Structural Characterization of the Li-Ion Battery Cathode Materials LiTi x Mn 2 − x O 4 (0.2 ≤ x ≤ 1.5): A Combined Experimental 7 Li NMR and First-Principles Study

Sydney, NSW 2006, Australia * S Supporting Information ABSTRACT: Titanium doping in lithium manganese oxide spinels was shown to be bene ﬁ cial for the structural stability of the potential Li-ion battery cathode materials LiTi x Mn 2 − x O 4 , 0.2 ≤ x ≤ 1.5, yet the distribution of Li/Ti/Mn in the structure and the cation oxidation states, both pivotal for the electrochemical performance of the material, are not fully understood. Our work investigates the changes in the local ordering of the ions throughout this series by using a combination of 7 Li NMR spectroscopy and ab initio density functional theory calculations. The 7 Li NMR shifts are ﬁ rst calculated for a variety of Li con ﬁ gurations with di ﬀ erent numbers and arrangements of Mn ions in the ﬁ rst metal coordination shell and then decomposed into Li − O − Mn bond pathway contributions to the shift. These Li − O − Mn bond pathways are then used to simulate and assign the experimental NMR spectra of di ﬀ erent con ﬁ gurations and stoichiometries beyond those in the initial subset of con ﬁ gurations via a random distribution model and a reverse Monte Carlo approach. This methodology enables a detailed understanding of the experimental 7 Li NMR spectra, allowing the variations in the local ordering of the ions in the structure to be identi ﬁ ed. A random distribution of Ti 4+ − Mn 3+/4+ sites is found at low Ti content ( x = 0.2); an inhomogeneous lattice of Mn 4+ rich and Ti 4+ -rich domains is identi ﬁ ed for x = 0.4, and single-phase solid solution is observed for x = 0.6 and 0.8. A mixed Li − Mn 2+ tetrahedral and Li − Mn 3+/4+ − Ti octahedral con ﬁ guration is determined for the x = 1.0 case. A speci ﬁ c cation ordering in the partially inverse LiTi 1.5 Mn 0.5 O 4 case is found, which transforms into a two-phase network of disordered Mn 3+ -rich and ordered Mn 2+ systems, we simulated independently two single-phase NMR spectra and summed them according to the fractions of the structure corresponding to each of the phases present in the system. In the Monte Carlo simulations, we built a single simulation box with two regions, each of the regions corresponding to a given phase.


■ INTRODUCTION
Spinel-type lithium metal oxides are interesting cathode materials for rechargeable Li-ion battery applications. The robust host structure of spinel oxides retains capacity for many cycles, and the three-dimensional network of interstitial sites allows high Li diffusion rates. 1,2 LiMn 2 O 4 spinel has been studied as a potential alternative to the more widely used LiCoO 2 because of its lower cost, lower toxicity, and higher thermal stability. 2,3 However, the application of Li x Mn 2 O 4 as a positive electrode is limited by the presence of Jahn−Teller active Mn 3+ ions, which accelerate structural degradation of the material upon cycling due to a cooperative Jahn−Teller distortion. 4 Moreover, the charge disproportionation of 2Mn 3+ → Mn 4+ + Mn 2+ results in the formation of Mn 2+ ions, which dissolve in the electrolyte at the surface of the particles and deplete the spinel framework of transition-metal ions. 5 The electrochemical properties and the structural stability of LiMn 2 O 4 were shown to be improved by introducing excess Li in the spinel structure to form Li 1+α Mn 2−α O 4 . 6−9 The inclusion of the excess lithium raises the average Mn oxidation state, thereby reducing the amount of Jahn−Teller active Mn 3+ ions. Another approach to limit the negative effects of the cooperative Jahn−Teller distortion has been to partially substitute the Mn ion with other transition metals (TM) such as Ni, Zn, and Ti, hence stabilizing the structural integrity of the electrode upon cycling. 10−13 Under-standing the structural ordering and the distribution of Li/TM ions in the structure is central to rationalizing the relationship between the electrochemical performance and the physical properties of the material. As an example, the kinetic and electrochemical properties of the high-energy LiNi x Mn 2−x O 4 cathode material have been analyzed in relation to the complex cation ordering throughout the series, unraveling the key role of compositional (dis)order in the electrochemical performance of this material. 14 Here, the randomization of Ni/Mn ions among the octahedral sites of the spinel lattice was proposed to lead to improved electrochemical performance.
