# Superstructure and Correlated Na + Hopping in a Layered Mgsubstituted Sodium Manganate Battery Cathode are driven by Local Electroneutrality

Euan N Bassey, Ieuan D Seymour, Joshua D Bocarsly, David A Keen, Guido Pintacuda, Clare P Grey

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To cite this version:

Euan N Bassey, Ieuan D Seymour, Joshua D Bocarsly, David A Keen, Guido Pintacuda, et al. Superstructure and Correlated Na + Hopping in a Layered Mgsubstituted Sodium Manganate Battery Cathode are driven by Local Electroneutrality. Chemistry of Materials, In press, 35 (24), pp.10564-10583. 10.1021/acs.chemmater.3c02180. hal-04301860

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Distributed under a Creative Commons Attribution 4.0 International License
Superstructure and Correlated
Na⁺ Hopping in a Layered Mg-
substituted Sodium Manganate
Battery Cathode are driven by
Local Electroneutrality

Euan N. Bassey,¹,ᵃ leuan D. Seymour,²,ᵇ
Joshua D. Bocarsly,¹,ᶜ David A. Keen,³ Guido
Pintacuda,⁴ Clare P. Grey¹,*,

¹Yusuf Hamied Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge, CB2 1EW, United Kingdom

²Department of Materials, Imperial College London, South Kensington Campus, London, SW7 2AZ, United Kingdom

³ISIS Facility, STFC Rutherford Appleton Laboratory, Harwell Oxford Campus, Didcot, OX11 0QX, United Kingdom

⁴Centre de RMN à Très Hauts Champs, FRE 2034 (CNRS/Université Claude Bernard Lyon 1/Ecole Normale Supérieure de Lyon), University of Lyon, 69100 Villeurbanne, France

Abstract.

In this work, we present a variable-temperature
²³Na NMR and variable-temperature and
variable-frequency EPR analysis of the local
structure of a layered P2 NIB cathode material,
Na₀.67[Mg₀.28Mn₀.72]O₂ (NMMO). For the first
time, we elucidate the superstructure in this
material using synchrotron X-ray diffraction and
total neutron scattering and show that this
superstructure is consistent with NMR and EPR
spectra. To complement our experimental data,
we carry out *ab initio* calculations of the
quadrupolar and hyperfine ²³Na NMR shifts, the
Na⁺ ion hopping energy barriers and the EPR
g-tensors. We also describe an in-house
simulation script for modelling the effects of
ionic mobility on variable-temperature NMR
spectra and use our simulations to interpret the
experimental spectra, available upon request.
We find long-zig-zag-type Na ordering with two
different types of Na sites, one with high
mobility and the other low mobility, and
reconcile the tendency towards Na⁺/vacancy
ordering to the preservation of local
electroneutrality. The combined magnetic
resonance methodology for studying local
paramagnetic environments from the

perspective of electron and nuclear spins will be
useful for examining the local structures of
materials for devices.

**Introduction.**

The development of cheap energy storage
systems with long lifetimes and high capacities
for grid-scale applications is critical to mitigating
the effects that arise from the mismatch
between use and production of electricity with
modern renewable energy sources. In this
regard, sodium-ion batteries (NIBs) are ideally
suited, owing to their sustainability and low
production costs.¹⁻⁴ If NIBs are to be used on
the grid, their capacities, lifetimes and rate
behaviours must, however, be optimised and
their degradation and failure mechanisms
understood.

At present, the electrochemical performance –
and in particular the capacity of a NIB is limited
primarily by the cathode material.
Understanding how the structures of these
cathode materials evolve during cycling—
ideally in a non-invasive manner, such that
metastable states can be probed without
damaging the structure—is critical to
developing new cathode materials with high
capacities, long lifetimes and fast rate
capabilities.

Layered sodium transition metal oxide materials, NaₓTMO₂ (TM = transition metal), similar to their related Li analogues, represent an important class of NIB cathode materials.⁵ As for the Li-containing materials, the structures of these materials may be described using the notation developed by Delmas et al.,⁶ which uses a letter to denote the coordination environment of the alkali metal cation—in NIB cathodes, this is either P for prismatic or O for octahedral—and a number to indicate the number of TMO₂ layers per unit cell. Many problems remain to be solved before layered sodium transition metal oxide materials, NaₓTMO₂, progress to widespread adoption; one of which is the large number of phase transformations that they undergo during charge and discharge, leading to poorer power performance and capacity retention problems.⁵

Arguably, one of the most promising class of
cathode materials are the P2-type cathodes,
where Na⁺ ions occupy prismatically
coordinated sites, known as either P(2d) sites—
where the edges of the Na⁺ prismatic coordination polyhedra share edges with the *TM* octahedra in the *TMO*₂ layers above and below—or P(2b) sites—where the Na⁺ prisms share faces with the *TM* octahedra [Figure 1(a) and (b), respectively]. Within the Na layer, the large, open faces through which Na⁺ ions in P2-type cathodes hop enable fast charge-discharge rates in these materials when cycled in a NIB. These rates, however, may be severely impacted by Na⁺/vacancy ordering phase transitions, where the development of ordered arrays of Na⁺ and vacancies increases the barrier to Na⁺ hopping,<sup>6,8–10</sup> resulting in a slower rate of Na⁺ extraction and insertion. Such ordering transformations are electrostatically driven and more prevalent in NIB cathodes than their Li⁺ ion counterparts (although ordering in Li materials is seen, such as in LiₓCoO₂ when x = 0.5).<sup>11,12</sup> The origins of this are the greater ionicity (i.e., greater charge separation and weaker covalent interactions) and size of Na⁺ ions compared to Li⁺, resulting in strong Na⁺–Na⁺ and Na⁺–*TM* Coulombic interactions and increased steric hindrance to hopping.<sup>7</sup> Such transitions may also be only partially reversible, resulting in losses in active material, or require that large overpotentials are applied to overcome the barrier to Na⁺ extraction (and to move phase boundaries between ordered and disordered regions through the solid<sup>13</sup>); all are deleterious to the electrochemical performance of the cathode. Therefore, understanding and removing these ordering processes is important if we are to develop new cathodes with good performances.

The ordering pattern adopted by Na⁺ ions depends intimately on the Na⁺ content in the cathode (x in NaₓTMO₂); in P-type materials, known ordering schemes include: row ordering at x = 0.5 and x = 3/5, where P(2b) Na⁺ are arranged in straight rows separated by small zig-zag rows of P(2d) centres; long zig-zag (LZZ, a term coined by Hinuma, Meng and Ceder<sup>9,10</sup>), at x = 2/3 and x = 5/7, where P(2b) Na⁺ surround ‘islands’ of P(2d) Na⁺ ions (note that if lines connected the P(2b) centres, one could describe the hexagonal set of lines as zig-zags; unlike the row ordering, these zig-zags span a larger portion of the unit cell); and ‘droplet’ ordering above x = 0.75 (also coined by Hinuma, Meng and Ceder<sup>9,10</sup>), where islands of P(2b) Na⁺ are surrounded by ‘pools’ of P(2d) Na⁺ [Figure 1].

In addition to the Na⁺ content, the presence of Jahn-Teller (JT) active *TM* ions (either present in the pristine material or created on cycling) can encourage Na⁺/vacancy ordering, for example in Na₅/₈MnO₂, where Na⁺ ions and their vacancies charge-order with the JT-distorted Mn³⁺ centres. In these systems, it was argued that the JT distortion, rather than electrostatics, drives Na⁺/vacancy ordering.<sup>14</sup> As such, several studies have sought to minimise the number of JT centres present in the pristine material and/or during cycling, most commonly achieved in NaₓMnO₂-based materials by doping low-valent cations into the *TM* sublattice and raising the average oxidation state of Mn.

Whilst, in many cases, dopants occupy the *TM* sites randomly, some NIB cathode compositions exhibit ordering on the *TM* sublattice, resulting in a superstructure.<sup>15</sup> This in turn influences the Na⁺ ion ordering patterns. In our earlier work on Na₀.67[Mg₀.28Mn₀.72]O₂ (henceforth NMMO),<sup>20</sup> we showed that the Mg and Mn were partially ordered across the *TM* layers. This material showed a surprisingly simple (considering the partial disordering on the Mg/Mn sublattice<sup>21</sup>) <sup>23</sup>Na NMR spectrum comprising three peaks. Using hybrid DFT calculations and a simplistic statistical averaging model, we attempted to reproduce the observed spectrum. We noted in that work the strong dependency of Na⁺ ion mobility on the state of charge (SOC) and the need for more complex models to accurately describe the effect of motion on the (variable-temperature) spectra. A similar method was also employed by Lin *et al*.<sup>22</sup> to examine the related material P2-Na₂/₃[Mg₁/₃Mn₂/₃]O₂. This description faithfully models these complex NMR spectra at fast Na-ion exchange rates, paving the way for future studies seeking to model paramagnetic NMR spectra in the presence of ion hopping as a means to extract useful information about ion dynamics in paramagnetic NIB cathode materials. These models do not, however, model all regimes of
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<figure><img src="image_3.png" /><figcaption>(b) P(2d)</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>(c) $x = 3/5$ (0.600)</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>(d) $x = 2/3$ (0.667)</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>(e) $x = 5/7$ (0.714)</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>(f) $x = 10/13$ (0.769)</figcaption></figure>

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>

**Figure 1:** Illustrations of the (a) P(2d) and (b) P(2b) sites in P-type Na<sub>x</sub>TMO<sub>2</sub> cathodes and the possible Na<sup>+</sup> ion/vacancy ordering schemes, as proposed by Hinuma, Meng and Ceder in refs. 9 and 10: (c) x = 3/5, corresponding to “row” ordering; (d) x = 2/3 and (e) x = 5/7, corresponding to long zig-zag ordering; and (f) x = 10/13, corresponding to droplet ordering. The black outline indicates the unit cell of the superstructure, whilst the dashed line shows the original parent unit cell (i.e., without Na<sup>+</sup>/vacancy order).

