J. Phys. Chem. B 2006, 110, 9413-9420
9413
Theoretical Analysis of Structural, Energetic, Electronic, and Defect Properties of Li2O Mazharul M. Islam,† Thomas Bredow,*,† and Christian Minot‡ Theoretische Chemie, UniVersita¨t HannoVer, Am Kleinen Felde 30, 30167 HannoVer, Germany, and Laboratoire de Chimie The´ orique, UMR 7616 CNRS, UniVersite´ P. et M. Curie, Case 137, Tour 23-22, 4 Place Jussieu, Paris 75252 Ce´ dex 05, France ReceiVed: NoVember 18, 2005; In Final Form: March 10, 2006
The structural, energetic, and electronic properties of stoichiometric and defective Li2O were studied theoretically. The reliability of the Perdew-Wang method in the framework of density functional theory (DFT), and of two DFT/Hartree-Fock hybrid methods (PW1PW and B3LYP), was examined by comparison of calculated and available experimental data. Atom-centered orbitals and plane waves were used as basis functions for the crystalline orbitals. For both cases, the basis set dependence of calculated properties was investigated. With most of the methods, good agreement with the experimental Li2O lattice parameter and cohesive energy was obtained. In accordance with experiment, the analysis of electronic properties shows that Li2O is a wide gap insulator. Among the considered methods, the hybrid methods PW1PW and B3LYP give the best agreement with experiment for the band gap. The formation of an isolated cation vacancy defect and an F center in Li2O were studied. The effect of local relaxation on the calculated defect formation energies and the defect-induced changes of electronic properties were investigated and compared to available experimental results. The migration of a Li+ ion in Li2O bulk was investigated. The activation energy for the migration of a Li+ ion from its regular tetrahedral site to an adjacent cation vacancy was calculated, including the effect of local relaxation. The calculated activation barriers, 0.27-0.33 eV, are in excellent agreement with experiment.
1. Introduction Lithium oxide has become a subject of considerable interest because of its potential applications. It is used in high-capacity energy storage devices for next-generation electric vehicles, in lightweight high-power-density lithium-ion batteries for heart pacemakers, mobile phones, and laptop computers,1,2 and as blanket breeding material for deuterium-tritium fusion reactors3. Diffusion and ionic conduction in Li2O are subjects of great interest due to the superionic behavior of this material. Both theoretical and experimental investigations have been performed for Li2O on the energetic4-6, electronic,6-15 and defect properties,12-25 and the ion conduction mechanism.17,18,26-33 Lithium oxide has antifluorite structure (space group Fm3m). The lattice consists of a primitive cubic array of Li+ ions with spacing a/2, where the O2- ions occupy alternating cube centers. The lattice parameter a has recently been measured at different temperatures in the range 293-1603 K by using inelastic neutron scattering on single crystals and polycrystals.10 An extrapolation to T ) 0 K gives a ) 4.573 Å, about 0.05 Å smaller than the room-temperature value, 4.619 Å.34 The experimental value of the heat of atomization is 1154 kJ/mol35 and the band gap (Eg) is 7.99 eV.15 It is well-known that defects have a large effect on the ion conductivity of ceramics. The dominant intrinsic defects in Li2O are point defects,16-18 either as cation vacancies or of cationFrenkel type, i.e., vacancies and interstitials in the Li sublattice. Schottky disorder is also observed, but it is not as predominant * Corresponding author. E-mail:
[email protected]. † Theoretische Chemie, Universita ¨ t Hannover. ‡ Laboratoire de Chimie The ´ orique, UMR 7616 CNRS, Universite´ P. et M. Curie.
as the cation-Frenkel defect.17 On the other hand, the dominant irradiation defects in Li2O are known as F centers12,19,20 and F+ centers.12-14,21,24,25 In a combined experimental and theoretical study of defects in Li2O, Chadwick et al.17 showed that Li+ ions migrate via cation vacancies. This study was performed in a combination of ac conductivity measurements and nonlinear least-squares computer simulation, where the activation energy for Li+ diffusion (EA) was investigated. Their experimental value of EA is 0.49 eV as compared to their calculated value of 0.21 eV. In more recent theoretical investigations,18,26,27 where density functional methods based on the local density approximation (LDA) were employed, it was also observed that Li+ ions were migrating through the cation vacancies. For this process, activation energies of 0.34 eV,18 0.30 eV,26 and 0.29 eV27 were calculated. Recently, Heitjans et al.28 presented two types of experimental approaches for the study of diffusion and ionic conduction in nanocrystalline ceramics. The tracer diffusion method is a macroscopic method, while NMR relaxation is known as a microscopic method. EA for Li ion diffusion in Li2O derived from NMR relaxation method is 0.31 eV, whereas 0.95 eV was obtained with the tracer diffusion method. The NMR relaxation method gives a smaller EA compared to that of the tracer diffusion method because it has access to short-range motions of the ions. The barrier heights correspond to a single jump process. On the other hand, the tracer diffusion method probes the long-range transport.28 In the present study, cation vacancy and F center defects in bulk Li2O are investigated theoretically. Unlike previous DFT studies that were based on the LDA, we employ DFT methods based on the generalized gradient approximation (GGA) and hybrid methods incorporating exact exchange from the HartreeFock (HF) method. The particular choice of these methods is
10.1021/jp0566764 CCC: $33.50 © 2006 American Chemical Society Published on Web 04/27/2006
9414 J. Phys. Chem. B, Vol. 110, No. 19, 2006 based on previous studies of other oxides where they provided accurate results. Complementary basis sets, atom-centered basis functions, and plane waves were employed in order to examine their effect on the calculated properties.
