Formation and Mobility of Li Point Defects in LiBO2: A First-Principles

ARTICLE pubs.acs.org/JPCC

Formation and Mobility of Li Point Defects in LiBO2: A First-Principles Investigation Mazharul M. Islam,*,† Thomas Bredow,† and Paul Heitjans‡ † ‡

Institut f€ur Physikalische und Theoretische Chemie, Universit€at Bonn, Wegelerstrasse 12, 53115 Bonn, Germany Institute of Physical Chemistry and Electrochemistry, and ZFM
Center for Solid State Chemistry and New Materials, Leibniz University Hannover, Callinstrasse 3a, 30167 Hannover, Germany ABSTRACT: The formation and mobility of Li point defects in lithium metaborate (LiBO2) are investigated theoretically with periodic quantum chemical methods. Calculated defect formation energies obtained with a density functional theory/Hartree
Fock hybrid method and with the Perdew
Wang density functional method are compared. The basis set effect is investigated by comparison of results obtained with atom-centered basis functions and plane waves. With both methods, only a moderate relaxation is observed for the atoms surrounding the Li defect position. The defect-induced change of electronic properties is investigated by calculating the density of states for the stoichiometric and defective supercells. Various pathways for Li diffusion are investigated using the climbing-image nudged elastic band (cNEB) approach. It is observed that the Liþ ion migrates along the c direction and in the xy plane. The calculated activation energies are in reasonable accordance with experiment.

1. INTRODUCTION In recent years, lithium metaborate (LMB) LiBO2 has drawn considerable interest because of its technical applications. Because of its good dissolvability, low melting temperature, and the resistance against transition-metal contamination, LMB is widely employed as a flux or solvent.1 It is an excellent basic flux for silicate analysis,2 for the synthesis of low-density γ-Al2O3 from high-density R-Al2O3,3 for the identification and characterization of resistant minerals containing uranium and thorium,4 and for the growths of single crystals.5
7 LMB is also used as a chemical modifier during the mechano-chemical synthesis processes for generating new compounds from clays and refractory materials.8 Because of its deep ultraviolet transparency combined with mechanical durability and high optical damage thresholds,9,10 LMB is one of the most attractive materials for wide-band-gap nonlinear optics. Hydrogen is undeniably an appropriate candidate to overcome key challenges associated with the future green energy sources.11,12 A recent study13 shows that lithium borohydride is an attractive potential hydrogen storage material whose dehydration reaction forms LiBO2 along with two molecules of water. LiBO2 is considered to be a congruent compound of the Li2O
B2O3 system.14 Two phases of anhydrous LiBO2 crystals exist in the literature,15 R-LiBO2 and γ-LiBO2. R-LiBO2 belongs to space group Pn21/c (monoclinic) and has 16 atoms per unit cell (number of formula units in unit cell Z = 4).16 The measured lattice parameters are a = 5.85 Å, b = 4.35 Å, c = 6.45 Å, and β = 115°. The structure (Figure 1a) contains one-dimensional chains of BO3 triangles. These chains are parallel to b, and the atoms of a chain are almost coplanar. The Li atom also lies r 2011 American Chemical Society

approximately in the same layer and is 5-fold coordinated by O atoms. At a pressure of 3.5 GPa and temperature of 850 °C, RLiBO2 undergoes a transformation from tricoordinated boron to tetracoordinated boron, forming dense tetrahedral γ-LiBO2, where both boron and lithium atoms are 4-fold coordinated with oxygen.17 The high-pressure phase, γ-LiBO2, is quenchable and has a tetragonal symmetry with lattice parameters a = 4.1961 Å and c = 6.5112 Å, space group I-42d, and a density of 2.882 g/cm3.18 Of these two phases, the ionic bonding between Li atoms and the anion subsystems in the R-LiBO2 may result in the appearance of ionic conductivity and of superionic properties16 and is, therefore, the subject of the present study. Lithium-based fast ionic conductors have attracted considerable attention due to their broad potential applications as advanced materials with controlled chemical and new physical properties, such as lithium-ion batteries, electrochromic displays, gas sensors, etc.19
21 Recently, Heitjans and coworkers21 have performed experimental studies on ion transport and diffusion in nanocrystalline and glassy ceramics of LiNbO3, LiAlSi2O6, and LiBO2 using the measurement of dc conductivities and 7Li nuclear magnetic resonance spin
lattice relaxation rates. Their measured activation energy for the Liþ ion long-range transport as derived from the dc conductivity in LiBO2 (with R-LiBO2 as the majority phase) is 0.80 eV to 0.71 eV when going from the microcrystalline via the nanocrystalline to the glassy state. Whereas the activation energy for the short-range Liþ ion Received: April 1, 2011 Revised: May 12, 2011 Published: May 19, 2011 12343

