Lithium Diffusion Mechanisms in Î²-LiMO2 (M = Al, Ga): A

Article Cite This: J. Phys. Chem. C 2017, 121, 27788−27796

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Lithium Diffusion Mechanisms in β‑LiMO2 (M = Al, Ga): A Combined Experimental and Theoretical Study Mazharul M. Islam,*,†,‡,§ Johanna Uhlendorf,∥ Elena Witt,‡,§ Harald Schmidt,∥,⊥,‡ Paul Heitjans,‡,§ and Thomas Bredow†,‡ †

Mulliken Center for Theoretical Chemistry, Institut für Physikalische und Theoretische Chemie, Universität Bonn, Beringstraße 4, 53115 Bonn, Germany ‡ ZFM − Zentrum für Festkörperchemie und Neue Materialien and §Institut für Physikalische Chemie und Elektrochemie, Leibniz Universität Hannover, Callinstrasse 3-3a, 30167 Hannover, Germany ∥ Institut für Metallurgie, AG Mikrokinetik and ⊥Clausthaler Zentrum für Materialtechnik, Technische Universität Clausthal, 38678 Clausthal-Zellerfeld, Germany ABSTRACT: Lithium diffusion mechanisms in β-LiMO2 (M = Al, Ga) were studied in a combined experimental and theoretical approach based on Li tracer diffusion experiments and climbing-image nudged-elastic-band (cNEB) calculations at density functional theory (DFT) level, respectively. Secondary ion mass spectrometry (SIMS) investigations were carried out for β-LiAlO2 and β-LiGaO2 polycrystalline films in the temperature range between 473 and 773 K. A thin layer of ion-beam sputtered isotope-enriched 6LiAlO2 or 6 LiGaO2 was used as a tracer source. The diffusivities of β-LiGaO2 polycrystalline films are in good agreement with those measured on single crystals of the same type. The diffusivities of β-LiAlO2 are higher than in βLiGaO2 by almost 2 orders of magnitude. This can be traced back to a lower activation energy for diffusion in β-LiAlO2. Our computational study shows that the formation energy of a Li vacancy (VLi) is much higher than that of the Li Frenkel pair (VLi + Lii) showing that Li vacancies are not abundant in both systems. Irrespective of the defect types, the defect formation energy values are smaller in β-LiAlO2 than in β-LiGaO2, indicating that Li ion migration could be facile for the former case. In both systems, the most likely Li migration pathways involve Li diffusion from a regular LiO4 tetrahedral location to the first and/or second nearest tetrahedral sites by octahedral interstitial sites. On the basis of calculated activation energies it is concluded that Li diffusion is faster in LiAlO2 than in LiGaO2. Our calculated data are in good accord with the experiments. β-LiAlO2 and β-LiGaO2 are isostructural. The unit cell dimensions for β-LiAlO2 are slightly smaller than β-LiGaO2 (vide supra) showing a comparatively compact structure for the former one. Thus, it might be expected that ion diffusion in βLiAlO2 is less favorable than in β-LiGaO2 due to increased repulsion in the transition states. However, the ionic radius of Al (67.5 pm) is smaller than that of Ga (76 pm) which may compensate the effect of the more compact structure. Therefore, it is interesting to compare the Li diffusion mechanisms in these two materials. In the present study, a theoretical investigation of various possible migration pathways and activation barriers in β-LiAlO2 and β-LiGaO2 is performed using a first-principles density functional theory (DFT) method and periodic supercell models. In addition, Li tracer diffusion in polycrystalline films was investigated in the temperature range between 473 and 773 K by secondary ion mass spectrometry. The activation energies obtained theoretically and experimentally are compared.

1. INTRODUCTION In recent years, β-LiMO2 (M = Al, Ga) compounds have attracted considerable attention due to their numerous practical applications. LiAlO2 is applied as a coating material in Li electrodes,1,2 as an additive in composite Li electrolytes,1,3 as a substrate material for epitaxial growth of III−V semiconductors like GaN,4 and as a candidate material for tritium breeder or fusion reactors.5,6 LiGaO2 is used as a lattice-matched substrate material for optoelectronic semiconductors, e.g., for heteroepitaxy of GaN,7−9 ZnO,8,10 and InN.11 Both types of materials may occur in a distorted wurtzite-type orthorhombic structure with space group Pna21 at ambient pressure.12,13 Due to the polarity, crystalline LiGaO2 shows piezoelectric properties.14 β-LiMO2 (M = Al, Ga) have orthorhombic unit cells with dimensions a = 5.28 Å, b = 6.3 Å, and c = 4.9 Å for β-LiAlO215 and a = 5.402 Å, b = 6.372 Å, and c = 5.007 Å for βLiGaO2.12,13 A unit cell contains four formula units (Figure 1a). The Li and Al/Ga atoms are located at the center of oxygen tetrahedrons forming an alternate stacking of two-dimensional arrays at the centers of LiO4 and (Al/Ga)O4 tetrahedrons as shown in Figure 1b,c. © 2017 American Chemical Society

Received: June 30, 2017 Revised: November 22, 2017 Published: November 23, 2017 27788

DOI: 10.1021/acs.jpcc.7b06460 J. Phys. Chem. C 2017, 121, 27788−27796

Article

The Journal of Physical Chemistry C

Figure 1. (a) Unit cell of β-LiAlO2. (b) Local structures showing the LiO4 and AlO4 tetrahedron shared by O corners. (c) Formation of 3D channel by corners sharing of LiO4−AlO4 pairs.

