Phys Chem Minerals (1999) 27: 138±143
Ó Springer-Verlag 1999
ORIGINAL PAPER
J.-P. Crocombette
Theoretical study of point defects in crystalline zircon
Received: 8 January 1999 / Revised, accepted: 14 May 1999
Abstract We present a numerical study of point defects in crystalline zircon (ZrSiO4). Vacancies and interstitials of all the constituents of zircon have been considered. For each defect, the structure and the formation energies have been calculated. Calculations, using the supercell method, are based on the Density Functional Theory in the Local Density Approximation. Empirical potentials have also been considered for comparison with electronic structure results. We ®nd a formation energy for the oxygen interstitial of 1.7 eV. This value is compatible with the experimental activation energy for oxygen diffusion in zircon, which proves an interstitial mechanism for the diusion of oxygen in zircon. For all other defects the calculated formation energies lead to negligible thermal concentration at equilibrium. Key words Zircon á Point defects á First-principles calculations
Introduction Zircon (ZrSiO4) is a common accessory mineral in igneous and metamorphic rocks. It is widely used in age determination studie, because of its very high radiogenic isotope retentiveness and high resistance to corrosion. For the same reasons it has been considered in the nuclear industry for high level waste disposal of actinides (Ewing et al. 1995). In both contexts the understanding of the diusion properties in this mineral is of crucial importance. Diusion properties depend highly on the point defects present in the material. In this paper we present the results of a theoretical study on point defects in pure crystalline zircon.
J.-P. Crocombette CEA-CEREM, SRMP CE Saclay 91191 Gif/Yvette Cedex France Fax: 33-1-69-08-68-67 e-mail:
[email protected]
We have considered vacancies and interstitials of all the constituents of zircon (zirconium, silicon and oxygen) and restricted the study to neutral defects only. To calculate the atomic arrangements around the point defects and their formation energies, we have used ab initio models based on the Density Functional Theory in the Local Density Approximation. Empirical potentials of the Born-Mayer-Huggins type have also been considered for comparison with the electronic structure results.
Implementation of the simulations The ®rst-principles calculations have been carried out in the framework of DFT-LDA. The exchange and correlation functional used was of the Cerperly-Alder type (Ceperley and Alder 1980). We used the pwscf (plane wave self consistent ®eld) code obtained via a collaboration with Pr. S. Baroni from CECAM (Centre EuropeÂen de Calculs Atomiques et MoleÂculaires). This code considers a plane wave basis and pseudopotentials (Pickett 1989). It allows the calculation of the relaxed structures via the calculation of the Hellman-Feynman forces. The ability of plane wave DFT-LDA method and pwscf code to reproduce crystalline zircon properties has been demonstrated in a previous study (Crocombette and Ghaleb 1998). The pwscf code has proved able to deal with large defective supercells in metallic systems (Satta et al. 1998). In our former study on the structure of crystalline zircon, we employed norm conserving pseudopotentials for all species. We therefore had to use a large energy cuto for the plane wave basis (95 ryd), which does not allow us to consider the large supercells required for defect studies. In order to decrease the energy cuto, we used in the present study ultrasoft pseudopotentials in the scheme developed by Vanderbilt (1990) except for silicon for which standard norm conserving pseudopotential was used (Car, unpublished work). The pseudopotential for oxygen atoms was taken from the literature (Laasonen et al. 1991). We have built
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the zirconium pseudopotential from the electronic con®guration of the isolated atom. Eight localised beta functions have been introduced: two for the p levels, two for the d levels and four for the s levels. This large number of beta functions is essential to describe accurately the diusion properties of the real potential, especially for its s component, for which one has to include both 4s and 5s states in the valence con®guration. The energy values of the beta functions are indicated in Table 1. The obtained pseudopotential was successfully tested on bulk zirconium and zircon structure. The results are very close to the one obtained with the norm conserving pseudopotentials (see Crocombette and Ghaleb 1998 for details). With this set of pseudopotentials, the energy cuto is reduced down to 25ryd. In its crystalline form, zircon has a tetragonal bodycentred Bravais lattice with twelve atoms in the unit cell (Robinson et al. 1971). The space group is I4=amd . Its structure is made of an arrangement of SiO4 tetraheda and ZrO8 dodecaheda. It has empty cylinders along the c axis into which the interstitials have been introduced. For the defect calculations, we considered supercells (built by repetition of the unit cell), in which a point defect is introduced. These supercells should be as large as possible to decrease the non-physical interactions between the periodically repeated defects. Due to computer limitations we have only