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Figure 1. (a) Unit cell of LiBO2. (b) Local structure containing the Li point defect. The blue, red, green, and yellow spheres represent Li, O, B, and Li vacancy, respectively.

migration obtained from the spin-lattice relaxation rates is 0.23 eV to 0.21 eV when going from the nanocrystalline to the glassy state. Based on these values, the experimental activation energy for short range Liþ ion migration in the microcrystalline state is estimated to be 0.30 eV. In this study, a theoretical investigation of R-LiBO2 bulk properties, such as lattice constants, bond distances, cohesive energy, and the electronic structure, is presented. The Li vacancy defect and the migration of a Liþ ion in LMB are studied using first-principles methods and periodic supercell models. The activation energy is calculated for the Liþ ion migration from its original position to an adjacent Li vacancy position.

2. COMPUTATIONAL METHODS Bulk and defect properties of LiBO2 were obtained with periodic calculations at the DFT level. The Perdew
Wang (PW91)22,23 correlation functional based on the generalized gradient approximation (GGA) was combined with two different PW91 exchange functionals in two approaches. The first one is the pure DFT approach, denoted as the PWGGA, where both exchange and correlation functionals are based on the PW91 functional.22,23 The second approach is a HF/DFT hybrid method, PW1PW,24 where the exchange functional is a linear combination of the Hartree
Fock expression (20%) and the PW91 exchange functional (80%). These two methods have been applied for calculations of bulk properties of Li2O,25 TiO2,26 B2O3,27 and Li2B4O7;28 surface properties of Li2O29 and B2O3;30 electronic properties of Li2O
B2O3 mixed

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compounds;31 and defect properties and Liþ ion migration in Li2O,25,29 Li2B4O7,32 and Li2O/B2O3 nanocomposite materials.33 In these studies, good agreement between calculated and experimental bulk properties was observed, in particular, for the PW1PW hybrid method. These DFT approaches were used as implemented in the crystalline orbital program CRYSTAL09.34 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 used extended basis sets for the elements, 7-11G (2d) on Li, 6-21G (2d) on B, and 8-411G* on O, which gave good reproduction of experimental structural and electronic properties of LiBO2 and other Li2O
B2O3 systems.31 The PWGGA exchange-correlation functional22,23 was also used with the plane-wave program VASP.35
37 In this way, the effect of different kinds of basis sets (atom-centered and delocalized plane waves) on the results obtained with the same density functional method was studied. In contrast to the LCAO approach, which allows the explicit treatment of all electrons, inner electrons are replaced by effective potentials in plane-wave methods. The projector-augmented wave (PAW) method38,39 was used for the core electron representation. Therefore, this method is denoted as PWGGA-PAW in the following. The transition-state search for the migration processes was conducted with the climbing-image nudged elastic band (cNEB)40 method as implemented in VASP. Vibrational analysis calculations were performed to verify the true local minima and saddle point character of the optimized geometries. No imaginary frequency arose for the local minima structures, whereas imaginary frequency is observed for the transition-state structures.