Islam et al. TABLE 1: Optimized Lattice Parameter a (Å), Heat of Atomization Per Li2O Unit Ea (kJ/mol) and Band Gap Eg (eV); Dependence from Basis Set (BS) and Cutoff Energy ECut BS, Ecut
2. Computational Methods Bulk and defect properties of Li2O were obtained from periodic calculations with three methods at DFT level. The Perdew-Wang correlation functional based on the generalized gradient approximation (PW91)36,37 was combined with two different exchange functionals. In the PW1PW hybrid method, the exchange functional is a linear combination of the HartreeFock expression (20%) and the Perdew-Wang exchange functional (80%).38 PW1PW is similar to the mPW1PW91 functional proposed by Adamo and Barone39 but uses the original PW91 exchange functional rather than the modified form introduced by these workers. The exact-exchange coefficient is also reduced from 25 to 20%. The second approach is the original PW91 DFT method.36,37 These two methods have been applied for calculations of bulk properties of MgO, NiO, CoO,38 TiO2,40 Li2B4O7,41 B2O3,42 and electronic properties of Li2O-B2O3 compounds.43 In these studies, good agreement between calculated and experimental bulk properties was observed, in particular for the PW1PW hybrid method. For comparison, we also used the well-known B3LYP hybrid method.44,45 These DFT approaches were used as implemented in the crystalline orbital program CRYSTAL03.46 In CRYSTAL, the Bloch functions are linear combinations of atomic orbitals (LCAO). The quality of the atomic basis sets determines the reliability of the results. Therefore, we have tested different basis sets in the present study. We started with a 6-1G basis47 for Li. In the second set, a 6-11G Li basis was used where the outer sp exponent has been optimized in Li(OH)H2O.48 The third Li basis set is 7-11G*.49 The 7-11G* basis for Li was further extended to 7-11G(2d) in our previous study.43 The inner 1s and 2sp shells remained unchanged, while the orbital exponents of the 3sp and d shells were optimized for bulk Li2O at PW1PW level.43 For O, first a 8-411G basis was used as optimized for Li2O by Dovesi et al.4 The second O basis set was 8-411G*.50 The 8-411G* basis set was further extended to 8-411G(2d) by adding one more d polarization function. Five combinations of these atomic basis sets (BS) were applied, BS A (Li: 6-1G, O: 8-411G), BS B (Li: 6-11G, O: 8-411G*), BS C (Li: 7-11G*, O: 8-411G*), BS D (Li: 7-11G(2d), O: 8-411G*), and BS E (Li: 7-11G(2d), O: 8-411G(2d)). We also employed the PW91 method implemented in the plane-wave program VASP.51-53 In this way, the effect of complementary types of basis sets, atom-centered functions, and delocalized plane waves, on the results obtained with the same density functional method, could be studied. In contrast to the LCAO approach, which allows the explicit treatment of all electrons, inner electrons are replaced by effective potentials in VASP. In this study, the projector-augmented wave (PAW) potentials54,55 are used for the core electrons representation. Accordingly, the approach is denoted as PW91-PAW. In planewave methods, the quality of the basis set is determined by a single parameter, the energy cutoff Ecut. Three energy cutoff values, E1 ) 400 eV, E2 ) 520 eV, and E3 ) 600 eV, were used. Here, E1 is the standard value obtained from the VASP guide,56 while E2 and E3 correspond to increased numbers of plane waves. 3. Results and Discussion In this section, we present the results for the structural, energetic, and electronic properties for stoichiometric and
a
Ea
Eg
PW1PWa
A B C D E
4.56 4.57 4.59 4.58 4.58
1107 1116 1130 1134 1134
10.19 8.37 8.66 7.95 7.96
B3LYPa
A B C D E
4.58 4.59 4.59 4.59 4.59
1106 1110 1122 1123 1123
10.23 8.49 8.82 8.11 8.12
PW91a
A B C D E
4.59 4.61 4.63 4.63 4.62
1131 1143 1160 1164 1164
8.00 6.24 6.53 5.82 5.83
PW91-PAWb
E1 E2 E3
4.58 4.64 4.63
1169 1176 1165
5.02 5.00 5.00
1154e
7.99f
4.573c,4.619d
expt a
b
c
CRYSTAL03 results. VASP results. Ref 10, extrapolated to T ) 0 K. d Ref 34, at room temperature. e Ref 35. f Ref 15.
defective Li2O and the energy barriers for Li+ ion migration in Li2O. A primitive unit cell containing one formula unit was used as a model for stoichiometric Li2O. Integration in reciprocal space was performed with a Monkhorst net57 by using shrinking factors s ) 8. To minimize direct defect-defect interaction between neighboring cells, we used a large supercell (Li64O32) as a model of the defective bulk. It was obtained from the primitive unit cell by a transformation with matrix L.