Na⁺ ion motion: they can only capture the high-
temperature, dynamically averaged (fast-
exchanged) regime.

In this work, we use synchrotron X-ray
diffraction (PXRD), powder neutron diffraction
(PND) and neutron pair distribution function
(PDF) analysis on pristine NMMO to establish

the superstructure. Variable-temperature X-
band EPR is used to examine the local
(super)structure of NMMO from the perspective
of the unpaired electron spins. The effect of this
superstructure is examined in the context of Na⁺
ion mobility by using variable-temperature <sup>23</sup>Na
NMR. These spectra are accurately simulated
by solving the Bloch-McConnell equations (see
Theoretical Background) for a set of independent two-site exchanges. These simulations enable the assignment of the NMR spectra to individual local Na⁺ environments and provide estimates of the Na⁺ ion hopping barriers, which are compared to ab initio barriers determined from climbing image nudged elastic band (CI-NEB) and against qualitative observations from molecular dynamics (MD) simulations. Our approach is unique in its ability to simulate fast-, intermediate- and slow-motion regimes, making it applicable to all states of charge and discharge.

## Background Theory.

*Theoretical Background: Hyperfine shifts.* The NMR spectrum of paramagnetic materials is generally dominated by the hyperfine interaction between an unpaired electron spin and nearby nuclear spins, via through-space (dipolar) and through-bond (Fermi contact) components.²² The effects of this interaction are: shortened nuclear longitudinal and transverse relaxation times, T₁ and T₂, respectively, in addition to large magnitude isotropic shifts, the latter usually dominated by Fermi contact.²³⁻²⁵

Previous studies have shown that the hyperfine shift, δ_hyp, of the nucleus of interest (here, ²³Na) may be determined by calculating the shifts induced by a single “bond pathway” between a nucleus and the unpaired spin (here, Na⁺–O–TM) and summing the bond pathway shifts, δ_path, for a given ²³Na environment—i.e.:

$$ \delta_{\text{hyp}} = \sum_{i} z_{i} \delta_{\text{path},i}, \quad (1) $$

where zᵢ is the number of paths with the shift δ_path,i associated and the sum is taken over all paths between the Na⁺ ion and the TM cations in the material.²⁶⁻²⁸ In practice, this sum is only computed over nearest and next-nearest neighbours, as TM spins in outer coordination spheres make negligible contributions to the observed shift. We note that, whilst Mg²⁺ is not a TM cation, we denote the sites occupied by both Mg and Mn as TM sites for the sake of clarity.

The layered P2 cathode has two crystallographic Na⁺ sites: P(2d) and P(2b)

[Figure 1(a)], but due to atomic site disorder, Na ions on a single crystallographic site can have different local environments (i.e., different number of nearby TM cations with different Na⁺–O–TM bond angles and distances), resulting in an NMR spectrum with more than just two resonances.²³,²⁵ In NMMO, three species can occupy the TM layer: Mn³⁺, Mn⁴⁺ or Mg²⁺. All P(2d) sites have six nearest neighbour TM cations and six next-nearest neighbours; all P(2b) sites have two nearest neighbours and 12 next-nearest neighbours. If a ‘random’ distribution of Mn³⁺, Mn⁴⁺ and Mg²⁺ is assumed, 1144 unique Na⁺ environments occur (784 P(2d) and 360 P(2b)); this number significantly increases if more distant shells are considered.

In our previous work, we used a model system, Na₂/₃[Mg₁/₃Mn₂/₃]O₂, to calculate the hyperfine bond pathways, as well as the quadrupole-induced shifts.²⁴ We note that Mg contributes a negligible amount to the total shift of Na as Mg²⁺ is diamagnetic. Attempts to calculate bond pathways involving Mn³⁺—either by adding a single Mn³⁺ centre or adding multiple Mn³⁺ centres (and charge compensating each by adding Na⁺ ions), or by doping a polaron into the system (making the cell charged)—were unsuccessful. Despite this, the bond pathways calculated for this model represent a good starting point with which to model the ²³Na NMR shifts of NMMO (considering that the ‘true’ composition is Na₀.₆₇[Mg₀.₂₈Mn³⁺₀.₁₁Mn⁴⁺₀.₆₁]O₂).

Previous attempts to model the NMR spectra using a random arrangement of Mg²⁺ and Mn⁴⁺ ions (a total of 88 unique local Na⁺ environments), or to model a system with a honeycomb-ordered arrangement of Mg²⁺ and Mn⁴⁺ ions generated spectra inconsistent with experimental spectra. Each of these models, however, assumed a static arrangement of Na⁺ ions—i.e., Na⁺ ions did not hop between sites during the NMR experiment. As observed in several layered NIB and lithium-ion battery (LIB) cathodes,²⁴,²⁸⁻³⁰ the effect of alkali ion hopping on the NMR timescale results in significant changes to the appearance of the spectrum.

The form of the NMR spectrum in the presence of Na⁺ ion motion (more generally known as chemical exchange) depends primarily on the difference in chemical shifts between the sites, Δω, and the hopping rate, k.³¹ Three regimes exist: (1) the fast hopping regime, where Na⁺ ions hop at a rate that is faster than the
difference in the sites' resonant frequencies (i.e. $k \gg \Delta\omega$); (2) the intermediate regime, where $Na^+$ ions hop at a rate which is comparable to the difference in resonant frequencies (i.e., $k \sim \Delta\omega$) and (3) the slow hopping regime, where $Na^+$ ions hop infrequently between sites compared to the frequency difference (i.e. $k \ll \Delta\omega$). In the fast hopping regime, a sharp resonance is observed at a frequency which is the population-weighted average of the sites' frequencies; the intermediate regime results in resonances that are broad; whilst the slow regime is characterised by two (or more) distinct resonances at the resonant frequencies of the different sites [Figure S1].

In our earlier work, we modelled the NMMO spectra by assuming a population weighted average of the $Na^+$ sites (in a similar fashion to Lin et al$^{21}$), using an exchange equilibrium constant, K, to obtain the $^{23}$Na NMR shifts for a Na ion hopping between a P(2d) and a P(2b) site.$^{25}$ This approach is appropriate for the fast regime, but does not capture the behaviour of the intermediate or slow regimes seen in NMMO. In this work, we construct a model which accounts for all three regimes and simulate the variable-temperature $^{23}$Na NMR spectra of NMMO using this model.

## Experimental.

**Synthesis.** The synthetic route used was based on previous reports (see SI, section S1 for further details).$^{16,32}$ Throughout this work, all samples were handled and prepared in an Ar-filled glovebox ($H_2O$ and $O_2 < 1$ ppm).

**Powder Synchrotron X-ray Diffraction.** A powder sample of NMMO was sealed in a borosilicate glass capillary using two-component epoxy resin inside the glovebox. Powder synchrotron X-ray diffraction (PXRD) patterns were acquired on beamline I11 at the Diamond Light Source.$^{33,34}$ The pattern was collected with an exposure time of 1 minute using a position sensitive detector (PSD, Mythen2; X-ray wavelength = 0.826866 Å) over the range $2\theta = 2.0 - 92^{\circ}$. Variable-temperature PXRD patterns were also recorded on heating from 100 K to 500 K (heating rate 3 K min$^{-1}$), with an exposure time of 12 s per pattern (i.e., 0.6 K temperature difference between the start and end of acquisition), using the same PSD and $2\theta$ range.

**Powder Total Neutron Scattering.** Powder neutron diffraction (PND) measurements were carried out at the ISIS Pulsed Neutron and Muon Source using the GEM instrument.$^{35}$ A powder sample of pristine $Na_{0.67}[Mg_{0.28}Mn_{0.72}]O_2$ (3.44 g) was loaded into a thin-walled vanadium canister (8 mm diameter) to a height of 59 mm to maximise scattering from the sample. Total scattering data were initially collected at 130 K, then 1.6 K, followed by a series of diffraction patterns acquired from 2 K up to 7 K in 1 K steps. The complete list of temperature steps and data collections may be found in the SI (section S4). The data were corrected for absorption effects using the Mantid software package$^{36}$ to provide data suitable for Rietveld refinement. To normalise the total scattering data, additional measurements of the empty vanadium canister, empty cryostat and vanadium rod were taken and all data were processed within the GudrunN software to produce absolute total scattering structure functions, $F(Q)$. The $F(Q)$ were then Fourier transformed to provide pair distribution functions, PDF, or $D(r)$, as defined in ref. $^{37}$.

The structure was determined by Rietveld refinement$^{37,38}$ against powder neutron diffraction data collected above the Néel temperature, $T_N$, using a model derived from the previously reported powder diffraction structure of $Na_{0.67}[Mg_{0.28}Mn_{0.72}]O_2$.$^{16}$ All refinements were carried out using TOPAS Academic 6.0.$^{39}$

**ISODISTORT and Refining Superstructures.** Candidate superstructures were generated using the software package ISODISTORT, part of the ISOTROPY suite.$^{40}$ Occupancies and displacements of ions in the structure are calculated based on the irreducible representations of the parent structure space group which are consistent with the ordering scheme provided (the ordering scheme may be deduced from the location of superlattice reflections). The standalone packages ISOVIS and ISOVISQ are especially useful for identifying symmetry-compatible distortion modes, as well as for visualising the effects of distortions on the structure and diffraction pattern.

