EUROPHYSICS LETTERS

1 June 2004

Europhys. Lett., 66 (5), pp. 680–686 (2004) DOI: 10.1209/epl/i2003-10247-3

Oxygen neutral defects in silica: Origin of the distribution of the formation energies L. Martin-Samos 1 , Y. Limoge 1 , N. Richard 2 , J. P. Crocombette 1 , G. Roma 1 , E. Anglada 3 and E. Artacho 4 1 Service de Recherche en M´etallurgie Physique, CEA-Saclay 91191 Gif-sur-Yvette, France 2 CEA-DIF - Bruy`ere-le-Chˆ atel, France 3 Departamento de F´ısica de la Materia Condensada, Universidad Aut´ onoma de Madrid Madrid, Spain 4 Department of Earth Sciences, University of Cambridge - Cambridge, UK (received 14 October 2003; accepted in final form 26 March 2004) PACS. 61.72.Bb – Theories and models of crystal defects. PACS. 61.43.Fs – Glasses.

Abstract. – We present a numerical study of the neutral oxygen defects in silica. This work was performed using two complementary approaches: molecular dynamics using an empirical potential and ab initio calculations. We show that the formation energy fluctuates from point to point. The dispersion is due to the fluctuations of the local stress before defect creation. Using lattice statics, we establish the relation between the local-stress fluctuations inherent in the glassy state and the formation energies.

Introduction. – Silica in its amorphous or crystalline phases is a very common material used in a variety of scientific and technological fields. For instance, SiO2 is a major constituent of electronic devices and of nuclear glasses [1]. The long-term behaviour of components is controlled by the native as well as radiation-created defects which produce induced phase transformation and diffusion. Nuclear waste glasses, for example, are heavily irradiated, leading possibly to large and detrimental structural evolutions. Our interest in diffusion processes is then directly related to the radiation damage and, more generally, to the aging of materials under irradiation. The characterization and understanding of defects related to glass formers is the first step in the study of the atomic diffusion processes. The structure, formation and migration energies of neutral oxygen vacancies and interstitials is well known for the crystalline phases of SiO2 [2–7]. Studies of charged vacancies have been developed in silica [8,9] mainly for micro-electronic purposes, like silicon oxidation. Under high enough oxygen partial pressure, the main diffusion mechanism at work involves molecular oxygen [10, 11]. For aging studies, however, as in the context of nuclear glasses, one needs a complete picture of defects and self-diffusion mechanisms in all conditions. Here, we present a first-principles study of the neutral oxygen vacancies and interstitials, putting a great emphasis on a careful determination of the energy distributions, needed for the calculation of the self-diffusion coefficient. The c EDP Sciences