L)
(
-2 2 2 2 -2 2 2 2 -2
)
(1)
To study the convergence behavior of calculated defect properties, we also considered the smaller supercell Li32O16. 3.1. Stoichiometric Li2O. Prior to the study of the defect properties, it is mandatory to perform method and basis set tests on the defect-free Li2O solid. A part of this testing has been done in our previous study.43 Here, we investigate the basis set effect in more detail on the computed properties. In a similar comparison of calculated and measured bulk properties of lithium tetraborate,41 it was found that the use of ultrasoft pseudopotentials (US-PP) leads to larger deviations from experiment compared to those of the PAW approach. Therefore, US-PP are not considered in the present study. The calculated results for the Li2O lattice parameter a, heat of atomization per formula unit Eu, and the optical band gap Eg, as obtained with PW1PW, B3LYP, PW91, and PW91-PAW, are presented in Table 1. The lattice parameter obtained with all methods is close to the range of experimental values. PW91-PAW and PW91 methods give close values of lattice parameter a to each other and to the experimental values. For the CRYSTAL calculations, it is found that the size of the atomic basis set has almost no effect on the structural properties. a is already converged with the BS B. The calculated lattice parameter is converged with energy cutoff Ecut ) E2 for the plane-wave-based approach (PW91-PAW). The atomization energy per Li2O unit (Ea) was calculated for the optimized values of lattice parameter a. For the
Properties of Li2O calculation of the atomic reference energies with CRYSTAL03, the basis sets of the free atoms were optimized by augmenting the basis sets of the periodic calculations with diffuse functions until convergence was achieved for the total energy. For the PW91-PAW implementation in VASP, atomic reference energies were calculated with PAW potentials by using pseudolattice constants of 13 Å for Li and 8 Å for O. All the methods give atomization energies within ( 30 kJ/mol (Table 1) of the experimental heat of atomization, 1154 kJ/mol.35 In the calculated atomization energies, contributions from zero-point energy and entropy are neglected. The two hybrid methods, B3LYP and PW1PW, give close agreement with experiment for Ea, with deviations of -31 and -20 kJ/mol (BS D), respectively. The basis set convergence for Ea is slower than for structural properties. With the three LCAO-based DFT implementations, Ea is converged with BS D. PW91-PAW shows an oscillation of Ea with Ecut. In a recent experimental investigation,15 7.99 eV was obtained for the optical band gap Eg of Li2O. This value is much higher than the results of photoemission and electron energy loss spectroscopy (EELS), 7-7.5 eV,8 of absorption spectroscopy at low temperature, 7.02 eV,12 and of optical absorption spectroscopy, 6.6 eV.11 Several theoretical studies on the electronic properties of Li2O have already appeared in the literature. Band gaps and bandwidths are generally overestimated by the ab initio Hartree-Fock method6,7 and underestimated by the DFT local density approximation (LDA).9,7 DFT methods based on the generalized gradient approximation (GGA) give closer agreement7 with the experiment, and a hybrid DFT method incorporating exact HF exchange7 further improves electronic properties. In our previous study,43 we have performed an investigation on the electronic properties of Li2O-B2O3 compounds with the PW1PW method by applying only two basis sets (BS B and BS D of the present study). To predict the effect of different types of basis functions on the electronic properties of Li2O, we performed a reinvestigation of band structure and density of state in the present study. The band structure was computed along the way that contains the highest number of highsymmetry points of the Brillouin zone (BZ),58 namely W f L f Γ f X f W. The calculated band gap (Eg) values are given in Table 1, and the minimal vertical transition (MVT) and minimal indirect transition (MIT) energies are given in Table 2. For the LCAO-based approaches (PW1PW, B3LYP, and PW91), the minimal energy gap is indirect in the Γ-X direction with all BS larger than BS A. On the contrary, with PW91PAW, the Γ-Γ transition is lower than Γ-X (Table 2), irrespective of the cutoff energy. To check if this discrepancy is due to the