3. RESULTS AND DISCUSSION 3.1. Bulk Properties of Stoichiometric LiBO2. The optimized lattice parameters, bond distances, cohesive energy Ecoh per LiBO2 formula unit, and band gap Eg, as obtained with PW1PW and PWGGA methods (PW1PW-LCAO and PWGGA-LCAO) using CRYSTAL and with PWGGA-PAW using VASP, are given in Table 1 together with the corresponding experimental values. With all the methods, the structural optimization was performed using a conventional unit cell of LiBO2. Among all the considered methods, PW1PW-LCAO gives the best agreement for lattice parameters with experimental values (Table 1), namely, the deviation is less than (0.03 Å for a, b, and c and 0.40° for the β. The results obtained with PWGGA-LCAO and PWGGA-PAW are relatively similar. There is reasonable agreement for a and b, whereas the c parameter is overestimated by 0.15 Å compared with experiment. For both methods, the deviation of β from the experimental value is less than 1°. The calculated bond lengths with all the methods are compared to measured values (Table 1). The deviation of the calculated bond distances is smaller than 1.5% in all cases. The PW1PW-LCAO approach gives the best agreement with experimental values for the boron
oxygen and lithium
oxygen bond distances where the deviation is less than 0.01 Å. Other methods give nearly similar values of bond distances. The calculated values of cohesive energy Ecoh per LiBO2 formula unit are compared with the experimental41 heat of atomization of crystalline LiBO2 (Table 1). Ecoh is calculated by a normalization of the obtained binding energies, which are differences of the total energies of the periodic system and the free atoms in their ground states with converged basis sets. For 12344

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Table 1. Comparison of Calculated and Experimental Lattice Vectors a, b, c (Å), and β (°); Bond Distances (Å); Cohesive Energy Ecoh per LiBO2 Unit (kJ/mol); and Band Gap Eg (eV) PW1PW-LCAO lattice parameters

bonds

Eg a

PWGGA-PAW

exptla,b

a

5.82

5.91

5.84

5.85

b c

4.37 6.48

4.35 6.60

4.39 6.60

4.35 6.45

β

114.89

114.67

114.38

115.09

B
O1

1.332

1.339

1.333

1.323

B
O2

1.397

1.405

1.403

1.392

B
O3

1.411

1.427

1.418

1.410

Li
O1

1.933

1.955

1.957

1.945

Li
O2

1.940

1.970

1.962

1.960

Li
O3 Li
O4

1.983 2.000

1.998 2.026

1.998 2.018

1.970 2.007

Li
O5 Ecoh

PWGGA-LCAO

2.518

2.606


2230

2.510


2276

8.19


2340

6.29

2.473
2238

5.74

Experimental reference for structural data.16 b Experimental reference for cohesive energy.41

the PWGGA-PAW approach, atomic reference energies were calculated with PAW potentials by using pseudo lattice constants of 13 Å for the Li atom, 15 Å for the B atom, and 8 Å for the O atom with a energy cutoff of E2 = 520 (eV). The VOSKOWN keyword42 was used for the better convergence of the groundstate energy of atoms as it is important, particularly, for GGAbased calculation.43 The experimental value of the heat of atomization of crystalline LiBO2 is
2238 kJ/mol.41 PW1PW-LCAO gives the closest agreement with experiment as it was also observed for Li2O,25 B2O3,27 and Li2B4O7.28 With this method, Ecoh is only 8 kJ/mol smaller than the experimental value. However, it has to be mentioned that zero-point energies and thermal contributions to the enthalpy were neglected in the theoretical calculations. Of the two PWGGA methods, the PWGGA-LCAO gives a slightly better agreement with experimental E coh , with a deviation of
38 kJ/mol. With PWGGA-PAW, the difference is larger, namely,
102 kJ/mol. This is in line with the previous investigation with another system containing Li, B, and O.28 One possible reason for these differences between the LCAO and plane-wave based cohesive energies is that the atomic reference energies obtained with plane waves are too high. In our previous study,31 the electronic properties, namely, the band structure and density of states (DOS), of the crystalline LiBO2 are investigated elaborately with the PW1PW-LCAO approach. Here, we have repeated this investigation with all three methods in order to compare them with defective LiBO2 in the next section. The band structure was calculated along the path that contains the highest number of high-symmetry points of the Brillouin zone (Z f C f Y f Γ f B f D f C).44 The calculated minimum vertical transition (MVT) and minimum transition (MT) energy values are compiled in Table 2. According to our results, LiBO2 is a wide-gap insulator.31 The VB is characterized by a rather small dispersion, and there are small internal gaps all over the VB region. This type of band structure signifies that the electronic states are more or less dictated by the localized B
O bonding units. As a result, the hole effective masses are very large, which is typical of wide-gap insulators.45 With all considered methods, the top of the VB is at point Γ, and the bottom of the CB is at point B. All methods indicate that the

