland
[ 141 found an activation energy of 2.55 eV for polycrystalline thoria, whereas Ring [ 151 and Matzke [ 161 reported much higher values of 6.48 eV and - 6.5 eV respectively. The latter values, in particular, confirm that ionic conduction involves the oxide ion and that comparisons between oxygen diffusion and electrical conductivity energies are meaningful. The electrical conductivity of doped thoria has been investigated in some detail, notably by Hammou [lo], Subbarao and Maiti [ 1I] and Schouler et al. [12]. For calcium doped solid solutions, Th(Ca),O,_, with x = 0.01-0.07, activation energies in the range 1.12 eV to 1.42 eV have been measured [ 1 l] and interpreted as the sum of a dopant-anion vacancy association energy and the migration energy for free vacancies [l]. Similar considerations apply to the activation energies of about 1.1 eV for yttrium-doped thoria [ 11. We have, then, what appear to be consistent transport data for thoria which can be interpreted in terms of oxygen vacancies and interstitials as the predominant disorder, with cation vacancies and holes as the minor defects. However, the assignment of specific defect processes to the measured Arrhenius energies is open to question. In particular energies of about 2 eV seem to be too small to account for intrinsic diffusion of oxygen. This implies a Frenkel formation energy of less than 3 eV which theoretical considerations suggest as being somewhat low for a 4-valent oxide [19]. Furthermore, the nature of the diffusing species is not determined. Likewise, the interpretation of the cation diffusion data is not fully resolved; for if it assumed that the energy reported by Hawkins and Alcock [4] is for extrinsic diffusion, and that by Ring [ 151 and Matzke [ 161 for intrinsic diffusion, this would lead to a Schottky forma-
51
EA. Colbown, W.C. Mackrodt ,I’ The calculated defect structure of thoria
tion energy of less than 6 eV, which, once again, seems rather too low for an oxide of this type [19]. In view of these uncertainties, then, calculations of the type previously reported [19-221 might have a useful role to play in a re-assessment of the experimental data. Catlow and coworkers have examined UO, 1231,CeO, [24] and ZrO, [25] using similar methods and have provided valuable insight into the defect structure of these materials. Here we extend this approach to thoria. We report energies for defect formation, migration and association with cation impurities; in addition we consider electronic energy levels related to thermal excitations and energies of oxidation and reduction. Our calculations give a unified account of the defect properties of pure and doped thoria which is in good accord with the reported data.
2. Theoretical methods The theoretical methods used in the present calculations have been discussed in full recently [26]. The treatment of lattice relaxation is that introduced by Lid&d and Norgett [27] and extended by Catlow et al. [28] to systems with arbitrary structure. It has been used previously to examine numerous other oxides [19], including UO, [23], CeO, [24] and Ca*+: ZrO, [25]. Our interatomic potentials are based on the electron-gas method 1291, with the inclusion of the shell-model [30] to account for defect-induced electronic polarisation of the lattice. However, in deriving these potentials we have introduced a modification to earlier procedures [29]: this we discuss in brief. For the most part, electron-gas potentials lead to zero-strain unit cells that are approximately 3-58 larger than the measured structures. In some cases the agreement is better, in others, notably the transition-mete oxides, it is much worse. The reason for this is that the ‘short-range’ potentials are too repulsive, largely because a fully ionic model is normally assumed. However, the overallform of these potentials has been shown to be very similar to those derived from accurate ab initio calculations which include electron correlation [31]. We use this observation to obtain zero strain unit cells at the correct lattice spacing by simply scaling the electron-gas potentials in the following way. We write
for the oxygen-oxygen potential can be used for a wide variety of oxides, including the transition metal oxides, and the shift for the metal-oxygen interaction then found by fitting to the appropriate lattice parameter [32]. In other words, we have a semi-empirical electron gas potential with a single fitted variable, namely the shift re. A list of relevant perfect lattice properties of thoria obtained from these potentials is given in table 1, while a more complete account of the general procedure will be given elsewhere [32]. Finally, in this section we consider the effective ionisation potentials/eI~tron affinities of oxygen in thoria, for these have an important influence on electronic energy levels and the energies of oxidation and reduction. Our theoretical methods are similar to those used previously for the alkaline earth oxides [20] and cu-Al,O, [21,28]. They are based on single-centre Hartree-Fock calculations of oxygen in spherical Coulomb potentials [29], the strength of which we estimate from defect lattice simulations. As before we deduce electron correlation contributions from published calculations for iso-electronic systems [29]. The results are collected in table 2. ‘eL(0), the optical ionisation potential, is used to calculate the corresponding optical ionisation energy, from which we deduce the position of the valence and conduction bands and the formation energy of holes. We note in particular the difference in energy between the electron affinities of oxygen at a ‘normal lattice site and in an interstitial position. This is due to a difference of approximately 30% in the Madelung potentials at these two sites in the fully relaxed lattice. Table 1 Perfect lattice properties of ThO, derived from modified electron gas potentials Calculated Zero strain lattice parameter (A) Lattice energy (eV/mole) High frequency dielectric constant Static dielectric constant Elastic constants (10” dyne/cm*) c,,
ir(r-r,)=
U(r),
in which 0 is the modified potential: that is to say, the electron gas potentials U(r) are shifted inwards by an amount, q,, to give a zero strain lattice with the observed structure. We have found that a “standard” shift
c,2
c ~~mpressibilit y Cation polarisability (A3) Anion polarisability (K)
Experimental
5.5997 [36] 5.5991 - 103.0; - 104.7 [37] - 104.39 4.4 18.96
4.8 [38) 18.9 [39]
49.53 13.88 6.25 0.039 2.122 2.186
36.7 [40] 10.6 7.97 -
E.A. Colbourn,
52
WC.
Mackrodr / The calculared deject struclure of thoria
Table 2 Effective ionisation potentials (eV) in ThO, First ionisation potential of lattice oxygen Sum of ionisation potentials of lattice oxygen First ionisation potential of interstitial oxygen Sum of ionisation potentials of interstitial oxygen Fourth ionisation potential of thorium
‘CL(O)
-8.8 -7.3
‘r,(O) - 5.6 28.8 [41]
From our atomic Hartree-Fock calculations we estimate the (internal) electronic energy of Of- to be - 1.7 eV less than that of Oi- .
Frenkel defects are predicted to predominate the intrinsic disorder with a formation energy of 6.01 eV, which is approximately half that to create Schottky defects (11.93 eV): cation Frenkel pairs require a further 8 eV. These values are close to those reported previously by Catlow [23] for UO,, based on similar calculations. We find an appreciable binding energy of 2.74 eV between cation and anion vacancies so that the principal cation disorder is predicted to consist of free vacancies and di-vacancy pairs. Cation interstitials would seem to play no part in the defect structure of thoria. We deduce ‘effective’ formation energies for intrinsic disorder from the electro-neutrality condition and the temperature dependence of the defect equilibria. As shown in table 2 they vary from 3.0 eV for anion vacancies and interstitials to 13.97 eV for cation interstitials. From these values we conclude that the Arrhenius energies for the intrinsic diffusion of oxygen and thorium are in excess of 3 eV and 6 eV respectively. We return to this point later.