Rietveld refinements of the superstructure using the candidate $Na^+$ ion occupation and Mg/Mn occupation irreducible representations (irreps) showed that the $M_2^+$ irrep structures
were consistent with the experimental data. On this basis, the superstructure was refined using the P3 mode of the $M_{2+}$ irrep. Here, P3 is distinct from the labels for cathode structures, the P here denoting a one-dimensional order parameter direction, and three an arbitrary label (i.e., the third such one-dimensional order parameter direction). All refinements were carried out by simultaneously refining against the pair distribution function data (obtained from a Fourier transform of the total scattering data in GudrunN), the diffraction data collected on banks 2 to 5 and the PXRD data collected.

For refinement against the PND and laboratory XRD data, the background was fit with a 12-term Chebychev polynomial. The lattice parameters and atomic positions were allowed to refine freely, aside from restraints on the Mn-O (ca. 1.85 Å) and Mg-O (ca. 2.00 Å) bond lengths. Separate sets of isotropic atomic displacement parameters were freely refined for Na, Mg, Mn and O atoms, whilst the occupancies of Na, Mg and Mn were refined using the symmetry modes of the irrep and under the constraint that the sample composition was $Na_{0.67}[Mg_{0.28}Mn_{0.72}]O_2$.

**Electrochemistry.** Electrodes of NMMO were prepared as described in the SI. All electrochemical tests were conducted using NMMO/Na metal half-cells in 2032 stainless-steel coin cells. Each cell was assembled from a stack of one cathode, one glass fibre separator (Whatman, GF/B, 0.68 mm thick, 16 mm diameter, 1.0 µm pore size) soaked with 150 µL of electrolyte (1.0 M $NaPF_6$ in propylene carbonate, PC), and one Na metal disc. The NMMO/Na cells were galvanostatically charged and discharged at a rate of 10 mA g⁻¹ (corresponding to approximately C/19, for a theoretical C rate determined from the time elapsed and current applied, assuming that x in $Na_x[Mg_{0.28}Mn_{0.72}]O_2$ varies between 0 and 1 and that no parasitic reactions take place during cycling) over a voltage window of 1.5–4.5 V (vs. Na/Na⁺). A slow cycling rate was chosen to minimise the effect of ionic concentration gradients and large overpotentials often seen at higher cycling rates.

**Solid-state Nuclear Magnetic Resonance Spectroscopy.** Electrochemically cycled cathodes of NMMO were prepared by cycling a cathode to a given cutoff voltage and allowing the cell to rest for at least one hour. The cell was opened inside an Ar-filled glovebox and the

cathode extracted, washed in dimethyl carbonate (DMC; approximately 1 cm³; Sigma-Aldrich, 99%, anhydrous) and dried *in vacuo* for at least 20 minutes. The cathode was then scraped off the Al foil current collector and either packed into a 1.3 mm diameter $ZrO_2$ magic angle spinning (MAS) rotor or centre-packed into a 2.5 mm diameter $ZrO_2$ MAS rotor, using polytetrafluoroethylene (PTFE) tape to fill the gap at either end of the rotor and Vespel caps at each end. No rotor spent longer than 10 minutes outside of the glovebox before being inserted into the magnet under a protective atmosphere of flushing nitrogen gas.

The majority of the $^{23}$Na NMR spectra were recorded on a Bruker Avance III 11.7 T spectrometer using a Bruker 2.5 mm MAS probe with MAS frequencies of 28 kHz or a Bruker 1.3 mm MAS probe at an MAS rate of between 30 and 60 kHz. In all cases, a rotor-synchronised Hahn-echo pulse sequence was used for quantitative measurements and the recycle delay (25 ms; at least $5T_1$) was set such that the bulk, paramagnetically shifted signal was recorded quantitatively, while the diamagnetic signal due to electrolyte decomposition products was suppressed. An effective $\frac{\pi}{2}$ pulse length of 1.15 µs (for the 2.5 mm probe) or 0.67 µs (for the 1.3 mm probe; this corresponds to $\frac{\pi}{6}$, which is longer than the selective $\frac{\pi}{8}$ pulse which would ensure all quadrupolar $^{23}$Na centres are in the quadrupolar liquid limit;⁴¹ a compromise was therefore selected between the linear quadrupolar regime and maximising the signal intensity). The 2.5 mm $ZrO_2$ MAS rotor was chosen so that a relatively wide range of temperatures could be accessed; the spinning speed chosen was 28 kHz, as this was sufficient to separate the spinning sideband manifold from its isotropic resonance whilst ensuring rotor stability. Spectra were also recorded on a Bruker Avance III 16.4 T spectrometer using a Bruker 1.3 mm MAS probe with an MAS frequency of 50 kHz and an effective $\frac{\pi}{2}$ pulse length of 0.58 µs (again corresponding to $\frac{\pi}{6}$). Additional experiments were carried out using a Bruker Avance III 9.4 T spectrometer using a Bruker 1.3 mm MAS probe with an MAS frequency of 35 kHz and an effective $\frac{\pi}{2}$ pulse length of 1.47 µs (again corresponding to $\frac{\pi}{6}$).
All spectra were referenced to solid NaCl at 7.21 ppm and scaled according to the sample mass and number of residuals recorded.

For all experiments, temperature calibration measurements were performed either *ex situ* using the ²⁰⁷Pb shift of Pb(NO₃)₂ (Alfa Aesar, 99%) or *in situ* by mixing pristine NMMO powder with KBr (Aldrich, 99% FTIR grade; dried at 473 K for 12 hours under dynamic vacuum) and recording the ⁷⁹Br spin-lattice T₁ relaxation time. Calibrations were run under the same heater powers, nitrogen gas flow rates, and driving and bearing pressures used for ²³Na NMR experiments; the calibration curves are given in the SI [Figures S2 and S3]. Using the *in situ* calibration curves obtained at 11.7 T for KBr at probe temperatures between 300 K and 325 K and MAS rates between 30 kHz and 50 kHz, sets of probe temperatures were selected at each MAS speed, so that, at each speed, the sample temperature remained constant (to within 2 K, corresponding to a change of approximately 15 ppm to the isotropic shifts of NMMO) and only the MAS rate changed.

*Continuous Wave X-Band Electron Paramagnetic Resonance Spectroscopy.* Continuous-wave X-band EPR measurements were performed on a Bruker E500 X-band spectrometer with an ER 4122SHQE cavity, tuned to 9.373 GHz. The external magnetic field was modulated at 100 kHz with a modulation amplitude of 0.3 mT. The microwave power was set to 0.6325 mW, sufficient to avoid saturation. The as-synthesised powder (prepared as above) was loaded into a quartz EPR tube (Aldrich, Wilmad® CFQ tubes, outer diameter 2 mm) inside an Ar-filled glovebox and then loaded inside an Oxford Instruments ESR900 cryostat with a temperature stability of 0.5 K. The sample was first cooled to base temperature (5 K) and spectra recorded on heating, with a five-minute equilibration period allowed once temperature stability was reached. All EPR spectra were fitted to a powder pattern lineshape with isotropic g tensors using the EasySpin toolbox for MATLAB.⁴²

*First Principles Calculations.* Three model systems were examined for calculation of NMR and Na⁺ ion diffusion parameters: firstly, a (2 × 1 × 2) supercell of honeycomb-ordered P2-Na₂/₃[Mg₁/₃Mn₂/₃]O₂ with 88 atoms (chosen to capture as many exchange interactions as

possible; structures available as part of the SI) for NMR shifts in a Mn⁴⁺-only system; secondly, a (2 × 2 × 1) supercell of P2-Na₂/₃[Mg₁/₃Mn₂/₃]O₂ for Na⁺ ion hopping barriers (chosen to minimise self-interactions between hopping Na⁺ centres) and lastly a (2 × 2 × 1) supercell of P2-Na[Mg₁/₃Mn₂/₃]O₂ for NMR calculations in a mixed Mn³⁺/Mn⁴⁺ system, in which there are equal number of Mn³⁺ and Mn⁴⁺ centres. Whilst this composition differs from pristine NMMO, it is assumed that the Mn³⁺ and Mn⁴⁺ bond pathways are independent and therefore additive, based on previous studies.²⁴,²⁶

*NMR Shift Calculations.* The ²³Na NMR shifts of different local Na⁺ ion environments were calculated using methods outlined previously.²⁶⁻²⁸,⁴³ An initial geometry optimisation was performed using the Vienna Ab Initio Software Package (VASP) code (energy tolerance 10⁻⁵ eV; force tolerance 0.02 eV Å⁻¹),⁴⁴⁻⁴⁶ employing the projector-augmented wave (PAW) method.⁴⁷,⁴⁸ Spin-polarised Perdew-Burke-Ernzerhof exchange-correlation functionals were used under the Hubbard U model⁴⁹,⁵⁰ within the rotationally invariant formalism proposed by Liechstenstein et al.⁵¹ to correct for known deficiencies of pure functionals in highly localised 3d states.⁵² A plane-wave energy cutoff of 520 eV was chosen, alongside an effective Hubbard U parameter on the Mn d states of U<sub>eff</sub> = U − J = 3.9 eV, where U and J are the effective on-site Coulomb and exchange parameters (J = 1 eV), respectively, in line with previous work on the parent material, NaₓMnO₂.⁵³ The Brillouin zone was sampled with a Monkhorst-Pack⁵⁴ k-point mesh of density < 0.5 Å⁻¹.

