Point Defects and Non-stoichiometry in Li2TiO3 - Chemistry of

Article pubs.acs.org/cm

Point Defects and Non-stoichiometry in Li2TiO3 Samuel T. Murphy*,† and Nicholas D. M. Hine‡ †

Department of Materials, Imperial College London, South Kensington, London SW7 2AZ, United Kingdom Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, United Kingdom

‡

ABSTRACT: The intermediate-temperature, monoclinic β-phase of Li2TiO3 shows a stoichiometry range from 47 to 51.5 mol % TiO2. This broad stoichiometric range may be exploited for industrial applications, such as breeder material in a fusion reactor or a microwave dielectric. Here, density functional theory is employed to calculate formation energies for the intrinsic defect species, allowing the identification of the mechanisms responsible for accommodating both excess Li2O and TiO2 across a wide range of temperatures and oxygen partial pressures. The results predict that while the exact mode of accommodating non-stoichiometry depends on factors such as the temperature and oxygen partial pressure, cation disorder plays a major role in the incorporation of non-stoichiometry and that oxygen defects are of relatively minor importance.

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excess lithium in Li2+xTiO3: first, creation of Li1+ i interstitials, 2+ and second, incorporation by V4− Ti defects with VO defects providing charge compensation. They conclude that the increase in cell volume observed in their X-ray diffraction profiles indicates the former mechanism for x < 0.08 and the latter for x > 0.1. The proposed formation of V4− Ti defects with V2+ for x < 0.2 is also proposed by Bian and Dong on the basis O of XRD supported in this case by Raman spectra.7 Mukai et al. interpreted their neutron diffraction data to suggest that the excess lithium is accommodated at either interstitial sites or as Li3− Ti antisite defects with the resulting charge compensated for 4+ by the reduction of Ti or site vacancies, i.e., Li2+xTi3+ 1−xTix O3 or 8 −3 Li2+2xTi1−xO3−x. Substitution of Li for Ti (i.e., LiTi ) was also observed by Vitins et al. in samples containing a slight Li-excess at temperatures in the range 770−850 °C analyzed using scattered reflectance spectroscopy.9 Hoshino et al. show that upon heating Li2TiO3 in a reducing atmosphere the color of their sample changed from white to dark blue, suggesting that an oxygen deficiency is accommodated by VO2+ defects compensated for by reduced Ti3+ cations.10 The conductivity studies of Vitins et al. suggest excess TiO2 in Li2TiO3 is incorporated by Ti3+ Li defects charge compensated −1 for by vacancy defects (presumably V−1 Li ). The presence of VLi is also predicted to facilitate Li diffusion in samples containing a slight Li-deficiency.11 At high temperatures the level of cation disorder in Li2TiO3 increases prior to the transformation from the monoclinic β-phase to the cubic γ-phase.12 In the structurally similar Li2MnO3 there have been a number of studies examining the incorporation of MnO2 excess due to delithiation.13−15 Wang et al. observe the removal of Li from both the Li6 and Li2Mn4 layers as well as the appearance

INTRODUCTION Lithium metatitanate, Li2TiO3, has a number of promising industrial applications, including use as a cathode material for lithium ion batteries, a breeder blanket material for a future fusion reactor, and a microwave dielectric. According to the Li2O-TiO2 phase diagram,1 Li2TiO3 crystallizes into three phases: at low temperatures it forms a metastable α-phase, before undergoing a transformation to the monoclinic β-phase at 300 °C. This is the stable phase up until 1215 °C, at which point it transforms into the cubic γ-phase, before finally melting at >1500 °C. Also evident from the phase diagram is the stoichiometry range of the Li2TiO3 phase, which varies from 47 to 51.5 mol % TiO2 even at relatively low temperatures. Varying the stoichiometry of the material has been shown to alter properties such as the ion dynamics, 2,3 thermal conductivity,4,5 and the dielectric constant.6 This stoichiometry range and the resulting physical properties may be exploited for industrial applications. In particular, Li2TiO3 with excess Li could be employed as an advanced breeder material in a fusion reactor as the increase in the lithium density offers enhanced tritium production. During reactor operation lithium ions will undergo transmutation, making the lattice Li deficient, and therefore it is desirable to understand how this nonstoichiometry will be accommodated by the lattice and how the material’s properties will be affected. Non-stoichiometry is accommodated in the crystal by intrinsic defects such as vacancies and interstitials, the relative concentrations of which then determine the material’s properties. Of particular interest is how the change in defect population due to Li burn-up may influence the tritium release characteristics of Li2TiO3 employed as a breeder material in a fusion reactor. The defect chemistry responsible for accommodating either excess Li2O or excess TiO2 is currently unclear. Hao et al.6 propose two possible mechanisms for the accommodation of © 2014 American Chemical Society