3. Calculated defect eneqjies 3.2. Impurity defects 3.1. Fundamental defects Calculated formation energies of fundamental lattice defects in ThO, are given in table 3. As expected anion
Table 3 Fundamental lattice defect energies in ThO, Defect
Energy (ev)
cation vacancy
84.66 15.83 - 64.86 - 9.82 91.15 2.14 112.04 4.28 11.93 (10.34) a 6.01 (5.31, 5.47) 19.80 (18.54)
anion vacancy cation interstitial anion interstitial divacancy divacancy binding trivacancy trivacancy binding Schottky (E,) anion Frenkel (E,_) cation Frenkel (El+)
Table 4 Defect energies Dopant
Mg2+ Ca*+ Sr’+ Ba*+
Ejjecrioe formation energies for intrinsic disorder anion vacancy 3.00 anion interstitial I cation vacancy 5.93 cation interstitial 13.87 divacancy 6.19 trivacancy 1.65 a values ~231
in brackets
are those calculated
by Catlow
Cation impurities play an important role in determining the defect properties of oxides [22]. In the case of thoria experimental investigations have been concerned largely with the electrical conductivity of solid solutions. Di- and trivalent impurities give rise to anion vacancies, which, in turn, are responsible for ionic conduction. Here we concentrate on the alkaline-earth ions and a series of trivalent substituents ranging from A13+ to Bi3+. Table 4 lists calculated defect energies associated with the former. As shown, substitution energies are high, due mainly to Coulombic interaction with the lattice, with differences in size accounting for
for UO,
associated
Substitution
52.15 54.59 56.44 58.83
with divalent
impurities
Energy (eV) vacancy binding
,=,‘a’
5
H’b’
1.54 1.20
4.33 2.03 1.73 2.21
2.19 0.83 0.62 1.13
1.11 1.11
s
(a) Calculated heat of solution in the absence of any aggregation based on the following lattice energies from ref. [20]: MgO ( - 40.74), CaO ( - 36.00), SrO ( and BaO (- 32.00). (b) Calculated heat of solution assuming complete defect gation.
defect taken 33.85) aggre-
E.A. Colbourn, W.C. Mackrodt / The calculated defect structure of thoria
approximately 15% of the total energy. The corresponding heats of solution range from 4.33 eV for MgO to 1.73 eV for BaO. We find vacancy binding energies in excess of 1 eV for all four dopants which suggests an appreciable degree of defect aggregation even at high temperatures. The variation in the binding energy with ion size follows that found previously for MgO [20] and CeO, [24]. The value for Ca2+ (1.20 eV) corresponds to an interaction which is almost totally Coulombic in nature since the ionic radius of Ca” (0.99 A) and Th4” (1.02 A) are practically identical. The enhanced binding of Mg2+ (0.34 ev) arises because the expansion of the lattice around the vacancy is compensated by the contraction around the dopant ion. For much the same reason the binding to Sr2+ and Ba2+ is slightly diminished. Trivalent impurities exhibit a similar pattern of behaviour to that found recently by Butler et al. [24] in their study of doped ceria. As shown in table 5 our values of 1.05 eV and 0.65 eV for the binding of Sc3+ and Y 3+ respectively to an anion vacancy are slightly larger than those reported for CeO, [24] but this is to be expected in view of the enlarged ThO, lattice. As before we note a marked variation in binding energy with dopant size, ranging from 1.81 eV for A13+ to 0.45 eV for Bi3+ which is just above the minimum value of 0.38 eV for Th3’. We return to the association of Ca2’ and Y 3+ with anion vacancies in relation to the measured activation energies for the electrical conduction later in section 4.
and electrical conductivity in thoria. The possible mechanisms for cation and anion migration are illustrated in figs. 1 and 2 respectively, and the corresponding energies collected in table 6. For anions we find vacancy migration to be the least energetic process with an activation energy of 0.78 eV. However, our value for the migration of interstitials by an interstitialcy mechanism is only slightly higher at 0.92 eV. Direct interstitial migration and the direct exchange of anions appear to be much more energetic and unlikely, therefore, to contribute to anion mobility. Recently, Kilner and Brook [33] have suggested a value of - 0.6 eV for the migration energy of anion vacancies in fluorite oxides, but in the case of thoria there seems to be no direct experimental data for comparison. Combining these migration energies with the “effective” formation energies in table 2 we deduce an Arrhenius energy for the intrinsic diffusion of oxygen in ThO, of 3.8-3.9 eV. This is appreciably higher than the value of 2.16 eV reported by Ando et al. [ 17,181, and later in section 4 we suggest an alternative interpretation for their results. As shown in table 6 we predict the migration of
(a)
3.3. Ion migration
53
0
MtQratin9 Cation
q
Cation Vacancy
Cl Anion Vaeoncy
The mechanisms and energetics of ion migration are central to an understanding of diffusion phenomena
Table 5 Defect energies associated
Substitution
AIs+ SC3+
In’+ Ys+ Bi3+ Ths+
21.54’*’ 25.17 27.12 28.04 31.32 34.27
with trivalent impurities
Vacancy binding
(b)
1.81 1.05 (0.63)‘b’ 0.63) 0.65 (0.27) 0.45 0.38
(a) The calculated heat of solution for a-Al,Os is 12.03 eV per mole in the absence of any defect aggregation. (b) Values in brackets are those calculated by Butler et al. [24] for the corresponding energies in CeO,.
(cl Fig. 1. Mechanisms
of cation migration in thoria.
W. C. Mackrodt / The calculated deject structure of thoria
E.A. Colboum,
54
3.4. Electronic defects and energy levels
(a)
(b)
m l’r------
---_
i
I’+----
’
T
-
-_----_L-
(cl
0
Migrating
0
Catlon
Anion
0
Anion Vacancy
Fig. 2. Mechanisms of anion migration in thoria.
cations to be considerably more energetic than anions, with activation energies ranging from 5.36 eV for a divacancy mechanism to 7.04 eV for the migration of free vacancies. These values are evidently high, but we note that they are comparable to those calculated by Catlow [23] for the migration of U4+ in UO,. Thus we predict Arrhenius energies for the intrinsic diffusion of Th4’ in thoria in excess of 11 eV, which suggests that it is unattainable even at the highest temperatures.
U = E;(O)
- E/‘(O).
The present value of 6.0 eV is similar to those calculated previously for MgO (5.0 eV), CaO (5.3 eV) and a-Al,O, (6.4 eV) [21], and consistent with photo-electron measurements for cubic oxides [34]. From our value of lJ we can estimate the difference in energy AE, between the large and small polaron hole. It is given by AE=E;(O)-E;(O)-22.
Table 6 Cation
We turn now to the calculation of energy levels and electronic defects, principally free electrons and holes, for they are involved in both the non-stoichiometry and the electrical conductivity of thoria. Our approach is similar to that used previously for the alkaline earth oxides [20], and, more recently, for (Y-AI~O~ [21,28] where full details of the relevant calculations are given. Lattice energies such as those for 0;) Thyi etc., are obtained from simulation procedures outlined in section 2, while the “effective” ionisation potentials of oxygen e,(O), e,(O), are estimated from electronic structure calculations also referred to in section 2. The results are collected in tables 7 and 8. With reference to fig. 3, we estimate the optical ionisation energy, Eo = E:(O) + ‘eL(0), to be 5.72 eV, and from the difference, e, = EP - EgO, in which EgOis the optical band gap, we find the conduction band edge to be close to the vacuum level. This compares with the range, - 0.7 eV to - 1.O eV, calculated for the alkaline earth oxides [20] and a measured value of - 1.0 eV for cr-Al,O, [21]. The corresponding thermal ionisation energy, E,’ = EL(O) + ‘tL(0), is found to be 4.6 eV, from which we deduce a thermal band-gap, Ei = (Et - e,), of 4.6 eV. To calculate the formation energy of free holes we need the O(2p) valence band width, U, which in the absence of experimental data we estimate, as before [21], from
and anion migration
energies
in ThO, Energy (eV)
Thus we predict the large polaron to be lower in energy by about 1.8 eV with a formation energy of 2.8 eV. For the excess free electron we obtain a value of 5.5 eV for
0.78 5.80 3.27 0.92 4.28
Table 7 Lattice energies
Anion migration Vacancy Direct exchange Direct interstitial Interstitialcy mechanism Direct interstitial migration of aim
of
Cation migration Vacancy Divacancy Trivacancy
7.04 5.36 6.35
of electronic
defects in ThO,
Defect
Energy (eV)
0, 0; 0; Td O-
17.49 14.52 12.89 34.27 -0.14
(perfect lattice), Et(O) (optical) E!(O) (thermal), EL(O) + Th interstitial
55
E.A. Colbourn, W.C. Mackrodt / The calculated defect structure of thoria Table 8 Calculated
electronic
defect energies
Table 9 Enthalpies
in ThO,
Conduction-band edge e, Optical band gap, Ei, (experimental) Thermal band gap, Ej O(2p) valence band width, U Small polaron hole formation Large polaron hole formation Th3T+h(localised electron)
EnergyWI
Process
- 0.0 5.8 [2]
fO,-+O; f0,
4.6
3.5. Enthalpies of oxidation and reduction
The results of the previous section, and in particular the energies of the excess free electron and hole, allow us to estimate the enthalpies of oxidation and reduction of thoria: these are collected in table 9. From this we draw a number of important conclusions. The most general is that “pure” or undoped thoria will exhibit a fairly narrow range of stoichiometry, comparable to CaO for example [35], with oxidation more favoured than reduction. The corresponding enthalpies are calculated to be approximately 4 eV and 6 eV respectively. Thus we would expect the incorporation of oxygen down to low partial pressures of oxygen. In the limit of high oxygen pressure, oxygen interstitials and 0(2p)-band large polaron holes are predicted to predominate the disorder, and here our calculations suggest that both singly- and doubly-charged lattice defects can be formed, with an energy difference of - 0.2 eV. Thus at low temperatures we would expect the majority defects to be 0; and the p-type conductivity to show a half power dependence on the oxygen partial pressure. At higher temperatures, however, O,?-
__---__ CONDUCTION
BAND
t
o,zp,
Fig. 3. Electronic
energy
levels in thoria.