Figure 2. Atomic fraction of 6Li as a function of depth for as-deposited samples and for (a) a β-LiGaO2 polycrystalline film annealed at 573 K for 100 and 420 min and (b) a β-LiAlO2 polycrystalline film annealed at 623 K for 5 min, respectively. Also shown are fitting curves according to eq 2 in red. The deviation for the as-deposited samples is discussed in the text.

pressure of 5 × 10−3 mbar. The base pressure was better than 5 × 10−7 mbar. LiAlO2 sputter targets were prepared by solidstate syntheses. In an agate mortar coarse Al2O3 (Alfa Aesar) was pestled to a fine powder and mixed with Li2CO3. For the production of the tracer layer 6Li2CO3 (96% 6Li, Eurisotop) was used. In contrast, for the synthesis of the polycrystalline layer where diffusion is studied in 7Li2CO3 (99.9% 7Li, Alfa Aesar) was used. After subsequent ball milling, the powder mixture in a SPEX 8000 M shaker mill, pellets of 2 cm in diameter were pressed and heated to 973 K with a rate of 2 K/ min. The reaction step was followed by a sintering process at 1173 K for 12 h, which yielded polycrystalline dense targets. LiGaO2 sputter targets were prepared accordingly with Ga2O3 (Alfa Aesar). Due to the fact that the tracer layer and diffusion samples have approximately the same chemical composition pure isotope interdiffusion is expected to be measured during the diffusion experiments. For the diffusion experiments the prepared samples were annealed in an argon atmosphere at temperatures up to 773 K using a commercial rapid annealing setup (AO 500, MBE, Germany).

2. METHODS 2.1. Tracer Diffusion Experiments. The polycrystalline films for diffusion studies were produced in the following way: first, an amorphous layer of isotope-enriched 7LiMO2 (M = Ga or Al) of about 1000 nm thickness was deposited on a sapphire substrate by ion-beam sputtering. A 7Li-enriched target is preferred over a natLi target to minimize the natural isotope background of the deposited films. Afterward, the amorphous films were annealed at 923 K for 2 h in Ar to realize complete crystallization. Experiments with grazing incidence X-ray diffractometry on a Bruker D5000 diffractometer done on the crystallized films revealed the presence of polycrystalline βLiGaO2 or β-LiAlO2 phase, respectively. For both types of materials, also about 10−15% unidentified impurity phase was found. The grain size was assessed to about 40 nm by the Scherrer method. Tracer deposition was carried out by depositing a 20−30 nm thin layer of isotope-enriched 6LiGaO2 or 6LiAlO2 on top of the samples under investigation by ion beam sputtering. Ion-beam sputtering was carried out using a commercial setup (IBC 681, Gatan) equipped with two penning ion sources. The deposition was done at 5 keV and at 180 μA in argon at an operation 27789


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The Journal of Physical Chemistry C Secondary ion mass spectrometry (SIMS) investigations were carried out using a Cameca ims 3f/4f machine and an O− ion beam (15 keV, 30 nA). In depth profiling mode, the secondary ion intensities of 6Li+ and 7Li+ ions were recorded as a function of sputter time. Because the two Li isotopes are chemically identical, for diffusion analysis the intensity of the signals is converted into 6Li atomic fractions c(x,t) according to c(x ,t ) =

I(6 Li) I(6 Li) + I(7Li)

(1)

Depth calibration was obtained by measuring the crater depth with a mechanical profilometer (Tencor, Alphastep). 2.2. Computational Methods. Bulk and defect properties of β-LiMO2 (M = Al, Ga) were obtained with periodic calculations at DFT level. The Perdew−Burke−Ernzerhof (PBE) functional,16,17 based on the generalized gradient approximation (GGA), was used as implemented in the plane-wave program VASP.18−20 The projector-augmented wave (PAW) method was used for the core electron representation.21,22 We used a cutoff energy value of Ecut = 400 eV, obtained from convergence tests for β-LiMO2 in the present study. The integration in reciprocal space was performed with a 4 × 4 × 4 Monkhorst−Pack grid.23 The energy convergence was achieved at 10−6 eV per cell with these values. The defective structures were simulated using Li4M4O8, Li32M32O64, and Li108M108O216 supercells. The transition-state search for the migration processes was conducted with the climbing-image nudged-elastic-band (cNEB) method24 as implemented in VASP.