description of core electrons by effective potentials, we performed CRYSTAL PW91 test calculations where the 1s electrons of Li and O were replaced by Stuttgart-Dresden (SDD) effective core potentials (ECP).59,60 As can be seen in Table 2, the difference between Γ-Γ and Γ-X is reduced in this way, whereas other transitions except L-Γ are not affected significantly. Therefore, we conclude that the qualitative difference between PW91 (LCAO) and PW91-PAW is caused by the inaccurate representation of core electrons by effective potentials. The present discussion is only of a qualitative nature because of the differences between the SDD-ECP and PAW approach. The best agreement for the experimental value of Eg is obtained with the PW1PW method (converged with BS D). Only direct (allowed) transitions are considered because indirect transitions should appear with much lower intensities in optical
J. Phys. Chem. B, Vol. 110, No. 19, 2006 9415 TABLE 2: Basis Set Dependence of Minimal Vertical Transition (MVT) and Minimal Indirect Transition (MIT) Energies (eV) Obtained with CRYSTAL and VASP MVT
MIT
BS, Ecut L-L W-W X-X Γ-Γ W-L L-Γ Γ-X PW1PWa
A B C D E
14.46 12.92 12.06 12.05 12.05
17.62 16.60 10.19 15.65 10.70 15.03 11.82 9.12 8.37 14.10 8.87 7.59 10.87 8.45 8.66 13.22 9.15 6.94 10.91 8.47 7.95 13.20 8.45 6.95 10.90 8.46 7.96 13.21 8.46 6.94
B3LYPa
A B C D E
14.47 12.88 12.30 12.24 12.25
17.43 16.45 10.23 15.63 10.74 14.91 11.96 9.29 8.49 14.03 8.99 7.78 11.11 8.70 8.82 13.44 9.31 7.20 11.11 8.67 8.11 13.38 8.60 7.19 11.10 8.67 8.12 13.38 8.61 7.19
PW91a
A B C D E SDDb
11.99 14.95 14.07 10.45 9.63 7.09 9.75 8.70 6.42 9.70 8.71 6.42 9.70 8.70 6.41 9.54 8.78 6.54
PW91-PAWc
E1 E2 E3
9.69 9.58 9.58
7.79 7.69 7.69
8.36 8.26 8.27
8.00 6.24 6.53 5.82 5.83 5.38
13.07 11.51 10.80 10.73 10.74 10.60
5.02 5.00 5.00
9.70 9.59 9.59
8.45 12.65 6.68 5.70 6.97 5.05 6.26 5.05 6.27 5.05 5.82 5.18 5.46 5.42 5.42
7.89 7.80 7.80
a CRYSTAL03 results (PW1PW data slightly deviate from previously published values43 because for the present study a higher numerical accuracy was applied in the numerical integration procedure). b Inner electrons of Li and O are described by Stuttgart-Dresden ECPs.59,60 c VASP results.
spectra. The calculated value of Eg is 7.95 eV, close to the experimental value of 7.99 eV.15 The second best agreement is obtained with the B3LYP method, 8.11 eV (BS D). The two pure Perdew-Wang implementations, LCAO-based PW91 and plane-wave-based PW91-PAW, underestimate the band gap. With PW1PW and B3LYP, the converged MIT energies (6.94 and 7.19 eV, respectively) closely resemble the measured absorption energy at low temperatures, 7.02 eV.12 The atomic basis set has a pronounced effect on the electronic structure. With the smallest BS A, the PW1PW Eg value is 10.19 eV. This is 2.23 eV larger than that with the largest BS E. The difference is mainly due to the inclusion of diffuse and polarization functions in the Li basis set. These orbitals are dominating at the lower part of the conduction band (CB), as shown below. The basis set effect is almost independent from the method, as can be seen by the difference obtained with BS A, B, C, D, and E for B3LYP and PW91 methods. In all cases, Eg is converged with BS D. A density of states (DOS) calculation with PW1PW shows that the valence band (VB) is mainly formed by the oxygen 2p orbitals with only small contributions from Li, whereas the conduction band (CB) is dominated by Li states. The calculated valence bandwidth is about 5.5 eV, which is in good agreement with the experimental value of 5 eV.8 All other methods give similar results. 3.2. Defective Li2O. Two defect types in Li2O were studied, the cation vacancy defect and the F center. On the basis of the optimized structural parameters for the perfect crystals, a Li64O32 supercell was constructed for defect calculations by using the transformation matrix L of eq 1. For comparison also the smaller Li32O16 supercell was studied. The defective systems are considered as a new crystal with an artificially introduced point defect periodicity. 3.2.1. Cation Vacancy. An experimental investigation17 showed that the most mobile species in lithium oxide is the Li+ ion and the most likely mechanism for its migration is via cation vacancies. Although there have been several experi-
9416 J. Phys. Chem. B, Vol. 110, No. 19, 2006
Islam et al.