+h
3.7
+2h
3.9
VALENCE
BAND
5.6
+$O,+V,+Ze
Vo + 40, tTh4rl
of ThO, Energy (eV)
+ Of-
Oi-
the localised state, Tlr$,, so that the present calculations suggest that both holes and electrons will exhibit band or large polaron characteristics.
EC
and reduction
fTh4,+,+f0,+fVrt,+2h+fTh0,
6.0 4.6 2.8 5.5
Th:+
VACUUM __ __--_____
of oxidation
-. O;-
+ Oi-
6.0 +2h
+ ;O,
+ e, + Th:’
-0.4 + iTht+
+ 2e
9.9 4.2
will be formed with an associated P(O,)f dependence of the electronic conductivity. Assuming the mobility of large polaron holes to be non-activated, therefore, our calculations predict a difference in the temperature dependence of ionic and electronic conduction. At high temperature the difference in Arrhenius energies will equal the migration energy of OF-, which for an interstitialcy mechanism we calculate to be - 0.9 eV. Thus we predict activation energies of - 1.3 eV and - 2.2 eV for electronic and ionic conduction respectively. With regard to doped thoria, our calculations indicate a change of defect structure, though not of the oxygen pressure dependence. For solid solutions of thoria and di- and tri-valent oxides such as CaO and Y,O,, we predict the quenching of anion vacancies (introduced by the dopant cations to form lattice ions and holes. The pressure dependence of the p-type coninterstitials
ductivity is once again P(O,)f, but we predict this ‘in-filling’ of vacancies to be exothermic, with a calculated enthalpy of -0.4 eV. It should, therefore, be less facile as the temperature is increased. In the event of very low oxygen pressure, we find anion vacancies rather than cation interstitials to be the major lattice defect, with an associated P(O,)-i dependence of the n-type semi-conductivity. As before we predict a difference in the Arrhenius energies for ionic and electronic conductivity, and here we predict this to be 0.8 eV which is the activation energy for vacancy migration. The activation energy for electronic conductivity is calculated to be - 2.0 eV and that for ionic conductivity - 2.8 eV. In between these limits there will be a region of oxygen pressure in which oxidation and reduction are more or less equally balanced in which case the disorder will consist entirely of oxygen vacancies and interstitials with no electronic charge carriers. The electrical conductivity, therefore, will be solely ionic. We return to a discussion of this in the next section.
56
E.A. Coibourn,
W.C. Mackmdt
/ The ealcuiated defect stmctwe
4. Comparison with experiment
of thoria
formed by the reaction, 0,2-+f02+VO+2e
Since diffusion and conductivity data are the major source of information on the defect structure of thoria it is to these data that we turn for detailed comparison with our results: we begin with diffusion. From measurements of oxygen self-diffusion coefficients Arrhenius energies of 2.85 eV 1131 and 2.16 eV [17,18] have been deduced; both have been interpreted in terms of an intrinsic mechanism involving Frenkel defects [ 17,181. However, the present calculations suggest that the intrinsic diffusion of oxygen would require an energy close to 4 eV - 3.8 eV for a vacancy process and 3.9 eV for diffusion by an interstitialcy mechanism - in which case we need an alternative explanation for the experimentaf values. We suggest, instead, that the measured Arrhenius energies correspond to separate extrinsic processes. The lower energy process involves the formation of oxygen interstitials and the creation of valence band holes. 40, + OF- + 2h, for which we cakulate an energy of 3.9 eV. From this we deduce an effective formation energy of 1.3 eV for 02- interstitials (and holes), which leads to an Arrhenius energy for oxygen diffusion of 2.2 eV, based on an interstitialcy mechanism for migration. Given the uncertainty in the calculated ionisation potentials of interstitial oxygen, which we estimate to be no better than f0.3 eV, our calculated energy of 2.2 eV is in good agreement with that of 2.16 eV reported by Ando et al. [ 17,181. Thus we interpret their high temperature (1200-165O’C) oxygen self-diffusion measurements in terms of an extrinsic mechanism involving the formation of 02- interstitials, rather than an intrinsic process. Furthermore, these authors have also reported a low temperature @SO-1250°C) activation energy of 0.76 eV and it is tempting to relate this to our calculated migration energy of 0.78 eV for oxygen vacancies which might result from residual trivalent impurities. We note that even if singly-charged interstitials, 0; , were the major lattice defect, our calculations indicate that these are most unlikely to be involved in oxygen seIf-diffusion for, as shown in table 6, we predict an activation energy of migration for these defects in excess of 4 eV and an Arrhenius energy close to 6 eV. The energy reported by Edwards et al. [ 131, 2.85 eV, would seem to be rather too high to be accounted for by this mechanism, though we cannot rule this out completely, and here we suggest that the measured coefficients correspond to the diffusion of oxygen vacancies
for which we calculate an energy of 6.0 eV. This leads to an Arrhenius energy of 2.8 eV which is close to the value reported by Edwards et al. [13]. In table 10 we collect the calculated Arrhenius energies for the various processes. With regard to cation diffusion our calculations have been confined solely to vacancy mechanisms, for interstitial defects would seem to be far too energetic to play any significant role in cation disorder in thoria. For vacancy migration we find activation energies ranging from 5.36 eV (for a divalency mechanism) to 7.04 eV (for free vacancies) which supports the view [ 15,161 that the diffusion coefficients measured by Hawkins and Alcock [ 141 do not relate to volume diffusion. King [ 151 and Matzke [16] have reported Arrhenius energies of - 6.5 eV and this can be interpreted in terms of any one of the three vacancy mechanisms we have considered. However, our calculations suggest that whatever the mechanism the reported diffusion coefficients correspond to extrinsic control. Writing the Arrhenius energy, EA, in the form: EA=Ef+E,,
where E, and E, are formation and migration energies respectively, our calculated values for E,, if they are correct, require effective formation energies for cation vacancy defects of 1.1 eV or less. This, we suggest, is possible only if cation vacancies result from the presence of 5-valent impurities by charge-compensation. If this is the case and we assume oxygen disorder to be governed by 40, + Of- + 2h, we deduce effective formation energies of 2.0 eV and 5. t eV respectively for divacancies and &i-vacancies. Thus our calculations favour either a free vacancy or a di-
Table 10 Arrhenius energies Mechanism
for oxygen
self-diffusion
Arrhenius Calculated
,.-
“0
.,
* “o +o;-
;02 -+ Of-
i2h
in ThO,
energy (ev) Experimental
3.8 (Vo) 3.9 (02 - ) 2.2
2.16 [‘” 2.85
O;-
+ :O, +Vo+2e
2.8
$0,
+ 0;
0.1
+h
[ 131
51
E.A. Colbourn, W.C. Mackrodi / The calculated defect structure of thoria