Received: November 20, 2013 Revised: January 21, 2014 Published: January 25, 2014 1629

dx.doi.org/10.1021/cm4038473 | Chem. Mater. 2014, 26, 1629−1638

Chemistry of Materials

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of Mn in the previously pure Li layer,15 again demonstrating the importance of cation disorder in the incorporation of nonstoichiometry in such materials. Point defects are difficult to observe directly from experiment and the presence of different defect types must be inferred from observations as above. Atomistic simulation techniques, especially first principles methods such as density functional theory (DFT), allow the study of point defects, specifically through the calculation of defect formation energies. Using simple thermodynamics, defect formation energies may be determined, parametrized by chemical potentials for each species that may be added or subtracted to create the defect. Specific values of the chemical potentials can then be linked to different stoichiometry regimes and environmental conditions, thus allowing the calculation of the concentrations of the defects thought to accommodate non-stoichiometry under different conditions. In our previous work we investigated the influence of finite size effects on the calculation of defect 3− formation energies in Li2TiO3, focusing on the V4− Ti , LiTi , and 16 2− Oi defects. The aim of this work is to examine the properties of all point defects in Li2TiO3 and to use this data to determine which defects are responsible for the accommodation of both Li2O and TiO2 excess.

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CRYSTALLOGRAPHY The crystal structure of β-phase Li2TiO3 was first determined by Lang17 and subsequently refined using X-ray diffraction of large single crystals by Kataoka et al.18 These works envisaged Li2TiO3 as a distorted rocksalt structure with alternating Li6, O6, Li2Ti4, O6 (111) planes.19 Within the Li2Ti4 layers the Ti ions form hexagons centered on the Li ions. The stacking sequences of these honeycomb structures then give rise to a number of different space groups for Li2TiO3 (i.e., C2/m, C2/c, and P3112).20 In this work we limit ourselves to the study of the C2/c structured material where the Li2Ti4 layers are stacked so that they create a glide plane. A pictorial representation of the Li2TiO3 unit cell is given in Figure 1, and the lattice parameters and symmetry reduced atomic coordinates are reported in Table 1.

Figure 1. Unit cell of Li2TiO3. Green, yellow, and red spheres represent lithium, titanium, and oxygen ions, respectively.

Table 1. Crystal Structure for Li2TiO3 with the C2/c Space Groupa atom

Wyckoff position

x

y

z

occupancy

Li1 Li2 Li3 O1 O2 O3 Ti1 Ti2 Int1 Int2

8f 4d 4e 8f 8f 8f 4e 4e 8f 8f

0.24064 0.25000 0.00000 0.14058 0.10516 0.13921 0.00000 0.00000 0.42500 0.75000

0.08004 0.25000 0.08436 0.26366 0.58444 0.90583 0.41746 0.74961 0.25000 0.40500

0.00055 0.50000 0.25000 0.13682 0.13695 0.13415 0.25000 0.25000 0.09200 0.32000

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.00b 0.00b

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METHODOLOGY Simulation Procedure. All DFT simulations presented here were performed using the plane-wave pseudopotential code CASTEP,22,23 where a crystal is described using the supercell approach and periodic boundary conditions with special point integration over the Brillouin zone. Exchangecorrelation is described using the generalized gradient approximation of Perdew, Burke, and Ernzerhof (PBE).24 Standard CASTEP ultrasoft pseudopotentials (USPs)25 were used, necessitating a kinetic energy cutoff of 550 eV for the plane-wave expansion. Γ-centered Monkhorst−Pack26 grids were used to sample the Brillouin zone, with the distance between sampling points maintained as close as possible to 0.05 Å−1 on each axis. The Fourier transform grid for the electron density is larger than that of the wave functions by a scaling factor of 2.0, and the corresponding scaling for the augmentation values was 2.3. These values were determined by performing convergence tests of the energy from selfconsistent single point simulations. To optimize the 48-atom Li2TiO3 unit cell, energy minimization under constant pressure conditions was performed, with the lattice parameters and angles free to change.

a

Lattice parameters are given in Å, angles are in deg, and the atom positions are given in fractional co-ordinates.21 Space group: C12/c1 (15). Cell: 5.09 Å, 8.83 Å, 9.80 Å, 90.00°, 100.19°, 90.00°. bInterstitial site is unoccupied in the perfect lattice.