considered supercells of 24 (1) or 48 (1) atoms, made of the repetition of two or four units cells respectively. In both cases only the G k point was considered. The 24 atom supercell is zircon's conventional tetragonal unit cell. To consider a larger cell the pattern is repeated along one of the three directions a, b or c. In tetragonal symmetry, the c direction is dierent from the a and b directions. If one considers the distances between point defects there is nearly no dierence between the dierent directions. If one thinks in terms of neighbouring shells a dierence appears for the oxygen vacancy. For this defect the repetition of the conventional cell along the c direction proves to be the one for which the ``self neighbouring'' of the oxygen vacancy is the smaller. So we chose this particular supercell for our calculations of all vacancies and interstitials. Calculation of the energy of the perfect crystal in the 48 atom supercell leads to a value per formula unit which diers from the one calculated in the unit cell by less than 4 meV. Such a small dierence evidences the ability of the pwscf code to deal with large supercells in insulating materials. The convergence of the calculated energy with respect to the supercell size has been checked for two sample point defects: oxygen and zirconium vacancies. The
dierence in formation energy (see below for the de®nition of this quantity) between the 48 and 24 atom supercells in the unrelaxed (resp. relaxed) con®guration for the oxygen vacancy is 0.6 eV (resp. 0.3 eV). For the zirconium vacancy the dierence is 0.3 eV (resp. 0.2 eV). These dierences are quite large. Unfortunately it was not possible, with our computers, to consider the next larger supercell made of 96 (1) atom. The precision of the energies given hereafter is therefore of the order of 0.5 eV. Nevertheless this large uncertainty does not alter the validity of the conclusions. To save computational time, calculations were made at constant supercell volume. The error introduced in the calculated formation energy has been estimated using crystal bulk modulus and assuming a relaxation volume of the order of 10 AÊ3. It is of the order of a few 10)2 eV. This small error proves negligible compared to the uncertainties associated with the supercell size.
Relaxed defect con®gurations Atomic relaxation was taken into account thanks to the calculation of the forces acting on the atoms in the plane wave formalism. For vacancies, relaxation leads to a decrease in energy of the supercell around 1 eV (Table 2). For cation vacancies relaxation involves only the ®rst shell of oxygen neighbours around the vacant site and is smaller than 0.2 AÊ. Ten atoms around the oxygen vacancy are displaced by more than 0.1 AÊ but only two of them are displaced by more than 0.2AÊ. For interstitials, atomic relaxations are much larger. Indeed ®ve (resp. 19, 17) atoms are displaced by more than 0.1 AÊ when an oxygen (resp. silicon, zirconium) interstitial is introduced. These large relaxations lead to energy gains up to 9 eV. Such large values are not only due to the large atomic relaxations but also to the fact that the initial positions of the interstitials were poorly guessed. The cationic interstitials remain along the c-axis cylinder. The silicon interstitial is surrounded by three oxygen atoms at approximately 2 AÊ (Fig. 1a). The usual SiO4 tetrahedra is not rebuild by the structure relaxation. The zirconium interstitial is surrounded by six atoms at 2.7 AÊ (Fig. 1b). Table 2 Energy gain between unrelaxed and relaxed supercell for zircon point defects in the 48 atoms supercell (units eV); V denotes a vacancy and I an interstitial Defects
VO
VSi
VZr
IO
ISi
IZr
Energy gain
0.99
0.61
1.72
4.05
2.10
9.29
Table 1 Energies of the localised wave functions (beta functions) for the zirconium ultrasoft pseudopotential (units eV). The corresponding electronic level (if any) is indicated in brackets
1
Energy
2
3
4
1
2
1
2
Es
Es
Es
Es
Ep
Ep
Ed
Ed
)54.5 (4s)
)32.5 (4p)
)4.62 (5s)
)3.67 (4d)
)32.5 (4p)
)13.6
)3.67 (4d)
13.6
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Fig. 1a±c Calculated structure of the interstitials in zircon. The oxygen, silicon and zirconium atoms are in dark, medium and light grey respectively. a silicon interstitial, b zirconium interstitial, c oxygen interstitial forming a dumbbell
EN EN 1 lX
Table 3 Distance between the two oxygen atoms forming a dumbbell and their three cations neighbours. The third line denotes the distances for an oxygen atom in its normal site in the numerically relaxed crystalline structure (units AÊ)
O1 O2 O
Si
Zr
Zr
1.83 1.78 1.62
2.24 2.20 2.09 2.15
2.57 2.75 2.35
The oxygen interstitial leaves the empty cylinder and combines with another oxygen atom to form a dumbbell (Fig. 1c). In the perfect structure without defects each oxygen atom can be associated with a (1 0 0) or (0 1 0) plane in which its three neighbouring cations are situated. The oxygen dumbbell axis is nearly perpendicular to this plane. The distance between the two oxygen atoms is 1.62 AÊ. As can be seen from Table 3 the distances between the two oxygen atoms and the three cations neighbours are nearly equal. This proves that the oxygen interstitial indeed forms a dumbbell and not a peroxide bridge, contrary to what is observed in quartz for instance (Hamann 1998).