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study of charged oxygen defects will be presented in a forthcoming publication. We show that these formation energies are characterized by a continuous Gaussian-like distribution. Moreover, combining first-principles and empirical potential calculations, we also show that the origin of this energy spread lies in the atomic-scale fluctuations of the stress before the introduction of the defect. The dispersion is a measure of the local elastic response of the sites submitted to the point defect distortion. In this study we are faced with a double difficulty. First, one needs a silica model truly representative of an actual material from both points of view of structure and nature of binding, including homo-polar bonds. The second difficulty lies in the glassy nature of silica. Indeed each possible defect site experiences a different neighborhood, which means that a proper characterisation of the structure and formation energies of defects is inherently statistical. Both characteristics imply an increase of the computational burden. In order to speed up the generation of the glass model, we choose to combine molecular dynamics using empirical potentials (EPMD) and first-principles calculations. This two-step procedure is justified by the close similarity of the structures obtained after a relaxation using ab initio methods or EPMD ones, particularly the one-to-one correspondence between the ground-state energies [12]. Within an EPMD approach it is possible to obtain, at a low computational cost, well-relaxed amorphous models whose structure is close to the structure of real silica. Then a relaxation step using first-principles calculations allows to get a structure even closer to the experimental one. Thanks to the realistic description of the homo-polar bonds in firstprinciples methods, the defect formation energies calculations can then be performed with a good precision. All EPMD calculations have been done using, as an interaction model, the empirical potential adjusted by van Beest [13], which is known to reproduce well the local structural and vibrational properties of SiO2 polymorphs [13,14]. With the aim to fulfil the requirement of statistical accuracy underlined above with the available computational resources, we choose to pursue the ab initio calculations using both a method based on an atomic functions basis, SIESTA code [15], and methods based on plane wave, VASP [16, 17] and PWSCF [18] codes. All are first-principles codes based on the density functional theory (DFT) within the local density approximation (LDA). The first has the capability of very rapid calculations, but needs a very careful optimization of the basis set with respect to the problem under study, while the other two are free of limitation concerning the basis, but more computer demanding. A key point of this work has therefore been the careful validation of the use of SIESTA in the context of defect studies in silica. Methods. – In order to generate a glass whithin the EPMD approach, we melt at approximatively 7000 K a crystal of SiO2 containing 108 atoms. Then, we quench it at a rate of 2.4 · 1014 K/s down to a temperature of 3000 K. At this temperature we anneal the system during 250 ps and quench it again down to 300 K. The annealing temperature and duration have been chosen in order to systematically obtain at 300 K well-connected glassy networks, containing neither two membered rings [19] nor coordination defects. The anneals are performed in the NV(T) ensemble. The resulting configuration is then fully relaxed with respect to atomic positions and cell parameters at zero pressure in EPMD, and used to calculate the local stress within the empirical potential interaction model. The same configuration is also fully relaxed using the ab initio approaches with respect to atomic positions and cell parameters by a conjugate gradient method; in VASP we relaxed only the atomic positions. Once relaxed, this initial configuration is used to perform the first-principles calculations of defect properties. For SIESTA, the electronic structure is accounted for by using a pseudoLCAO double-zeta basis set, orbitals sp for oxygen and sp-polarized for silicon, as designed

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Table I – Density of the amorphous models vs. experiment.

Density (g/cm3 )

Empirical MD 2.14

SIESTA 2.19

PWSCF 2.18

Experimental 2.20

by Anglada [20] for α-quartz bulk properties. In order to test the capabilities of the basis set in the study of oxygen defects, defect calculations have also been performed in the planewave approach, using PWSCF with norm-conserving pseudo-potentials and an energy cutoff of 80 Ry, and VASP, using the PAW method and an energy cutoff of 36 Ry. The vacancies are created by swiching off an oxygen of the silica network. The interstitials are created by inserting an extra oxygen near a network oxygen, the resulting structure Si-O-O-Si being close to a peroxy bridge configuration, according to the expected interstitial structure from the known results in α-quartz. These defect configurations are then relaxed for atomic positions and cell parameters as concerns calculations with SIESTA and PWSCF, here too with VASP atomic positions only. The formation energies are calculated using half the energy of the oxygen molecule as reference state for oxygen [7], according to the following equations: 1 SiO2   (SiO2 )V + O2 , 2 1 Interstitials: SiO2 + O2   (SiO2 )I , 2

Vacancies:

1 E fV = E(SiO2 )V + E(O2 ) − E(SiO2 ), 2 1 fI I E = E(SiO2 ) − E(SiO2 ) − E(O2 ), 2

where E(SiO2 ) is the ground-state energy of silica, E(SiO2 )V and E(SiO2 )I are the groundstate energies of silica with one oxygen defect and E(O2 ) is the ground-state energy of the magnetic oxygen molecule. Owing to the large size of the super-cell used, in our calculations the Brillouin zone is sampled only at the Γ point. Local stress calculations. – We use the definition of the atomic stress tensor as proposed by Vitek and Egami [21], based on the work of Martin and Nielsen [22]. Within a first-order approximation in strain, and for pair interaction potentials, the atomic stress tensor may be written as
 N 1 −2Piα Piβ  α β αβ + fij rij , (1) σi = 2Ωi mi j where rij is the relative coordinate of particles i and j, Pi is the conjugate moment of particle i and fij is the force between i and j. V and N are, respectively, the volume and number of atoms in the super-cell; Ωi , the volume associated with the atomic site i, is chosen here to be Ωi = V /N , i.e. the mean atomic volume. The scalar quantities which can be extracted from the atomic stress tensor are, for instance, the octahedral shear stress and the hydrostatic pressure. Procedure validations. – The two-step generation of the glass results in a quite satisfactory model of silica with respect to density, pair distribution function, bond angles, rings statistics. See table I for the density results; full details will be given in a forthcoming publication [23]. For the purpose of validation of the SIESTA basis set in the present perspective, we have performed calculations of defect formation energies using SIESTA, PWSCF and VASP codes in parallel. On the one hand, we have compared results obtained on a 108 atoms α-quartz cell and on the former silica. In table II SIESTA vs. PWSCF results are shown. The structural characteristics and formation energies of the neutral oxygen defects obtained with both methods are very close. Given the high cutoff used for PWSCF calculations the gain in