The resulting relaxed lattice parameters and symmetry-reduced atomic coordinates were in excellent agreement with the X-ray diffraction data of Kataoka et al.18 Further demonstrations of the efficacy of our simulation model can be found in previous work.21 As would be expected from DFT simulations employing a semilocal exchange correlation functional, our simulations underestimate the bandgap by ∼15% (3.27 eV compared to an experimental bandgap of 3.9 eV27). Employing a hybrid functional, such as that of Heyd, Scuseria, and Ernzerhof (HSE),28 may well correct the band gap and consequently modify the calculated formation energies. However, given the large number of defects considered and the large supercells required by the extrapolation scheme used to mitigate for the interaction of defect images, it is not currently computationally 1630



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function of oxygen partial pressure and temperature and μLi2TiO3(s) is the chemical potential of solid Li2TiO3. For a solid, μ (pO° 2, T°) ≈ μ(0, 0), therefore the temperature and pressure dependencies have been dropped. At equilibrium, the chemical potentials of the constituent elements in a crystal have an upper bound that is the Gibbs free energy of the relevant element in its natural state; for example, the chemical potential of Li2O in Li2TiO3 cannot exceed that of solid Li2O, otherwise a Li2O precipitate would form.30 We can vary our chemical potentials to represent different stoichiometric regimes: for growth conditions with excess Li2O (referred to here as Li2O-rich), we have μLi2O(pO2, T) = μLi2O(s) and μTiO2(pO2, T) = μLi2TiO3(s) − μLi2O(s). Similarly, for growth conditions with excess TiO2 (i.e., TiO2-rich conditions), we have μTiO2(pO2, T) = μTiO2(s). We can further decompose μLi2O(s) as follows:

feasible to employ a hybrid functional here. Furthermore, the idea that defect formation energies calculated using hybrid functionals can act a reference to which standard PBE must be compared has recently been challenged.29 Supercells for the defect simulations were constructed by creating l × m × n multiples of the relaxed unit cell. A range of different values for l, m, and n were employed in order extrapolate the formation energies in the dilute limit as discussed below. Defect simulations were performed under constant volume conditions (i.e., lattice parameters and cell angles were fixed during minimization) in order to represent the dilute limit. Within the Li2TiO3 unit cell described by the C2/c space group there are seven unique atom positions as shown in Table 1. Vacancy defects were created at each of the unique lattice sites for charge states from the neutral to the fully charged defect for the ion occupying that site. For clarity, the labels given in column one of Table 1 will be used to label the sites in 1− the defect notation, e.g., VLi1 . Two interstitial sites are considered here: the first is situated between the Li6 and O6 layers, and the second is situated between the mixed cation Li2Ti4 layer and the O6 layer. Defect Formalism. The concentrations of defects in a crystal are controlled by the equilibrium conditions under which it is formed. The Law of Mass Action shows that the concentration, ci, of species i can be related to the change in the Gibbs free energy, ΔGif, to form the defect, i: ⎛ −ΔG i ⎞ f ci ∝ mi exp⎜ ⎟ k T ⎝ B ⎠

μLi O = 2μLi (pO , T ) + μ1/2O (pO , T ) 2 (s)

where mi is the multiplicity of equivalent sites, kB is the Boltzmann constant, and T is the temperature. At the temperatures relevant to solids, the difference in vibrational contributions to the free energy between perfect and defect supercells can be safely neglected, so we can approximate ΔGif by ΔEif, calculated from the total energies of the perfect and defective cells according to the formalism of Zhang and Northrup.30 The defect formation energy is then given by

∑ nαμα + qμe

ΔGfLi 2O(pO° , T °) = μLi O − 2μLi − μ1/2O (pO° , T °) 2

where and are the DFT total energies of the system with and without the defect, nα is the number of atoms added/ removed of each atomic species α, μα is the chemical potential of each species, q is the charge on the defect and μe = EVBM + εF is the electron chemical potential. EVBM is the energy of the valence band maximum (VBM), and εF is the electron chemical potential above the VBM. Chemical Potentials. The starting point for determining the chemical potentials is to assume that Li2TiO3 can be formed from the reaction of Li2O and TiO2 via reaction 3 (we note that this is not the usual route for Li2TiO331 synthesis):

2(g )

2

2

2

2

3(s)

2

2(g )

⎛p ⎞ O log⎜⎜ 2 ⎟⎟ ⎝ pO°2 ⎠

2

2

1 kBT 2

(7)

and the rigid-dumbbell ideal gas for Δμ(T) can be given by: ⎛T ⎞ 1 Δμ(T ) = − (SO°2 − C P°)(T − T ◦) + C P°T log⎜ ⎟ ⎝ T° ⎠ 2

(8)

where S°O2 is the molecular entropy at standard temperature and pressure and C°P is the constant pressure heat capacity of oxygen gas. Values for these two properties are taken from the literature with S°O2 = 0.0021 eV/K and C°P = 7kB = 0.000302 eV/ K. An analogous scheme can be devised to determine the chemical potentials under TiO2-rich conditions. Gibbs free energies of formation for Li2O and TiO2 were −5.82 and −9.21 eV, respectively.33 For intermediate compositions the chemical