Point defect formation energies Extrinsic point defects From the energies of the defective supercells it is possible to calculate the formation energies of the corresponding isolated point defects thanks to the following formula: VX EF EN 1 ÿ EN lX IX In this expression: VX EF is the formation energy of the vacancy or inIX terstitial of the X species;
is the calculated energy for the non defective supercell (with 48 atoms); is the calculated energy for the supercell containing the defect; is the chemical potential of the X species.
To calculate the point defect formation energies of an atom X one has to specify a value for the chemical potential of the X atom, but there is an uncertainty on the chemical potentials of the elements in zircon; indeed when zircon exists they have to verify the equation: lSi lZr 4lO GZrSio4 ; where GZrSio4 is the free enthalpy of zircon at the given temperature and pressure. When this equation stands alone there are many sets of chemical potential that can verify it. Specifying the chemical potential of an element corresponds to specifying its reference state. For each element one should choose a state that can be calculated with our plane wave code to guarantee the coherence of the calculations. This precludes the choice of the isolated atoms as reference states because DFT-LDA is known to describe isolated atoms very poorly. Moreover such a choice of chemical potentials would lead to values that could not be directly compared to experimental values. Indeed, creating an isolated monoatomic point defect involves the process of taking away the defective atom from the crystal, in the case of the creation of a vacancy, or bringing it in the crystal, in the case of an interstitial creation. Thus specifying a chemical potential for the X element corresponds to specifying that there is an equilibrium of the X atoms, between the zircon phase and another phase outside of the material in which the chemical potential is uniquely de®ned. Thermodynamical equilibrium ensures that the chemical potential of the elements are equal in the two phases. The outer phase is the reference that sets the chemical potential, as it is the reservoir in which the atom is put or from which it is taken. We ®rst considered as reference states the elements in their standard states: bulk silicon and zirconium and the diatomic oxygen molecule. This choice leads to the values indicated in Table 4. The energies of the three reference states have been calculated with the pwscf code. For the oxygen molecule the in¯uence of pressure on the
141 Table 4 Point defects formation energies in zircon (unit eV); references are to the elements in their standard states Ef
O
Si
Zr
Vacancies Interstitials
5.6 1.7
15.8 7.0
18.7 5.2
chemical potential calculated in the ab initio framework is neglected. Nevertheless, as the calculations are carried out at 0 K, there is no kinetic pressure. Another choice of reference states can be built from the assumption that for each cation there is an equilibrium between zircon, the simple cation dioxide and oxygen vapour. For instance the silicon atom chemical potential may be ®xed by assuming the equilibrium zircon with quartz (SiO2) and oxygen vapour. With this choice silicon is always in an oxidised state. The equations ®xing the chemical potentials are then 2lO GO2
lSi 2lO GSiO2 :
