DFT Study of the Role of Al3+ in the Fast Ion-Conductor Li7–3xAl3+xLa3Zr2O12 Garnet

We investigate theoretically the site occupancy of Al3+ in the fast-ion-conducting cubic-garnet Li7–3xAl3+xLa3Zr2O12 (Ia-3d) using density functional theory. By comparing calculated and measured 27Al NMR chemical shifts an analysis shows that Al3+ prefers the tetrahedrally coordinated 24d site and a distorted 4-fold coordinated 96h site. The site energies for Al3+ ions, which are slightly displaced from the exact crystallographic sites (i.e., 24d and 96h), are similar leading to a distribution of slightly different local oxygen coordination environments. Thus, broad 27Al NMR resonances result reflecting the distribution of different isotropic chemical shifts and quadrupole coupling constants. From an energetic point of view, there is evidence that Al3+ could also occupy the 48g site with its almost regular octahedral coordination sphere. Although this has been reported by neutron powder diffraction, the NMR chemical shift calculated for such an Al3+ site has not been observed experimentally.


■ INTRODUCTION
The fast Li-ion conductor with the nominal composition Li 7 La 3 Zr 2 O 12 (LLZO) is receiving much scientific attention since its discovery in 2007. 1 It has a garnet-based structure, and it occurs in at least two structural modifications. 2,3 At room temperature, LLZO is tetragonal (I4 1 /acd) while the cubic modification (Ia-3d) is stable above approximately 150°C. 4 Geiger et al. argued that the better conducting cubic phase can be stabilized at room temperature (RT) through the incorporation of small amounts of Al 3+ . 4 The stabilizing effect of Al 3+ has now been confirmed by a number of subsequent investigations. 5−19 The exact role Al 3+ plays in cubic Al-bearing LLZO is important because LLZO shows a high ionic conductivity of about 10 −4 S/cm at RT. This is approximately 2 orders of magnitude higher than that for the lower symmetry Al-free tetragonal LLZO phase. LLZO also has good chemical and thermal stability, as well as a wide energy potential window making it an excellent candidate for use as an electrolyte in an all-solid-state lithium-ion battery. 1,20 As has been shown recently, ionic conductivity also seems to depend on the amount of Al 3+ incorporated during synthesis. 6,8 Further work, however, is needed to quantify this effect. For this purpose, the chemical and physical properties governing Li + diffusion have to be understood in detail; in particular this includes the important question as to which crystallographic sites the Al 3+ ions preferably occupy in the cubic phase of LLZO.
Considerable experimental research has been undertaken to obtain information about Al 3+ in LLZO, including its local coordination and site partitioning behavior. In this context, 27 Al magic angle spinning (MAS) nuclear magnetic resonance (NMR) spectroscopy is a key method. Several NMR studies have proposed that the resonance observed at a chemical shift ranging from 64 to 68 ppm corresponds to Al 3+ located at the "standard garnet" tetrahedral site. 4−7 There is uncertainty, however, about the interpretation and assignment of the other measured NMR lines that have chemical shifts ranging from approximately 78 to 82 ppm. The interpretations given so far include Al 3+ residing at nontetrahedral Li + sites (4-fold to possibly 6-fold coordinated) and tetrahedrally coordinated sites in the neighborhood of La 3+ or Zr 4+ vacancies in Al-rich LLZO. 4,5 In particular, mechanochemically prepared LLZO samples with a high amount of Al 3+ , but reduced in La 3+ and Zr 4+ content, even indicate two magnetically inequivalent tetrahedral sites. 6 Neutron powder diffraction (NPD) measurements were interpreted as indicating that Al 3+ is located at an octahedrally coordinated site in the garnet framework. 11 Summarizing the various published experimental results, there is no clear understanding as to which sites are occupied by Al 3+ in cubic LLZO. This also concerns the detailed interpretation of 27 Al NMR MAS spectra and diffraction results.
To address the important role of Al 3+ in LLZO and to obtain a better understanding of the various experimental results, we undertook a density functional theory (DFT) investigation of Al-bearing cubic LLZO with the aim to calculate (relative) 27 Al NMR parameters such as chemical shifts and electric quadrupole coupling constants.

■ COMPUTATIONAL METHODS
In order to understand the computational models used in this investigation, we provide a short description of the crystal structure of cubic LLZO garnet ( Figure 1).