In this work, we focus on the series of materials LiTi x Mn 2−x O 4 with 0.2 ≤ x ≤ 1.5 and study the structural changes of the Mn-oxide spinel framework as an effect of Ti doping. In spinel materials of general formula AB 2 O 4 , the oxygen anions form a face-centered-cubic sublattice, within which octahedral (Oh) and tetrahedral (Td) interstitial sites are present in a 2:1 ratio. In a normal spinel, the A cations occupy the Td sites, and the B cations occupy the Oh sites (denoted as A[B 2 ]O 4 ), while in an inverse spinel, the B cations occupy all of the Td sites and half of the Oh sites, and the A cations occupy the other half of the Oh sites (denoted as B[AB]O 4 ). In the case of LiTi x Mn 2−x O 4 , the partial substitution of Mn with Ti results in a mixed cation occupancy on both the Oh and Td sites of the spinel lattice. 15,16 In the LiTi x Mn 2−x O 4 series for 0.2 ≤ x ≤ 1.0, the disordered Fd3̅ m cubic spinel preferentially forms (Figure 1a) with proposed partial occupancy of Li + /Mn 2+ and Li + /Ti 4+ /Mn 3+/4+ on the tetrahedral 8a and the octahedral 16d sites, respectively, and with the Li(Oh)/Mn 2+ (Td) fraction increasing with increasing Ti content. For x > 1.0, the more ordered P4 3 32 cubic spinel also starts to form (Figure 1b), in which it was proposed that the two inequivalent octahedral sites, 4b and 12d, are occupied by a mixture of Li + /Mn 3+/4+ and a mixture of Li + /Ti 4+ /Mn 3+/4+ ions, respectively, while the tetrahedral 8c sites are partially occupied by Li + /Mn 2+ ions. 15,16 A detailed study using synchrotron X-ray, neutron powder diffraction, and XANES spectroscopy further investigated the effects of different sintering temperatures and cooling regimes during synthesis on the phase behavior of LiTiMnO 4 . 17 Although it is a challenge to provide an accurate description of the coordination site disorder throughout the LiTi x Mn 2−x O 4 series because of the presence of multiple mixed-valence transition metals, a detailed characterization of the structure provides the fundamental basis with which to understand and monitor the electrochemical properties of the material. Analysis based on X-ray diffraction (XRD) showed limited accuracy in determining the distribution of Li, Ti, and Mn ions in the lattice due to the difficulty of detecting Li in the presence of heavier elements and of distinguishing between Ti and Mn, which have similar X-ray scattering factors. 15,16 Electron spin resonance (ESR) studies were also used in combination with XRD analysis, but the presence of multiple Mn oxidation states made the assignment of the TM distribution challenging. 16 6/7 Li NMR has been successfully used to characterize the local Li environments and the cation ordering in similar systems such as LiMn 2 O 4 , 18,19 23,24 The dominant interaction leading to the observed 6/7 Li NMR shift in this class of paramagnetic materials is the isotropic Fermi contact (FC) hyperfine interaction, 25−27 which results from the coupling between the nuclear moment of the Li and the time-average of the local field due to the unpaired d electrons present on the neighboring TM ions. 19,28 In the series of systems studied here, Ti is present as Ti 4+ throughout (i.e., d 0 ion); hence, the only paramagnetic centers are the Mn 2+/3+/4+ sites (i.e., d 5 /d 4 /d 3 ions, respectively). The Fermi contact interaction is proportional to the unpaired spin density transferred from the d orbitals of Mn to the s orbitals of Li. This transfer can occur either directly through the overlap of the involved orbitals or, more prominently in the systems studied here, indirectly via the bridging oxygen p orbitals which form Mn−O−Li bond pathways. The observed 7 Li Fermi contact shift is hence given by the sum of the individual Mn− O−Li pathway contributions. 29 The sign and magnitude of the shift depend on the geometry and covalency of the pathway as well as on the Mn oxidation state and on the magnetic susceptibility of the material. In mixed cation systems, the variety of Li environments often results in a multitude of paramagnetic shifts and a significant broadening of the resonances, making the spectra difficult to interpret. 30 Computational predictions and ab initio calculations of paramagnetic NMR parameters constitute a robust and invaluable aid for the understanding of experimental NMR spectra of paramagnetic solids.
In this work, we report a detailed 7 Li magic-angle spinning (MAS) NMR spectroscopy investigation of the LiTi x Mn 2−x O 4 spinel series using state of the art spectroscopic methods for paramagnetic materials. 31 The interpretation of the experimental spectra is supported by first-principles density functional theory (DFT) calculations of the magnetic interactions   29 To compare these shifts with the experimental spectra obtained at finite temperature, we evaluate the possible Mn−Mn magnetic interactions, determine the Curie−Weiss magnetic factors, and use these to scale the shifts obtained with density functional theory (DFT) to the paramagnetic regime of the NMR experiments (performed at room temperature). 32 16 Solid-State MAS 7 Li NMR. Solid-state NMR spectra of the LiTi x Mn 2−x O 4 samples (0.2 ≤ x ≤ 1.5) were acquired on a Bruker 200 Avance III spectrometer using a 1.3 mm probe with a MAS frequency of 60 kHz. The one-dimensional 7 Li spectra were recorded using a double-adiabatic spin−echo sequence, 31 employing a pair of 50 μs tanh/tan short high-powered adiabatic pulses (SHAPs) of 5 MHz sweep width 34,35 and a 1.025 μs 90°excitation pulse. All pulses used a radiofrequency (RF) field strength of 244 kHz. For each spectrum, 32 768 scans were acquired using a recycle delay of 30 ms. The experimental 7 Li NMR spectra were fitted using the DMFIT software. 36 An initial model was set up with components based on the hyperfine shifts predicted with DFT calculations, and the fitting of the isotropic region and the sideband pattern was then obtained by optimizing the shift and the amplitude of the deconvoluting regions. between Li, Ti, and Mn were enumerated. For the x = 1.25 and 1.5 cases, additional configurations were considered: for larger cells of 56 atoms, all configurations were enumerated starting from LiTi 2 O 4 and replacing (2−x) Ti ions with Mn ions. Additionally, swaps between Li, Mn, and Ti were allowed to include inverse spinel lattices in the analysis, i.e. networks of mixed Ti−Mn−Li occupancy on both the tetrahedral and the octahedral sites were generated.