The ®rst equation sets the chemical potential of oxygen, the second sets the one of silicon. The same equations can be written for zirconium atoms in the case of an equilibrium between zircon, zirconia (ZrO2) and oxygen. We have calculated with the pwscf code the energies of SiO2 and ZrO2. From these calculations we have deduced the new set of chemical potentials of the cations. They come out to be more than 10 eV smaller than the one calculated for elements in their standard states. The values of the formation energies of the related point defects are then modi®ed by more than 10 eV (see Table 5). In the oxide the chemical potential of the cations are smaller than in the bulk material, so it is much easier to subtract one cation from zircon and put it in its oxide than in the standard state; and the reverse holds for the interstitials. Intrinsic point defects Intrinsic composite defects are association of point defects that can be created without taking matter away from the material nor bringing matter into it. In zircon, elementary intrinsic point defects are the Frenkel pairs of each component, the Schottky defect made of six vacancies and the corresponding defect made of interstitials, which we should call the anti-Schottky defect. The formation energies of these defects are given by the following equations (for the Frenkel pairs, the Schottky defect and the anti-Schottky defect respectively): Table 5 Point defects formation energies in zircon (unit eV); references are to the cations oxides in equilibrium with oxygen vapour Ef
O
Si
Zr
Vacancies Interstitials
5.6 1.7
5.8 17.0
5.9 18.0
EFX EVNXÿ1 EINX1 ÿ 2 EN
N ÿ1 EN N N 1 EAS EINZr1 EINSi1 4 EINO1 ÿ 6 EN N In these equations, the point defects are supposed to be perfectly dissociated. As these defects are intrinsic their formation energies do not involve the external chemical potentials of the elements. The values obtained are given in Table 6. ES EVNZrÿ1 EVNSiÿ1 4 EVNOÿ1 ÿ 6
Empirical potential calculations In our previous study on the zircon structure (Crocombette and Ghaleb 1998) we compared the abilities of ®rst-principles electronic structure calculations and empirical potentials to reproduce the zircon structure. In order to supplement this comparison we have used the same empirical potentials to describe point defects. These potentials are pair potentials of the BornMayer-Huggins type which include long range coulombic interactions between ions bearing their formal charges. These crude but robust potentials reproduce quite satisfactorily the structure and equilibrium volume of zircon. For the defect calculations we used supercells of 5184 1 ions with a constant volume taking away or adding an ion to the perfect supercell. The structures predicted around vacancies with the empirical potentials involve little relaxations and are thus in agreement with what is obtained with ab initio calculations. The structures of interstitial defects are more poorly reproduced especially for the oxygen interstitial for which the dumbbell con®guration does not come out of the empirical potentials calculations. The oxygen interstitial is found to lie in a (1 0 0) plane in a position for which the distances with the other oxygen atoms are maximum. With the ionic empirical potentials we have used, it not possible to calculate the energy of non ionic systems and therefore the energy of the oxygen molecule cannot be estimated with our potentials. So it is not possible to obtain a set of chemical potentials for the elements. Therefore the formation energies of extrinsic point defects cannot be estimated. On the other hand, the formation energies of intrinsic composite defects, as they do not involve the description of the exterior of the material, can be calculated with empirical potentials. The Table 6 Formation energy of intrinsic defects in zircon for the empirical potential and DFT-LDA approaches (units eV). FP is for Frenkel pair, AS for Anti-Schottky defect Ef
OFP
SiFP
ZrFP
Schottky
AS
Empirical potentials DFT-LDA Ratio
14.1
33.3
36.3
61.7
64.6
22.9 1.45
24.0 1.51
34.1 1.79
41.8 1.53
7.3 1.93
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values thus obtained are given in Table 6. Compared to the ab initio values, the values calculated with empirical potentials are always larger, the ratio between the two ®gures ranging between 1.5 and 1.9.