Periodic spin-polarised DFT calculations of the hyperfine and quadrupole-induced shifts were performed in CRYSTAL17.⁵⁵ Hyperfine parameters were calculated with B3LYP⁵⁶,⁵⁷ and a modified B3LYP hybrid functional containing 20% and 35% Hartree-Fock exchange, referred to as Hyb20 and Hyb35, respectively. These weights were chosen based on the success of these functionals in calculating the properties of TM compounds and have been previously reported to provide an upper and lower bound on experimental shifts.²⁶⁻²⁸ Additional computational details, including the number of Gaussian primitives and the contraction scheme used for each basis
set, alongside details of convergence criteria used, are provided in the SI (section S1).

*Climbing Image Nudged Elastic Band Calculations.* The energy barriers to Na⁺ ion hopping were determined using the climbing image nudged elastic band (CI-NEB) double-ended transition state searching method implemented in VASP.⁵⁸ In all cases, a geometry optimisation of the initial and final states was carried out using PBE functionals, with the same *U*<sub>eff</sub> parameter and plane-wave energy cutoff as above, but a Gamma-centred *k*-mesh sampling of < 0.2 Å⁻¹ and energy tolerance of 10<sup>-7</sup> eV. To obtain the energy barriers, five linearly interpolated image states were generated between the initial and final states using the VTST packages,⁵⁹ and the forces perpendicular to the path connecting them were minimised to < 0.01 eV Å⁻¹. Linear interpolation of the forces between these images was then carried out to obtain the energies between each pair of images. The composition of each cell is given in the SI (section S9).

*Molecular Dynamics Simulations.* Molecular dynamics (MD) simulations—used to assess Na⁺ ion mobility—were also carried out in VASP. To ensure the calculations were affordable, the plane wave kinetic energy cutoff was lowered to 400 eV and Gamma-centred *k*-point sampling was kept at < 0.5 Å⁻¹. The MD simulations were carried out in the NVT ensemble using a Nosé-Hoover thermostat, with a time step of 2 fs. To ensure diffusion could be observed over the timescales used, elevated temperatures were required,⁶⁰,⁶¹ with simulations being carried out at a range of temperatures from 400 to 800 K. The systems were equilibrated for a period of 5 ps and the simulations then run for at least 160 ps. Final configurations of MD runs were visually inspected and then optimised to ensure the *TM* framework had not changed. The self-diffusion coefficient, *D*<sup>+</sup>, could not be obtained, owing to highly correlated Na⁺ ion motion (see results).

## Results.

*Synchrotron X-Ray Diffraction, Neutron Diffraction and Pair Distribution Function Analysis of Cation Ordering.* NMMO was synthesised *via* a high-temperature solid-state reaction between Na₂CO₃, MgO and Mn₂O₃ using the route reported by us²⁵ and by Maitra

*et al.* Simultaneous Rietveld refinements³⁷,³⁸ of the synchrotron X-ray diffraction (PXRD) and powder neutron diffraction (PND) data showed that the majority of reflections could be indexed within the previously reported structure of NMMO (space group *P*6₃/mcm),¹⁶,³² in which the Mg and Mn centres are randomly distributed across the *TM* sublattice and Na⁺ ions occupy the Na sublattice with a random occupancy [Figure 2(a), Table 1]. Additional reflections (for example, at *Q* = 0.91 Å⁻¹, 1.99 Å⁻¹, 2.21 Å⁻¹ and 2.83 Å⁻¹, amongst others, see Figure 2 inset) were also present; these have been observed and qualitatively ascribed by several authors (studying both NMMO and other P2 NIB cathode materials)¹⁶,⁶² to ordering on the *TM* and/or Na sublattices, but the nature of the ordering was not determined in these previous studies. The low-intensity reflection at *Q* = 1.88 Å⁻¹ was assigned to a Na₂CO₃ impurity phase (consistent with ²³Na NMR; see later).

Further refinements of the PXRD and PND data to identify any potential superstructures proved challenging as the additional reflections could be indexed to both (2 × 2 × 1) and (√3 × √3 × 1) ordering vectors. To test possible ordering schemes, several hundred candidate structures based on these vectors with different combinations of symmetry distortions of Na and/or O displacements, as well as Na, Mg and Mn occupancies, were generated using the ISODISTORT software package.⁴⁰ Careful consideration of the low-Q reflection at *Q* = 0.91 Å⁻¹ and the maximal non-isomorphic subgroups of the parent *P*6₃/mcm space group indicated that only a handful of space groups allowed this low-Q reflection (see supporting information, SI). Of these, the best fit was obtained with both the parent structure (35 wt.%) and the P3 mode under the *M*₂⁺ irrep. (i.e., a (2 × 2 × 1) expansion of the parent structure with space group *P*6₃/m indicated by the bold cell in Figure 1(d), 65 wt.%). This refinement produced a better match to the observed diffraction patterns than refinement to the parent alone, indicating that a superstructure is required to capture the 'true' structure of NMMO [Figure 2(b)]. Note that, in the refinements, the Na, Mg and Mn occupancies were allowed to refine separately (within the constraints dictated by the symmetry
<figure><img src="image_10.png" /><figcaption>**Figure 2:** Rietveld refinement of the PXRD pattern collected for pristine NMMO at room temperature against (a) the parent phase structure only ($R_{wp}$ = 11.85%) and (b) the parent and superstructure phases together ($R_{w.p.}$ = 3.93%). Insets show the fits in the regions $Q = 0.85 - 0.95 \AA^{-1}$ and $Q = 1.75 - 2.25 \AA^{-1}$, where superstructure reflections are prevalent. In (c), Rietveld refinements of neutron pair distribution function data recorded at 130 K over the region $r = 1 - 40 \AA$, against, in the same model, both the parent (35 wt.%) and superstructure (65 wt.%) are shown; (d) shows the same refinement in the region $1 - 10 \AA$. In (e), (f) and (g), views of the Rietveld-derived ($2 \times 2 \times 1$) superstructure and expanded view of the ordering in the Na and Mg/Mn layers, respectively, are shown. Note that the Mg and Mn occupancies have been set to be perfectly honeycomb ordered and that all Na occupancies have been set to 1, for clarity. For the true occupancies, please see site occupancies in Table 1.</figcaption></figure>

mode distortions), such that each Mg/Mn site was fully occupied and the overall stoichiometry of the sample was preserved.

The superstructure Mg/Mn ordering reflects the ordering seen in the parent material—Mg at the corners, centre and centres of edges of the unit cell, with Mn surrounding them (Figure 2g). This cell, however, has more pronounced ordering than in the parent (see site occupancies in Table 1). The superstructure generates eight different Na sites—four that are P(2d)- and four P(2b)-like. In Figure 2(f) we have relabelled the Na sites as Na$_{a}$ and Na$_{b}$, the two different

subgroups each containing both P(2d) and P(2b) coordination, so as to illustrate the Na ion ordering and its relationship to the LZZ ordering seen in Na$_{0.67}$CoO$_{2}$, with Na ions either in the 'long zig-zags' (to use Hinuma, Meng and Ceder's description$^{9,10}$) or in the 'islands' between the long zig-zags. Some of these sites are clearly too close together to be simultaneously occupied, as discussed later.
Refinement of both the (2 × 2 × 1) superstructure and parent structure against, simultaneously, the synchrotron and neutron diffraction, and the neutron pair distribution function (PDF) data gave an excellent match, suggesting it also provides an accurate description of the local structure [Figure 2(c)]

and (d)]. We note, however, that the fit to the long-range structure from PXRD and PND is not perfect, with some of the calculated reflections having intensities that are lower than the observed reflections—for example, the (200), (210) and (212) at Q = 1.44 Å⁻¹, 1.99 Å⁻¹, 2.21 Å⁻¹, respectively [Figure 2(b)]