Figure 3. Diffusivities of Li in a β-LiGaO2 polycrystalline film and in a single crystal (from ref 26) as a function of reciprocal temperature as obtained by tracer diffusion measurements with SIMS. The straight line is a guide to the eye.

seen, the diffusivities of both types of materials are in good agreement. That means than for the polycrystalline films also volume diffusion was measured. The presence of an impurity phase as detected by XRD is not expected to influence the results because of the following reasons: due to the low amount of 10−15 vol %, a percolation path is not formed and consequently no long-range diffusion is expected. The percolation threshold in the case of a random distribution of the two phases would typically show up at a volume fraction of about 30% (see, e.g., ref 27). If the two phases were not randomly distributed, but the impurity phase would show instead some clustering, the distances between the individual clusters (with the total volume fraction of at most 15%), assumed to enable local ion conduction, would be larger than in the random case. As a consequence, percolating pathways, leading to long-range diffusion, would be expected even less (i.e., be the more suppressed). The grain size of the main phase was assessed to about 40 nm by XRD. The grain size of the impurity phase is roughly the same. The diffusion lengths of LiGaO2 range from 50 to 450 nm and those of LiAlO2 from 250 to 500 nm, respectively. For the first case the diffusivities of the single crystal and the polycrystalline film coincide, indicating that diffusion in β− LiGaO2 is measured and not in the impurity phase. The diffusion lengths are the same or higher than the grain size. For the second case the diffusion lengths are higher than the grain size by a factor of 5−10. This means during our experiment we probe always several grains. In Figure 4, the diffusivities of both types of polycrystalline films are compared. The diffusivities of β-LiAlO2 are higher than that of β-LiGaO2 by almost 2 orders of magnitude. The diffusivities of each compound obey the Arrhenius law

3. RESULTS AND DISCUSSION 3.1. Experimental Results. Figure 2 shows characteristic examples of isotope depth profiles as obtained by SIMS. The 6 Li atomic fraction as a function of sputter depth is given for selected as-deposited and diffusion annealed samples for both types of polycrystalline films. With increasing annealing time the 6Li tracer penetrates into the sample typically to depths of some hundred nanometers. Experimentally determined depth profiles broadened after annealing can be described by the following solution of Fick’s second law for tracer diffusion across an interface25 c(x ,t ) = c∞ +

⎛ h − x ⎞⎤ (c0 − c∞) ⎡ ⎛ h + x ⎞ ⎟ + erf⎜ ⎟ ⎢erf⎜⎝ ⎠ ⎝ R ⎠⎥⎦ ⎣ 2 R (2)

where c∞ is the relative isotope fraction of 6Li in the 7LiGaO2 or 7LiAlO2 polycrystalline film and c0 that in the 6LiGaO2 or 6 LiAlO2 tracer layer. The original thickness of the as-deposited tracer layer is h = 10−30 nm, depending on the single experiment. The quantity R describes the broadening of the tracer profile. The self-diffusivity D is determined from the difference in R of the diffusion profile and of a starting profile, R0, according to D = (R2 − R02)/4t, where t is the annealing time. The quantity R0 describing the starting profile is due to initial diffusion/interface broadening during sputtering as well as due to ion-beam mixing and interface roughening during SIMS analysis. In Figure 2 also fitting curves are given. In Figure 3 the determined tracer diffusivities of LiGaO 2 polycrystalline films are plotted as a function of reciprocal temperature together with literature data on single crystals, also measured in our group.26 More than one diffusivity at a given temperature corresponds to different annealing times. As can be

D = D0 exp( −Ea /kBT )

(3)

where EA is the activation energy of diffusion and D0 is the preexponential factor. To elucidate the reason for the higher diffusivities of β-LiAlO2, the data were fitted by eq 3 with the same activation energy (Figure 4a) and alternatively with the 27790


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Figure 4. Diffusivities of Li in β-LiAlO2 and β-LiGaO2 polycrystalline films as a function of reciprocal temperature as obtained by tracer diffusion measurements with SIMS. In (a) the experimental values (dots) are fitted with the same activation energy and in (b) with the same pre-exponential factor according to eq 3. The fitting results are indicated.