"Garnet" is the common name for a large number of natural and synthetic metal oxide and fluoride phases. 21 Conventional oxide garnets have the general formula A 3 B 2 C 3 O 12 and crystallize with cubic symmetry Ia-3d. In the case of LLZO, the O 2− ions, located at general crystallographic positions, 96h, form an oxygen-ion framework with interstices occupied by the A cations (La 3+ ) at an 8-fold coordinated position 24c (point symmetry 222), by B cations (Zr 4+ ) at a 6-fold coordinated position 16a (point symmetry −3), and by C cations (Li + ) at a 4-fold coordinated 24d position (point symmetry −4). In addition to these cation sites, there are other interstices within the oxygen framework that are empty in the conventional garnet structure, 22 such as the 6-fold coordinated 48g positions (point symmetry 2) and a general 96h position, sometimes described as 4-fold coordinated with two additional longer bonds greater than 2.8 Å in length, 23 5-fold 4 coordinated or 6-fold 11 coordinated (point symmetry 1). These interstices can be filled by "extra" cations (such as Li + ), giving rise to compositions with nonstandard garnet stoichiometry. An important property is the partial occupancy of the structural sites (24d, 48g, and 96h) where Li + is located and also delocalization of Li + ions throughout.
Several theoretical DFT studies on LLZO have been published before. 24−29 They concentrate on the basic structural properties and on the Li + diffusion, but none of them considers Al 3+ ions explicitly.
For our first-principle calculations, the ions in LLZO were arranged on the basis of crystal structure descriptions in the literature. 24 Three different structural models were used with all having a body centered (I-type) Bravais lattice with a = b = c = 12.972 Å and α = β = γ = 90°.  25 No significant effect on the results was observed by choosing a different Li + arrangement. The highest symmetry possible was maintained to save computational costs. The first model was used to understand the behavior of Al 3+ at various possible sites occupied by Li + . The calculations were made with 17 different local Al 3+ positions located among the 24d, 96h, and 48g sites. To avoid cation repulsion, all Li + ions close to Al 3+ were removed. The various Al 3+ positions were fixed during relaxation and all Al 3+ ions are equivalent in the unit cell.
All calculations are based on DFT methods using the all-electron full potential linearized augmented plane wave (LAPW) method as implemented in the Wien2k code. 30,31 The Perdew Burke Ernzerhof (PBE) generalized gradient approximation (GGA) was employed. 32 The atomic positions were optimized by minimization of the forces (below 2 mRy/au) acting on the atoms simultaneously with the selfconsistent-field cycle as implemented by Marks. 33 The eigenvalue problem was solved by an "iterative diagonalization" using an efficient preconditioning (inverse of H -λS) and the block-Davidson method. 34 The radii of the atomic spheres (R MT ) for the Li + , La 3+ , Zr 4+ , O 2-, and Al 3+ ions were chosen to be 1.46, 2.27, 1.93, 1.71, and 1.40 au, respectively. The cutoff for the plane wave R MT K max = 6.0 and the maximum Fourier expansion of charge density cutoff G max = 12 (au) −1 were applied. The separation parameter between the valence and core states was chosen to be −6.0 Ry. We used 1 k-points in the irreducible Brillouin zone. The computational considerations were checked by increasing R MT K max and the number of k-points, but no significant changes with respect to the energy, geometry, and the electric field gradient (EFG) around Al 3+ were observed.
The behavior of Al 3+ at different crystallographic sites was analyzed qualitatively by weighing the various Al 3+ positions, x i , with their corresponding energy, E(x i ). The various local energies for Al 3+ at and around the crystallographic positions where Li can also be located, were described by fitting a polynomial, describing a possible diffusion pathway. The ocuppation probabilities for an Al 3+ ion along this path are described using a normalized Boltzmann factor given by where k is Boltzmann's constant, and where x i+1 indicates the states next to x i , x i−1 indicates the states before x i with an interval of 0.01 Å, and T is 298.15 K. The site preference energy, ΔE, is defined as where E global is the global energy minimum and corresponds to Al 3+ located at 24d. It has the lowest total energy for all the calculated arrangements of Al in LLZO.