<table><caption>**Table 1:** Rietveld-derived lattice parameters, atomic coordinates and occupancies for the parent structure of P2-NMMO at room temperature, R<sub>w.p.</sub> 9.00%, and the (2 × 2 × 1) superstructure, R<sub>w.p.</sub> 3.93%. Standard errors are shown in parentheses, whilst asterisks denote values which were not refined. The coordination of the Na⁺ site (P(2d) or P(2b)) and designated name in the superstructure (Na<sub>a</sub> or Na<sub>b</sub>) is indicated next to the Wyckoff site label.</caption><thead><tr><th colspan="2">Parent</th><th></th><th></th><th></th><th></th><th></th><th></th></tr><tr><th colspan="2">Space group</th><th colspan="4">P6<sub>3</sub>/mcm</th><th></th><th></th></tr><tr><th>a (Å)</th><th>5.02634(2)</th><th>α (°)</th><th>90*</th><th></th><th></th><th></th><th></th></tr><tr><th>b (Å)</th><th>5.02634(2)</th><th>β (°)</th><th>90*</th><th></th><th></th><th></th><th></th></tr><tr><th>c (Å)</th><th>11.1929(10)</th><th>γ (°)</th><th>120*</th><th></th><th></th><th></th><th></th></tr><tr><th></th><th>Site</th><th>x</th><th>y</th><th>z</th><th>Occupancy</th><th>B<sub>eq</sub> (Å<sup>2</sup>)</th><th></th></tr></thead><tbody><tr><td>Mg1</td><td>2b</td><td>0</td><td>0</td><td>0</td><td>0.49(2)</td><td>0.509(10)</td><td></td></tr><tr><td>Mn1</td><td>2b</td><td>0</td><td>0</td><td>0</td><td>0.51(2)</td><td>0.509(10)</td><td></td></tr><tr><td>Mg2</td><td>4d</td><td>1/3</td><td>2/3</td><td>0</td><td>0.174(2)</td><td>0.509(10)</td><td></td></tr><tr><td>Mn2</td><td>4d</td><td>1/3</td><td>2/3</td><td>0</td><td>0.826(2)</td><td>0.509(10)</td><td></td></tr><tr><td>O</td><td>12k</td><td>1/3</td><td>1/3</td><td>0.0880(3)</td><td>1*</td><td>0.569(10)</td><td></td></tr><tr><td>Na1</td><td>6g (P(2d))</td><td>0.301*</td><td>0</td><td>1/4</td><td>0.413(6)</td><td>0.26(6)</td><td></td></tr><tr><td>Na2</td><td>4c (P(2b))</td><td>1/3</td><td>2/3</td><td>1/4</td><td>0.386(6)</td><td>0.26(6)</td><td></td></tr><tr><th colspan="2">(2 × 2 × 1)</th><th></th><th></th><th></th><th></th><th></th><th></th></tr><tr><th colspan="2">Space group</th><th colspan="4">P6<sub>3</sub>/m</th><th></th><th></th></tr><tr><th>a (Å)</th><th>10.05266(5)</th><th>α (°)</th><th>90*</th><th></th><th></th><th></th><th></th></tr><tr><th>b (Å)</th><th>10.05266(5)</th><th>β (°)</th><th>90*</th><th></th><th></th><th></th><th></th></tr><tr><th>c (Å)</th><th>11.1929(10)</th><th>γ (°)</th><th>120*</th><th></th><th></th><th></th><th></th></tr><tr><th></th><th>Site</th><th>x</th><th>y</th><th>z</th><th>Occupancy</th><th>B<sub>eq</sub> (Å<sup>2</sup>)</th><th></th></tr><tr><td>Mg1_1</td><td>2b</td><td>0</td><td>0</td><td>0</td><td>0.82(5)</td><td>0.51(10)</td><td></td></tr><tr><td>Mg1_2</td><td>6g</td><td>1/2</td><td>0</td><td>0</td><td>0.82(5)</td><td>0.51(10)</td><td></td></tr><tr><td>Mn1_1</td><td>2b</td><td>0</td><td>0</td><td>0</td><td>0.18(5)</td><td>0.51(10)</td><td></td></tr><tr><td>Mn1_2</td><td>6g</td><td>1/2</td><td>0</td><td>0</td><td>0.18(5)</td><td>0.51(10)</td><td></td></tr><tr><td>Mg2_1</td><td>12i</td><td>1/3</td><td>1/3</td><td>0</td><td>0.02(3)</td><td>0.51(10)</td><td></td></tr><tr><td>Mg2_2</td><td>4f</td><td>1/3</td><td>2/3</td><td>1/2</td><td>0.02(3)</td><td>0.51(10)</td><td></td></tr><tr><td>Mn2_1</td><td>12i</td><td>1/6</td><td>1/3</td><td>0</td><td>0.98(3)</td><td>0.51(10)</td><td></td></tr><tr><td>Mn2_2</td><td>4f</td><td>1/3</td><td>2/3</td><td>1/2</td><td>0.98(3)</td><td>0.51(10)</td><td></td></tr><tr><td>O1_1</td><td>12i</td><td>0.1529(5)</td><td>0</td><td>0.5962(7)</td><td>1*</td><td>0.57(10)</td><td></td></tr><tr><td>O1_2</td><td>12i</td><td>0.1529(5)</td><td>1/2</td><td>0.5962(7)</td><td>1*</td><td>0.57(10)</td><td></td></tr><tr><td>O1_3</td><td>12i</td><td>0.6529(5)</td><td>0*</td><td>0.5962(7)</td><td>1*</td><td>0.57(10)</td><td></td></tr><tr><td>O1_4</td><td>12i</td><td>0.6529(5)</td><td>1/2</td><td>0.5962(7)</td><td>1*</td><td>0.57(10)</td><td></td></tr><tr><td>Na1_1</td><td>6h (P(2d), Na<sub>a</sub>)</td><td>0.8648(12)</td><td>0.0227(12)</td><td>1/4</td><td>0.40(4)</td><td>0.26(6)</td><td></td></tr><tr><td>Na1_2</td><td>6h (P(2d), Na<sub>a</sub>)</td><td>0.8031(12)</td><td>1/2</td><td>1/4</td><td>0.53(14)</td><td>0.26(6)</td><td></td></tr><tr><td>Na1_3</td><td>6h (P(2d), Na<sub>b</sub>)</td><td>0.3421(12)</td><td>0.9773(12)</td><td>1/4</td><td>0.40(4)</td><td>0.26(6)</td><td></td></tr><tr><td>Na1_4</td><td>6h (P(2d), Na<sub>b</sub>)</td><td>0.4039(12)</td><td>1/2</td><td>1/4</td><td>0.26(13)</td><td>0.26(6)</td><td></td></tr><tr><td>Na2_1</td><td>6h (P(2b), Na<sub>b</sub>)</td><td>0.1348(4)</td><td>0.3652(4)</td><td>1/4</td><td>0.41(5)</td><td>0.26(6)</td><td></td></tr><tr><td>Na2_2</td><td>2d (P(2b), Na<sub>a</sub>)</td><td>2/3</td><td>1/3</td><td>1/4</td><td>0.41(5)</td><td>0.26(6)</td><td></td></tr><tr><td>Na2_3</td><td>6h (P(2b), Na<sub>a</sub>)</td><td>0.8970(4)</td><td>0.6985(4)</td><td>1/4</td><td>0.41(5)</td><td>0.26(6)</td><td></td></tr><tr><td>Na2_4</td><td>2c (P(2b), Na<sub>b</sub>)</td><td>1/3</td><td>2/3</td><td>1/4</td><td>0.41(5)</td><td>0.26(6)</td><td></td></tr></tbody></table>
inset]. To identify possible discrepancies in our structural model that caused these differences, a Fourier difference map was synthesised,

which suggested that Mg and Mn partially occupy tetrahedral sites within the *TM* layer. Given the good fit of Mn and Mg *K*-edge EXAFS

<figure><img src="image_11.png" /><figcaption>**Figure 3:** Variable-temperature PXRD and neutron PDF data: (a) shows the expansion, $\epsilon$, in the lattice parameters, *a* and *c*, of the superstructure, and (b) Na site occupancies for both the parent and (2 × 2 × 1) superstructure. Note that the *a* lattice parameter for the parent structure is one half of that for the superstructure, whilst the *c* axis is the same in both the parent and superstructures. The expansion is calculated in the following way: $\epsilon = (x-x_{295K})/(x_{295K}-x)$ where *x* is *a* or *c* and $x_{295K}$ the value of the parameter at 295 K, to compare to lattice parameters under battery operating conditions. The errors for the lattice parameters and occupancies are given by the sizes of the markers. In (c), the variable-temperature neutron PDF data is indicated, with the *r* = 4.1 Å to 4.35 Å and *r* = 2.1 Å to 2.6 Å regions highlighted. In (d), an expansion of the Na layer of the (2 × 2 × 1) superstructure is shown, with the positions of selected Na and Mg centres highlighted to show exemplar $Na^+$ ions with short $Na^+-Na^+$ distances that cannot be simultaneously occupied and are likely undergoing rapid exchange.</figcaption></figure>
data published in work of Boivin *et al*. and us to octahedral Mg and Mn, tetrahedral Mg/Mn was deemed unlikely.¹⁷,²⁰ Additional refinements including domains of O2- or OP4-NMMO indicated that neither was present in the pristine material.

It therefore appears likely that the mismatched intensities arise from additional displacement and/or occupational distortion modes; the fit was not improved by incorporating simple preferred orientation models or without introducing a large number of additional degrees of freedom.

Overall, our refinement demonstrated the presence of both Mg/Mn ordering and Na ordering: the Mg and Mn centres show a strong tendency towards honeycomb order, with Mg ions preferentially occupying the centre, corners, centres of edges and faces of the unit cell and Mn centres occupying the remaining sites [Figure 2(e), (g)].

The effect of Na⁺ ion ordering on the local structure is observed in the PDF data, where the peaks between r = 2.06 Å and r = 2.76 Å correspond to Na-Na distances in the (2 × 2 × 1) supercell; these are not captured by the parent only, as seen in additional refinements in the SI [Figure S7]. Note that the total number of Na sites in each Na<sub>a</sub>/Na<sub>b</sub> group is equal, with a 2:3 ratio of P(2b) and P(2d) sites, respectively [Table 1].

Variable-temperature PXRD (between 100 K and 500 K, on heating) and PND patterns and PDF data (between 1.6 K and 295 K, again on heating) were collected [Figure 3] to explore Na mobility and changes in ordering with temperature. As expected, on increasing the sample temperature, the refined *a* and *c* lattice parameters increased, with *c* increasing approximately twice as much as *a*, due to the greater flexibility in the *c* axis [Figure 3(a)].

The variable-temperature PXRD also revealed changes in Na⁺ site occupancies; in P2-NMMO, the ratio of P(2b) to P(2d) sites is 2:3, and in a truly "random" occupation of the Na⁺ sublattice, this ratio should be reflected in the site occupancies. Thermodynamically, however, the P(2d) sites are anticipated to lower in energy, owing to the lower Coulombic repulsions (between Na⁺ and the Mg and Mn cations in the adjacent TMO₂ layers) in these sites. The trend observed for the P(2d) (Na1_1 to Na1_4 in Table 1) and P(2b) (Na2_1 to

Na2_4 in Table 1) Na⁺ ions in the superstructure follows the expected trend: P(2d) has a higher occupancy at low temperature and this occupancy decreases on increasing temperature due to thermally activated occupancy of the P(2b) sites [Figure 3(b)]. Refinements carried out with the parent structure do not follow this trend, with occupancies tending to 1:1 at low temperatures [Figure 3(b)], further suggesting that the parent structure does not accurately capture Na⁺ ion ordering.