Details of GGA+U Calculations of Formation Energies. The unit cell parameters and atomic positions of each generated configuration were relaxed with DFT using the PBE 40 spin-polarized generalized gradient approximation (GGA) functional within the VASP code. 41 The projector augmented waves (PAW) 42 method was used with a plane-wave cutoff of 500 eV and an energy tolerance of 10 −6 eV, resulting in a convergence of the energy of approximately 4 meV/ atom. A force tolerance of 10 −5 eV/Å was used for ionic relaxations, resulting in a convergence of the energy of approximately 4 meV/ atom. A force tolerance of 10 −5 eV/ \AA was used for ionic relaxations. The electronic energy of each relaxed structure was calculated with a single-point energy minimization. The reciprocal space sampling was performed with a k-point grid of 8 × 8 × 8 points for the smaller cells (14 atoms) and 4 × 4 × 4 points for the larger cells (56 atoms). To correct for the self-interaction error in the GGA formalism, a Hubbard U parameter was included for the Mn ions to treat the 3d correlations. 43 In this work, the approach proposed by Liechtenstein was used, 44 where the Coulomb matrix (U) and the exchange matrix (J) are combined to give an overall effective value U eff = U − J. The value of J was fixed to 1 eV throughout. In a previous study by Wang et al., 45 values of U eff = 4.5, 4.0, and 3.5 eV were calculated for Mn 2+ , Mn 3+ , and Mn 4+ ions, respectively. Because in our systems the Mn ions are present in multiple oxidation states, an average U eff value of 3.9 eV was chosen for all the Mn sites.
Paramagnetic Shift Calculations. The calculation of the Fermi contact shift, δ FC , adopted in this work follows the methodology presented by Kim et al., 46 which is summarized here. In this method, the hyperfine coupling constant, A iso , is first calculated from the system in the ferromagnetic state nominally at 0 K and then scaled using a Curie−Weiss factor, Φ, to match the paramagnetic regime typical of NMR experiments. and where h is the Planck constant, ν is the Larmor frequency, |ψ N α−β | 2 is the unpaired spin density at the Li nuclear position, B 0 is the static magnetic field, μ eff is the effective electronic magnetic moment, μ B is the Bohr magneton, S is the formal electronic spin of the paramagnetic center(s), k B is the Boltzmann constant, g e is the free-electron g-value, g I is the nuclear g-factor, T is the temperature used in the experiments, here estimated to be 320 K to account for frictional heating due to MAS NMR, and θ is the Weiss constant. In this work, μ eff is taken to be the spin-only value of μ . This is considered a good approximation for the class of systems studied here. As an example, in the case of LiMn 2 O 4 with an average oxidation state of Mn 3.5+ , S = 1.75, and the calculated spin-only value is μ eff = 4.39 μ B , in good agreement with the experimental μ eff range of 4.33−4.36 μ B reported by Masquelier et al. 47 Methodology for Calculating the Magnetic Parameters θ and Φ. Values of θ were obtained ab initio by calculating the magnetic exchange coupling constants, J, by a multivariate linear regression of the DFT-calculated energies of systems with different magnetic configurations of coupled spins. More details of the method are presented in the Supporting Information. The calculations of the various exchange coupling constants were performed on selected structures, i.e., the lowest energy configurations for the LiTi 0.5 Mn 1.5 O 4 , LiTiMnO 4 , LiTi 1.25 Mn 0.75 O 4 , and LiTi 1.5 Mn 0.5 O 4 stoichiometries, containing networks of Mn 3+/4+ , Mn 3+ , Mn 2+/3+ , and Mn 2+ ions, respectively. When equivalent Mn−Mn interactions are present in different lattices, the corresponding J values were found to differ by less than 3%, differences arising from local distortions of the optimized geometries. All of the considered structures containing Mn 3+ ions were found to exhibit a cooperative Jahn−Teller distortion.
Methodology of Hyperfine Coupling Constant, A iso , Calculation. The isotropic value of the hyperfine tensor A iso , in eq 2, was calculated with DFT by integrating the unpaired electron spin density, |ψ N α−β | 2 , directly at the Li nuclear position in the ferromagnetic state, which was then scaled to the paramagnetic regime by multiplying it by the scaling factor Φ (eq 1). 46 The bond pathway decomposition method presented by Middlemiss et al. 29 was followed to obtain the Mn− O−Li Fermi contact bond pathway contribution from each Mn ion to the total Li shift using the computed site-specific scaling factor, Φ i .