TABLE 3: Effect of Relaxation and Basis Set Size on the Cation Vacancy EDe(V) and F Center EDe(F) Formation Energy (kJ/mol) (Li64O32 Supercell) PW1PW BS, Ecut Ede(V) (unrelaxed) Ede(V) (relaxed) Ede(F) (unrelaxed) Ede(F) (relaxed)
A 682 577 1003 1001
B
B3LYP C
A
642 671 576 567 878 853 1010 873 848 1009
B
PW91 PW91-PAW C
A
E1
E2 E3
629 614 593 558 566 525 517 480 893 894 950 1016 975 888 890 949 1001 957
555 477 966 948
mental17,28-30 and theoretical16-18,26,27 studies of ionic transport in Li2O, the defect formation energy of a cation vacancy and the relaxation effect for defective systems are still not known. In the present section, the formation energy of a cation vacancy in Li2O, the effect of relaxation for this type of defect, and the influence of this defect on the electronic properties are studied. The vacancy was created by removing one Li atom from the Li64O32 supercell, keeping the system neutral. Thus the Li63O32 cell contains an odd number of electrons and its ground state is a doublet. The calculations were performed with the spinpolarized method (unrestricted Kohn-Sham, UKS). The Li defect concentration is 1.6%, and the shortest distance between two defects is 9.1 Å in this model. This was assumed to be sufficient to model isolated defects. The formation energy of cation vacancy Ede(V) is calculated as:
Ede(V) ) E(Li63O32) + E(Li) - E(Li64O32)
TABLE 4: Distances r (Å) of Neighboring Atoms from the Vacancy and Changes of the Distances ∆r (%) for Relaxed Atoms for the Cation Vacancy and F Center (Li64O32 Supercell, Method PW1PW) defect types cation
vacancya
F centerb
a
atom
r
unrelaxed
relaxed
∆r
O(4) Li(6) Li(12) O(12) Li(8) Li(6)
r1 r2 r3 r4 r5 r6
1.98 2.29 3.23 3.79 3.98 4.57
2.11 2.06 3.26 3.78 4.00 4.61
+6.6 -10.0 +0.9 -0.3 +0.5 +0.9
Li(8) O(12) Li(24) O(6) Li(24) O(24)
r1 r2 r3 r4 r5 r6
1.99 3.24 3.80 4.59 5.00 5.62
2.02 3.25 3.80 4.59 5.00 5.62
+1.5 +0.2 0.0 0.0 0.0 0.0
PW1PW results with BS B. b PW1PW results with BS C.
(2)
Here E(Li63O32) and E(Li64O32) denote the total energy of the supercell with and without vacancy, respectively, and E(Li) is the energy of the free Li atom. The relaxation energy, ER, is calculated by subtracting the energy of the relaxed system from that of the system where all remaining atoms occupy positions of the defect-free bulk. In Table 3, calculated cation vacancy formation energies are presented. Ede(V) is calculated with eq 2 for unrelaxed and fully relaxed systems. The values of Ede(V) with PW1PW and B3LYP obtained with the larger BS B are 576 and 566 kJ/mol, respectively. Because the basis set dependence of Ede(V) is only 1 kJ/mol for the relaxed structures, no calculations with larger BS were performed. The relaxation energy, ER, is ≈65 kJ/mol for both methods (BS B). For the plane-wave-based DFT method PW91-PAW, Ede(V) is converged with energy cutoff E2. Compared to the results of the PW1PW method, the PW91-PAW method gives smaller values of Ede(V) by 96 kJ/mol (477 kJ/mol, Ecut ) E3). Both the pure Perdew-Wang implementations (PW91 and PW91-PAW) give a similar trend for the calculation of Ede(V) values (525 kJ/ mol, PW91 with BS A, and 517 kJ/mol, PW91-PAW with Ecut ) E1). PW91 calculations with the larger BS B failed due to SCF convergence problems. It can be concluded that inclusion of exact exchange leads to an increase of the defect formation energy. The structural relaxation effects are investigated by measuring the changes of distances of the relaxed atoms from the Li defect position. PW1PW results for the relaxation effects are shown in Table 4. All other methods give the same trend. The four O atoms in the first coordination shell (first nearest neighbors, 1-NN) show an outward relaxation from the vacancy, namely by 6.6% with the PW1PW method. This is reasonable because the electrostatic attraction by the Li+ cation is missing. The removal of one neutral Li atom creates a hole in the valence band. One of the surrounding four O atoms (formally O2-) in 1-NN becomes O-, and spin density is localized on this O atom. Six Li atoms in the second coordination shell (2-NN) show a
Figure 1. Density of states (DOS) for Li vacancy in Li64O32 obtained with PW1PW. EF denotes the Fermi level.