Under any given conditions, the sum of the chemical potentials per formula unit of the constituent species must equal the total Gibbs free energy of Li2TiO3: 2

2(g)

μ1/2O (pO , T ) = μ1/2O (pO° , T °) + Δμ(T ) +

(3)

μTiO (pO , T ) + μLi O(pO , T ) = μLi TiO

(s)

Here we assume the temperature dependence of the Gibbs free energy of the solid species can be neglected, but this is not a safe assumption for oxygen. Instead, the oxygen chemical potential at the desired formation conditions, μ(1/2)O2(g)(pO2, T), is extrapolated from μ(1/2)O2(g)(pO° 2, T°) using the ideal gas relations. The full expression for the oxygen chemical potential is given by eq 7:

ETperf

Li 2O(s) + TiO2(s) → Li 2TiO3(s)

2 (s)

(6)

(2)

α

ETdefect

(5)

2

where, μLi(pO2, T) and μ1/2O2(g)(pO2, T) are the respective chemical potentials of lithium and oxygen atoms in μLi2O(s) in equilibrium with O2(g). A difficulty then arises due to the wellknown problems of semilocal forms of DFT when describing the O2 dimer. To sidestep this, we adopt an approach first suggested by Finnis et al.,32 which removes the necessity of using DFT to obtain the chemical potential of oxygen, by instead referencing the known experimental formation energy 2O of the oxide, ΔGLi (pO2, T). Following this methodology, the f chemical potential of oxygen at standard temperature and pressure can be obtained via

(1)

T T ΔEf = Edefect − Eperf +

2(g)

2

(4)

where μTiO2(pO2, T) and μLi2O(pO2, T) are the chemical potentials of TiO2 and Li2O within lithium metatitanate as a 1631



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potentials are taken as fractions, s, of the chemical potentials in the Li2O and TiO2-rich regimes, i.e., for μLi(pO2, T)

i

(11)

μLis (pO , T ) = (1 − s)μLiLi 2O‐rich (pO , T ) + sμLiTiO2‐rich 2

where the first term is simply the sum of the charges of the ionic defects and the second and third terms correspond to the electron and hole concentrations, respectively. Nc and Nv are the effective conduction band and valence band density of states, and Eg is the bandgap, which in our DFT simulations is predicted to be 3.27 eV.16 Boltzmann rather than Fermi-Dirac statistics have been employed for the electron and hole populations as in an insulating material they are expected to be sufficiently low that this approximation holds accurately within the stoichiometry and temperature ranges considered. As shown in eq 11, the formation energy for an electron in the conduction band is approximated to be equal to Eg − εF. In reality the formation energy will be slightly lower due to selftrapping of the electrons;36 however, the use of Eg is a reasonable approximation in a wide bandgap insulator such as Li2TiO3 . Likewise, a reasonable approximation to the energy cost to create a hole is the energy level εF of the valence band edge. Using eq 11 it is possible to determine the value of εF that ensures overall charge neutrality for any chosen TiO2 fraction, oxygen partial pressure, and temperature. The concentrations of the different defect species for any given conditions then follow from eq 1. In regions where the electron chemical potentials εF required for charge neutrality did not fall in the region 0 − Eg, the formation conditions were deemed to be inaccessible. Similarly, the defect formation energies for all point defects were required to be positive, as a negative value would imply that the crystal is not stable under these growth conditions and a phase transition or transformation would occur. Kröger−Vink or Brouwer diagrams can then be constructed by plotting the defect concentrations as a function of the oxygen partial pressure at any given TiO2 fraction and temperature. We note that all defects studied here are considered to be in the dilute limit and thereby defect−defect interactions are neglected except in the mean-field sense that their charges interact via εF. In insulating systems such as Li2TiO3 the Coulombic attraction between oppositely charged point defects can lead to the formation of small bound clusters of defects. Point defects in bound clusters will often have lower defect formation energies than their constituents, but clusters will only tend to remain bound at elevated temperatures if their binding energies are very large compared to kBT. This could have an influence on the mechanisms of incorporating nonstoichiometry,37 but we do not address this in the current work.