The most noticeable differences in the variable-temperature neutron PDF data occur in the region r = 4.1 Å to 4.3 Å: at 1.6 K, two distinct peaks at 4.12 Å and 4.31 Å are observed, whilst at 295 K, the intensity of these peaks drops and an additional (negative) shoulder peak at approximately 4.20 Å appears [Figure 3(c)]. Based on the scattering factors of Mn and Mg (-3.73 fm and +5.375 fm⁶⁴), these distances correspond to the distance between a Mg²⁺ cation in the *TM* layer and a next-nearest neighbour Na⁺ centre (e.g., Mg1_1 and Na1_3 or Na2_1 or Mg1_2 and Na1_2 or Na2_2; Figure 3(d)), suggesting that the time-averaged distance between these sites is changing, likely due to Na⁺ ion motion and changes in Na site occupancies. Note that on increasing temperature, the lattice expansion would drive an increase in both distances (by approximately 0.005 Å), rather than an averaging of the two distances. A small decrease in intensity of the peak at 2.40 Å is also seen from 1.6 K to 295 K, alongside a more distinctive, better resolved peak is seen at 2.45 Å; both are consistent with Na⁺-Na⁺ distances (e.g., Na1_2 and Na2_3 to Na1_4; Figure 3(d)). The changes in intensity are consistent Na⁺ ion hopping and occupancy changes.

Finally, it is noted that NMMO exhibits diffuse magnetic scattering in the neutron total scattering data below 40 K, but no new Bragg peaks appear, suggesting spin-glass-like behaviour, as seen in other layered cathode materials [Figure S10].⁶³ This highlights the absence of long-range magnetic ordering in NMMO as explored below using variable-
<figure><img src="image_12.png" /><figcaption>**Figure 4:** Variable-temperature X-band EPR of NMMO. (a) shows the spectra acquired between 5 K and 295 K, with inset showing the subtle changes in the point at which the spectrum crosses the x-axis (indicating a change in *g* with temperature), whilst the fitted *g* values for the two resonances, A and B, are plotted in (b). The fit to the 295 K spectrum is displayed in the inset. The fitted variable-temperature intensities, (c), and linewidths, (d), of the X-band EPR spectra for NMMO. Intensities are arbitrary, so no scale is shown. In (c), a line has been added between data points as a guide to the eye; the inset shows the paramagnetic region where the Curie-Weiss law was fit to the data. Error bars are shown in the inset but omitted in the main plot for clarity. In (d), the linewidth is plotted against the model described in the text with three different curves, each with different critical exponents, $\beta$. Inset shows the overall variation in linewidth.</figcaption></figure>

temperature electron paramagnetic resonance (EPR) spectroscopy.

Overall, the PXRD, PND and PDF results are consistent with the presence of ordering on both the *TM* and $Na^+$ sublattices, as well as being consistent with a change in $Na^+$ ion hopping rates/occupancies with temperature.

To investigate the *TM* ordering scheme and local environments of the *TM* centres in NMMO further, variable-temperature, variable-frequency EPR spectroscopy of NMMO was carried out.
Variable-Temperature X-Band EPR Spectroscopy. At X-band frequencies (ca. 9 GHz) at room temperature, pristine NMMO displays a broad resonance spanning the entire field sweep (0 – 1 T; Figure 4(a)). This peak-to-peak linewidth, $\Delta H_{pp}$, is dominated by the strong electron-electron dipolar interaction, but also has contributions from a distribution of environments and exchange coupling (which at room temperature has a negligible effect). Only $Mn^{4+}$ species are observed: transitions involving the electron spin microstates of $Mn^{3+}$ cannot be seen, due to the large zero-field splitting of $Mn^{3+}$ ($S = 2$) centres.$^{66}$

The simplest fit to the spectrum is to fit to two isotropic *g*-factors; fitting to a single isotropic *g*-factor yielded poorer results, whilst an anisotropic *g*-tensor or more than two resonances over-fit the spectra with little improvement in the quality of the fit. The two resonances (labelled A and B, Figure 4(b) inset) most likely originate from $Mn^{4+}$ ions with different numbers of $Mn^{3+}$, $Mn^{4+}$ and $Mg^{2+}$ nearest neighbours. A more detailed assignment of these two features is presented later (see Discussion). We again note that, in reality, the spectrum is made up of several overlapping resonances, each corresponding to a unique local $Mn^{4+}$ environment.

On cooling NMMO from 290 to 50 K, the X-band EPR spectrum increases in intensity, the *g*-factors of both resonances decrease—this is more clearly seen in on close inspection of the spectra and their fits [Figure 4(a) inset and (b)]—and $\Delta H_{pp}$ generally increases (i.e., the resonance initially narrows, then broadens; Figure 4(a) to (d)). The increase in intensity is ascribed to the increase in the static susceptibility of the sample, whilst the decrease in *g* and narrowing of signals on cooling can be ascribed to an exchange interaction effect.$^{65,66}$ As temperature decreases further, spin fluctuations occur less frequently (due to less thermal randomisation) and the spatially anisotropic magnetic exchange interactions become poorly averaged and local magnetic environments with different anisotropic exchange interactions start to be observed and spin clusters develop, causing a broadening of the resonances.$^{65}$ Below 50 K, the observed signals arise from transitions between the microstates of spin clusters (these are a distinct set of spin microstates from the localised, single-centre spin states seen at higher temperature).$^{66}$ Cooling the sample therefore

enables new magnetic environments to be observed whose resonant frequencies differ from those in the paramagnetic state and are thus difficult to detect.

The temperature variation in the maximum intensity of the overall spectrum (i.e., the combined intensity of resonances A and B together) was fit to a Curie-Weiss law [Figure 4(c)] between 50 and 300 K, yielding a localised mean-field Weiss constant, $\theta_i$, of –36 K, reflecting weak antiferromagnetic coupling; the measured value from bulk magnetic susceptibility measurements in the literature is also negative but is smaller (–9 K$^{25}$). The EPR-derived Weiss constant only probes local interactions, rather than the bulk average, likely accounting for the difference between the two values. Among the exchange interactions in pristine NMMO, the antiferromagnetic $Mn^{3+}$–$Mn^{3+}$ interaction (mediated via $e_g$ orbitals and by direct overlap between $t_{2g}$ orbitals due to shortening of the JT-distorted Mn–O bonds) should dominate in strength, but will likely have a low probability of occurrence (since, per formula unit, there are only ca. 0.11 equivalents of $Mn^{3+}$); the $Mn^{3+}$–$Mn^{4+}$ and $Mn^{4+}$–$Mn^{4+}$ interactions will be either weakly ferromagnetic or weakly antiferromagnetic, owing to the competing ferromagnetic superexchange (from $t_{2g}$ to $t_{2g}$ via intervening O 2p orbitals; see SI, Figure S13) and antiferromagnetic direct exchange interactions; the former will have a slightly higher probability of occurrence than the $Mn^{3+}$–$Mn^{3+}$ interaction; the antiferromagnetic $Mn^{4+}$–$Mn^{4+}$ interaction will be the most abundant, accounting for the overall negative Weiss constant.

Extending the local Weiss constant method to the individual resonances, we obtain $\theta_A = –41$ K and $\theta_B = –18$ K. This also matches well to the assignment and expected magnitudes of the exchange interactions between $Mn^{3+}$ and $Mn^{4+}$. Plots of 1/intensity vs temperature revealed an approximately straight-line trend, as expected from Curie-Weiss [Figure S14].

The variation in $\Delta H_{pp}$ as a function of temperature displays three regimes [Figure 4(d)]: an initial, approximately linear, decrease from room temperature down to 200 K; an approximately constant linewidth between 200 K and 140 K; finally, a rapid increase in the linewidth down to 50 K. Cooling below 50 K sharpens the signal, a consequence of
exchange narrowing and the loss of severely broadened signals.

The temperature dependence of $\Delta H_{pp}$ was fit to the equation derived by Mori and Kawasaki to describe the temperature dependence of EPR linewidths for low- and three-dimensional antiferromagnetic materials:

$$ \Delta H_{pp} = \Delta H^* + P \left( \frac{T_N}{T - T_N} \right)^\beta + \alpha T, \quad (2) $$

where $\Delta H^*$ is a parameter describing the exchange-narrowed linewidth, $P$ and $\alpha$ are constants of proportionality, $T_N$ is the Néel temperature (6.5 K from our previous work$^{20}$) and $\beta$ is the critical exponent. The second term in the equation describes the critical behaviour between the paramagnetic regime and the (locally) ordered regime. The critical exponent, $\beta$, in eq. (2) may be expressed in terms of a spin correlation length, $\eta$, its divergence $v$ and the divergence of the specific heat capacity of the material, $\zeta$:

$$ \beta = \frac{(7 + \eta)v}{2} - 2(1 - \zeta). \quad (3) $$

For three-dimensional antiferromagnetic materials, $\beta = 1/3$, whilst two-dimensional antiferromagnets have $\beta = 3/2$. $^{69-72}$ These quantities are in contrast to the conventional critical exponent obtained from magnetization data; the origin of this difference is complex, but primarily stems from magnetization data requiring no wave-vector-dependent sums over the dipolar auto-correlation function; EPR data does require this. For a more detailed explanation, please see refs. $^{69-71}$.