same pre-exponential factor (Figure 4b). It can be seen that both Arrhenius lines describe the experimental data very well within error limits (although χ2 is slightly better for Figure 4a). The fitting results are indicated in Figure 4. To elucidate which of the two quantities, EA or D0, is responsible for the higher diffusivities in β-LiAlO2, the results are discussed in section 4 below. Note that fits with both EA and D0 as free parameters gives values of 1.23 ± 0.08 eV for β-LiGaO2 and 1.15 ± 0.12 eV for β-LiAlO2, respectively. This is in good agreement with the activation energies of Figure 4b within error limits for both compounds. 3.2. Theoretical Results. 3.2.1. Stoichiometric β-LiMO2. The calculated lattice parameters (a, b, and c), bond lengths, and bond angles are compared with the experimental values12,13,15 in Table 1. The calculated lattice parameters are in reasonable agreement with the experiment with the maximum deviation of b parameter of 0.05 Å (+0.79%) for βLiAlO215 and a parameter of 0.03 Å (+0.55%) for βLiGaO2.12,13 In both cases, all the atoms are tetrahedrally coordinated. The structure consists of alternating two-dimensional arrays of distorted tetrahedrons, with aluminum/gallium atoms at the centers and oxygen atoms at the vertices. These distorted tetrahedrons are composed of two pairs of inequivalent oxygens, O(I) and O(II). According to the experimental data, the bond distances in β-LiAlO2 are slightly smaller than in βLiGaO2, indicating a more compact structure for the former case. Our calculated bond distances and bond angles are in good agreement with the experimental values12,13,15 with the largest deviation of ≈0.05 Å and ≈0.7° (Table 1). Our calculated fractional coordinates for β-LiAlO2 and β-LiGaO2 are also reasonably close to the experimental values12,15 (Table 2). 3.2.2. Electric Field Gradient. To identify magnetically inequivalent atomic sites, the electric field gradient (EFG) parameters for β-LiAlO2 and β-LiGaO2 were calculated. The asymmetry parameter η and the quadrupole coupling constant CQ were calculated according to the following equations as discussed in our previous study for β-Li2TiO3.28 η = (Vyy − Vxx)/Vzz

Table 1. Comparison of Calculated Lattice Parameters, Bond Distances and Bond Angle with Available Experimental Data12,13,15 β-LiAlO2 lattice parameters (Å) a b c bond distances (Å) tetrahedron about Al/Ga (Al/Ga−O)av (O−O)av tetrahedron about Li (Li−O)av (O−O)av tetrahedron about O Al−Al/Ga-Ga Li−Li Al−Li/Ga-Li bond angles (deg) O−Al/Ga-O O−Li−O Al/Ga−O−Li Al/Ga−O−Al/Ga Li−O−Li

CQ = eQ /h × Vzz

β-LiGaO2

calc

exp

calc

exp

5.253 6.254 4.879

5.28 6.3 4.9

5.429 6.366 5.018

5.395 6.367 5.005

1.775 2.916

1.814 2.968

1.852 3.066

1.844 3.016

1.98 3.183

1.951 3.156

1.974 3.200

1.98 3.239

3.026 3.063 3.14

3.051 3.059 3.148

3.12 3.09 3.150

3.11 3.1 3.18

107.3 106.9 106.5 115.1 104.2

107 107.2 106.6 115.8 104.5

107.020 107.220 109.900 116.800 105.1

107.1 107.3 109.7 116.1 104.8

(5)

Here Vxx, Vyy, and Vzz are elements of the traceless EFG tensor V, e is the proton charge, Vzz corresponds to the largest principal component of the EFG, h is the Planck constant, and Q = −4.01 × 10−30 m2 is the electric quadrupole moment of 7 29 Li. The results are compared with the experimental data in Table 3. The calculated η and CQ show that there is only one type of Li and Al/Ga and two distinct types of O in β-LiAlO2 and βLiGaO2. The calculated EFG parameters are in good agreement with the experimental range. The experimental 27Al NMR investigation of β-LiAlO2 shows that the measured CQ and η for

(4) 27791


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The Journal of Physical Chemistry C Table 2. Comparison of Calculated Atomic Coordinates with Available Experimental Data12,15 x β-LiAlO2

y

β-LiGaO2

β-LiAlO2

β-LiAlO2

β-LiAlO2

atoms

calc

calc

exp

calc

calc

exp

calc

calc

exp

Al/Ga O(I) O(II) Li

0.082 0.4063 0.0691 0.4185

0.0822 0.4069 0.0682 0.418

0.0821 0.4066 0.0697 0.4207

0.1261 0.1389 0.1112 0.1262

0.1263 0.138 0.1117 0.1271

0.1263 0.1388 0.1121 0.1267

0 0.9 0.3701 0.4943

0 0.8931 0.3695 0.5

0 0.8927 0.3708 0.4936

The calculated values of vacancy formation energy are given in Table 4. It is observed that for both β-LiAlO2 and β-LiGaO2,

Table 3. Comparison of Calculated and Experimental Asymmetry Parameter (η) and Quadrupole Coupling Constant (CQ) in β-LiAlO2 and β-LiGaO2 η calc β-LiAlO2

β-LiGaO2

a

z

β-LiGaO2

Li Al O(I) O(II) Li Ga O(I) O(II)

0.384 0.61 0.72 0.87 0.44 0.426 0.996 0.946

Table 4. Calculated Li Vacancy Formation Energy Ede(Li) and Frenkel Pair Formation Energy Ede(Fr) Values (eV) in β-LiAlO2 and β-LiGaO2

CQ(MHz) exp 0.55a

0.4b

calc 0.076 1.8 1.7 1.58 0.049 4.338 4.212 4.16

exp

Ede(Li)

1.8a β-LiAlO2 0.04b 3.8b β-LiGaO2

Reference 30. bReference 31.