Table 2. Values of Minimum Vertical Transition (MVT) and Minimum Transition (MT) Energies ΔE (eV) for LMB Calculated with Different Methods

ΔE ΔE

PW1PW-LCAO

PWGGA-LCAO

PWGGA-PAW

Γ
Γ( MVT)

Γ
Γ (MVT)

Γ
Γ (MVT)

8.25

6.30

5.81

Γ
B (MT)

Γ
B (MT)

Γ
B (MT)

8.19

6.29

5.74

LiBO2 crystal has an indirect (Γ
B) band gap (Eg). However, the direct Γ
Γ transition energy is only slightly larger. The difference does not exceed 0.08 eV. The values of Eg vary from 5.74 eV (PWGGA-PAW) to 8.19 eV (PW1PW-LCAO) due to different amounts of self-interaction error. There is no experimental value of Eg of LiBO2. In our previous investigations on Li2B4O7, the PW1PW-LCAO method gave the best agreement for Eg with the experimental value.28,31 Therefore, PW1PW-LCAO can be taken as an internal reference for the comparison with other methods. The band gap obtained with the PW1PW-LCAO approach is 8.19 eV. The two PWGGA implementations, PWGGA-LCAO and PWGGA-PAW, give similar values of the transition energies except for the Γ
Γ transition. For PWGGA-PAW, the Γ
Γ transition energy is 5.81 eV, which is considerably smaller than 6.30 eV obtained with PWGGALCAO. The difference is ≈0.5 eV, which is responsible for the difference in the value of the band gap (5.74 eV for PWGGAPAW and 6.29 eV for PWGGA-LCAO). PWGGA-LCAO gives a closer agreement with the PW1PW-LCAO than the plane-wavebased PWGGA-PAW approach. The density of states (DOS) (Figure 2a) were calculated at the PW1PW-LCAO level using the Fourier
Legendre technique46 with a Monkhorst net47 using shrinking factors s = 8. The valence band is composed of O 2p states and B
O bonding states of BO3 triangles. Li has almost no contribution in the VB. The lower valence band region lies between
25 and
30 eV and is composed of O 2s states. The bottom of the CB is dominated 12345

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Table 3. Formation Energy of a Li Defect, Ede(V) (kJ/mol), as a Function of the Defect Concentration c (%)a PW1PW-LCAO

PWGGA-LCAO PWGGA-PAW

c (%)

unrel

rel

unrel

rel

unrel

rel

Li4B4O8

6.25

709

701

682

669

677

663

Li32B32O64

0.78

717

698

694

672

678

667

702

673

supercell

Li108B108O216 0.23 a

Figure 2. Density of states of stoichiometric LiBO2 (a) and Li defective LiBO2 (b), obtained with the PW1PW-LCAO approach. EF denotes the Fermi energy level.