2

(pO , T ) 2

(9)

where s = 0 represents the limit beyond which no further Li2O can be incorporated in Li2TiO3 without phase separation occurring. Similarly, s = 1 represents maximally TiO2-rich Li2TiO3. Finite Size Effects. In a supercell calculation, such as those presented here, the interactions between the defect and its periodic images provide an unphysical contribution to the defect formation energy, which is often referred to as finite size error. This source of error is particularly acute in the case of charged defects due to the slow decay of the Coulomb interaction with distance. Here, due to the layered crystal structure and the resulting anisotropic dielectric properties of Li2TiO3,9,16 we employ a recently developed method of extrapolation based on the Madelung potential, which we have shown is able to correct for this error to obtain formation energies converged to ≃0.1 eV.16,34 In this extrapolation procedure the defect formation energy is calculated in a range of supercells made from l × m × n repetitions of the unit cell in a, b, and c, respectively. The defect formation energies are plotted as a function of the screened scr Madelung potential, vscr M , where vM is calculated by Ewald summation with the charge screening of the host lattice incorporated via the dielectric tensor, ϵ.̅ A function of the form scr Ef (vM )

=−

qi2 2

scr vM + Ef∞

⎛ Eg − εF ⎞ ⎛ ε ⎞ ⎟ + Nv exp⎜ − F ⎟ = 0 kBT ⎠ ⎝ ⎝ kBT ⎠

∑ qici − Nc exp⎜−

(10)

is fitted to the data by optimizing the elements of ϵ̅ and the desired defect formation energy in the dilute limit, E∞ f , using a Nelder−Mead Simplex algorithm.35 A demonstration of this extrapolation procedure for the Ti2+ Li1 defect is presented in Figure 2. Constructing the Kröger−Vink diagram. In a crystal, the sum of the concentrations of ionic defects and electronic defects, each multiplied by their charges, must balance to produce overall charge neutrality. This can be expressed as

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RESULTS AND DISCUSSION Defect Formation Energies. Initially we examine the defect formation energies calculated under Li2O-rich conditions with an oxygen partial pressure of 0.2 atm at a temperature of 1000 K. Figures 3, 4, and 5 show the defect formation energies as a function of the Fermi level for the vacancy, interstitial, and antisite defects, respectively. The electron chemical potential is shown extending to the experimental bandgap of 3.9 eV, rather than the calculated bandgap of 3.27 eV, so that the transition energies to all relevant charge states can be seen. It has been suggested that such an extension is not valid as the defect states no longer fall within the bandgap predicted by DFT; however, while this is important when determining the exact positions of

Figure 2. Convergence of the defect formation energy with vscr M for the Ti2+ Li1 antisite defect. The desired dilute limit defect formation energy corresponds to vscr M = 0. 1632



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Figure 3. Defect formation energies as a function of the Fermi energy, εF, for all vacancy defects. The dashed vertical line represents the bandgap determined using DFT.

oxygen lattice sites all falling into the same layer, there is nevertheless some variation in the oxygen vacancy formation energies: the formation energy for the V2+ O3 defect is 0.2 eV 2+ and V defects. greater than for the V2+ O1 O2 Titanium antisite defects (i.e., TiLi) show some variation also: the lithium site in the mixed layer (Li3) produces the highest energy when in the full 3+ charged state (see Figure 5a), whereas the Li1 and Li2 sites are 0.18 eV lower. This behavior can be explained using a simple charge argument: substitution of a Ti4+ ion onto a lithium site in the mixed cation layer will bring the Ti3+ Li defects closer to the other highly charged Ti ions than substitution into the pure Li6 layer. When εF is near the conduction band minimum, which is where the Ti0Li state becomes the most thermodynamically stable, the formation energies are insensitive to the choice of substitutional site. The defect formation energies for lithium substitution onto the titanium sublattice are presented in Figure 5b. The results suggest that there is very little difference in the formation energies for Li substitution onto the different Ti sites. Two possible interstitial sites were considered: the first is between the Li6 and O6 layers, while the second falls between the pure Li2Ti4 and O6 layers. The relaxed geometry of the Li1+ i1 defect residing between the Li6 and O6 layers is shown in Figure 6a. The interstitial ion lies very close to the Li6 plane due to the repulsion of the Ti4+ cation at the apex of the distorted cube. Figure 6b shows the geometry that is obtained when a lithium ion is placed between the oxygen ions and the mixed cation layer (the Li1+ i2 defect). In this case, the strong electrostatic