Al are 1.8 MHz and 0.55, respectively.30 Our calculated values for the corresponding parameters are 1.8 MHz and 0.61, respectively. Similarly, the calculated 7Li NMR quadrupole parameter in β-LiGaO2 is 0.049 MHz, which is in good agreement with the experimental value of 0.042 MHz.31 Our 71 Ga NMR calculations for β-LiGaO2 give CQ and η values of 4.338 MHz and 0.426, which are in good accord with the experimental values of 3.8 MHz and 0.4, respectively.31 The calculated assymmetry parameters for Al and Ga presented in Table 3 are within 10% of the measured value. This is within typical error bars of DFT methods.32 The same holds for the quadrupole coupling constant, where there is even a perfect match for Al and a small deviation for Li. 3.2.3. Calculation of Defect Formation Energy. 3.2.3.1. Li Vacancy. One neutral Li atom was removed from Li4M4O8, Li32M32O64, and Li108M108O216 supercells to create the defective systems. This leads to an open-shell electronic structure with one unpaired electron per cell. The calculations were therefore conducted using the spin-polarized method. 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 starting structure for the defective system. This artificial “frozen” geometry will be denoted as 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 structure optimization will be denoted as relaxation energy ER. The formation energy of a Li vacancy Ede(Li) is calculated according to eq 6 Ede(Li) = ESCM(V) + E(Li) − ESCM

Ede(Fr)

supercell

unrelaxed

relaxed

unrelaxed

relaxed

Li4Al4O8 Li32Al32O64 Li108Al108O216 Li4Ga4O8 Li32Ga32O64 Li108Ga108O216

7.03 7.21 7.22 7.20 7.41 7.39

6.77 6.98 6.97 6.93 7.17 7.15

4.77 5.01 5.04 5.21 5.31 5.36

2.65 2.78 2.8 2.94 3.01 3.03

the vacancy formation energy values are converged for a middle sized supercell (Li32M32O54). The converged Ede(Li) for βLiAlO2 and β-LiGaO2 is 6.98 and 7.17 eV, respectively. The relaxation energy ER is very small, namely, 0.23 eV, which is on the order of 3.5% of the vacancy formation energy. 3.2.3.2. Li Frenkel Pair. The formation energy of the Li Frenkel pair Ede(Fr) is calculated according to eq 7 Ede(Fr) = ESCM(Fr) − ESCM

(7)

Here ESCM(Fr) and ESCM denote the total energy of the supercell model with and without Li Frenkel pair, respectively. The calculated Li Frenkel formation energy for β-LiAlO2 and βLiGaO2 is 2.8 and 3.0 eV, respectively (Table 4). The relaxation energy ER is 2.3 eV, which shows that there is a huge relaxation in the structure due to the Li Frenkel pair. The calculated defect formation energy values reveal two important facts. The Frenkel pair formation energy is much smaller than the Li vacancy formation energy. Therefore, Frenkel pairs are more likely to be observed in β-LiAlO2 and βLiGaO2 compared to Li vacancies. This is in line with our previous investigation in γ-LiAlO2, which shows that lithium vacancies cannot be abundant defects under ambient conditions in LiAlO2.33,34 Another important aspect is that, irrespective of the defect types, defect formation energy values are slightly smaller in β-LiAlO2 than in β-LiGaO2. For both systems, the defect energetics are converged with a middle sized supercell (Li32M32O64); therefore, this supercell was used to study the Li diffusion. 3.2.4. Li Migration Pathways. To the best of our knowledge, the exact diffusion pathways for Li diffusion in βLiAlO2 and β-LiGaO2 are not clarified yet. According to our theoretical study, the Li diffusion processes can occur either by Li vacancy or by Li Frenkel pairs which are discussed below. 3.2.4.1. Li Migration by Li Vacancy. As depicted in Figure 5, Li can migrate from its regular LiO4 tetrahedral location to the nearest (1-NN) and second nearest (2-NN) LiO4 tetrahedral

(6)

Here ESCM(V) and ESCM denote the total energy of the supercell model with and without vacancy, respectively, and E(Li) is the energy of the free Li atom. 27792


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Figure 5. Li migration pathway for a Li jump from a regular tetrahedral site to the first nearest tetrahedral location by Li vacancy, (a) initial structure and (b) transition state, and to the second nearest tetrahedral location by Li vacancy, (c) initial structure and (b) transition state. The red, gray, pink, and yellow spheres represent oxygen, aluminum/gallium, lithium, and lithium vacancies, respectively.