In NMR spectroscopy, the chemical shift, δ, of a nucleus, i, describes the nuclear shielding effect of an applied (external) static magnetic field and the locally induced magnetic field arising from the surrounding electrons with a certain probability of presence near the corresponding nuclear site i. The magnitude of the resulting effective field, B eff , is given by B eff = B 0 (1 − σ i ), where σ i is a second-rank nuclear shielding tensor, 1 is the unit matrix and B 0 is a uniform external field along the z-axis. The resonance NMR frequency, ν i , is then given by v i = (γ i /2π) B eff , where γ i is the magnetogyric ratio of the nucleus under observation. The isotropic chemical shift, δ iso (henceforth δ), describes the relation between the NMR resonance frequency for the nucleus of interest, ν i , and the corresponding resonance frequency for a reference compound, ν ref , giving δ = 10 6 Because of the orientation in systems with periodic boundary conditions, all anisotropic interactions causing line broadening are projected onto the axis of rotation where they collapse at the magic angle. 35 To compute δ, we used the relaxed geometry and applied the NMR module 36−38 of the Wien2k code. 31 These chemical shift calculations are based on an all-electron linear response method where one obtains the induced current density considering the perturbation of the ground state wave functions due to the external magnetic field. The resulting magnetic shielding is then obtained by integration of the all-electron current according to Biot−Savart's law without further approximations. Because there is agreement in the literature 4−7 that the 27 Al NMR resonance at 66 ± 2 ppm represents tetrahedrally coordinated Al 3+ at 24d in LLZO (δ 24d = 68 ppm), we calculate the difference of the chemical shifts (Δδ) for Al 3+ at other positions x i with a δ xi as follows Besides the chemical shift, the interaction of the quadrupole moment, Q, of the 27 Al nucleus (spin-quantum number I = 5/2) with a sufficiently large electric field gradient (EFG), which is produced by a nonspherical charge distribution around the nuclear site, also affects the central NMR transition. Second order quadrupolar effects, influencing the shape of the NMR central line, cannot be (completely) eliminated by magic angle spinning. From the simulation of spectra recorded under MAS conditions, the quadrupole coupling constant and the corresponding asymmetry parameter can be estimated. Here, e is the positive elementary charge, h denotes Planck's constant, and V xx , V yy and V zz are the elements of the traceless EFG tensor V with |V zz | ≥ |V xx | ≥ |V yy |. V zz = eq denotes its principal component. The shape of the NMR central line depends sensitively on the asymmetry of the electronic charge density close to the nucleus. The parameter η describes the deviation of the EFG from axial symmetry and it can take values between 0 and 1. Second order quadrupole interactions also affect the position of the NMR line in terms of a quadrupolar shift. A larger external magnetic field, however, lowers the effect. Here, we compute the EFG from the all-electron charge distribution without further approximation and C Q is obtained using the nuclear quadrupole moment Q( 27 Al) = 1.616 × 10 −29 m 2 determined from the slope of the linear regression proposed by Body et al. 39 The data visualization were performed using the program VESTA. 40 ■ RESULTS Crystal Chemistry of Al-bearing LLZO and Al 3+ Partitioning Behavior. The most energetically favorable position of Al 3+ in LLZO is at the tetrahedrally coordinated crystallographic special 24d position. Al 3+ can be displaced toward a neighboring vacant general 96h site, which leads to a distortion of its oxygen coordination polyhedron. Calculated Al−O bond lengths for this coordination at 0 K are between 1.76 Å and 1.79 Å and the tetrahedral volume is 2.86 Å 3 (see Table 1).
Our calculations show that Al 3+ at 96h is 4-fold coordinated, and distorted from tetrahedral coordination, with Al−O bond lengths varying from 1. The calculated site preference energies, ΔE, for Al 3+ in LLZO are as follows: 24d > 96h > 48g > 16a ≫ 24c. Al 3+ at the 24d site has the global energy minimum and is 1.38 eV more stable than Al 3+ at 96h and 1.60 eV more stable than Al 3+ at 48g. The latter two sites are energetically similar with an energy difference of just 0.22 eV. The site preference energy for Al 3+ at 16a and 24c is 0.41 and 1.74 eV higher, respectively, when referenced to the global energy minimum. The site preference energies for Al 3+ at various structural positions between 48g and 24d are shown in Figure 3.