the ionization levels in the gap, it is less so in the determination of defect concentrations. In general, the simulations predict that with the exception of the oxygen interstitial, all defects have their lowest formation energy when they are in their full ionic charge state, across most of the bandgap. This is consistent with other studies of wide band gap insulators. Crucially, such fully charged defects are less affected by any error in the bandgap resulting from the use of semilocal functionals, as there are no occupied states in the bandgap.38 By contrast, when there are occupied states in the bandgap, the states involved may be excessively delocalized due to spurious self-interaction, resulting in incorrect defect formation energies. The reduced self-interaction present in hybrid functionals may be able to improve the prediction of defect formation energies in such cases.39 Figure 3a shows the defect formation energies of the two Li vacancy defects in the pure Li layer (Li1 and Li2): they are very similar and are both 0.1 eV greater than in the mixed LiTi2 layer (Li3) across the whole bandgap. This is in qualitative agreement with the empirical potential simulations of Vijayakumar et al.,2 who predict that the defect energy of a Li vacancy in the pure Li layer is 0.2 eV higher than in the LiTi2 layer. Both titanium sites in Li2TiO3 are in the LiTi2 layer as shown in Figure 1, and therefore it is not surprising that the defect formation energies for the VTi defects are similar. The defect formation energies for lithium substitution onto the two different Ti sites (thus forming LiTi defects) are also almost identical. By contrast, despite the three symmetrically distinct 1633



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Figure 4. Defect formation energies as a function of the Fermi energy, εF, for all interstitial defects. The insets show the region of the bandgap where the transitions between charge states occur in more detail. The dashed vertical line represents the bandgap determined using DFT.

large, and this defect will not be highly prevalent. These oxygen dimers, which are sometimes referred to as peroxides or superoxides depending on their charge state, have been identified in a number of other DFT studies of oxygen interstitials in oxides.40−43 A peroxide-like species has also been observed in Li2Ru1−ySnyO3 using X-ray photoelectron spectroscopy.44 Defect Concentrations and Stoichiometry. Using the defect formation energies presented above it is possible to determine the electron chemical potential that ensures charge neutrality for a given value of s, temperature, and oxygen partial pressure. Figure 8 shows the resulting electron chemical potential as a function of both the stoichiometry and the oxygen partial pressure, at 1000 K. The electron chemical potential is shown to explore a significant portion of the bandgap in order to ensure charge neutrality. It reaches a minimum of 0.62 eV under Li2O-rich and high oxygen partial pressure conditions and a maximum of 2.16 eV under TiO2-rich conditions with very low oxygen partial pressures. With the electron chemical potential for any given set of conditions, we have access to the defect concentrations and hence can describe the defect chemistry that defines the different stoichiometric regimes. In the following sections we examine the resulting point defect concentrations as a function of the temperature, oxygen partial pressure, and stoichiometry, s. For clarity we only plot defects with concentrations greater than 8 × 10−8 per formula unit. As mentioned previously, defect formation energies have been calculated in the dilute

interaction pushes the interstitial ion onto a neighboring interstitial site between the Li6 and O6 layers. As seen in Figure 4a, the latter configuration is lower in energy than the former. The origin of this energy difference is increased Coulombic interactions due to the presence of a nearest neighbor Ti4+ close to the interstitial (compare Figure 6a and b). For the Ti4+ i defect located between the Li6 and O6 layers the optimized geometry is very similar to that shown for the Li1+ i defect in Figure 6a, although the increased charge on the titanium leads to an increased repulsion of the surrounding cations and greater Coulombic attraction to the surrounding O2− ions. As a result, the titanium interstitial resides closer to 4+ the O6 plane than was the case for the Li1+ i defect. When a Tii defect is placed close to the mixed cation layer, the electrostatic repulsion is so great that the interstitial site is no longer stable, and therefore this defect will not be considered in any further analysis. The atomic configuration for the O2− i1 defect near the Li6 layer was presented in Figure 6b in our previous study.16 A similar configuration is also predicted for the singly negatively charged O1− i1 defect. However, in the charge-neutral case the simulations predict the formation of an O2− 2 dimer with a bond length of 1.45 Å centered on a neighboring oxygen lattice site. The relaxed geometry of an example of this configuration is shown in Figure 7. When the interstitial is placed between the mixed cation and oxygen layers, the dimer configuration is predicted to have the lowest energy for all charge states. However, the formation energy for the fully charged O2− i2 is 1634



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Figure 7. Atomic configuration surrounding the O0i1 defect in the 2 × 2 × 2 supercell.

Figure 5. Defect formation energies as a function of the Fermi energy, εF, for all antisite defects. The insets show the region of the bandgap where the transitions between charge states occur in more detail. The dashed vertical line represents the bandgap determined using DFT.