by boron orbitals with a small contribution from Li. All other methods qualitatively show the same behavior, only with a shift of VBs and CBs that reduce the band gap. 3.2. Cation Vacancy in LiBO2. A systematic investigation was performed for the cation vacancy formation energy Ede(V), the effect of relaxation, and the electronic properties of the defective LiBO2. Supercells (Li4B4O8, Li32B32O64, and Li108B108O216) were used for defect calculations. The lowest vacancy concentration that we studied here is, therefore, 0.23%. One neutral Li atom was removed from the cell to create the defective system. This leads to an open-shell electronic structure with one unpaired electron per cell. The calculations were, therefore, performed using the spin-polarized method. 3.2.1. Energetics and Structural Relaxation. A full optimization of atomic fractional coordinates was performed taking the optimized lattice parameters from the bulk optimization. The optimized fractional coordinates of the nondefective supercell were taken as the starting structure for the defective system. This artificial “frozen” geometry will be denoted as an unrelaxed structure in the following. In the next step, a full optimization of all remaining atoms of the defective cell was performed without symmetry constraints. The final structure will be referenced as relaxed. The energy lowering due to structural optimization will be denoted as the relaxation energy ER. For both geometries, the formation energy of a Li vacancy Ede(V) is calculated according to the following equation Ede ðVÞ ¼ ESCM ðVÞ þ EðLiÞ
ESCM Here, ESCM(V) and ESCM denote the total energy of the supercell model with and without a vacancy, respectively, and E(Li) is the energy of the free Li atom. As described above, extended basis sets and cutoff energies were used for the atomic reference calculations. In Table 3, the calculated Ede(V) are presented for the unrelaxed and relaxed supercells. To our knowledge, there is no previous experimental or theoretical value of the Li vacancy formation energy of LiBO2. Therefore, the calculated Ede(V) values obtained with different methods are compared with each other in the following. Because PW1PW-LCAO gives the best reproduction of the experimental bulk and defect properties of Li2O25 and Li2B4O7,28,32 this

Abbreviations: unrel, unrelaxed; rel, relaxed.

method is taken as an internal reference. Ede(V) for the fully relaxed system obtained with PW1PW-LCAO is 698 kJ/mol. As for the Li vacancy defect in Li2O25 and Li2B4O7,32 pure DFT approaches give a smaller value of Ede(V) (672 and 673 kJ/mol with PWGGA-LCAO and PWGGA-PAW, respectively) compared with PW1PW-LCAO. The relaxation energies ER, 19
29 kJ/mol, are in the order of 3
5% of the defect formation energies. The absolute values of ER are smaller than those obtained for Li2O25 and Li2B4O7,32 indicating less relaxation effect on the defect formation in LiBO2. Here, it should be noted that employment of the largest supercell, Li108B108O216, with the LCAO-based PW1PW and PWGGA methods was not possible due to the huge CPU time requirement. However, the calculated Ede(V) has already converged with the medium-sized supercell, Li32B32O64, within 3
4 kJ/mol with all the methods. The effect of relaxation is further investigated by measuring the changes of distances of the nearest oxygen atoms, boron atoms, and lithium atoms with respect to the defect position during geometry optimization. In nondefective LiBO2, the Li atom is surrounded by four oxygen atoms in a distorted tetrahedral arrangement.16 The four lithium
oxygen distances range from 1.93 to 2.00 Å,16 thereupon follows a fifth lithium
oxygen distance of 2.52 Å, forming an oxygen five-vertex polyhedron.16 In the following, the five nearest oxygen atoms are considered to show the effect of relaxation (Figure 1b). Three nearest boron atoms (B1, B2, and B3) and two nearest lithium atoms (Li1 and Li2) from the vacancy are also considered. In Table 4, the calculated distances of O, B, and Li atoms from the vacancy before and after relaxation are shown. Here, r1
r10 denote the distances of O, B, and Li atoms from the vacancy. The numbering follows that in Figure 1b. With all the methods, an increase of the nearest-neighbor oxygen-defect position distance is obtained. This is due to the fact that the electrostatic attraction by the Liþ ion is missing. The fifth oxygen atom shows only a small relaxation, þ0.2% (PW1PW-LCAO), þ0.6% (PWGGA-LCAO), and þ0.4% (PWGGA-PAW), indicating that relaxation is mainly restricted to the nearest neighbors of the vacancy. All the boron atoms move toward the vacancy with a very small amount. Also, the two nearest lithium atoms show an inward relaxation. This behavior can be explained by the reduced electrostatic repulsion of the positively charged boron and lithium ions after removal of a Li. The movement of the nearest Li neighbors around the vacancy in LiBO2 is in line with the corresponding geometry changes in Li2
xO25 and Li2
xB4O7,32 where the nearest Li atoms show strong inward relaxation. 3.2.2. Electronic Properties. The removal of a neutral Li atom creates a hole in the valence band. One of the surrounding oxygen atoms, which was formally O2
in stoichiometric LiBO2, 12346