strong inward relaxation of -10.0% with the PW1PW approach. Because of the reduced electrostatic repulsion, the 2-NN Li atoms tend to move toward the vacancy. The 12 Li atoms in the third coordination shell (3-NN) show an outward relaxation. All other atoms with larger distances to the defect position show only small relaxation. Thus the relaxation is mainly restricted to the nearest and the second nearest neighbor atoms. The DOS for a defective Li63O32 supercell obtained with PW1PW (BS B) is shown in Figure 1. All the other methods show qualitatively the same behavior. It can be seen that the neutral Li vacancy introduces an extra unoccupied level in the β ladder roughly 0.33 eV above the Fermi level (EF), which is
Properties of Li2O
J. Phys. Chem. B, Vol. 110, No. 19, 2006 9417
marked by an arrow (Figure 1). The unoccupied defect level consists of oxygen states. All the other methods give nearly the same energetic position of the unoccupied defect level above EF (B3LYP 0.31 eV and PW91-PAW 0.31 eV). 3.2.2. F Center. Li2O is proposed as a blanket material in deuterium-tritium fusion reactors,3 where it is exposed to strong radiation. One of the predominant irradiation defects is known as the F center, an oxygen vacancy trapping two electrons. Only a limited number of theoretical and experimental investigations12,19,20 of the F center defect in Li2O could be found in the literature. Tanigawa et al.20 performed supercell calculations at the HF level by using CRYSTAL95 and at the GGA level by using the plane-wave program CASTEP. The effect of relaxation on F centers in Li2O was investigated only for eight 1-NN Li atoms and 12 2-NN O atoms, and relaxations were too small to estimate the accuracy. The optical transition energy of F centers was calculated as 4.82 eV19 with the embedded molecular cluster model by using the semiempirical INDO-type calculation scheme. In a photoluminescence study of Li2O under excitation with UV light in the fundamental absorption region at low temperature,12 the optical transition energy of F centers was approximated as 3.70 eV. No experimental or theoretical investigations were found in the literature that predict the defect formation energy of an isolated F center, Ede(F). In the present study, a systematic investigation is performed for the calculation of F center formation energy, Ede(F), the effect of relaxation on F center, and the optical transition energy of the F center. To create the F center, one neutral oxygen atom was removed from the supercell. The defect formation energy of F center, Ede(F), is calculated as:
Ede(F) ) E(Li64O31) + E(O) - E(Li64O32)
(3)
Here, E(Li64O31) and E(Li64O32) denote the total energy of the supercell with and without defect, respectively, and E(O) is the energy of the free O atom in its ground state. The calculations were performed with the UKS method. In Table 3, Ede(F) obtained with PW1PW, B3LYP, PW91, and PW91-PAW are presented for unrelaxed and fully relaxed Li64O32 cells. For the CRYSTAL03 calculations, the basis functions of the oxygen ion were left at the defect position. Calculations were performed for the closed-shell singlet state. A test calculation for the triplet state was performed with the PW1PW method by using BS B. It was observed that Ede(F) is much higher (by 539 kJ/mol) for the triplet state than for the closed-shell singlet state. The converged value of Ede(F) with PW1PW by using BS C is 848 kJ/mol. The B3LYP method gives a slightly larger value of Ede(F), 890 kJ/mol. In both methods, the relaxation energy is very small, ≈5 kJ/mol. For the plane-wave-based DFT method, PW91-PAW, the converged value of Ede(F) is 948 kJ/mol. Compared to the PW1PW method, this value is overestimated by 100 kJ/mol. Similar to Ede(V), PW91-PAW and PW91 give similar values also for Ede(F). The differences between hybrid and pure DFT methods demonstrate the importance of exact exchange in the calculation of defect energies. With all methods, the relaxation energy ER (1-18 kJ/ mol) is small compared to the absolute value of Ede(F). The structural relaxation effects of the F centers are investigated by measuring the changes of distances of the relaxed atoms from the defect position. Only PW1PW results (BS C) are presented in Table 4. All other methods give qualitatively the same trend. The F center is surrounded by eight Li atoms in the first coordination shell (1-NN). The Li atoms in 1-NN show an outward relaxation from the vacancy, namely by 1.5% with PW1PW. This is reasonable because the 1-NN Li atoms
Figure 2. Density of states (DOS) for F center in Li64O32 obtained with PW1PW.