Figure 8. Fermi energy, εF, as a function of the stoichiometry and the oxygen partial pressure at 1000 K. The variation in the chemical potentials due to the change in TiO2 ratio, oxygen partial pressure, and temperature leads to significant change in value of εF required to ensure charge neutrality.

concentrations plotted are the sum of all like defects, i.e., cV1− = Li 1− 1− 1− cVLi1 + cVLi2 + cVLi3, etc. For all different stoichiometric regions, the concentrations of defects are predicted to increase as the temperature is increased as would be expected of a thermally activated process. In the Li2O-rich regime at 500 K (Figure 9a), excess lithium is accommodated mainly by Lii1+ defects, whose charge is compensated for by Li3− Ti defects for oxygen partial pressures lower than ∼10−10 atm. At higher oxygen partial pressures, as Lii becomes less favorable, the excess lithium is instead accommodated mainly by Li3− Ti antisite defects, whose charge is compensated for by holes in the valence band. Similarly at 1000 K the Li3− Ti antisite defects are expected to be important irrespective of the oxygen partial pressure, as shown in Figure 9b. The observation by Vitins et al. of Li substitution onto Ti sites in the temperature range 1040−1120 K supports our prediction of the importance of the LiTi defect in samples with excess Li at these temperatures.9 Also at 1000 K a number of other defect concentrations have increased to greater than 8 × 10−8 per formula unit, including conduction 1+ 3+ electrons and the V2− O , VLi , and TiLi defects. Figure 9c shows

1+ Figure 6. Atomic configurations surrounding the (a) Li1+ i1 and (b) Lii2 defects calculated in the 2 × 2 × 2 supercell. The interstitial ions reside at the center of the cube.

limit. However, once the defect concentrations increase to levels as high as 10−2 per formula unit, this approximation will become increasingly invalid. Point Defects in Li2O-Rich Li2TiO3. Figure 9 presents Kröger−Vink diagrams showing the concentrations of the point defects as a function of the oxygen partial pressure at 500, 1000 and 1500 K for Li2O-rich conditions. For clarity the defect 1635



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excess Li.10 In these experiments these Li2O-rich samples were prepared by sintering at 1323 K under a reducing atmosphere, implying a low pO2. During sintering, the defect equilibrium for that specific temperature is attained and is then frozen in during cooling. Therefore, the low-pO 2 regime of the defect concentrations at 1500 K shown in Figure 9c) offers the most relevant comparison to this particular experimental regime and backs up the idea of reduced Ti3+ ions being present in such samples. In general, oxygen defects are predicted to play only a very minor role in the accommodation of Li2O excess, so the 4− defects compensated for by VO2+ (as mechanism of VTi discussed by Hao et al.6) is not anticipated to play an important role in the defect chemistry of Li2TiO3. Point Defects in TiO 2 -Rich Li 2 TiO 3 . For growth conditions with excess TiO2, the defect chemistry at 500 K is 3+ dominated by V1− Li defects and TiLi defects, as shown in Figure 10a. Both of these defects incorporate excess TiO2 and are mutually charge compensating. Within the range examined here, the oxygen partial pressure appears to have no influence on the concentration of these defects. At 1000 K the defect chemistry is still dominated by these same defects; however, both the electronic defects and a number of defects not displaying their formal charge states have increased concen0 trations, i.e., the Ti2− Li and VLi defects (Figure 10b). This 1− predicted dominance of the Ti3+ Li and VLi defects is in good agreement with the available experimental data9 as well as in observations of the delithiation of Li2MnO3.15 The defect chemistry at 1500 K is more complex, as shown in Figure 10c. At oxygen partial pressures lower than 10−16 atm, the excess TiO2 is incorporated by Ti3+ Li with the excess positive charge compensated for by conduction electrons. As the oxygen partial pressure is increased, the V1− Li becomes the dominant still plays an important role by defect. However, the Ti3+ Li providing charge compensation. For partial pressures greater than 102 atm the concentration of Ti antisite defects decreases and the V1− Li charge is accommodated by hole defects. Point Defects at Intermediate Stoichiometries. It is very difficult to pinpoint chemical potential values for which near-exact stoichiometry is reached, since all of the defect concentrations are also functions of both the temperature and the oxygen partial pressure. Figure 11 shows the defect concentrations as a function of stoichiometry, s, at 1000 K and oxygen partial pressures of 10−20, 1, and 105 atm. As expected, the defect concentrations are highest at the extremes of stoichiometry and decrease at intermediate compositions. At the extremes of stoichiometry, the point defects incorporate the non-stoichiometry in addition to the thermal defect population, while at stoichiometry only the relevant thermally activated population is present. Stoichiometric conditions are expected close to where the point defect with the highest concentration changes from one that accommodates Li2O excess to one that accommodates TiO2 excess. At an oxygen partial pressure of 10−20 atm there is 3+ a transition between Li3− Ti and TiLi defects at s ≃ 0.3 (Figure 11a). As the oxygen partial pressure rises, the point at which the change from Li2O-rich to TiO2-rich occurs (in both cases from 5 1− Li3− Li to VLi ) shifts to s ≃ 0.5 at 1 atm and ≃ 0.55 at 10 atm (Figure 11b and c, respectively). These results highlight the sensitivity of the point defect concentration to the conditions present during material fabrication, particularly around the