Discussion First-principles calculations show that the major cationic defects are either vacancies or interstitials depending upon the choice of the reference chemical potential, which therefore proves crucial for comparison with experiments or prediction of behaviour in natural samples. The results obtained with the chemical potentials of the elements in their standard states are technically correct, even if it is not obvious that thermodynamical equilibrium between ZrSiO4 and bulk Si or Zr may ever be reached experimentally. Nevertheless the second set, in which the elements in zircon are supposed to be in equilibrium with their component oxides, gives a much better insight into the possible real situation in the mineral. With this set of chemical potentials it appears that the most energetically favourable defects are the interstitials for the oxygen ions and the vacancies for the cations. For cations, both interstitial and vacancy formation energies are very high. Boltzman distributions built from these formation energies lead to negligible concentration of cationic thermal defects at any temperature in the material. The formation energy of the oxygen vacancy (5.6 eV) is also quite high and makes thermal concentration of this defect negligible. Note that this value of the formation energy of the oxygen vacancy in zircon is of the same order but a bit smaller than the one calculated in the two component oxides, quartz (SiO2) and zirconia (ZrO2). Indeed DFT calculations comparable to ours made by Carbonaro et al. (1997) for SiO2 give a value of 6.97 eV for the formation energy of the oxygen vacancy. In zirconia, Jomard et al. (1998) found a value of 8.03 eV with an atomic reference for the oxygen. As for the oxygen interstitial, the obtained energy (1.7 eV) is smaller. At 700 K, which is a possible temperature for a high level nuclear waste (Meldrum 1998), a concentration of 2.10)13 is found, which is not negligible. Therefore, apart from the oxygen interstitial, the point defects that would be present in zircon would not originate from thermal equilibrium. This does not mean that no point defects may be found in zircon, as other sources of defects could exist. For instance in the context of actinide doping for nuclear waste, defects may be associated with the presence of doping species or with alpha decay damage. As far as diusion is concerned, little can be said about cationic diusion apart from the fact that diusion is not due to thermal defects as their concentration is negligible. Oxygen diusion in zircon has been studied experimentally by Watson and Cherniak (1997). They mea-
sured the oxygen diusion coecient in dry zircon at temperatures ranging from 765 °C to 1500 °C at atmospheric pressure. They proceeded by encapsulating zircon samples with a ®ne powder of 18O-enriched quartz together with small particles of zirconium, and annealing the sealed capsules in air. An activation energy for diusion of 4.7 eV was found. When diusion takes place by a thermal point defect mechanism, the activation energy is the sum of the formation and the migration energies for the defect responsible for diusion. The values we have calculated for the formation energies of the oxygen vacancies (5.6 eV) and interstitials (1.7 eV) indicate that the oxygen diusion takes place via an interstitial mechanism. Comparing our calculated formation energy with the experimental activation energy leads to an estimation of the oxygen interstitial migration energy of 3 eV which appears a reasonable value for such an ionic oxide. To compare, for that matter, zircon with related oxides, we can note that an interstitial diffusion mechanism has recently been proposed for SiO2 based on quantum mechanical calculations (Hamann 1998). Thus we predict in zircon the same oxygen diffusion mechanism as in SiO2. The formation energies calculated with empirical potentials are always larger than the ones calculated with the ab initio approach. Nevertheless empirical potentials formation energies are shown to be insightful. First they are all positive and not completely out of range. Moreover the fact that the ratio between the values calculated with the two methods is roughly constant (Table 6) is also satisfactory. Crude potentials such as the ones we used are able to reproduce roughly the trends predicted by ab initio calculations even if it is not possible to make quantitative prediction with such potentials. It should be noted that one major drawback of these potentials is the fact that they deal with ions bearing their formal charges (Si4+,Zr4+and O2)). More complex empirical potentials allowing for partial interatomic charge transfer may lead to better results.
Conclusion We have calculated the point defect formation energy in zircon for the three constituents of this mineral. Two theoretical schemes have been employed, namely DFTLDA electronic structure calculations in the plane wave formalism and Born-Mayer-Huggins empirical potentials. The calculation of extrinsic point defect formation energies requires the speci®cation of an exterior reference state to de®ne the reference chemical potential of the defective element. The formation energies thus obtained depend strongly on the choice of these reference states. Nevertheless, independently of the choice of the reference states, electronic structure calculations lead to very high values for the formation energies of all defects (apart from the oxygen interstitial). Thus defect concentration at thermal equilibrium in the crystal is com-
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pletely negligible. Intrinsic point defects energies estimated with our empirical potentials are larger than the one calculated from ®rst-principles by a factor ranging from 1.5 to 1.9, which highlights the limitations of these empirical potentials. The value calculated for the oxygen interstitial formation energy leads to a non negligible concentration for this defect at thermal equilibrium. This value is consistent with the experimental diusion activation energy for oxygen in zircon. We thus predict that the diusion of oxygen in zircon takes place by an interstitial point defect mechanism. Acknowledgements The author acknowledges many fruitful discussion with Dr Yves Limoge.
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