are positively charged and should, therefore, repel each other as the central oxygen ion is removed. But the effect is much smaller than for the oxygens surrounding the Li defect. The 12 2-NN O atoms show an outward relaxation of 0.2%, indicating that the positions of the oxygen atoms are almost unchanged. These results agree well with the outward relaxation of 1-NN Li atoms and 2-NN O atoms for the F centers in Li2O obtained by Tanigawa et al.20 In that study, the displacements were even smaller, namely by 0.05% for 1-NN Li atoms and 0.01% for 2-NN O atoms. Further relaxation shows that 24 3-NN Li atoms, six 4-NN O atoms, 24 5-NN Li atoms, and 24 6-NN O atoms are unchanged. The experimental value of the optical transition energy for oxygen-deficient Li2O is 3.7 eV,12 indicating a location of a doubly occupied defect level about 4.3 eV above the valence band maximum (VBM). This is reflected by the calculated density of states (DOS) of the defective supercells. The DOS curve for a defective Li64O31 supercell using the PW1PW approach (BS C) is shown in Figure 2. The doubly occupied defect level (marked with an arrow in Figure 2) is located 3.9 eV below the CB edge. The obtained transition energy is in good agreement with the experimental value of 3.7 eV.12 Other methods show similar behavior. The optical absorption energy is 3.8 eV at B3LYP level, whereas the PW91-PAW approach is giving a too-small value of the optical absorption energy, namely 2.7 eV. This can be related to the artificial selfinteraction in DFT methods, which is not completely removed in the GGA. 3.2.3. Migration of Li+ Ion. For the study of Li migration from a regular lattice site to an adjacent vacancy, PW1PW and
9418 J. Phys. Chem. B, Vol. 110, No. 19, 2006
Islam et al. TABLE 5: Comparison of Calculated Activation Energy for Li Migration EA (eV) for Unrelaxed Systems, Relaxation of Nearest Neighbors (1-NN), and Fully Relaxed Systems with Experimental Value EA PW1PW B3LYP PW91-PAW
Figure 3. The Li+ ion migration process in Li2O. (a) Li+ ion and vacancy (V) are in their original position, (b) the transition state, (c) the migrating Li+ ion is on the vacancy position, while the vacancy is on the original position of Li+ ion. The light (dark) gray spheres denote oxygen atoms below (above) the Li-V plane.
B3LYP calculations were performed by using BS A and PW91PAW calculations were performed by using energy cutoff E1. Basis sets larger than BS A and energy cutoffs larger than E1 were not considered because of the high expense of CPU time and severe SCF convergence problems. The latter prevented the use of PW91 in the CRYSTAL03 implementation. In a previous section, it was documented that the basis set has only a small effect on the structural and energetic properties. Therefore, BS A and energy cutoff E1 are a compromise between accuracy and computational cost for these calculations. In Figure 3, the migration process is illustrated. Figure 3a shows the unrelaxed structure of defective Li2O where the migrating Li+ ion and the vacancy (V) are in their original positions. Figure 3b shows the symmetric transition structure. The lithium ion is centered between two oxygen ions. The activation energy EA for the migration process was calculated from the energy difference between the transition structure (Figure 3b) and the initial structure (Figure 3a). The final structure (Figure 3c), where the migrating ion has accessed the position of the vacancy, is isoenergetic with the starting structure. The experimental hopping distance for the migration of Li+ ion from its original tetrahedral site to the vacancy is the nearest Li-Li distance in Li2O,10 namely 2.29 Å. The calculated hopping distances slightly differ for the considered methods due to the different optimized lattice constants. They are 2.28, 2.29, and 2.29 Å for PW1PW, B3LYP, and PW91-PAW, respectively. For the calculation of the potential curve, the migration path of the Li+ ion was divided into 10 equidistant steps. Our theoretical model corresponds to a single Li ion hop. This process is comparable to that studied with NMR relaxation by Heitjans et al.28 Therefore, the calculated activation energies EA are compared with this experimental value, 0.31 eV.28 We performed the study of Li+ migration in three steps. First, the potential curve for the movement of Li from its regular position to the nearest defect position was calculated without relaxation of the neighboring atoms. In the second step, the nearest-neighboring atoms around the defect and the migrating Li were relaxed. The vacancy and the migrating ion are surrounded by four oxygens each in the first coordination shell. Together, they have six oxygen atoms as nearest neighbors (1NN). Two oxygens are bridging the two tetrahedra (Figure 3a). In the defective structure, one unpaired electron is localized on one of these bridging oxygens, mainly in the 2p orbitals. This situation was also observed in previous DFT-LDA investigations for the Li+ ion diffusion in Li2O.18 It can be expected that this oxygen atom undergoes pronounced changes in its coordination geometry and that it is important to include it in the relaxation. In the final step, all atoms of the cell were allowed to relax
unrelaxed
1-NN
full relaxation
0.45 0.63 0.49
0.17 0.16 0.17
0.33 (0.39d) 0.29 (0.38d) 0.28 (0.27d)
expta LDA
0.31 0.34b, 0.30 c
Ref 28. b Ref 18. c Ref 26. d Obtained with a smaller Li32O16 supercell. a