vacancy mechanism for thorium diffusion in ThO, with Arrhenius energies of 7.0 eV and 7.4 eV compared with the experimental value of - 6.5 eV. We turn now to the electrical conductivity of thoria and begin by considering the pure material. There seems to be wide-spread agreement that the activation energy for bulk ionic conduction is approximately 2 eV [8,9], which as Ando et al. [ 17,181 have pointed out, is close to their measured Arrhenius energy for oxygen self-diffusion. We interpret both, therefore, in terms of the same extrinsic mechanism involving the formation of oxygen interstitials. Lower energies, in most cases at lower temperatures, have also been reported, notably by Lasker and Rapp (1.47 eV) [6], Bransky and Tallan (1.02 eV) [7] and Maiti and Subbarao (0.98-1.52 eV) [ll]. These, we suggest, might be reasonably accounted for by grain boundary transport. If migration at grain boundaries can be assumed to be largely non-activated, the activation energy for ionic conduction would simply comprise the formation energy of the charge carrier, which in the case of 02- interstitials, we calculate to be 1.3 eV. This compares with the value of 1.47 eV found by Lasker and Rapp [6] and that of 1.52 eV by Maiti and Subbarao [ 111. Alternatively, the lower activation energies, particularly those of about 1 eV, could result from the transport of residual impurity-controlled defects, which are implicated in the low temperature Arrhenius energy of 0.76 eV found by Ando et al. [ 17,181. We now consider electronic conduction in thoria, which our calculations indicate will be n-type at very low oxygen pressures and p-type at higher pressures, in agreement with experiment [5,9,11]. There seems to be little data on n-conductivity [9] and no reports of the activation energy with which we can compare our calculated value of 2 eV. However, from the graphical results reported by Choudhury and Patterson [9] we estimate the activation energy (at log P(0,) - - 36 atm) to be approximately that for the ionic conductivity, i.e. 1.80-1.92 eV, which lends support for our results. Maiti and Subbarao [ll] have reported activation energies of 2.20 eV and 1.10 eV for hole conductivity at low ( < 7OO’C) and high (> 7OO’C) temperatures respectively. Now, assuming large polaron conductivity to be nonactivated, our calculations suggest that these energies correspond to the formation of holes by to,
+ 0;
40, --, Of-
+ h, and + 2h
for which our calculated hole formation energies are 1.9 eV and 1.3 eV respectively. Overall, therefore, we find good agreement with such data that exist for the activa-
Table 11 Arrhenius energies for thorium self-diffusion in ThO, Mechanism
Arrhenius energy (eV) Calculated
Extrinsic control (M&!, ), and ;O, -+ OF- + 2h - Free vacancy
7.0
- Divacancy
7.4
- Trivacancy
11.5
Experimental
6.48 [15] - 6.5 [ 161
tion energy associated with electronic conduction in thoria. Finally, we consider the electrical conductivity of doped thoria. Maiti and Subbarao [ 1 l] have reported Table 12 Activation energies for electrical conductivity doped ThO, Mechanism
Activation energy (eV) Calculated
Pure ThO, o;+vo+oy - Ionic
of pure and
Observed
3.8 (‘Jo) 3.9 (OF- ) Ionic
0.98-1.80 [7,1 l] 1.47 [6]; 1.90 (81 1.80-1.92 [9] $4”’ + 0; + h - Ionic - Electronic ;Oj8’+ 02- +2h - Ionic - Electronic
6.1 1.9 2.2 1.3 Electronic
n-type - 2 eV [9] p-type 1.10-2.20
1111
O;- +f02(g)+Vo+2e - Ionic - electronic
2.8 2.0
CaO-ThO,
Ionic
1.38
1.12-1.42 [ll]
1.43
0.85-1.38 [4,IO,I21
Y,O,-ThO,
Ionic
E.A. Colbourn, W. C. Mackrodf / The calculated defect structure of thoria
58
extensive measurements for CaO-ThO, solid solutions and from these have obtained numerous activation energies. The least ambiguous are those in the high temperature region where vafues from 1.12 eV to 1.42 eV are found for solid solutions ranging from 1 mole % to 7 mole % CaO. Kilner and Steele [l] have interpreted these values in terms of the migration energy for oxygen vacancies plus half the binding energy to Cayh. Our present results give an energy of 1.38 eV for this sum which ties in well with the experimental results. The electrical conductivity of Y,O,-ThO, solid solutions has been investigated by numerous workers, with reported activation energies varying from 0.85 eV [7] to 1.38 eV [4]. Here our calculated value is 1.43 eV, which suggests that the lower range of experimental values might well correspond to grain-boundary or some other facilitated transport mechanism.
5. Summary and conclusions The present paper has been concerned with calculated energies of lattice and electronic disorder in thoria. From these we have deduced the fundamental defect structure, electronic energy levels and oxidation-reduction potentials. This has enabled us to re-assess the reported diffusion and electrical conductivity data and to suggest alternative defect mechanisms for these processes. In particular, we suggest that all the reported data correspond to extrinsically-controlled defect processes, the most important being the oxidation reaction, +0,-O;-
+2h,
in which oxygen interstitials and valence band holes are formed. We find quantitative agreement with the majority of measured activation energies for diffusion and conduction, which taken as a whole, confirms the overall validity of our defect model for thoria.
References [1] J.A. Kilner and B.C.H. Steele, in: Non-Stoichiometric Oxides, Ed. O.T. Sorensen (Academic Press, New York, 1981) p. 233. [2] W.H. Strehlow and E.L. Cook, J. Phys. Chem. Ref. Data 2 (1973) 163. [3] W.A. Lambertson, M.H. Mueller and F.H. Grunzel, J. Amer. Ceram. Sot. 36 (1953) 397. [4] E.C. Subbarao, P.H. Sutter and J. Hrizo, .I. Amer. Ceram. Sot. 48 (1965) 443. [S] J.E. Bauerle, J. Chem. Phys. 45 (1966) 4162.
[6] M.F. Lasker and R.A. Rapp, Z. Phys. Chem. N.F. 49 (1966) 198. 171 I. Bransky and N.M. Tallan, J. Amer. Ceram. Sot. 53 (1970) 625. [8] H.M. Lee, J. Nucl. Mater. 48 (1973) 107. (91 N.S. Choudhury and J.W. Patterson, J. Amer. Ceram. Sot. 57 (1974) 90. [lo] A. Hammou, J. de Physique 72 (1975) 43 1. [ll] H.S. Maiti and E.C. Subbarao, J. Electrochem. Sot. 123 (1976) 1713. [12] E. Schouler, A. Hammou and M. KIeitz, Mat. Res. Bull. 11 (1976) 1137. [13] H.S. Edwards, A.F. Rosenberg and J.T. Bittel, Aeronautical Systems Division, Wright-Patterson Air Force Base, Report No. ASD-TDR-63-635 (1963). [14] R.J. Hawkins and C.B. Alcock, J. Nucl. Mater. 26 (1968) 112. [ 151 A.D. King, J. Nucl. Mater. 38 (1971) 347. [16] Hj. Matzke, J de Physique C7 (1976) 452. (171 K. Ando, Y. Oishi and Y. Hidaka, J. Chem. Phys. 65 (1976) 2751. [18] K. Ando and Y. Oishi, Solid St. Ionics 3/4 (1981) 473. [19] W.C. Mackrodt, in: Computer Simulation of Solids, Eds.: C.R.A. Catlow and W.C. Mackrodt (Springer-Verlag, Berlin, 1982). [20] W.C. Mackrodt and R.F. Stewart, J. Phys. Cl2 (1979) 5015. [21] E.A. Colbourn and W.C. Mackrodt, Solid St. Comm. 40 (1981) 265. [22] E.A. Colboum and W.C. Mackrodt, J. Mater. Sci. 17 (1982) 3021. [23] C.R.A. Catlow, Proc. Roy. Sot. A353 (1977) 533. [24] V.E. Butler, C.R.A. Catlow, B.E.F. Fender and J.H. Harding, Solid State Ionics 8 (1983) 109. [25] V. Butler, C.R.A. Catlow and B.E.F. Fender, Solid State Ionics 5 (1981) 539. [26] C.R.A. Catlow and W.C. Mackrodt, in: Computer Simulation of Solids, Eds.: C.R.A. Catlow and WC. Mackrodt (Springer-Verlag, Berlin, 1982). [27] A.B. Lidiard and M.J. Norgett, in: Computational Solid State Physics, Eds.: F. Herman, N.W. Dalton and T.R. Koehler (Plenum, New York, 1972). [28] C.R.A. Catlow, R. James, W.C. Mackrodt and R.F. Stewart, Phys. Rev. B25 (1982) 1006. [29] W.C. Mackrodt and R.F. Stewart, J. Phys. Cl2 (1979) 431. [30] B.G. Dick and A.W. Overhauser, Phys. Rev. 112 (1958) 90. [31] J. Kendrick and W.C. Mackrodt, Solid State Ionics 8 (1983) 247. [32] E.A. Colboum, J. Kendrick and W.C. Mackrodt, to be submitted for publication. [33] J.A. Kilner and R.J. Brook, Solid State Ionics 6 (1982) 237. [34] S.P. Kowalczyk, F.R. McFeely, L. Ley, V.T. Gritsyna and D.A. Shirley, Solid State Comm. 23 (1977) 161. [35] W.C. Mackrodt, Solid State Ionics, to be published. [36] R.W.G. Wyckoff, Crystal Structures, 2nd edition (Interscience, New York, 1963).