For NMMO, a value of $\beta = 1/3$ produces a relatively good fit ($r^2 = 0.6$); $\beta = 3/2$ or 1 gives a poor fit ($r^2 = 0.1$ or 0.2, respectively). However, all these fits appear qualitatively similar, suggesting a behaviour somewhere between a two- and three-dimensional antiferromagnet, as seen in other, layered honeycomb-ordered materials.$^{71,72}$ Overall, we may conclude that the variable-temperature X-band EPR data is consistent with Mn$^{4+}$ centres surrounded by Mn$^{3+}$, Mn$^{4+}$ and Mg$^{2+}$ neighbours in a honeycomb ordered array.

*Room-Temperature $^{23}$Na NMR Spectroscopy.* To investigate the consequences of cation ordering on the Na$^+$ mobility in NMMO, variable-

temperature $^{23}$Na NMR spectroscopy was carried out. We chose “intermediate” magnetic field strengths (11.7 T and 16.4 T) in this work as a compromise between the strong quadrupolar and hyperfine interactions [see Figure S20 for spectra acquired at different field strengths].

As shown in our previous work,$^{20}$ the $^{23}$Na NMR spectrum of pristine NMMO contains four isotropic resonances: one at 0 ppm, corresponding to diamagnetic Na-containing phases (most likely surface species, such as carbonates, as seen in PXRD, Figure 2), and, at “room temperature” (approximately 318 K under 60 kHz magic angle spinning, MAS, due to frictional heating), three overlapping signals between approximately 1350 ppm and 1550 ppm, denoted I, II and III for the low-frequency, central and high-frequency resonances, respectively [Figure 5(a)]. The isotropic resonances were identified using pjMATPASS experiments [Figure S15].

The shifts of the resonances are controlled by the strong electron-nuclear hyperfine interaction between unpaired electrons on Mn and the nuclear spins of $^{23}$Na, and as such, are temperature-dependent, due to the inherent Curie Weiss paramagnetism of the sample (i.e., hyperfine shifts are inversely proportional to temperature).$^{22,26-28}$ On decreasing the MAS frequency—which also decreases sample temperature—the shifts increase [Figure 5(a), Figure S18]. The width of the sideband manifold is dominated by the dipolar coupling between the $^{23}$Na nuclei and the time-averaged magnetic moments centred on the Mn ions, which for nuclei within the layers of the $\alpha$-NaMnO$_2$-type structure gives rise to a characteristic asymmetric lineshape.$^{75-78}$

The breadth and intensity of the individual isotropic resonances varies with spinning: higher spinning speeds (and thus temperatures) give rise to sharper and more intense signals, while lower spinning speeds and thus temperatures yield broader, weaker signals [Figure 5(a) and Figure S18].$^{31,79}$ This, as discussed in our prior work, indicates that mobility of Na$^+$ ions within the layers enters the “chemical exchange” timescale at temperatures below approximately 318 K.$^{31,79}$ When the Na$^+$ ion hopping rate is high compared to the difference in frequencies between the resonances arising from the sites Na$^+$ hops between, a sharp signal is seen, while slower
hopping (compared to the site frequency
difference) gives broader, less intense signals.
The relative intensities of resonances I, II and
III change as a function of spinning speed/
temperature. At 60 kHz, II is most intense,
followed by I and then III, whilst at 40 and 45
kHz (296 and 300 K, respectively), I is most

intense, while II and III have broadened out and
decrease in intensity, again indicating that
motion involving the ions that give rise to these
resonances is entering the intermediate regime.
Fixed-temperature, variable MAS speed
experiments at 316K and above are consistent
with this [Figure S18]. T₂ experiments indicate

<figure><img src="image_13.png" /><figcaption>**Figure 5:** (a) "Room temperature" <sup>23</sup>Na NMR spectra for P2-NMMO, acquired at different MAS rates under a field of 11.7 T; the isotropic resonances are highlighted in bold. All spectra are scaled by sample mass and number of scans. Note that "room temperature" here refers to the ambient sample temperature generated by frictional heating under MAS. The low-frequency, central and high-frequency resonances are labelled I, II and III, respectively, as shown in the 60 kHz spectrum. In (b), the variable-temperature 50 kHz MAS <sup>23</sup>Na NMR Hahn-echo spectra at a field of 16.4 T are shown.</figcaption></figure>
that each resonance experiences a different T₂ due to their unique local environments; sites experiencing shorter T₂s will generally be less intense than those with longer T₂s.

The most surprising observation that we need to rationalise is that there are three discrete peaks, and yet multiple Na-local environments are present in the static structure, even with a degree of cation ordering. While motion can explain this to a degree, it is not immediately obvious why three peaks remain.

*Variable-Temperature* ²³Na *NMR Spectroscopy*. To investigate the motion further, we carried out variable-temperature ²³Na NMR experiments at a fixed spinning speeds of 50 kHz over a much wider temperature range [Figure 5(b)]. Between 250 K to 391 K, the same three observations as above are noted—i.e., an increase in the isotropic shifts with decreasing temperature; a changeover in intensities between resonances I and II; and a broadening and overall decrease in intensity of the resonances as temperature falls, although there are some key differences. These spectra were obtained at a field of 16.4 T, the higher field strength used to determine what (if any) effect changing the quadrupolar interaction strength would have on the spectrum, as well as to increase resolution and identify any additional peaks which may be hidden underneath the sideband manifolds. Additional variable-temperature data at 28 kHz and 50 kHz at 11.7 T are presented in Figures S15 and S16; these results are consistent with those at 16.4 T, no change in lineshapes from 11.7 T to 16.4 T and no additional peaks were identified [Figure 5].

At 250 K, a relatively broad, low intensity, featureless isotropic resonance is observed with a very broad spinning sideband manifold, suggesting slow or no Na⁺ hopping. Curie-Weiss scaling of the shifts of the resonance between 250 K and 275 K to the shifts of resonances in the higher temperature spectra reveals that the low-temperature broad feature corresponds to resonance I [Figure 5(b)].

On heating the sample to 275 K, the peaks broaden; at 281 K, two peaks of approximately equal integrals become resolved [Figure S22]. Curie-Weiss scaling indicates that this second signal corresponds to resonance II [Figure S21].

On further heating to 293 K resonance II becomes slightly more intense than resonance I; further heating increasing the intensity of II more than I [Figure 5(b)]. At 299 K, resonance III becomes distinguishable from the baseline. Further heating increases the integral of peak II but decreases the integral of I; the integral for III remains approximately constant. I and II remain significantly sharper than III, suggesting that Na⁺ ions corresponding to III are in the intermediate regime, even at high temperatures—diagnostic of either slow hopping and/or a large difference in the exchanging sites' shifts. The decrease in signal I's integral (despite increasing signal height) suggests population transfer from the Na species in I to those in II. Additional attempts to reach higher temperatures than those in Figure 5(b) to see whether the three resonances would merge (suggesting exchange across all sites) proved unsuccessful, as the lower MAS speeds dictated by the larger rotor used to reach higher temperatures, resulted in poorer resolution. Spectra acquired at lower temperatures (242 K and 100 K) were also consistent with Na⁺ hopping between P(2d) and P(2b) environments: see SI for further details [Figure S23].

*Modelling and Simulating* ²³Na NMR Spectra. To account for the effect of Na⁺ ion mobility on the observed NMR spectra, the variable-temperature spectra for NMMO were simulated using the Bloch-McConnell model described in the SI and assuming Na⁺ ions undergo isolated hops (i.e., hops which are not influenced by other nearby Na⁺ ions hopping) between pairs of sites. Our model builds on that in previous work, where spectra were computed assuming a single hopping rate for all sites,²⁹ or where shifts are computed in the high-temperature (fast-exchange) regime.²²,⁸⁰ Since only three peaks appear in the high temperature regime, it is reasonable to assume that three exchanging pairs will be sufficient to describe the system. For the sake of simplicity, we elected to model hops between sites with Mg²⁺ and Mn⁴⁺ neighbours only (i.e., no Mn³⁺ neighbours). Examining the probabilities of the different honeycomb-ordered environments reveals that
the most probable site exchanges occur between these Mn⁴⁺/Mg²⁺ only sites [Table S3, S4 and S5]. The presence of a dynamic Jahn-Teller distortion as well as the strong tendency of the system to order towards Na₂/₃[Mg₁/₃Mn₂/₃]O₂ (a Mn⁴⁺-only system) makes modelling with Mn³⁺ challenging (see SI, section S8, for further details). Additional simulations which include Mn³⁺ are included in the SI to investigate what effect, if any, Mn³⁺ has on the simulations and resulting spectra. Owing to the fluctuating Jahn-Teller distortion of Mn³⁺

centres, it is expected that any Na⁺ nearby Mn³⁺ will undergo rapid relaxation, leading to a broad signal which would be challenging to model.

The simulation results and extracted hopping frequencies are shown in Figure 6. The simulations were iterated by varying the hopping frequencies of each site while keeping the hopping attempt rate constant and the shifts of static resonances fixed to those obtained from calculations (scaled to the experimental

<figure><img src="image_14.png" /><figcaption>**Figure 6:** Simulations of variable-temperature <sup>23</sup>Na NMR data obtained for NMMO. In (a), the simulated spectra (calculated from the Bloch-McConnell model; note that the simulations only recreate the isotropic peaks) are shown, with the experimental spectra in (b). The hopping rates for 'forward' (P(2d) to P(2b)) and 'reverse' (P(2b) to P(2d)) hops are shown in (c) and the final assignment of the spectra is shown in (d).</figcaption></figure>
sample temperature using the Curie-Weiss law).