Details of Hybrid DFT/Hartree−Fock Calculations of Paramagnetic Shifts. All calculations of magnetic and hyperfine parameters were performed in CRYSTAL09, 48 a solid-state DFT code using a Gaussian-type basis set to describe core states accurately. Because of the high dependence of the calculated paramagnetic shifts on the quality of the Gaussian basis sets, two types were utilized: a smaller basis set for geometry optimizations, and a more extended basis set for hyperfine and magnetic single-point calculations. More details are given in the Supporting Information. All calculations were performed with hybrid functionals in the spin polarized state. Previous ab initio studies on 7 Li paramagnetic NMR shifts show that values obtained using 20 and 35% Hartree−Fock (HF) exchange provide the upper and lower bounds for the experimental shifts. 29,30 Hence, separate calculations were performed with the B3LYP functional with 20% HF exchange 49 (denoted HYB20) and a modified B3LYP with 35% of HF exchange (denoted HYB35). The convergence of the energy and the spin density were checked with respect to the number of sampled points in the reciprocal space. The reciprocal space sampling was performed with a k-point grid of 4 × 4 × 4 points in the simulated cells, which contain 56 atoms. Self-consistent field cycles were converged to an energy difference of 2.7 × 10 −6 eV.
Simulation and Fitting of the 7 Li NMR Spectra. Random Solution Model of a Single Phase. To use the Fermi contact bond pathway contributions calculated from DFT to simulate model NMR spectra, one needs to know the possible cationic environments around the lithium ions and the population distribution among these environments. The simplest approximation that can be made to obtain such distributions of environments is to consider that there is no cation ordering, and thus, the cations are randomly distributed in the sites available to them. This approach corresponds to the random solid solution model. In the regular spinel structure, Li centers have the following neighboring cations: (1) each Li in a Td site has 12 neighbor cations located on Oh sites, which are each bound via an oxygen bridge and considered to contribute to the overall Fermi contact shift of this lithium; and (2) each Li in an Oh site has 12 neighbor cations, 6 of which are in Td sites and 6 of which are in Oh sites, which are each bound via an oxygen bridge and considered to give (different) contributions to the overall Fermi contact shift of this lithium. To simulate the NMR spectrum, we thus need to know (i) how the Li ions are distributed between Td and Oh sites and (ii) what ions are present in each Li neighboring shell. Once we know these possible configurations, we can calculate the corresponding Fermi contact shifts and estimate the probability of these environments to simulate the NMR spectrum. In the random solution model, for each Li environment with a given number of Mn and Ti neighbors, the number of all possible configurations is calculated with the corresponding probability p modulated by the stoichiometric ratio of the ions in the structure. For each environment, a Gaussian distribution is then generated of the form G = p·exp[−(δ − ∑ δ path ) 2 /(2ω 2 )]. In this formula, p is the probability associated with the environment considered, δ is the range of resonance values for which the distribution is calculated, ∑ δ path is the sum of all relevant bond pathway contributions to the shift involved in the particular environment, and ω is the Gaussian width. An approximate Gaussian peak width of 15 ppm was used to model the individual environments based on previous NMR studies on LiMn 2 O 4 . 19 The simulated NMR spectrum is then obtained as the sum of the Gaussian plots corresponding to the various environments present in the system.
Reverse Monte Carlo Simulations of a Single Phase. While in some cases, the random solution model provides a good agreement with the experimental data, in other cases, such as the LiTi 1.5 Mn 0.5 O 4 material studied here, it does not, and there is a need to calculate the populations of lithium environments according to different conditions. Here, for this purpose, we use a simulation method inspired by reverse Monte Carlo approaches. The idea of a reverse Monte Carlo method is to build a large simulation box that is representative of the system under study and explore the effect of configurational changes. Starting from an initial configuration, i.e. a large number of ions with defined positions in space, we allow certain moves which are accepted or rejected depending on their agreement with chosen constraints.