except the migrating Li atom, which was fixed at each predefined position on the migration path. It can be seen from Table 5 (column 2) that all methods significantly overestimate the experimental EA if relaxation is not taken into account. The values range from 0.45 eV (PW1PW) to 0.63 eV (B3LYP). With all considered methods, the barrier is drastically reduced when relaxation of the first neighbors (1-NN) is performed (Table 5, column 3). Surprisingly, the calculated barriers at this level of relaxation (0.160.17 eV) are even smaller than those with full relaxation of all atoms in the cell (Table 5, column 4). One reason for the small barriers obtained with 1-NN relaxation is that the movement of the oxygen atoms surrounding the vacant Li lattice site in Figure 3a and c is hindered by the repulsive interaction with their nearest oxygen neighbors, which are fixed. The stabilization of the defect structure was 2-3 times larger for the relaxation of the second nearest neighbors compared to the 1-NN relaxation. Apparently, this effect is less pronounced for the transitionstate structure. A good agreement with experiment was achieved only with the full relaxation. The calculated barriers range from 0.28 to 0.33 eV (Table 5, column 4). To study the dependence of EA from the supercell size, we also studied the smaller cell Li32O16 (see values in parentheses in Table 5). With the hybrid methods, slightly larger barriers (0.39 and 0.38 eV, respectively) were obtained, whereas for PW91-PAW, there is essentially no change (0.27 eV). Similar activation energies (0.34 and 0.30 eV) were obtained in plane-wave DFT studies of the Li32O16 supercell based on the local density approximation by using the norm-conserving pseudopotentials18 and ultrasoft pseudopotentials,26 respectively. The possibility of a different migration path was also investigated where the Li+ ion moves in a two-dimensional curve rotation rather than in a straight line. Here, only the PW91PAW method was employed. The energy hypersurface was scanned point-by-point. The lowest energy was found at a location almost identical to the central symmetric position of the straight line. Also, the energy barrier EA for the curve rotation (0.2763 eV) is virtually identical to the value of 0.2766 eV obtained for the migration in straight line. Therefore, it can be concluded that the migration of Li+ ion occurs in an almost straight line. 4. Summary and Conclusions The reliability of four quantum-chemical approaches was tested for the structural, energetic, and electronic properties of stoichiometric and defective Li2O by comparison of calculated and available experimental data. For the lattice parameter, the hybrid PW1PW method gives the best agreement among the considered methods with a deviation of 0.4%. Pure DFT methods (PW91 and PW91-PAW) give similar deviations from
Properties of Li2O the experimental data, with an error of less than 1.2%. All methods give reasonable agreement with experiment for the calculated atomization energy. The deviation is smaller than ( 30 kJ/mol. Fast convergence of calculated structure parameters with the size of the atomic basis sets was observed. The basis set convergence is slower for the atomization energy. A comparison of the calculated properties with two Perdew-Wang implementations (PW91 and PW91-PAW) showed that the effect of different types of wave functions (LCAO and plane waves) on the structural and energetic properties is small. PW91 and PW91-PAW give close values of the optimized lattice parameter to each other, and the deviation for calculated Ea from the experiment is similar. As for the structural and energetic properties, the hybrid methods PW1PW and B3LYP give the best agreement with experiment (7.99 eV) for the calculated band gap (7.95 and 8.11 eV, respectively). Pure DFT methods PW91 and PW91-PAW underestimate the band gap by 2.22.9 eV. A qualitative difference between all-electron LCAO and valence-only plane-wave approaches was observed for the Li2O band structure. With the former, the indirect Γ-X transition appears at lower energies than the direct Γ-Γ transition, while PW91-PAW predicts the smallest transition as direct at the Γ point. This discrepancy has been attributed to the approximate description of core electrons by effective potentials. The description of inner electrons has a notable impact on the calculation of bulk properties. A theoretical investigation of the isolated cation vacancy and F center defects was performed. There are no literature values for the formation energies of both types of defect in lithium oxide. The formation energy of a cation vacancy is 576 kJ/mol with the PW1PW approach. B3LYP gives very close agreement to PW1PW. All considered methods give the same trend for structural relaxation. Relaxation around the cation vacancy is mainly restricted to the first and the second nearest-neighbor atoms. The analysis of electronic properties showed that Li+ ion vacancy introduces an extra unoccupied level below the bottom of the conduction band. The calculated values of the F center formation energy range from 848 to 949 kJ/mol. The structural relaxation around the F center is rather short-ranged. It is restricted to the first nearestneighbor atoms. The removal of a neutral oxygen leads to formation of a doubly occupied defect level above the valence band. The best agreement for the calculated absorption band to the experiment was obtained with B3LYP and PW1PW, whereas PW91-PAW gives a too-small value. The ionic conductivity in Li2O was investigated by calculating the activation energy for the Li+ ion migration from a regular lattice site to the nearest vacancy site. It was shown that Li migrates in an almost straight line. Different density functional methods based on complementary types of basis sets give activation energies in close agreement with each other and with experiment, provided that all structural degrees of freedom are allowed to relax. This result gives confidence that more complex systems can be also studied with the presented approaches. Acknowledgment. This work was supported by the State of Lower Saxony, Germany, by a “Georg Christoph Lichtenberg” fellowship (M. M. Islam). References and Notes (1) Keen, D. A. J. Phys.: Condens. Matter 2002, 14, R819. (2) Noda, K.; Ishii, Y.; Ohno, H.; Watanabe, H.; Matsui, H. AdV. Ceram. Mater. 1989, 25, 155.
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