Figure 6. Li migration pathway for a Li jump from a regular tetrahedral site to the first nearest tetrahedral location containing Li Frenkel pair, (a) initial structure and (b) transition state. (c) Migration energy pathway (MEP) for the Li diffusion with the smallest activation energy. The red, gray, pink, and yellow spheres represent oxygen, aluminum/gallium, lithium, and lithium vacancies, respectively.

site by a Li vacancy (the yellow atom in Figure 5a,c. Due to slight differences in lattice parameters between β-LiAlO2 and βLiGaO2, the 1-NN and 2-NN migration distances (dLi−Li) vary; i.e, 3.01 and 4.34 Å for β-LiAlO2 and 3.1 and 4.45 Å for βLiGaO2, respectively. In these cases, the Li passes through the channel by a LiO4 tetrahedral site in the transition state (Figure 5b,d). Our calculated activation energy values for the 1-NN and 2-NN migration pathways are 0.53 and 0.74 eV and 1.82 and 1.88 eV for β-LiAlO2 and β-LiGaO2, respectively (Table 5). Table 5. Calculated Activation Energy EA for Li Diffusion by Li Vacancy in β-LiAlO2 and β-LiGaO2 β-LiAlO2 β-LiGaO2

dLi−Li

EA (eV)

3.01 4.34 3.1 4.45

0.53 1.82 0.74 1.88

Figure 7. Li migration pathway for a Li jump from a regular tetrahedral site to the second nearest tetrahedral location by Li interstitial, (a) initial structure and (b) transition state. The red, gray, pink, and yellow spheres represent oxygen, aluminum/gallium, lithium, and lithium vacancies, respectively.

possibilities. This means that, in principle, all VLi−Lii distances may exist at ambient conditions, and all those pathways have a similar probability. The calculated activation energies for these pathways are 1.0, 0.76, 0.57, 0.60, 0.58, and 0.50 eV, respectively (Table 6). Similarly, for the 2-NN Li diffusion (dLi−Li = 4.34 Å) in β-LiAlO2, six migration pathways have been considered for LiMig−Lii distances of 3.91, 4.41, 4.44, 5.39, 5.44, and 6.65 Å. The corresponding activation energies are 1.84, 1.80, 1.85, 1.86, 1.80, and 1.78 eV, respectively (Table 6). In the case of β-LiGaO2, the 1-NN Li migration (dLi−Li = 3.10 Å) processes have been modeled for seven LiMig−Lii distances such as 4.84, 4.94, 5.12, 5.61, 5.88, 6.04, and 6.22 Å, respectively. The calculated activation energies for these pathways are 0.92, 0.89, 0.88, 0.79, 0.81, 0.73, and 0.70 eV, respectively (Table 6). For the 2-NN Li diffusion (dLi−Li = 4.45 Å), the four investigated migration pathways have LiMig−Lii distances of 4.25, 4.58, 5.40, and 6.74 Å. The corresponding activation energies are 2.07, 1.92, 1.87, and 1.80 eV, respectively (Table 6).

3.2.4.2. Li Migration by Li Interstitials. We have modeled various Li diffusion processes for β-LiAlO2 and β-LiGaO2 containing Li Frenkel pairs, VLi + Lii. Li can jump from its regular LiO4 tetrahedral location to the first (1-NN) and second (2-NN) nearest LiO4 tetrahedral sites through the large channel (as shown in Figures 6 and 7, respectively). To study the effect of electrostatic repulsion between the migrating Li (LiMig) and the interstitial Li (Lii), the position of Lii is varied with respect to the migrating Li. For 1-NN Li migration (dLi−Li = 3.01 Å) in β-LiAlO2, six migration pathways are studied where LiMig−Lii distances are 4.17, 4.22, 4.62, 4.99, 5.92, and 6.28 Å, respectively. The differences in the formation energies of these Frenkel pairs are within 0.3 eV, indicating that the interaction between VLi and Lii is small. These calculations were performed to find out defect clustering effects. If a certain VLi/ Lii configuration would have been particularly stable, the corresponding structure would have been the most likely starting point for the migration study. Because the structures are relatively similar in energy, we had to consider all 27793