E.A. Colboum,
[37] G.C. Benson, P.I. Freeman and E. Dempsey, J. Amer. Ceram. Sot. 46 (1963) 43. [38] G.V. Samsonov, The Oxide Handbook (IFI/Plenum, 1973). [39] K.F. Young and H.P.R. Data 2 (1973) 313.
Frederikse,
59
W. C. Mackrodt / The calculated defect structure of thoria
J. Phys. Chem.
Ref.
[40] P.B. Macedo, W. Capps and J.B. Wachtman, Ceram. Sot. 47 (1964) 651. [41] R.C. Weast, Handbook of Physics edition (C.R.C. Press, 1979).
J. Amer.
and Chemistry,
60th
50
THE CALCULATED
E.A. COLBOURN
DEFECT
STRUCTURE
of Nuclear Materials 118 (1983) 50-59 North-Holland Publishing Company
OF THORIA
AND W.C. MACKRODT
ICI PLC, New Science Group, PO Box 11, The Heath, Runcorn, Cheshire WA7 4QE, UK Received
3 March
1983; accepted
11 April
1983
The present paper is concerned with calculated energies of lattice and electronic disorder in thoria. From these we deduce the fundamental defect structure, electronic energy levels and oxidation-reduction potentials. This enables us to re-assess the reported diffusion and electrical conductivity data and to suggest alternative defect mechanisms for these processes.
1. Introduction Thoria is an important nuclear material, the properties of which are of interest in other areas of application such as solid state electrolytes [ 11. It is a wide band-gap (5.75 eV) [2], high melting point (33OO“C) [3] oxide, whose defect structure has been inferred largely from conductivity [4- 1 l] and diffusion [ 13- 181 measurements. In common with other fluorite-structured materials, Frenkel defects are thought to predominate the intrinsic disorder [ 17,181, while at high temperatures (- 1OOO’C) and oxygen pressures down to - IO-l4 atm, the observed electronic conductivity is p-type ]5,9,111. Self-diffusion coefficients of oxygen in the temperature range 800-16OO“C were determined by Edwards et al. [ 131, and, more recently, by Ando et al. [ 17,181, who found a break in the temperature dependence at approximately 1100°C. The results have been interpreted in terms of intrinsic and extrinsic mechanisms. Arrhenius energies of 2.85 eV [13] and 2.16 eV [17,18] (from 1 lOO-1600°C) have both been attributed to intrinsic diffusion, the latter being in broad agreement with activation energies of electrical conduction determined by Lee (1.91 eV) [8] and by Choudhury and Patterson (1.80 eV and 1.92) [9] at the same temperatures. Lasker and Rapp [6], on the other hand, have reported a somewhat smaller value of 1.34 eV for electrical conduction. Between 800- 1100°C Ando et al. [ 17,181 derived an Arrhenius energy of 0.78 eV, which they ascribed to extrinsic diffusion, possibly dislocation-enhanced, while Bransky and Tallan [7] obtained a similar value (0.88 eV) for ionic conduction. Self-diffusion coefficients of Th4’ have also been measured. Hawkins and Alcock
0022-3115/83/0000-0000/$03.00
0 1983 North-Holland
[ 141 found an activation energy of 2.55 eV for polycrystalline thoria, whereas Ring [ 151 and Matzke [ 161 reported much higher values of 6.48 eV and - 6.5 eV respectively. The latter values, in particular, confirm that ionic conduction involves the oxide ion and that comparisons between oxygen diffusion and electrical conductivity energies are meaningful. The electrical conductivity of doped thoria has been investigated in some detail, notably by Hammou [lo], Subbarao and Maiti [ 1I] and Schouler et al. [12]. For calcium doped solid solutions, Th(Ca),O,_, with x = 0.01-0.07, activation energies in the range 1.12 eV to 1.42 eV have been measured [ 1 l] and interpreted as the sum of a dopant-anion vacancy association energy and the migration energy for free vacancies [l]. Similar considerations apply to the activation energies of about 1.1 eV for yttrium-doped thoria [ 11. We have, then, what appear to be consistent transport data for thoria which can be interpreted in terms of oxygen vacancies and interstitials as the predominant disorder, with cation vacancies and holes as the minor defects. However, the assignment of specific defect processes to the measured Arrhenius energies is open to question. In particular energies of about 2 eV seem to be too small to account for intrinsic diffusion of oxygen. This implies a Frenkel formation energy of less than 3 eV which theoretical considerations suggest as being somewhat low for a 4-valent oxide [19]. Furthermore, the nature of the diffusing species is not determined. Likewise, the interpretation of the cation diffusion data is not fully resolved; for if it assumed that the energy reported by Hawkins and Alcock [4] is for extrinsic diffusion, and that by Ring [ 151 and Matzke [ 161 for intrinsic diffusion, this would lead to a Schottky forma-
51
EA. Colbown, W.C. Mackrodt ,I’ The calculated defect structure of thoria
tion energy of less than 6 eV, which, once again, seems rather too low for an oxide of this type [19]. In view of these uncertainties, then, calculations of the type previously reported [19-221 might have a useful role to play in a re-assessment of the experimental data. Catlow and coworkers have examined UO, 1231,CeO, [24] and ZrO, [25] using similar methods and have provided valuable insight into the defect structure of these materials. Here we extend this approach to thoria. We report energies for defect formation, migration and association with cation impurities; in addition we consider electronic energy levels related to thermal excitations and energies of oxidation and reduction. Our calculations give a unified account of the defect properties of pure and doped thoria which is in good accord with the reported data.
2. Theoretical methods The theoretical methods used in the present calculations have been discussed in full recently [26]. The treatment of lattice relaxation is that introduced by Lid&d and Norgett [27] and extended by Catlow et al. [28] to systems with arbitrary structure. It has been used previously to examine numerous other oxides [19], including UO, [23], CeO, [24] and Ca*+: ZrO, [25]. Our interatomic potentials are based on the electron-gas method 1291, with the inclusion of the shell-model [30] to account for defect-induced electronic polarisation of the lattice. However, in deriving these potentials we have introduced a modification to earlier procedures [29]: this we discuss in brief. For the most part, electron-gas potentials lead to zero-strain unit cells that are approximately 3-58 larger than the measured structures. In some cases the agreement is better, in others, notably the transition-mete oxides, it is much worse. The reason for this is that the ‘short-range’ potentials are too repulsive, largely because a fully ionic model is normally assumed. However, the overallform of these potentials has been shown to be very similar to those derived from accurate ab initio calculations which include electron correlation [31]. We use this observation to obtain zero strain unit cells at the correct lattice spacing by simply scaling the electron-gas potentials in the following way. We write
for the oxygen-oxygen potential can be used for a wide variety of oxides, including the transition metal oxides, and the shift for the metal-oxygen interaction then found by fitting to the appropriate lattice parameter [32]. In other words, we have a semi-empirical electron gas potential with a single fitted variable, namely the shift re. A list of relevant perfect lattice properties of thoria obtained from these potentials is given in table 1, while a more complete account of the general procedure will be given elsewhere [32]. Finally, in this section we consider the effective ionisation potentials/eI~tron affinities of oxygen in thoria, for these have an important influence on electronic energy levels and the energies of oxidation and reduction. Our theoretical methods are similar to those used previously for the alkaline earth oxides [20] and cu-Al,O, [21,28]. They are based on single-centre Hartree-Fock calculations of oxygen in spherical Coulomb potentials [29], the strength of which we estimate from defect lattice simulations. As before we deduce electron correlation contributions from published calculations for iso-electronic systems [29]. The results are collected in table 2. ‘eL(0), the optical ionisation potential, is used to calculate the corresponding optical ionisation energy, from which we deduce the position of the valence and conduction bands and the formation energy of holes. We note in particular the difference in energy between the electron affinities of oxygen at a ‘normal lattice site and in an interstitial position. This is due to a difference of approximately 30% in the Madelung potentials at these two sites in the fully relaxed lattice. Table 1 Perfect lattice properties of ThO, derived from modified electron gas potentials Calculated Zero strain lattice parameter (A) Lattice energy (eV/mole) High frequency dielectric constant Static dielectric constant Elastic constants (10” dyne/cm*) c,,
ir(r-r,)=
U(r),
in which 0 is the modified potential: that is to say, the electron gas potentials U(r) are shifted inwards by an amount, q,, to give a zero strain lattice with the observed structure. We have found that a “standard” shift
c,2
c ~~mpressibilit y Cation polarisability (A3) Anion polarisability (K)
Experimental
5.5997 [36] 5.5991 - 103.0; - 104.7 [37] - 104.39 4.4 18.96
4.8 [38) 18.9 [39]
49.53 13.88 6.25 0.039 2.122 2.186
36.7 [40] 10.6 7.97 -
E.A. Colbourn,
52
WC.