Figure 9. Kröger−Vink diagrams for Li2TiO3 under Li2O-rich, conditions at (a) 500 K, (b) 1000 K and (c) 1500 K. At 500 K (a) and 1000 K (b) the defect chemistry of Li2O-rich Li2TiO3 is and Li antisite defects at low oxygen partial dominated by Li1+ i pressures and Li3− Ti defects compensated for by holes in the valence band at high oxygen partial pressures. At 1500 K (c) a region where Li1+ i are compensated for by electrons in the conduction band appears at low oxygen partial pressures.

that at 1500 K for very low oxygen partial pressures, less than 10−15 atm, excess lithium is accommodated again by Lii1+ defects. However, the charge is now compensated for by electrons in the conduction band. Since the conduction band is predominantly formed of titanium d-states,16 these electrons are likely to manifest themselves as reduced Ti3+ ions where the electron is trapped. This mechanism of Li1+ i defects with their charge compensated for by Ti4+ ions reduced to Ti3+ has been suggested from experimental observations of Li2TiO3 with 1636



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Figure 11. Kröger−Vink diagrams for Li2TiO3 at 1000 K and oxygen partial pressures of (a) 10−20 atm, (b) 1 atm, and (c) 105 atm. The results suggest that cation disorder is important for all TiO2 fractions at a range of oxygen partial pressures at 1000 K. Oxygen defects are only shown to be present in concentrations of >10−8 under Li2O-rich conditions at low oxygen partial pressures.

Figure 10. Kröger−Vink diagrams for Li2TiO3 under TiO2-rich, conditions at (a) 500 K, (b) 1000 K and (c) 1500 K. The defect 1+ chemistry in TiO2-rich Li2TiO3 is dominated by V1− Li and TiLi defects for all temperatures studied here. At 1500 K (c) there is a region at low oxygen partial pressures where the dominant defects are Ti antisites with electrons in the conduction band provide charge compensation and at high partial pressures excess TiO2 is incorporated by V1− Li compensated for by holes in the valence band.

temperatures, and therefore under most conditions the change arising due to the use of the experimental bandgap is negligible. Under Li2O-rich conditions (Figure 9c) the region in which the excess Li2O is accommodated by Li1+ i charge compensated for by electrons is moved to a lower oxygen partial pressure; otherwise the defect chemistry is unchanged. Similarly, in the TiO2-rich case the region in which the electrons compensate for Ti3+ Li has moved to such a low oxygen partial pressure that it is now outside the range of interest and the dominance of the charge compensating V1− Li is extended.

stoichiometric composition, where there are a number of competing defects. Influence of Bandgap. All of the Kroger−Vink diagrams presented so far have been constructed using the theoretical bandgap 3.27 eV, which represents an underestimate of the experimental bandgap. Consequently, the formation energy for the electrons is underestimated, resulting in an elevated concentration. As shown in Figures 9−11, free electrons play a prominent role in the defect chemistry only at high 1637



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CONCLUSIONS In this paper we have used DFT formation energies to calculate defect concentrations for the intrinsic point defects in Li2TiO3 for a range of different temperatures and oxygen partial pressures, allowing an exploration of the material’s defect chemistry. We have identified the point defects responsible for accommodating both excess Li2O and TiO2. Excess Li2O is accommodated in the crystal by Li i1+ defects charge compensated for by either electrons or Li3− Ti defects depending on the oxygen partial pressure. Conduction electrons are expected to localize on titanium atoms, reducing them to the Ti3+ charge state, as has been predicted on the basis of experimental observations.10,8 By contrast, excess TiO2 will be 3+ accommodated by mutually charge-compensating V1− Li and TiLi defects, in good agreement with experiment, except at low oxygen partial pressures where the defect chemistry is dominated by Ti antisite defects charge-compensated by conduction electrons. The limiting step to tritium extraction in solid breeder materials is diffusion to the surface of the pebble. During operation the stoichiometry of the Li2TiO3 will be modified due to the transmutation of Li. Our results show that the type of defects present to accommodate and subsequently facilitate diffusion of tritium to the pebbles surface changes as a consequence of this shift in the crystal stoichiometry. This implies that there will be a change in the tritium release rate during the breeders lifetime, which may have implications for recovery of tritium from the blanket.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Computational resources were provided by the Imperial College High Performance Computing Centre. NDMH acknowledges the support of the Leverhulme Trust and the Winton Programme for the Physics of Sustainability. Prof. Robin Grimes, Prof. Mike Finnis, Prof. Matthew Foulkes, Patrick Burr, Hassan Tahini, and Mostafa Youssef are thanked for useful conversations.

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REFERENCES

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