In the present case, we built the large simulation box by replicating an initial spinel structure corresponding to LiTi 1.5 Mn 0.5 O 4 in the P4 3 32 space group. The initial structure contained 56 atoms corresponding to (i) 8 Li ions in Td sites (8c), (ii) 12 Ti ions and 4 Mn ions in Oh sites (12d and 4b, respectively), and (iii) 32 oxygen sites (8c and 24e). This simulation box was replicated 10 times in all 3 dimensions, leading to a large simulation box containing 8000 Li ions, 12 000 Ti ions, 4000 Mn ions, and 32 000 oxygen ions. We checked that this 10 × 10 × 10 system is large enough by simulating some of the NMR spectra with a larger box of 15 × 15 × 15 repeat units. The results from the two system sizes showed no significant differences, and so the size of the 10 × 10 × 10 cell was considered to be sufficient. Once the initial simulation box is built, the Monte Carlo method proceeds via the following steps: (1) swap two cations, (2) characterize the new Li environments, and (3) accept or reject the move according to a set of chosen rules. The swaps can be done between Li, Mn, and Ti belonging to octahedral sites (or to tetrahedral sites) or between cations belonging to different structural sites. If the simulations are done without constraints, all the swaps will be done randomly. If the simulations are done under constraints, the swaps which do not lead to Li environments conflicting with the set constraints will always be accepted, while the swaps leading to unwanted Li environments will be accepted with a small probability. The acceptance probability of unwanted moves is P acc = exp(−E a /k B T). Here, for all simulations, E a was taken to be 10 eV, i.e. the probability of having unwanted Li environments is very close to zero. In this work, the constraints are set based on the experimental 7 Li NMR isotropic shifts. For example, if the spectrum of a particular composition does not show peaks at negative shifts, this is used as a constraint in the Monte Carlo simulation, which associates a small probability to the Li environments calculated to have negative shifts.
In addition to the swaps between the cations present in the structure, the Monte Carlo moves can include changes between different cation types. This is needed to reach some of the intermediate x values in the series. For example, on going from LiTi 1.5 Mn 0.5 O 4 to LiTi 1.4 Mn 0.6 O 4 , a number of Ti ions need to be replaced by Mn ions and some Mn 2+ ions (the only Mn oxidation state possible for LiTi 1.5 Mn 0.5 O 4 ) will be converted to Mn 3+ ions. These changes in cation types are also made with or without constraints. Once a satisfactory representation of the system has been reached, in terms of (i) Li distribution in Td and Oh sites and (ii) stoichiometry of the structure, the number of Li cations in each environment is simply counted, the corresponding shifts are calculated and, as in the random solution model representation, the simulated NMR spectrum is then obtained by summing the Gaussian plots corresponding to the various environments present in the system.
Model for Two Phase Systems. The random solution model and Monte Carlo approaches described above were first used to simulate homogeneous (single-phase) systems. In the LiTi x Mn 2−x O 4 series, this single phase representation is able to depict only part of the series of materials. To investigate the possibility of having inhomogeneous systems, we simulated independently two single-phase NMR spectra and summed them according to the fractions of the structure corresponding to each of the phases present in the system. In the Monte Carlo simulations, we built a single simulation box with two regions, each of the regions corresponding to a given phase.

■ RESULTS AND DISCUSSION
Geometry Optimization and Energy Profile. Figure 2 compares the formation energies of a number of structures in the LiTi x Mn 2−x O 4 series with formation energies below 0 eV, which are all obtained using the search strategy described in the previous section. The full set of simulated structures is shown in Figure S1 of the Supporting Information. The convex hull (tie-line in Figure 2)  Our results show that this ordering does not correspond to the lowest energy structure of this composition. We find that the thermodynamically favorable ordering for x = 1.0 is the partially inverse spinel with mixed Li−Mn 2+ occupancy of the tetrahedral (8a) sites and mixed Li−Mn 3+ −Mn 4+ −Ti 4+ occupancy of the octahedral (16d) sites, in agreement with previous X-ray diffraction studies. 15,16 The mixed Mn oxidation state in the inverse spinel results from the charge disproportionation of Mn (8a) 3+ +Mn (16d)

3+
→ Mn (8a) 2+ +Mn (16d) 4+ , in agreement with the preferential occupancy of the Td (8a) sites by Mn 2+ over Mn 3+ . 15,16 Our results show that configurations with mixed Li− Ti occupancy of the tetrahedral sites and mixed Li−Mn 3+ −Ti occupancy of the octahedral sites are thermodynamically unfavorable. This is in agreement with the XRD studies of Petrov et al. 16 and Krins et al. 15 A recent work by Murphy et al. 17 based on synchrotron X-ray and neutron powder diffraction, however, indicated partial occupancy of Ti in the tetrahedral sites of the LiTiMnO 4 lattice, which was not reported in the previous studies 15, 16 nor reproduced by our DFT results.
According to the presented analysis based on the calculated formation energies, we can gain some insights into the more stable cation distributions in the spinel lattice, and the results we work with are gathered in Table 1.