Mackrodr / The calculared deject struclure of thoria
Table 2 Effective ionisation potentials (eV) in ThO, First ionisation potential of lattice oxygen Sum of ionisation potentials of lattice oxygen First ionisation potential of interstitial oxygen Sum of ionisation potentials of interstitial oxygen Fourth ionisation potential of thorium
‘CL(O)
-8.8 -7.3
‘r,(O) - 5.6 28.8 [41]
From our atomic Hartree-Fock calculations we estimate the (internal) electronic energy of Of- to be - 1.7 eV less than that of Oi- .
Frenkel defects are predicted to predominate the intrinsic disorder with a formation energy of 6.01 eV, which is approximately half that to create Schottky defects (11.93 eV): cation Frenkel pairs require a further 8 eV. These values are close to those reported previously by Catlow [23] for UO,, based on similar calculations. We find an appreciable binding energy of 2.74 eV between cation and anion vacancies so that the principal cation disorder is predicted to consist of free vacancies and di-vacancy pairs. Cation interstitials would seem to play no part in the defect structure of thoria. We deduce ‘effective’ formation energies for intrinsic disorder from the electro-neutrality condition and the temperature dependence of the defect equilibria. As shown in table 2 they vary from 3.0 eV for anion vacancies and interstitials to 13.97 eV for cation interstitials. From these values we conclude that the Arrhenius energies for the intrinsic diffusion of oxygen and thorium are in excess of 3 eV and 6 eV respectively. We return to this point later.
3. Calculated defect eneqjies 3.2. Impurity defects 3.1. Fundamental defects Calculated formation energies of fundamental lattice defects in ThO, are given in table 3. As expected anion
Table 3 Fundamental lattice defect energies in ThO, Defect
Energy (ev)
cation vacancy
84.66 15.83 - 64.86 - 9.82 91.15 2.14 112.04 4.28 11.93 (10.34) a 6.01 (5.31, 5.47) 19.80 (18.54)
anion vacancy cation interstitial anion interstitial divacancy divacancy binding trivacancy trivacancy binding Schottky (E,) anion Frenkel (E,_) cation Frenkel (El+)
Table 4 Defect energies Dopant
Mg2+ Ca*+ Sr’+ Ba*+
Ejjecrioe formation energies for intrinsic disorder anion vacancy 3.00 anion interstitial I cation vacancy 5.93 cation interstitial 13.87 divacancy 6.19 trivacancy 1.65 a values ~231
in brackets
are those calculated
by Catlow
Cation impurities play an important role in determining the defect properties of oxides [22]. In the case of thoria experimental investigations have been concerned largely with the electrical conductivity of solid solutions. Di- and trivalent impurities give rise to anion vacancies, which, in turn, are responsible for ionic conduction. Here we concentrate on the alkaline-earth ions and a series of trivalent substituents ranging from A13+ to Bi3+. Table 4 lists calculated defect energies associated with the former. As shown, substitution energies are high, due mainly to Coulombic interaction with the lattice, with differences in size accounting for
for UO,
associated
Substitution
52.15 54.59 56.44 58.83
with divalent
impurities
Energy (eV) vacancy binding
,=,‘a’
5
H’b’
1.54 1.20
4.33 2.03 1.73 2.21
2.19 0.83 0.62 1.13
1.11 1.11
s
(a) Calculated heat of solution in the absence of any aggregation based on the following lattice energies from ref. [20]: MgO ( - 40.74), CaO ( - 36.00), SrO ( and BaO (- 32.00). (b) Calculated heat of solution assuming complete defect gation.
defect taken 33.85) aggre-
E.A. Colbourn, W.C. Mackrodt / The calculated defect structure of thoria
approximately 15% of the total energy. The corresponding heats of solution range from 4.33 eV for MgO to 1.73 eV for BaO. We find vacancy binding energies in excess of 1 eV for all four dopants which suggests an appreciable degree of defect aggregation even at high temperatures. The variation in the binding energy with ion size follows that found previously for MgO [20] and CeO, [24]. The value for Ca2+ (1.20 eV) corresponds to an interaction which is almost totally Coulombic in nature since the ionic radius of Ca” (0.99 A) and Th4” (1.02 A) are practically identical. The enhanced binding of Mg2+ (0.34 ev) arises because the expansion of the lattice around the vacancy is compensated by the contraction around the dopant ion. For much the same reason the binding to Sr2+ and Ba2+ is slightly diminished. Trivalent impurities exhibit a similar pattern of behaviour to that found recently by Butler et al. [24] in their study of doped ceria. As shown in table 5 our values of 1.05 eV and 0.65 eV for the binding of Sc3+ and Y 3+ respectively to an anion vacancy are slightly larger than those reported for CeO, [24] but this is to be expected in view of the enlarged ThO, lattice. As before we note a marked variation in binding energy with dopant size, ranging from 1.81 eV for A13+ to 0.45 eV for Bi3+ which is just above the minimum value of 0.38 eV for Th3’. We return to the association of Ca2’ and Y 3+ with anion vacancies in relation to the measured activation energies for the electrical conduction later in section 4.
and electrical conductivity in thoria. The possible mechanisms for cation and anion migration are illustrated in figs. 1 and 2 respectively, and the corresponding energies collected in table 6. For anions we find vacancy migration to be the least energetic process with an activation energy of 0.78 eV. However, our value for the migration of interstitials by an interstitialcy mechanism is only slightly higher at 0.92 eV. Direct interstitial migration and the direct exchange of anions appear to be much more energetic and unlikely, therefore, to contribute to anion mobility. Recently, Kilner and Brook [33] have suggested a value of - 0.6 eV for the migration energy of anion vacancies in fluorite oxides, but in the case of thoria there seems to be no direct experimental data for comparison. Combining these migration energies with the “effective” formation energies in table 2 we deduce an Arrhenius energy for the intrinsic diffusion of oxygen in ThO, of 3.8-3.9 eV. This is appreciably higher than the value of 2.16 eV reported by Ando et al. [ 17,181, and later in section 4 we suggest an alternative interpretation for their results. As shown in table 6 we predict the migration of
(a)
3.3. Ion migration
53
0
MtQratin9 Cation
q
Cation Vacancy
Cl Anion Vaeoncy
The mechanisms and energetics of ion migration are central to an understanding of diffusion phenomena
Table 5 Defect energies associated
Substitution
AIs+ SC3+
In’+ Ys+ Bi3+ Ths+
21.54’*’ 25.17 27.12 28.04 31.32 34.27
with trivalent impurities
Vacancy binding
(b)
1.81 1.05 (0.63)‘b’ 0.63) 0.65 (0.27) 0.45 0.38
(a) The calculated heat of solution for a-Al,Os is 12.03 eV per mole in the absence of any defect aggregation. (b) Values in brackets are those calculated by Butler et al. [24] for the corresponding energies in CeO,.
(cl Fig. 1. Mechanisms
of cation migration in thoria.