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Table 4. Distance r (Å) of the Nearest Oxygen, Boron, and Lithium Atoms from the Li Vacancy (V) and Changes Δr (%) of the Distancesa PW1PW-LCAO

a

PWGGA-LCAO

PWGGA-PAW

distance

V
atom

unrel

rel

Δr

unrel

rel

Δr

unrel

rel

Δr

r1

V
O1

1.933

1.947

þ0.72

1.955

2.008

þ2.71

1.961

2.002

þ2.09

r2

V
O2

1.939

1.963

þ1.24

1.970

2.011

þ2.08

1.963

2.006

þ2.20

r3 r4

V
O3 V
O4

1.983 1.999

2.019 2.023

þ1.82 þ1.20

1.998 2.026

2.040 2.066

þ2.10 þ1.99

1.997 2.019

2.030 2.050

þ1.65 þ1.55

r5

V
O5

2.518

2.522

þ0.15

2.454

2.469

þ0.63

2.510

2.521

þ0.42

r6

V
B1

2.636

2.617


0.73

2.606

2.574


0.47

2.645

2.637


0.30

r7

V
B2

2.751

2.739


0.44

2.750

2.702


1.75

2.986

2.934


1.74

r8

V
B3

2.860

2.851


0.31

2.876

2.827


1.70

2.878

2.857


0.73

r9

V
Li1

2.589

2.503


3.32

2.611

2.508


3.95

2.623

2.504


4.54

r10

V
Li2

2.773

2.745


1.01

2.779

2.709


2.52

2.798

2.668


4.65

Abbreviations: unrel, unrelaxed; rel, relaxed.

Figure 3. (a) Local structure showing various migration pathways. The blue, red, and green spheres represent Li, O, and B, respectively. (b) Local structures for the Liþ ion migration along the xy plane, where, in the starting structure, two oxygen atoms are bridging between the migrating Li (blue sphere) and the Li vacancy (yellow sphere, marked with V). In the transition state, the migrating Li passes through two bridging oxygen atoms. (c) Local structures for the Liþ ion migration along the c direction, where, in the starting structure, one oxygen atom is bridging between the migrating Li (blue sphere) and the Li vacancy (yellow sphere, marked with V). In the transition state, the migrating Li passes through a triangle formed by three oxygen atoms.

becomes O
. One unpaired electron is localized on the 2p orbital of one of those oxygen atoms. The study of electronic properties is performed by calculating the density of states (DOS) of the defective supercells. The DOS for a defective Li32B32O64 supercell obtained with PW1PWLCAO is shown in Figure 2b. Pure DFT approaches show qualitatively the same behavior. The main difference to PW1PW-LCAO is that the energetic difference between occupied and unoccupied bands is smaller with PWGGA-LCAO and PWGGA-PAW methods. The Liþ vacancy introduces an extra unoccupied level 1.0 eV above the Fermi level EF, which is marked in Figure 2b. With PWGGA-LCAO and PWGGA-PAW, this energy difference is 0.54 and 0.40 eV, respectively. This band is mainly composed of oxygen p orbitals from atoms surrounding the vacancy site. In the analysis of the electronic structure obtained with PW1PW-LCAO, it is found that the p orbitals of one of the four nearest oxygen atoms have much larger contributions than those of the other atoms. This corresponds to the simplified picture of a change from O2
to O
for a single ion. A similar picture is obtained for the LCAO-based PWGGA