Calculated 27 Al NMR Parameters. Calculated 27 Al NMR parameters for Al 3+ at the different structural sites in LLZO are shown in Table 2. The corresponding polyhedra (Figure 4) around Al 3+ , slightly displaced from 24d and 96h (Figure 3),

■ DISCUSSION
In this DFT study, we investigated the crystal-chemical role of Al 3+ in LLZO. Specifically, we address the question of the coordination and the site distribution behavior of Al 3+ . The results of the calculations can be used, together with published experimental results, to achieve a better understanding of the nature of Al 3+ incorporation in LLZO. We find that Al 3+ at the 24d site is energetically the most stable state, followed by Al 3+ at 96h and 48g, whereby the site preference energies for the latter two are similar. Al 3+ at both the 16a and 24c sites is energetically unfavorable. Based on our DFT calculations, Al 3+ should be exclusively located at 24d at 0 K, which would give rise to a single NMR resonance. Thus, the question arises why additional resonances are observed in the 27 Al MAS NMR spectra ( Figure 5). 4−7 Before discussing our results, we briefly outline the NMR results from the various studies with respect to the Al 3+ site occupation in LLZO. In all recent studies the resonance at approximately 66 ± 2 ppm is assigned to Al 3+ at the 24d site. 4−7  a The position corresponds to Al 3+ with the coordination polyhedra in Figure 4 and the site preference energies in Figure 3.  Other NMR lines also indicate 4-fold coordinated Al 3+ ions, 4−7 because their chemical shift values of 78 to 82 ppm are usually indicative of tetrahedral coordination. Geiger et al. assigned the resonance at 81 ppm to Al 3+ residing at the 96h sites. 4 In particular, the spectra presented in ref 6 indicate that several NMR lines contribute to a signal occurring between 75 to 85 ppm. It should be noted that the latter has been reported for LLZO samples with a high amount of Al 3+ and lower than stoichiometric La 3+ and Zr 4+ contents. 6 Such samples were prepared by mechanochemical activation combined with subsequent annealing at moderate temperatures. Duvel et al. observed a stronger NMR signal at 64 ppm and a less intense line at 78 ppm for garnets with Al 3+ contents up to 0.30 per formula unit (pfu). However, at concentrations above 0.60 Al 3+ pfu the resonance at 78 ppm becomes the most intense. At even higher Al 3+ contents a new NMR line at 81 ppm emerges, which notably appears in La-and Zr-deficient LLZO samples.
Geiger et al. made, in addition to 27 Al MAS NMR experiments, also 27 Al MQMAS NMR measurements. 4 The latter spectra show a resonance at δ = 81 ppm, a coupling constant, C Q , of 3.3 MHz and an asymmetry parameter, η, of 0.7. These parameters indicate a strongly distorted local coordination geometry. The second resonance observed at δ = 68 ppm shows a stronger quadrupolar interaction with C Q = 5.0 to 5.2 MHz and a lower η value between 0.0 and 0.1 that suggests axial symmetry at the site. It should be noted that the 27 Al MAS NMR spectrum was recorded at 156.26 MHz, thus providing a sufficiently high resolution to simulate its line shape. Such a simulation yields a resonance with δ = 81 ppm, C Q = 3.3 MHz and η = 0.7, and for the second signal values of δ = 70 ppm, C Q = 5.5 MHz and η = 0.5.
The calculated NMR parameters (Table 2) are in good agreement with an assignment of Al 3+ to 96h for the resonance at 81 ppm. The calculated NMR parameters for the NMR line at 68 ppm that is assigned to Al 3+ at 24d, however, disagree with those determined from the MQMAS NMR experiments. For the latter, it has been proposed that possibly more than one Al 3+ site is reflected by this broad resonance. This possibility was underlined by Hubaud et al. 7 They made 27 Al MQMAS NMR spectroscopic measurements to investigate this resonance at 68 ppm for an Al-doped LLZO garnet synthesized at 850°C. Their line shape analysis yielded two NMR lines with the first having δ = 68.4 ppm and C Q = 4.8 MHz and the second δ = 73.5 ppm and C Q = 6.9 MHz. Both resonances were assigned to tetrahedrally coordinated Al 3+ at 24d in two different garnet phases. The two tetrahedral sites have slightly different distortions and thus different δ and C Q values. Hubaud et al. report that this interpretation is consistent with their highresolution XRPD results, and they proposed that slow Al 3+ diffusion within the lattice is responsible for the disordering over the two sites.