Solid-State NMR. Full one-dimensional 7 Li doubleadiabatic spin−echo 31 spectra of the LiTi x Mn 2−x O 4 powder samples (0.2 ≤ x ≤ 1.5) are given in Figure 3a. The doubleadiabatic spin−echo sequence was chosen to obtain an efficient inversion of the whole spinning-sideband pattern, here of more than 400 kHz width. 50 The corresponding central regions of the spectra are shown in Figure 3b. The intensity ratio between the centerband and the sidebands did not change across the spectra, and for this reason we fit only the isotropic resonances. The variety of Li environments occurring in each phase leads to multiple distinct resonances and a broad isotropic region. The resonances, on average, range from around 500 to 30 ppm with the increase in Ti concentration. We also note that the spectrum is significantly broader for intermediate   the average Mn oxidation state and the decrease in the overall concentration of paramagnetic Mn ions. 28 A more detailed understanding of the structural differences across the series is now presented based on the DFT analysis of the Fermi contact interaction. DFT Calculation of Magnetic and Hyperfine Parameters. As described previously (eq 3), the magnetic scaling of the hyperfine interaction was modeled via a mean field approach based on the exchange coupling interaction between Mn pairs, as shown in Figure 4. Mn 3+ has a (t 2g ) 3 (e g *) 1 electronic configuration, which makes it Jahn−Teller active. Consequently, different exchange couplings and Fermi contact interactions were identified, depending on whether the pathway involves the Jahn−Teller lengthened or shortened Mn−O bond. The calculated J n values are presented in  32 Additionally, the accuracy of the calculated exchange integrals is tested for the LiTi 0.4 Mn 1.6 O 4 case by comparing the magnetic scaling factors obtained with the mean field approach and using the experimental magnetic susceptibility. The computational and experimental results obtained are in good agreement, and the details of the comparison are presented in the Supporting Information. The magnitude of the exchange interactions is sensitive to the distance between the coupled ions as well as to the coupling mechanism between the involved orbitals. As an example, we compare the exchange interactions between Mn 3+ −Mn 3+ in octahedral sites. The J 4 coupling (Figure 4, center) involves four Jahn−Teller shortened Mn−O bonds which allow direct overlap between the Mn t 2g orbitals, leading to a strong antiferromagnetic exchange interaction. The J 3 coupling (Figure 4, center) involves two Jahn−Teller shortened and two Jahn−Teller elongated Mn−O bonds, the latter ones reducing the direct overlap between t 2g orbitals and hence the strength of the direct exchange interaction. The superexchange interaction between the orbitals along the Jahn−Teller axis (d z 2 −d xz/yz ) results in a combination of weak ferro-and antiferromagnetic interactions, resulting in a significantly smaller J 3 coupling that is antiferromagnetic overall.
The possible Mn−O−Li bond pathways were identified for Li occupying Td as well as Oh sites. All the Li environments and corresponding pathways are described in Figure 5 and are presented in Table 3. As rationalized by Carlier et al., 51 the transfer of paramagnetic electron spin from the Mn t 2g /e g * orbitals to the s orbitals of the Li occur primarily via a superexchange-like mechanism. As presented in eq 1, the sign and magnitude of the Fermi contact shift are determined by a combination of factors. The extent of the transferred spin density as well as the strength of the magnetic interaction discussed previously strongly depend on the bond distances between the involved sites and on the orbitals involved in the interaction. 51 The Mn−O−Li pathways in these systems   a For the couplings involving Mn 3+ ions, the nature of Jahn−Teller shortened or lengthened bond involved in the interaction is also specified. The results obtained with the HYB20 and HYB35 hybrid functionals are shown separately. Different J n types are labeled as in Figure 4. involve intermediate angles (neither exactly 90°nor exactly 180°); hence, the spin-density transfer deviates from a pure delocalization/polarization mechanism, and a complex combination of both processes is expected to occur. Nonetheless, we take the Li−O−Mn 3+ pathways P 1 and P 2 ( Simulation and Fitting of the Experimental 7 Li NMR Spectra. We now demonstrate how the bond pathway contributions calculated in Table 3 can be used to model the 7 Li NMR spectra of the LiTi x Mn 2−x O 4 series, allowing us to extract detailed local structural information. The shift values calculated with the HYB20 and HYB35 functionals give the upper and lower bounds; however, in the following analysis, we use a pragmatic approach in which the HYB20 and HYB35 results are averaged to give a single value for each pathway P, as shown in Table 3. For all the systems containing Mn 3+ , we model the Jahn−Teller distortion as a dynamic process in which the time scale of the changes in local Mn−O bond lengths is much faster than the typical NMR time scale. 29 As a consequence, we assume that the shifts corresponding to Mn 3+ −O−Li are well-represented by a weighted average with 2 / 3 of Jahn−Teller-short and 1 / 3 of Jahn−Teller-long pathway contributions. 29 For all the systems containing exclusively Mn 3+ and Mn 4+ ions, i.e. the 0.2 ≤ x ≤ 0.8 stoichiometries, we make an additional assumption. In this case, we consider that the time scale of electronic conduction is fast compared to the NMR time scale. This means that we can consider only one shift corresponding to an average oxidation state of the Mn ions. The weighted average depends on the stoichiometry of the system, as the average oxidation state is a function of the Ti content of the material. For example, for LiTi 0. 