W. C. Mackrodt / The calculated deject structure of thoria
E.A. Colboum,
54
3.4. Electronic defects and energy levels
(a)
(b)
m l’r------
---_
i
I’+----
’
T
-
-_----_L-
(cl
0
Migrating
0
Catlon
Anion
0
Anion Vacancy
Fig. 2. Mechanisms of anion migration in thoria.
cations to be considerably more energetic than anions, with activation energies ranging from 5.36 eV for a divacancy mechanism to 7.04 eV for the migration of free vacancies. These values are evidently high, but we note that they are comparable to those calculated by Catlow [23] for the migration of U4+ in UO,. Thus we predict Arrhenius energies for the intrinsic diffusion of Th4’ in thoria in excess of 11 eV, which suggests that it is unattainable even at the highest temperatures.
U = E;(O)
- E/‘(O).
The present value of 6.0 eV is similar to those calculated previously for MgO (5.0 eV), CaO (5.3 eV) and a-Al,O, (6.4 eV) [21], and consistent with photo-electron measurements for cubic oxides [34]. From our value of lJ we can estimate the difference in energy AE, between the large and small polaron hole. It is given by AE=E;(O)-E;(O)-22.
Table 6 Cation
We turn now to the calculation of energy levels and electronic defects, principally free electrons and holes, for they are involved in both the non-stoichiometry and the electrical conductivity of thoria. Our approach is similar to that used previously for the alkaline earth oxides [20], and, more recently, for (Y-AI~O~ [21,28] where full details of the relevant calculations are given. Lattice energies such as those for 0;) Thyi etc., are obtained from simulation procedures outlined in section 2, while the “effective” ionisation potentials of oxygen e,(O), e,(O), are estimated from electronic structure calculations also referred to in section 2. The results are collected in tables 7 and 8. With reference to fig. 3, we estimate the optical ionisation energy, Eo = E:(O) + ‘eL(0), to be 5.72 eV, and from the difference, e, = EP - EgO, in which EgOis the optical band gap, we find the conduction band edge to be close to the vacuum level. This compares with the range, - 0.7 eV to - 1.O eV, calculated for the alkaline earth oxides [20] and a measured value of - 1.0 eV for cr-Al,O, [21]. The corresponding thermal ionisation energy, E,’ = EL(O) + ‘tL(0), is found to be 4.6 eV, from which we deduce a thermal band-gap, Ei = (Et - e,), of 4.6 eV. To calculate the formation energy of free holes we need the O(2p) valence band width, U, which in the absence of experimental data we estimate, as before [21], from
and anion migration
energies
in ThO, Energy (eV)
Thus we predict the large polaron to be lower in energy by about 1.8 eV with a formation energy of 2.8 eV. For the excess free electron we obtain a value of 5.5 eV for
0.78 5.80 3.27 0.92 4.28
Table 7 Lattice energies
Anion migration Vacancy Direct exchange Direct interstitial Interstitialcy mechanism Direct interstitial migration of aim
of
Cation migration Vacancy Divacancy Trivacancy
7.04 5.36 6.35
of electronic
defects in ThO,
Defect
Energy (eV)
0, 0; 0; Td O-
17.49 14.52 12.89 34.27 -0.14
(perfect lattice), Et(O) (optical) E!(O) (thermal), EL(O) + Th interstitial
55
E.A. Colbourn, W.C. Mackrodt / The calculated defect structure of thoria Table 8 Calculated
electronic
defect energies
Table 9 Enthalpies
in ThO,
Conduction-band edge e, Optical band gap, Ei, (experimental) Thermal band gap, Ej O(2p) valence band width, U Small polaron hole formation Large polaron hole formation Th3T+h(localised electron)
EnergyWI
Process
- 0.0 5.8 [2]
fO,-+O; f0,
4.6
3.5. Enthalpies of oxidation and reduction
The results of the previous section, and in particular the energies of the excess free electron and hole, allow us to estimate the enthalpies of oxidation and reduction of thoria: these are collected in table 9. From this we draw a number of important conclusions. The most general is that “pure” or undoped thoria will exhibit a fairly narrow range of stoichiometry, comparable to CaO for example [35], with oxidation more favoured than reduction. The corresponding enthalpies are calculated to be approximately 4 eV and 6 eV respectively. Thus we would expect the incorporation of oxygen down to low partial pressures of oxygen. In the limit of high oxygen pressure, oxygen interstitials and 0(2p)-band large polaron holes are predicted to predominate the disorder, and here our calculations suggest that both singly- and doubly-charged lattice defects can be formed, with an energy difference of - 0.2 eV. Thus at low temperatures we would expect the majority defects to be 0; and the p-type conductivity to show a half power dependence on the oxygen partial pressure. At higher temperatures, however, O,?-
__---__ CONDUCTION
BAND
t
o,zp,
Fig. 3. Electronic
energy
levels in thoria.
+h
3.7
+2h
3.9
VALENCE
BAND
5.6
+$O,+V,+Ze
Vo + 40, tTh4rl
of ThO, Energy (eV)
+ Of-
Oi-
the localised state, Tlr$,, so that the present calculations suggest that both holes and electrons will exhibit band or large polaron characteristics.
EC
and reduction
fTh4,+,+f0,+fVrt,+2h+fTh0,
6.0 4.6 2.8 5.5
Th:+
VACUUM __ __--_____
of oxidation
-. O;-
+ Oi-
6.0 +2h
+ ;O,
+ e, + Th:’
-0.4 + iTht+
+ 2e
9.9 4.2
will be formed with an associated P(O,)f dependence of the electronic conductivity. Assuming the mobility of large polaron holes to be non-activated, therefore, our calculations predict a difference in the temperature dependence of ionic and electronic conduction. At high temperature the difference in Arrhenius energies will equal the migration energy of OF-, which for an interstitialcy mechanism we calculate to be - 0.9 eV. Thus we predict activation energies of - 1.3 eV and - 2.2 eV for electronic and ionic conduction respectively. With regard to doped thoria, our calculations indicate a change of defect structure, though not of the oxygen pressure dependence. For solid solutions of thoria and di- and tri-valent oxides such as CaO and Y,O,, we predict the quenching of anion vacancies (introduced by the dopant cations to form lattice ions and holes. The pressure dependence of the p-type coninterstitials
ductivity is once again P(O,)f, but we predict this ‘in-filling’ of vacancies to be exothermic, with a calculated enthalpy of -0.4 eV. It should, therefore, be less facile as the temperature is increased. In the event of very low oxygen pressure, we find anion vacancies rather than cation interstitials to be the major lattice defect, with an associated P(O,)-i dependence of the n-type semi-conductivity. As before we predict a difference in the Arrhenius energies for ionic and electronic conductivity, and here we predict this to be 0.8 eV which is the activation energy for vacancy migration. The activation energy for electronic conductivity is calculated to be - 2.0 eV and that for ionic conductivity - 2.8 eV. In between these limits there will be a region of oxygen pressure in which oxidation and reduction are more or less equally balanced in which case the disorder will consist entirely of oxygen vacancies and interstitials with no electronic charge carriers. The electrical conductivity, therefore, will be solely ionic. We return to a discussion of this in the next section.