approach. With plane-wave-based PWGGA-PAW, the contributions to the defect band are more evenly distributed; therefore, the hole is less localized. 3.3. Migration of Liþ Ion. In our previous studies for the Liþ ion migration in Li2O25 and Li2B4O7,32 it was observed that both the hybrid and the pure DFT approaches give similar agreement with respect to the experimental values. Because the NEB method has not yet been implemented in CRYSTAL, we only employ the VASP PWGGA-PAW approach for calculating the Li ion migration in LiBO2. There are various possibilities of Liþ migration in LiBO2 as presented in Figure 3a. Liþ can migrate along the xy plane, such as (a) migration of Li A to Li B1 (distance = 2.623 Å), (b) migration of Li A to Li B2 (distance = 2.798 Å), (c) migration of Li A to Li B3 (distance = 2.798 Å), and (d) migration of Li A to Li B4 (distance = 3.120 Å). Another possibility of Liþ migration is in the c direction indicated by positions A and C in Figure 3a. In this case, the distance between two lithium atoms is 3.314 Å. 12347

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Table 5. Comparison of Calculated Activation Energy EA (eV) Values with Available Experimental Results migration type

calcd

exptl21

A
B1

0.43

0.71

A
B2 (B3) A
B4

0.43 0.54

0.21
0.23 0.30

A
C

0.55

0.71
0.80

The structural analysis of the migration pathways along the xy plane and the c direction shows that there are clear differences among the neighboring sites of migrating lithium and vacancy positions. Compared to the migration along the c direction, the distance between the migrating Li and Li vacancy is shorter for the migrations along the xy plane. In the latter case, two oxygen atoms are bridging between the migrating Li and Li vacancy. For the migration along the c direction, one oxygen atom is bridging between the migrating Li and Li vacancy. For simplicity, here we present the local structures of only one migration pathway along the xy plane (migration of Li A to Li B1) in Figure 3b and compare these with the local structures of migration along the c direction (migration of Li A to Li C) in Figure 3c. As discussed in the previous section, the lithium atom (or lithium vacancy as marked by “V” in Figure 3b,c) is surrounded by five oxygen atoms as nearest neighbors: O1 (O10 ), O2 (O20 ), O3 (O30 ), O4 (O40 ), and O5 (O50 ). In the case of migration along the xy plane, the migrating Li (blue sphere in Figure 3b) is in the middle of two bridging oxygen atoms in the transition-state structure. In the final structure, the distance between the migrating Li and the O atoms has increased slightly (Δd = þ0.10 to þ0.20 Å) due to structural relaxation. For the Liþ ion migration along the c direction, the migrating Li (blue sphere in Figure 3c) passes through a triangle formed by three oxygen atoms. Here, three Li
O distances are 1.83, 1.93, and 2.18 Å. Because of strong relaxation, the Li
O distance has increased (Δd = þ0.34 Å). The investigation of all the structures in the migration pathways reveals that an unpaired electron is localized on the p orbitals of the nearest oxygen atoms. As stated before, the contributions to the defect band are more evenly distributed with the plane-wave-based PWGGA-PAW method. Therefore, the hole is less localized. The same situation was observed in our previous studies of Liþ migration in Li2O,25,29 Li2B4O7,32 and Li2O/B2O333 nanocomposite materials. As in the case of Li2O,25 the migration path is symmetric as the initial and final positions in the migration pathway are energetically equivalent. In Table 5, the calculated activation energies (EA) are compared with the experimental values.21 Our calculated EA in the xy plane ranges between 0.43 and 0.54 eV, whereas EA along the c direction is 0.55 eV. This shows that the migration of the Liþ ion along the xy plane would be easier than that in the c direction. The calculated EA (0.43
0.55 eV) is in the range of experimental EA value for R-LiBO2 (0.21
0.23 eV, 0.30 eV and 0.71
0.80 eV).21