2 18 the NMR spectrum of LiMn 2 O 4 with Li in the tetrahedral 8a position is dominated by one major resonance with an isotropic shift of 512−520 ppm, the exact value varying between samples and likely the temperature of the measurement. In this stoichiometry, Mn is present in an average oxidation state of 3.5+. Combining the shift contributions that we obtained with DFT, summarized in    7 Li NMR spectrum was simulated assuming a random distribution of ions in the Li local coordination shell. All Li ions were placed on the tetrahedral sites, following our observation that the presence of Ti 4+ , Mn 3+ , and Mn 4+ on tetrahedral sites is energetically unfavorable. The total DFT predicted spectrum is shown in Figure 6a with the individual peaks shown in Figure  6b Figure 6a. Moreover, the relative intensities of the different peaks are accurately reproduced by the model that considers a random distribution of Ti 4+ and Mn 3.44+ among the octahedral sites. The weak peak at 595 ppm is also predicted by DFT with the model that does not consider a dynamic exchange between Mn 3+ and Mn 4+ . In particular, the shift of the Li environment coordinated with 3 Mn 3+ and 9 Mn 4+ is calculated to be 599 ppm, in good agreement with the experimental shift of 595 ppm. Figure S2 in the Supporting Information shows the results of the model that does not consider a dynamic exchange between Mn 3+ and Mn 4+ . The line shape of the spectrum simulated in this way is not in as good of agreement with the experimental NMR spectrum, confirming that the fast Mn 3+/4+ hopping rate in the experimental NMR conditions is a good assumption for these systems. 0.4 ≤ x < 1.0 Stoichiometries. Following the successful description of the cation ordering for LiTi 0.2 Mn 1.8 O 4 with the random distribution model, we apply the same approach for the 0.4 ≤ x < 1.0 stoichiometries. As shown in Figure S3 of the Supporting Information, the deconvolution of the isotropic region for LiTi 0.4 Mn 1.6 O 4 deviates significantly from the spectrum simulated for the random distribution model. The disagreement between the experimental NMR spectra and the simulations based on the random distribution model persists for all the 0.4 ≤ x < 1.0 cases, as shown in Figure S4   concentration, indicative of a continuous solid solution behavior. 16 LiTiMnO 4 . As described in Figure 2 and in Table 1, for the x = 1 case, our calculations predicted that the configuration with mixed Li−Mn 2+ tetrahedral and Li−Mn 3+/4+ −Ti octahedral occupancies is more favorable than the normal spinel (Li[Mn 3+ Ti]O 4 ). Studies based on X-ray diffraction reported 20−30% Li occupancy of the Oh site (or, equivalently, 20−30% Mn 2+ on the Td site). 15,16 In an attempt to model the spectrum for LiTiMnO 4 Figure 8.
The spectrum simulated for the partially inverse case ( Figure  8b) shows shifts over the entire 0−700 ppm region of the experimental spectrum, while the simulation for the normal spinel (Figure 8a) shows shifts only between 20−300 ppm. This confirms the presence of both Li(Td) and Li(Oh) environments in the structure, as predicted by our calculations (Table 1). However, the line shape of the experimental spectrum is not well reproduced by the simulation for the disordered Li 0.7 Mn 0.3 2+ [Li 0.3 Mn 0.4 3+ Mn 0.3 4+ Ti]O 4 lattice, particularly between 0−100 and 350−450 ppm. This indicates that the cations are not distributed randomly in the structure, and cation ordering determines the preferential presence of certain environments over others. Due to the configurational complexity of the system, which contains Li/Mn 2+,3+,4+ /Ti ions and mixed occupancy of the Td and Oh sites, further work would be needed to systematically analyze the configurational energies of different orderings 52 and possibly account for the electroneutrality principle. 53 LiTi 1.5 Mn 0.5 O 4 . We now turn to the interpretation of the NMR spectra for the Ti-rich part of the series, starting with the end member LiTi 1.5 Mn 0.5 O 4 . In this system, we expect Li + , Mn 2+ , and Ti 4+ cations to be present. The formation energies calculated with DFT shown in Figure 2 indicate that it is energetically favorable for Mn 2+ ions to be on tetrahedral sites, leading to (at least) a fraction of the Li ions on the octahedral sites, also suggested by previous X-ray studies. 16 This fraction is referred to as the inverted fraction of Li, y, in the notation Li 1−y Mn y [Li y Ti 1.5 Mn 0.5−y ]O 4 . The first model we test is a random distribution of Mn 2+ /Li + ions on tetrahedral sites and Li + /Mn 2+ /Ti 4+ on octahedral sites. The simulated NMR spectra are shown in Figure S5  resonances at 0 ppm and some at negative frequencies, which are clearly not present in the experimental NMR spectrum. To use this information, we thus turn to another strategy and follow an approach similar to reverse Monte Carlo, where constraints are imposed. We use as a starting point a spinel structure with a P4 3 32 symmetry, as this was suggested by X-ray studies on this material. 16 In our starting P4 3 32 structure, the lattice is characterized by a full Li occupancy on the tetrahedral sites and a full Ti/Mn occupancy of the octahedral sites (i.e., a regular spinel structure). This ordering corresponds to the  presence of only one type of Li environment with 9 Ti 4+ and 3 Mn Oh 2+ neighbors. The calculated shift is 3 × 28 ppm = 85 ppm, in clear disagreement with the experimental NMR (Figure 9a). Starting from the fully regular structure, the Monte Carlo simulation then allowed for some fraction of Li to move onto the octahedral environment with the consequent swap of Mn (and Ti) onto the tetrahedral sublattice. After each swap, the Li coordination environments are checked, and the corresponding shifts are calculated. If the swap leads to a "wrong" shift (a shift ≤0 ppm), it will be accepted but only with a very low probability.