56
E.A. Coibourn,
W.C. Mackmdt
/ The ealcuiated defect stmctwe
4. Comparison with experiment
of thoria
formed by the reaction, 0,2-+f02+VO+2e
Since diffusion and conductivity data are the major source of information on the defect structure of thoria it is to these data that we turn for detailed comparison with our results: we begin with diffusion. From measurements of oxygen self-diffusion coefficients Arrhenius energies of 2.85 eV 1131 and 2.16 eV [17,18] have been deduced; both have been interpreted in terms of an intrinsic mechanism involving Frenkel defects [ 17,181. However, the present calculations suggest that the intrinsic diffusion of oxygen would require an energy close to 4 eV - 3.8 eV for a vacancy process and 3.9 eV for diffusion by an interstitialcy mechanism - in which case we need an alternative explanation for the experimentaf values. We suggest, instead, that the measured Arrhenius energies correspond to separate extrinsic processes. The lower energy process involves the formation of oxygen interstitials and the creation of valence band holes. 40, + OF- + 2h, for which we cakulate an energy of 3.9 eV. From this we deduce an effective formation energy of 1.3 eV for 02- interstitials (and holes), which leads to an Arrhenius energy for oxygen diffusion of 2.2 eV, based on an interstitialcy mechanism for migration. Given the uncertainty in the calculated ionisation potentials of interstitial oxygen, which we estimate to be no better than f0.3 eV, our calculated energy of 2.2 eV is in good agreement with that of 2.16 eV reported by Ando et al. [ 17,181. Thus we interpret their high temperature (1200-165O’C) oxygen self-diffusion measurements in terms of an extrinsic mechanism involving the formation of 02- interstitials, rather than an intrinsic process. Furthermore, these authors have also reported a low temperature @SO-1250°C) activation energy of 0.76 eV and it is tempting to relate this to our calculated migration energy of 0.78 eV for oxygen vacancies which might result from residual trivalent impurities. We note that even if singly-charged interstitials, 0; , were the major lattice defect, our calculations indicate that these are most unlikely to be involved in oxygen seIf-diffusion for, as shown in table 6, we predict an activation energy of migration for these defects in excess of 4 eV and an Arrhenius energy close to 6 eV. The energy reported by Edwards et al. [ 131, 2.85 eV, would seem to be rather too high to be accounted for by this mechanism, though we cannot rule this out completely, and here we suggest that the measured coefficients correspond to the diffusion of oxygen vacancies
for which we calculate an energy of 6.0 eV. This leads to an Arrhenius energy of 2.8 eV which is close to the value reported by Edwards et al. [13]. In table 10 we collect the calculated Arrhenius energies for the various processes. With regard to cation diffusion our calculations have been confined solely to vacancy mechanisms, for interstitial defects would seem to be far too energetic to play any significant role in cation disorder in thoria. For vacancy migration we find activation energies ranging from 5.36 eV (for a divalency mechanism) to 7.04 eV (for free vacancies) which supports the view [ 15,161 that the diffusion coefficients measured by Hawkins and Alcock [ 141 do not relate to volume diffusion. King [ 151 and Matzke [16] have reported Arrhenius energies of - 6.5 eV and this can be interpreted in terms of any one of the three vacancy mechanisms we have considered. However, our calculations suggest that whatever the mechanism the reported diffusion coefficients correspond to extrinsic control. Writing the Arrhenius energy, EA, in the form: EA=Ef+E,,
where E, and E, are formation and migration energies respectively, our calculated values for E,, if they are correct, require effective formation energies for cation vacancy defects of 1.1 eV or less. This, we suggest, is possible only if cation vacancies result from the presence of 5-valent impurities by charge-compensation. If this is the case and we assume oxygen disorder to be governed by 40, + Of- + 2h, we deduce effective formation energies of 2.0 eV and 5. t eV respectively for divacancies and &i-vacancies. Thus our calculations favour either a free vacancy or a di-
Table 10 Arrhenius energies Mechanism
for oxygen
self-diffusion
Arrhenius Calculated
,.-
“0
.,
* “o +o;-
;02 -+ Of-
i2h
in ThO,
energy (ev) Experimental
3.8 (Vo) 3.9 (02 - ) 2.2
2.16 [‘” 2.85
O;-
+ :O, +Vo+2e
2.8
$0,
+ 0;
0.1
+h
[ 131
51
E.A. Colbourn, W.C. Mackrodi / The calculated defect structure of thoria
vacancy mechanism for thorium diffusion in ThO, with Arrhenius energies of 7.0 eV and 7.4 eV compared with the experimental value of - 6.5 eV. We turn now to the electrical conductivity of thoria and begin by considering the pure material. There seems to be wide-spread agreement that the activation energy for bulk ionic conduction is approximately 2 eV [8,9], which as Ando et al. [ 17,181 have pointed out, is close to their measured Arrhenius energy for oxygen self-diffusion. We interpret both, therefore, in terms of the same extrinsic mechanism involving the formation of oxygen interstitials. Lower energies, in most cases at lower temperatures, have also been reported, notably by Lasker and Rapp (1.47 eV) [6], Bransky and Tallan (1.02 eV) [7] and Maiti and Subbarao (0.98-1.52 eV) [ll]. These, we suggest, might be reasonably accounted for by grain boundary transport. If migration at grain boundaries can be assumed to be largely non-activated, the activation energy for ionic conduction would simply comprise the formation energy of the charge carrier, which in the case of 02- interstitials, we calculate to be 1.3 eV. This compares with the value of 1.47 eV found by Lasker and Rapp [6] and that of 1.52 eV by Maiti and Subbarao [ 111. Alternatively, the lower activation energies, particularly those of about 1 eV, could result from the transport of residual impurity-controlled defects, which are implicated in the low temperature Arrhenius energy of 0.76 eV found by Ando et al. [ 17,181. We now consider electronic conduction in thoria, which our calculations indicate will be n-type at very low oxygen pressures and p-type at higher pressures, in agreement with experiment [5,9,11]. There seems to be little data on n-conductivity [9] and no reports of the activation energy with which we can compare our calculated value of 2 eV. However, from the graphical results reported by Choudhury and Patterson [9] we estimate the activation energy (at log P(0,) - - 36 atm) to be approximately that for the ionic conductivity, i.e. 1.80-1.92 eV, which lends support for our results. Maiti and Subbarao [ll] have reported activation energies of 2.20 eV and 1.10 eV for hole conductivity at low ( < 7OO’C) and high (> 7OO’C) temperatures respectively. Now, assuming large polaron conductivity to be nonactivated, our calculations suggest that these energies correspond to the formation of holes by to,
+ 0;
40, --, Of-
+ h, and + 2h
for which our calculated hole formation energies are 1.9 eV and 1.3 eV respectively. Overall, therefore, we find good agreement with such data that exist for the activa-
Table 11 Arrhenius energies for thorium self-diffusion in ThO, Mechanism
Arrhenius energy (eV) Calculated
Extrinsic control (M&!, ), and ;O, -+ OF- + 2h - Free vacancy
7.0
- Divacancy
7.4
- Trivacancy
11.5
Experimental
6.48 [15] - 6.5 [ 161
tion energy associated with electronic conduction in thoria. Finally, we consider the electrical conductivity of doped thoria. Maiti and Subbarao [ 1 l] have reported Table 12 Activation energies for electrical conductivity doped ThO, Mechanism
Activation energy (eV) Calculated
Pure ThO, o;+vo+oy - Ionic
of pure and
Observed
3.8 (‘Jo) 3.9 (OF- ) Ionic
0.98-1.80 [7,1 l] 1.47 [6]; 1.90 (81 1.80-1.92 [9] $4”’ + 0; + h - Ionic - Electronic ;Oj8’+ 02- +2h - Ionic - Electronic
6.1 1.9 2.2 1.3 Electronic
n-type - 2 eV [9] p-type 1.10-2.20
1111
O;- +f02(g)+Vo+2e - Ionic - electronic
2.8 2.0
CaO-ThO,
Ionic
1.38
1.12-1.42 [ll]
1.43
0.85-1.38 [4,IO,I21
Y,O,-ThO,
Ionic
E.A. Colbourn, W. C. Mackrodf / The calculated defect structure of thoria
58
extensive measurements for CaO-ThO, solid solutions and from these have obtained numerous activation energies. The least ambiguous are those in the high temperature region where vafues from 1.12 eV to 1.42 eV are found for solid solutions ranging from 1 mole % to 7 mole % CaO. Kilner and Steele [l] have interpreted these values in terms of the migration energy for oxygen vacancies plus half the binding energy to Cayh. Our present results give an energy of 1.38 eV for this sum which ties in well with the experimental results. The electrical conductivity of Y,O,-ThO, solid solutions has been investigated by numerous workers, with reported activation energies varying from 0.85 eV [7] to 1.38 eV [4]. Here our calculated value is 1.43 eV, which suggests that the lower range of experimental values might well correspond to grain-boundary or some other facilitated transport mechanism.
5. Summary and conclusions The present paper has been concerned with calculated energies of lattice and electronic disorder in thoria. From these we have deduced the fundamental defect structure, electronic energy levels and oxidation-reduction potentials. This has enabled us to re-assess the reported diffusion and electrical conductivity data and to suggest alternative defect mechanisms for these processes. In particular, we suggest that all the reported data correspond to extrinsically-controlled defect processes, the most important being the oxidation reaction, +0,-O;-
+2h,
in which oxygen interstitials and valence band holes are formed. We find quantitative agreement with the majority of measured activation energies for diffusion and conduction, which taken as a whole, confirms the overall validity of our defect model for thoria.
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