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Table II – Calculated oxygen vacancy (V) and interstitial (I) formation energies, Si-Si (V) and O-O (I) bond lengths in a 108 atoms α-quartz super-cell. Defect type V

Formation energy (eV) Si-Si bond length (˚ A) Formation energy (eV) O-O bond length (˚ A)

SIESTA 5.59

PWSCF 5.58

V

2.39

2.39

I

1.80

1.84

I

1.47

1.49

computation time is high. On the other hand, we have been able to perform an exhaustive comparison between VASP and SIESTA calculations for the whole of the 72 oxygen defects as shown in fig. 1. The high level of agreement is depicted by the correlation coefficient of 0.98 and slope of 1.005 for vacancies as well as 0.86 and 1.02 for interstitials. VASP calculations are performed at constant volume, so this explains the values higher by 0.2 eV in the case of vacancies, due to the volume relaxation energy, also possibly an effect of the standard pseudopotentials vs. PAW methods. Some discrepancies appear in the interstitial case which are due to the existence of meta-stable positions, already predicted by Stoneham et al. [8]. The SIESTA basis set we are using is thus proved to be of a high quality for the study of defects in silica and in the following we will focus on those sole calculations. Defect properties. – The formation energies of the neutral oxygen vacancy and interstitial are distributed in a continuous way following a Gaussian-like distribution as shown in fig. 2. In all cases, the resulting structure for the vacancies was found as a Si-Si bond with Si-Si distances ranging from 2.2 ˚ A to 2.6 ˚ A and a mean value of 2.36 ˚ A. The average value of the energy is 5.44 eV, vs. 5.59 eV for α-quartz, the standard deviation amounts to 0.3 eV. The resulting structure of the interstitials was also a Si-O-O-Si peroxy bridge with O-O bond distances ranging from 1.3 ˚ A to 1.5 ˚ A and a mean value of 1.46 ˚ A. The formation energies of the neutral oxygen interstitial have an average value of 1.46 eV, vs. 1.80 eV for α-quartz, and a standard deviation of 0.3 eV. The formation energy of the oxygen Frenkel pair is then, on average, smaller by about 0.5 eV in silica than in α-quartz.

2.5 V

6.0

VASP Ef (eV)

VASP Ef (eV)

6.5

5.5 5.0 4.5

I

2.0 1.5 1.0 0.5

4.5

5.0

5.5

6.0 f

SIESTA E (eV)

6.5

0.5

1.0

1.5

2.0

2.5

f

SIESTA E (eV)

Fig. 1 – Formation energies of the oxygen vacancies (V) and interstitials (I) in a 108 atoms silica glass, VASP vs. SIESTA.

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EUROPHYSICS LETTERS

0.35

0.25

Frequency

α-quartz

0.15 0.10 0.05

4.5

5.0

α-quartz

0.20 0.15 0.10 0.05

✻

0

I

0.25

✻

Frequency

0.30

V

0.20

0

5.5

6.0

0.5

6.5

1.0

Energy (eV)

1.5

2.0

2.5

Energy (eV)

Fig. 2 – Formation energy distribution of oxygen vacancy (V) and interstitial (I) in a 108 atoms silica glass.