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calculated activation energy values for 1-NN migration pathways are 0.5 and 0.7 eV and those for 2-NN pathways are 1.78 and 1.81 eV in β-LiAlO2 and β-LiGaO2 respectively. There is a discrepancy between the experimental and theoretical data (for the shortest migration pathways) as our calculated EA are much smaller than the experimental ones. This can be partially attributed to the different diffusion range studied in both approaches. Although energy barriers for local atomic jump processes are calculated with the cNEB method, the tracer diffusion method is a macroscopic method that probes the long-range transport yielding high activation barriers.35 However, if the macroscopic diffusion occurs due to a large number of individual jumps, the activation energies for microscopic and macroscopic diffusion processes could be same. Moreover, the tracer diffusion coefficient is the combination of the vacancy diffusivities and the differences in the site fraction of defects which may vary with temperature. In our present theoretical investigation, the migration energy and defect formation energy values are calculated at 0 K DFT approaches. Therefore, the temperature effect on the variation of site fraction for defects is beyond the scope of the present work. However, if we consider the sum of absolute values of calculated defect formation energy (irrespective of defect type, Li vacancy or Li Frenkel pair) and migration energy, we observe the same trend as with the tracer diffusion method. Another error source could be associated with the employed theoretical method as we know that pure DFT approaches underestimate the EA compared to HF/DFT hybrid approaches.35,36 To check this aspect, we have redone the calculation of EA for a migration pathway of β-LiAlO2 using the HF/DFT hybrid approach HSE06,37 which is expected to provide a higher accuracy but is at the same time computationally much more demanding than the pure DFT method. The EA obtained with HSE06 for the 1-NN pathway with shortest hopping distance is 0.75 eV, which is higher than that obtained with pure PBE based DFT (0.5 eV). Thus, the difference between theoretical and experimental activation barriers is reduced, but not completely removed with the higher rung method. Due to extremely large computational time requirement, the HSE06 approach could not be employed for all migration pathways. It is expected that all PBE-derived activation energies will be increased in a similar way as shown above. It has to be noted, however, that the difference in activation energy between the two systems, 0.2 eV, is nicely reproduced with PBE. It can therefore be expected that the main effects leading to the different Li diffusivity in β-LiAlO2 and β-LiGaO2 are included in the theoretical models. We will therefore attempt to give an explanation for the different activation barriers based on structural data obtained for the present supercell models. In particular, we analyzed the structural changes for the transition-state (TS) structures. We summarize the data obtained for the migration pathways with smallest activation barriers in Table 7. In all cases, the TS correspond to distorted LiO6 octahedra. For the 1-NN migration pathway in β-LiAlO2, two Li−O distances are smaller than 2.0 Å, two Li− O bonds are in the range 2.1−2.3 Å, whereas the remaining two Li−O distances are very large (3.2−3.3 Å) compared to the normal Li−O bond distance (∼2 Å). For the 2-NN migration pathway, three Li−O distances are below 2 Å and the remaining three are between 2.23 and 2.64 Å. However, for the 1-NN migration pathway in β-LiGaO2, one Li−O distance is 1.8 Å and the remaining five Li−O distances are in the range

Table 6. Calculated Activation Energy EA (eV) for Li Diffusion in β-LiAlO2 and β-LiGaO2 Containing a Frenkel Pair in Dependence of the Distance dLiMig−Lii between Interstitial Li (Lii) and the Migrating Li (LiMig) (Å) β-LiAlO2

dLi−Li (Å)

dLiMig−Lii (Å)

EA (eV)

3.01 (1-NN)

4.17 4.22 4.62 4.99 5.92 6.28 3.91 4.41 4.44 5.39 5.44 6.65 4.84 4.94 5.12 5.61 5.88 6.04 6.22 4.25 4.58 4.82 6.66

1 0.76 0.57 0.6 0.58 0.50 1.84 1.80 1.85 1.86 1.80 1.78 0.92 0.89 0.88 0.79 0.81 0.73 0.70 2.07 1.92 1.87 1.81

4.34 (2-NN)

β-LiGaO2

3.1 (1-NN)

4.45 (2-NN)

These findings reveal that there is a substantial repulsive interaction between the migrating Li and the interstitial Li when they are at a close distance, which increases the activation barrier, whereas at very large LiMig−Lii distances, the repulsion is screened and the activation energies converge to smaller values approaching the situation of a Li vacancy as observed by the similar values of activation energy by Li vacancies. According to our calculations, the activation energy for the 1NN migration pathway is slightly smaller for β-LiAlO2 (0.5 eV) than for β-LiGaO2 (0.7 eV), whereas the 2-NN migration barriers are similar in both cases. Therefore, β-LiAlO2 is a slightly faster ion conductor than β-LiGaO2. Although the activation energy are similar for both vacancy and interstitial mechanisms, due to larger values of vacancy formation energies compared to the interstitial formation energies, the vacancy mechanisms for Li diffusion can be discarded. 3.3. Comparison of Experimental and Theoretical Data. As discussed in the upper section, the experimentally determined diffusivities revealed that diffusion in β-LiAlO2 is higher than in β-LiGaO2. From the experimental results it cannot be clearly decided whether this is due to a lower activation energy or a higher pre-exponential factor (Figure 4). The computational results reveal two important aspects. First, the defect formation energies are smaller in β-LiAlO2 than in βLiGaO2, indicating that ion diffusion should be easier in βLiAlO2. Second, the activation energy for Li diffusion in βLiAlO2 is lower. Therefore, by comparison of computational and experimental results, the main reason for the faster diffusion in β-LiAlO2 is mainly due to a lower activation energy. Quantitatively, the tracer diffusion measurements with SIMS show that the activation energy for Li migration in β-LiAlO2 and β-LiGaO2 is 1.08 and 1.29 eV, respectively, when the same pre-exponential factors are used according to eq 3. The 27794