Our DFT calculations permit a somewhat different interpretation of the experimental results. For the first step, we calculated the NMR parameters for Al 3+ at all crystallographic sites, via 24d, 48g, 96h, 16a, and 24c. According to our results, the resonance at 80 ± 2 ppm should be assigned to Al 3+ residing at the 96h sites and the resonance at 66 ± 2 ppm to Al 3+ occupying 24d. These results agree with interpretations of experimental NMR spectra. 4−6 There is good agreement between the calculated C Q value for Al 3+ at 96h, but disagreement between C Q values for Al at 24d. Hubaud et al.
proposed a large C Q value for Al 3+ at 24d, but it cannot be easily explained why the symmetric 24d site should yield a C Q value approximately two times larger than Al 3+ at the more distorted 96h site. 7 To understand this issue better, we calculated the NMR parameters for Al 3+ at all positions at and around 24d and 96h that are occupied by a certain probability. Shifting Al 3+ away from 24d and 96h leads to a distribution of slightly different local oxygen coordination environments. Thereby, broad 27 Al MAS NMR resonances could result, which would reflect a distribution of slightly different δ and C Q values. We note that our calculations for η for Al 3+ at 24d do not agree with experiment, but η was obtained from the experimental spectra by using only one 4 or two 7 Al 3+ resonances to simulate this broad feature at 66 ± 2 ppm.
The NMR signal located at 78 to 82 ppm is also asymmetric in shape. This is possibly due to the slightly different geometries of local Al 3+ coordinations, as given by their calculated probabilities, toward the 48g minimum versus those in the direction of 24d. This situation could produce two overlapping resonances having similar values of δ and C Q (see also the agreement with experimental results and the discussion in ref 6).
Although experiment and calculations are in broad agreement in terms of the crystal chemical role of Al 3+ in LLZO, it is not clear based on the DFT calculations alone, why Al 3+ is not located exclusively at 24d 6 as suggested by the site energies. Al 3+ at 96h and 48g must also be considered, which, however, would lead to an additional NMR line which is usually not observed experimentally. (For the sake of completeness, let us note that an additional NMR line has been observed at 93 ppm for Al-doped Li 6.5 La 2.5 Ba 0.5 ZrTaO 12 [42] and for some of the Al-doped LLZO samples prepared via mechanosynthesis [6]; the prominent NMR line at 65 ppm, however, is absent in these cases.) Here, it must be stated that the exact thermodynamic state of LLZO, as obtained in the various sintering experiments, is not known. It is possible that metastable structural states are obtained that can depend on a number of experimental factors (dopant concentration, sintering temperature and time, heating rate, grain sizes, starting materials, etc.). Thus, Al 3+ could potentially be incorporated metastably at 96h and 48g as well in cubic LLZO, because both sites have similar site energies. The different site energies for Al 3+ at 24d and 96h could provide an explanation for differences in published NMR spectra of LLZO. The relationship between Al 3+ concentration and variations in the intensities of the different resonances (and thus Al 3+ site occupancies), as observed by Duvel et al., can be interpreted crystal chemically. When an Al 3+ ion is located at 24d, because of its large effective charge radius, it could create a larger inaccessible region around 24d compared to the situation for Al 3+ at 96h. This region cannot be occupied by other ions. This leads to a reduction of entropy and a loss of energy possibly making the 96h site energetically more accessible for Al 3+ with increased Al 3+ contents in contrast to the 24d site. Similar considerations were made by Bernstein et al. 36 They showed by using molecular dynamics simulations that the introduction of vacancies in LLZO does indeed reduce the free energy. This indirectly supports our suggestion.
Lastly, we consider the coordination geometry around the 96h site in LLZO, which has been described differently. 4,11,23 It is, of course, a matter of definition as to what constitutes a bond in a first coordination sphere. According to Li et al. in their description of Al at 96h, the average Al−O bond distance is d [4] = 2.08 Å, with a difference of 0.38 Å between the length of the shortest and the longest bonds. There are two further O 2− ions that give Al−O = 2.69 Å and 2.80 Å, thus being 0.46 Å and 0.57 Å longer compared to the longest bond in strict 4-fold coordination. Based on our calculations, Al 3+ at 96h has a locally 4-fold distorted coordination (Figure 4). It is worth mentioning that long-range structural properties, as determined by the diffraction experiment, such as neutron powder diffraction, can differ from those measured via spectroscopy, for example, NMR, which probes structure at shorter length scales.

■ CONCLUSION
Based on our DFT results, we propose that Al 3+ could have a number of slightly different local 4-fold coordinations around the crystallographic 24d and 96h sites in cubic LLZO garnet. The calculations are in general agreement with published experimental 27 Al NMR spectra. It should be noted further that Li et al., 11 in a neutron diffraction study of cubic LLZO garnet, proposed that Al 3+ is located at the 48g site in octahedral coordination. In terms of calculated site energies, this site could be partially occupied. However, the calculated Al NMR chemical shift value for such coordination has not been observed in experimental NMR spectra.