4. SUMMARY AND CONCLUSIONS The structural, energetic, and electronic properties of stoichiometric and defective LMB are investigated with the HF/ DFT hybrid method PW1PW-LCAO and with two density functional methods PWGGA-LCAO and PWGGA-PAW. For the lattice parameters, PW1PW-LCAO gives the best agreement

among all considered methods with a deviation of 0.4%. Pure DFT methods give similar deviations from the experimental data, with an error of less than 2.0%. All methods give reasonable agreement with experiment for the calculated atomization energy. LiBO2 is a wide-gap insulator with an indirect (Γ
B) band gap (Eg). The calculated Eg values vary from 5.74 eV (PWGGAPAW) to 8.19 eV (PW1PW-LCAO). The formation energy of a Li vacancy, the effect of structural relaxation around the Li vacancy, and the effect of the vacancy defect on the electronic properties of the LMB crystal are studied. With all the methods, defect formation energies around 700 kJ/mol are obtained. Local relaxation has a moderate effect on the formation energy. Significant structural changes are only observed for the first shell of oxygen atoms and the closest boron and lithium atoms surrounding the Li vacancy site. The Li defect introduces an unoccupied level mainly composed of oxygen 2p orbitals slightly above the top of the valence band. Four possible Liþ ion migration pathways are studied in the LMB crystal using the cNEB approach with the PWGGA-PAW method. The calculated activation energy for the migration in the xy plane is 0.43
0.54 eV and that along the c direction is 0.55 eV, indicating a slight preference for Li diffusion in the plane. These values are in reasonable agreement with experimentally measured values from the literature.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT M.M.I is grateful to Deutschen Forschungsgemeinschaft (DFG) for the postdoctorate funding of DFG-Forschergruppe 1277 molife “Mobilit€at von Li-Ionen in Festk€orpern” project. ’ REFERENCES (1) Huang, C.; Wang, S.; Ye, N. J. Alloys Compd. 2010, 502, 211. (2) Ingamells, C. O. Anal. Chim. Acta 1970, 52, 323. (3) He, K.; He, D.; Lei, L.; Zou, Y.; Qin, J.; Wang, S. Solid State Commun. 2010, 150, 2106. (4) Motabar, P.; Inn, G. W.; Davis, K.; LaRosa, J. J. Radioanal. Nucl. Chem. 2009, 282, 335. (5) Smith, R. W.; Keszler, D. A. J. Solid State Chem. 1991, 93, 430. (6) Moshopoulou, E. G. J. Am. Ceram. Soc. 1999, 82, 3317. (7) Kageyama, H.; Onizuka, K.; Yamauchi, T.; Ueda, Y. J. Cryst. Growth 1999, 206, 65. (8) Santos, M. C.; Nogueira, A. R. A.; Nobrega, J. A. J. Braz. Chem. Soc. 2005, 16, 372. (9) Becker, P. Adv. Mater. 1998, 10, 979. (10) Mcmillen, C. D.; Giesber, H. G.; Kolis, J. W. J. Cryst. Growth 2008, 310, 299. (11) Gardner, D. Renewable Energy Focus 2009, 9, 34. (12) Conte, M.; Iacobazzi, A.; Ronchetti, M.; Vellone, R. J. Power Sources 2001, 100, 171. (13) Goudon, J. P.; Bernard, F.; Renouard, J.; Yvart, P. Int. J. Hydrogen Energy 2010, 35, 11071. (14) Liang, J.; Chen, X.; Min, J.; Chai, Z.; Zhao, S.; Cheng, X.; Zhang, Y.; Rao, G. Phys. Rev. B 1995, 51, 756. (15) Lei, L.; He, D.; He, K.; Qin, J.; Wang, S. J. Solid State Chem. 2009, 182, 3041. (16) Kirfel, A.; Will, G.; Stewart, R. F. Acta Crystallogr., Sect. B 1983, 39, 175. (17) Marezio, M.; Remeika, J. P. J. Phys. Chem. Solids 1965, 26, 2083. 12348

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