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We have studied the structure, electric field gradients, Li defect formation energetics, and Li diffusion processes in βLiAlO2 and β-LiGaO2 theoretically with periodic quantumchemical DFT methods. The calculated structural parameters such as lattice constants, bond lengths, and bond angles are in close agreement with the experimental reference data. Our calculated electric field gradient parameters such as η and CQ show that there is only one type of Li and Al/Ga and two distinct types of O in β-LiAlO2 and β-LiGaO2, in agreement with the available experimental data. The calculated defect formation energy values are slightly smaller in β-LiAlO2 than βLiGaO2, indicating that β-LiAlO2 may contain a higher concentration of defects and consequently ion diffusion could be faster. Competing pathways for Li diffusion in β-LiAlO2 and β-LiGaO2 are investigated using the climbing-image NudgedElastic-Band approach. Li can migrate from a regular LiO4 tetrahedral location to the first and second nearest tetrahedral sites by octahedral interstitial sites. The calculated EA for the 1NN pathway is 0.5 and 0.7 eV for β-LiAlO2 and β-LiGaO2 respectively. The absolute values differ from the measured activation energies, but the same trend is obtained. Both results suggest that the faster diffusivities observed for β-LiAlO2 compared to β-LiGaO2 are associated with the lower activation energy of LiAlO2.

Table 7. Calculated Li−O Bond Distances of LiO6 TS for the 1-NN and 2-NN Migration Pathways with Smallest Activation Energies β-LiAlO2 β-LiGaO2

migration type/dLiMig−Lii (Å)

Li−O (Å)

1-NN/6.28 2-NN/6.65 1-NN/6.22 2-NN/6.66

1.8, 1.97, 2.1, 2.3 3.2, 3.3 1.85, 1.95, 2.0, 2.23, 2.51, 2.64 1.8, 2.0, 2.44, 2.52, 2.53, 2.8 1.9, 2.02, 2.05, 2.3, 2.5, 2.64

2.41 and 2.8 Å. For the 2-NN migration pathway, three Li−O distances are in the range from 1.9 to 2.05 Å and the remaining three are between 2.3 and 2.64 Å. This analysis shows that the Li−O distances are similar for 2-NN migration pathways in both systems; therefore, both migration pathways have similar activation energies for Li diffusion. The Li−O bond lengths are much larger in the TS of the 1-NN pathway of β-LiAlO2 than in the corresponding TS of β-LiGaO2 (at least two bond lengths are above 3 Å). In the TS, the LiO6 octahedron is much more distorted for β-LiAlO2 than for β-LiGaO2, as shown in Figure 8. Therefore, we assume that Li migration in the former pathway faces less hindrance from Li−O interaction than the latter case, which makes the activation energy smaller.

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AUTHOR INFORMATION

Corresponding Author

*M. M. Islam. E-mail address: [email protected]. ORCID

Mazharul M. Islam: 0000-0002-5638-8265 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Dr. R. Ü cker and Dr. C. Vinod Chandran for fruitful discussion. Financial support from the Deutsche Forschungsgemeinschaft (DFG) in the framework of the Research Unit 1277 “molife” (Schm 1569/18-2, He 1574/13-2, BR1768/5-2) is gratefully acknowledged. P.H. is grateful for a Niedersachsen Professorship.

Figure 8. LiO6 TS for the 1-NN Li migration pathway with lowest activation energy in (a) LiAlO2 and (b) LiGaO2. The red and pink spheres represent oxygen and lithium atoms, respectively.

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REFERENCES

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4. CONCLUSIONS A comparative study for Li diffusivity in β-LiAlO2 and βLiGaO2 was performed by a combined experimental and theoretical approach. The Li tracer diffusion measurements by SIMS show that the diffusivities of β-LiGaO2 single crystals and polycrystalline films are in good agreement for a wide temperature range (between 473 and 773 K). We have employed polycrystalline films of β-LiAlO2 and β-LiGaO2 to compare the diffusivity. Our study shows that the diffusivities of β-LiAlO2 are higher than in β-LiGaO2 by almost 2 orders of magnitude. To elucidate the reasons for this difference, we have employed the Arrhenius law (eq 3) by considering either same activation energy (EA) or same pre-exponential factor D0. It is observed that the higher diffusivities in β-LiAlO2 compared to β-LiGaO2 are associated with lower activation energy for Li diffusion in the former case. The experimental activation energy for Li diffusion for polycrystalline β-LiAlO2 and β-LiGaO2 is 1.08 and 1.29 eV, respectively. 27795


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Lithium Diffusion Mechanisms in Î²-LiMO2 (M = Al, Ga): A

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