Discussion. – As shown in fig. 2, the formation energies of the neutral oxygen defects are spread around an average value. The origin of this distributed character lies in the fluctuation of the local structure from point to point. We have searched for the origin of this dispersion. We tried first to detect correlations with various structural properties: Voronoi volumes, the Si-O-Si angles, “very strong” ring statistics [24] at the defect site. We did not find any clear connection, confirming then the results already found by [8, 11] for the Si-O-Si angle. Next, in the spirit of the former work by Delaye [25], we searched for the role of the local stress at the defect site prior to the defect introduction, and found a very good correlation. In fig. 3 we have plotted the formation energies of vacancies and intestitials vs. the initial local pressure. Similar plots can be found as a function of the octahedral shear stress. This representation makes it possible to predict the shape, average value and standard deviation, of

6.5

2.5 V

I

2 Ef (eV)

Ef (eV)

6 5.5 5

1.5 1

4.5

0.5 30

40

50

60

70

80

Local pressure (GPa)

90

30

40

50

60

70

80

90

Local pressure (GPa)

Fig. 3 – Neutral oxygen defects formation energies vs. local pressure; V: vacancies and I: interstitials.

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685

the formation energy distributions starting with a perfect silica. Here, the formation energies come from first-principles methods while the local stresses are from EPMD, which is clearly a drawback. Indeed, the definition of local stresses in ab initio calculations, although possible, poses several questions which are not all clarified. We are presently working in order to be able to propose reasonable solutions. However, since the atomic stress reflects the strain of the neighbourhood of each atom as well as the interaction of this atom with its surrounding, and since the structure of the EPMD model of silica has been proved to be quite close to actual silica [12], we expected the use of empirical stresses to be satisfactory enough. Indeed, the present results confirm this hypothesis. The very origin of the correlation between defect formation energies and initial local stress may be understood using lattice statics and the concept of defect dipole tensor [26–29]. In this framework, the energy E strain of the system containing a point defect when submitted to a homogeneous external strain δ is, neglecting anharmonic terms (Hardy approximation),   E strain = E + P H ∗ δ + O δ2 , (2) where E is the energy reference for the system containing the point defect and P H is the defect dipole tensor in the Hardy approximation. P H is given by the outer product of the forces exerted on the atoms by the defect and of their equilibrium coordinates [26]. The * means the inner product of tensors P H and δ. In a glass, each atomic site experiences its own local strain related to the internal stress fluctuations. When a defect is introduced on the site i the defect forces, characterized by the defect dipole tensor, have to work against this local strain. Assuming that the range of the defect dipole tensor is at most of the same order as the correlation length of the stress fluctuations, we get a formation energy as a function of the local stress tensor (σiνµ ) at site i according to    Eif = Ei + PiH ∗ C −1 ∗ σi + O δ2 − E0 , (3) where C −1 is the inverse of the elastic constants tensor, Eif is the formation energy of the defect at site i and E0 is the total energy of the system without defect. The range of the dipole tensor defines the volume where the atomic stress has to be homogeneous for the model to be strictly valid. The main assumption here is that the defect-system interaction range is local and of the same range as the stress fluctuations. Here we plotted the formation energies vs. the atomic level stresses, so to be rigorous the defect Kanzaki forces [26] should not go beyond first neigbours. Looking at fig. 3, indeed beyond the clear global agreement, we observe a residual fluctuation of the formation energy at a given atomic stress level which cannot be accounted for by the present model. This residual fluctuation can have several origins, fluctuations of the dipole tensor PiH , which could depend on the site, failure of the range hypothesis, but also from the empirical nature of the energy model used for the stress calculations. The generalization of this formalism to first-principles calculations is in progress. Conclusion. – In the prospect of aging studies of silica-based glasses, we have studied the formation of the oxygen point defect in silica combining empirical potential and ab initio methods. After a careful preparation of the glass, we have calculated the formation energies of the neutral oxygen vacancies and interstitials. The values are distributed according to a Gaussian-like distribution. On average, the formation energy of the oxygen Frenkel pair is 0.5 eV smaller in silica than in α-quartz. The origin of the fluctuation of the formation energy is shown to lie in the fluctuations of the local stress. Knowing the latter, it is possible to predict the distribution of the defect formation energies in very large systems using low-cost methods like molecular dynamics with empirical potentials (see ref. [23]).

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