The results of the reverse Monte Carlo approach are shown in Figure 9a Figure 9b. The Li environments present in the lattice are also shown with the associated NMR peaks. The good agreement between the model and the experimental spectrum allows us to conclude that the spinel network of the LiTi 1.5 Mn 0.5 O 4 system, of P4 3 32 symmetry as previously reported, 16 contains mixed cation occupancy of Li, Mn 2+ , and Ti in both tetrahedral and octahedral environments. The presented analysis also allows us to derive a specific cation ordering within P4 3 Figure S6 of the Supporting Information. It is clear from the comparison with experiments that this model is not sufficient to describe the system. In particular, there is a broad peak with a large shift (∼200 ppm) which cannot be explained by this model, while the region of the spectrum between 0−100 ppm decreases in intensity but does not vary in shift compared to the LiTi 1.5 Mn 0.5 O 4 case.
To use this information, we modify the model to retain the P4 3 32 ordering of the Mn 2+ -rich phase, as shown for the x = 1.5 case, while including a random distribution of cations in the Fd3̅ m Mn 3+ -rich phase, as shown for the 0.2 ≤ x ≤ 1.0 cases. In these simulations, a Gaussian width of 50 ppm was found to be required to model the Fd3̅ m domain, while a Gaussian width of 12 ppm was again found to be sufficient to model the P4 3 32 domain. This difference between the peak widths may be rationalized in terms of the higher degree of disorder among the Li environments present in the Fd3̅ m phase compared to the more ordered P4 3 32 phase. The approach is followed for the entire 1.1 ≤ x ≤ 1.4 series. Furthermore, for each composition, the Monte Carlo simulation also optimized the inverted fraction, y, within the P4 3 32 phase, and so we obtain for x = 1.  Figure 10. The good agreement between the simulated and the experimental NMR spectra throughout the 1.1 ≤ x ≤ 1.5 series suggests that as Mn 2+ starts to be formed in the system, it preferentially occupies the tetrahedral environment, determining a partial ordering between Li−Mn 2+ Td−Oh sites in the P4 3 32 symmetry, while the regular Mn 3+ -rich component retains a random distribution of cations consistent with the Fd3̅ m symmetry. As NMR probes structural short-range ordering, it does not allow us to distinguish the sizes of the Fd3̅ m and the P4 3 32 phases. Because previous diffraction studies, 16 which are sensitive to long-range ordering, reported a single  series may hinder a cooperative Jahn−Teller distortion in the bulk and facilitate the mechanical stability during electrochemical cycling. On the other hand, the observed increasing ratio of Li(Oh)/Mn 2+ (Td) mixing with increasing x may hinder the extraction of Li + from the structure, effectively resulting in a lower capacity.

■ CONCLUSIONS
A detailed solid-state 7 Li NMR and first-principles DFT study of the cation ordering and the structural changes in the LiTi x Mn 2−x O 4 series (0.2 ≤ x ≤ 1.5) was presented. 7 Li MAS NMR spectra were obtained for the LiTi x Mn 2−x O 4 series using state of the art spectroscopic methods for paramagnetic solids. The methodology used to analyze the NMR results involved the ab initio calculation of the magnetic and hyperfine parameters, obtaining a breakdown of the possible contributions to the 7 Li NMR shift. These were combined into random distribution and reverse Monte Carlo models to simulate the 7 Li NMR spectra of the LiTi x Mn 2−x O 4 series. For x = 0.2, a random distribution of octahedral Mn 3+/4+ /Ti 4+ cations in the Fd3̅ m structure was determined, evolving into an inhomogeneous lattice of Mn 3+ -rich/Mn 4+ -rich phases for x = 0.4 and a single-phase solid solution for x = 0.6 and 0.8. The x = 1.0 case showed partially inverse ordering of Mn 3+/4+ −Ti 4+ −Li (Oh) and Mn 2+ −Li(Td) sites. In the 1.1 ≤ x ≤ 1.5 structures, the results showed the preferential formation of coexisting disordered Mn 3+ -rich (Fd3̅ m) and ordered Mn 2+ -rich (partially inverse P4 3  Additional computational details, synthesis procedure, complete convex hull, additional lattice simulations, and details and results of magnetic SQUID measurements (PDF) Figure 10.      with the neighbouring spins, is expressed as a mean field, which is felt by spin i, as well as by the neighbouring spins. The time-averaged spin of a site i, S z,i , is obtained by calculating the Brillouin function at a particular temperature and external magnetic field, and the mean field equations are solved self consistently for all the spins in the cell. From the calculated S z,i , one can evaluate the magnetic scaling factor either for the total unit, or, like in this work, for a specific site, i, such as Φ i =