The spectra were modelled using all possible combinations of $Na^+$ sites in the honeycomb $Na_{2/3}[Mg_{1/3}Mn_{2/3}]O_2$ model. Of these, only one model reproduced the observed data; resonance I corresponded to $Na^+$ ions hopping between a P(2d) site and a P(2b) site with one $Mg^{2+}$ nearest neighbour and one $Mn^{4+}$ nearest neighbour; resonance II corresponded to $Na^+$ ions hopping between a P(2d) site and a P(2b) site with two $Mg^{2+}$ nearest neighbours; and III corresponded to $Na^+$ ions hopping between a P(2d) site and a P(2b) site with two $Mn^{4+}$ nearest neighbours [Figure 6(d)]; see discussion for the connection between each of the peaks and the environments in the supercell obtained from XRD. Simulations using the same parameters also provided a good fit to the lower field (11.7 T) spectra acquired at 28 kHz and 50 kHz, further validating the model [Figures S19 and S20].

The hopping rates extracted from the simulations reveal an approximately linear trend in the log of the hopping rates, *k*, as a function of the inverse of sample temperature, $T$, between 250 K and 312 K [Figure 6(c)], as expected for a constant activation energy barrier to hopping with respect to temperature. An Arrhenius fit of this line yielded energy barriers in between 260 meV and 380 meV [Figure S26]. Deviations from an ideal straight line likely arise from small changes to the barrier due to changes in bond angles and lengths (causing small changes to the bond pathways). It is also anticipated that the effect of the dipolar and quadrupolar interactions (not accounted for here) will influence the lineshape; accounting for these contributions and the effect of correlated motion is beyond the scope of this work but presents an interesting avenue of future research.

*Ab initio Calculations of the Na⁺ Ion Hopping Barriers.* To compare the extracted hopping rates from <sup>23</sup>Na NMR simulations, the energy barriers to $Na^+$ ion hopping in NMMO were determined using a series of climbing image nudged elastic band (CI-NEB) calculations on a set of model systems, namely: P2-$Na_{1/2}MnO_2$, P2-$Na_{1/2}Mg_{1/6}Mn_{5/6}O_2$ and honeycomb-ordered P2-$Na_{2/3}Mg_{1/3}Mn_{2/3}O_2$; see section S9 of the SI for further details and results. The gradual increase in Mg content was applied to see what effect Mg has on the energy barriers associated

with hopping between P(2d) and P(2b) sites. In each case, a (2 × 2 × 1) supercell was constructed; the P2-$Na_{1/2}MnO_2$ and P2-$Na_{1/2}Mg_{1/6}Mn_{5/6}O_2$ systems achieved good *k*-mesh and plane wave energy cutoff convergence, but the P2-$Na_{2/3}Mg_{1/3}Mn_{2/3}O_2$ system could not be converged, regardless of cell size or distribution of Mg and Na ions. For all $Na^+$ hops investigated, the hopping $Na^+$ ion begins in a P(2d) site and ends up in a P(2b) site with a different local environment (i.e., a different number of $Mg^{2+}$, $Mn^{3+}$ and $Mn^{4+}$ nearest neighbours); in all cases, the P(2b) site was higher in energy than the P(2d), as expected on Coulombic repulsion grounds.

In the case of $Na_{0.5}MnO_2$, the energy barrier to hopping was approximately 180 meV, whilst that for $Na_{1/2}Mg_{1/6}Mn_{5/6}O_2$ was approximately 270 meV [Figure S32]. These barriers are consistent with those extracted from Arrhenius fits from NMR simulations. The increased hopping barrier to a site with $Mg^{2+}$ nearest neighbours is perhaps unsurprising: to keep a net charge of +2 from the nearest neighbour cations around $O^{2-}$ (i.e., to remain locally electroneutral), the oxidation state of Mn must increase from +3 to +4 when $Mg^{2+}$ is introduced. The increase in Mn oxidation state increases Coulomb repulsion with nearby $Na^+$, raising the barrier to $Na^+$ hopping.

Intriguingly, CI-NEB runs with Mg doped into the $TMO_2$ layer resulted in more cooperative $Na^+$ ion motion: whilst, in all systems, when one $Na^+$ ion was set to hop, others also changed position, when $Mg^{2+}$ was introduced, more $Na^+$ ions nearby the hopping $Na^+$ centre moved and/or the distance moved by nearby $Na^+$ ions increased. It is anticipated that this cooperative motion results in a relatively shallow potential energy surface, meaning that converging on a transition state becomes challenging. The cooperative motion can also be ascribed to a tendency towards electroneutrality around $O^{2-}$: if the $O^{2-}$ centre loses a $Na^+$ nearest neighbour, the nearby $Na^+$ ions will hop towards the vacancy to minimise the change in local charge (removing one $Na^+$ from the $O^{2-}$ first coordination shell decreases the positive charge delivered to $O^{2-}$ by +1/6, as each $Na^+$ is bound to six $O^{2-}$). As a result, a single $Na^+$ ion hopping will encourage nearby ions to hop in such a way that preserves electroneutrality and therefore the local ordering of $Na^+$ ions.
<figure><img src="image_15.png" /><figcaption>**Figure 7:** *Ab initio* molecular dynamics results for Na⁺ ions diffusing in P2-Na₂/3[Mg₁/3Mn₂/3]O₂ (a ≈ 10.2 Å, b ≈ 10.2 Å, c ≈ 11.4 Å), showing the root-mean-square displacement of Na⁺ ions, 〈r²(t)〉¹/², between 400 K and 800 K.</figcaption></figure>

This strongly cooperative motion is also reflected in *ab initio* molecular dynamics (AIMD) simulations of (2 × 2 × 1) supercells (a = b ~ 10 Å, c ~ 11 Å) of Na₂/₃Mg₁/₃Mn₂/₃O₂ [Figure 7]: hops from one Na⁺ to the next are not independent, but instead are cooperative, with one hopping Na⁺ causing its neighbours to also (begin to) hop. Instead of a linear trend in the root-mean-square (rms) displacement, 〈Δr(t)²〉, with time, an initial sharp rise in 〈Δr(t)²〉 is seen, followed by an oscillatory rms displacement about an approximately constant value (ca. 40 Å²), suggesting that, after an initial perturbation, Na⁺ ions do not necessarily displace in a way to explore a large region of space, but instead remain constrained to a small region of space. It is acknowledged, however, that this picture is limited, as the simulation time is relatively short and only a few hundred Na⁺ hops are observed.

This variation in 〈Δr(t)²〉 with time has been previously seen in similar cooperative motion systems, for example in polymer-in-solution systems⁷⁴,⁷⁵ and other ionic systems,⁷⁶ and has

also been explored in other NIB cathodes, where Na⁺ ion motion was seen to be significantly faster at or near anti-phase boundaries between Na⁺/vacancy ordered regions. Our results therefore indicate that Na⁺ ion motion in NMMO is cooperative and likely driven by local electroneutrality.

## Discussion

The tendency of pristine NMMO to adopt an ordered structure in which Mg and Mn are approximately honeycomb ordered and the Na sublattice is also ordered can be understood by a tendency for local electroneutrality, as seen in other cathode systems.⁷⁷ Ideally, each O²⁻ centre would be coordinated by three species in the *TM* layer (Mg²⁺, Mn³⁺ or Mn⁴⁺) and zero, one, two or three Na⁺ ions in the adjacent layer. From the perspective of an O anion (charge -2), there are always three *TM*S in the first coordination sphere (ignoring vacancy defects) and either zero, one, two or three Na⁺ neighbours. Since each of the *TM* and Na⁺ centres share their charge with six O centres (assuming no O²⁻ vacancies), each O receives one sixth of a metal centre's charge.

Local electroneutrality can therefore only be achieved in six ways (assuming a random distribution of all cationic species; Table 2). Based on the 2/3 occupancy of the Na⁺ sublattice, the probability of an O²⁻ centre having no Na⁺ nearest neighbours is approximately 4%, whilst the probability of having one Na⁺ nearest neighbour is 22%, of two Na⁺ nearest neighbours is 44% and of three Na⁺ nearest neighbours is 30%. When combined with the probability of the occupancies of the *TM* sites, we find the most probable environments are environments (1), (3) and (5); O(1) has no Na⁺ neighbours, so is very unlikely given the high Na⁺ content in NMMO, while O(3) and O(5) correspond to locally honeycomb-ordered environments and are closest in composition to the bulk stoichiometry. These two constitute over half of the electroneutral environments.

The presence of (at least local) honeycomb order on the *TM* sublattice is also evident from X-band EPR, where two resonances, A and B, were observed. In previous work, Stoyanova *et al.* observed larger *g*-shifts (i.e., greater deviation of the *g*-factor from the free-electron *g*-value, *g*<sub>e</sub>) for centres in Na<sub>y</sub>[Co<sub>1-2x</sub>Ni<sub>x</sub>Mn<sub>x</sub>]O<sub>2</sub> (x = 1/2, 1/3; y = 1/2, 1/3) with strong local
exchange couplings, whilst species with weaker exchange couplings showed a g-value closer to $g_e$.⁷⁸ This may be rationalised by considering that strong exchange coupling augments the applied field significantly, resulting in a large apparent g-factor.⁷⁹,⁸⁰ In the case of $Mn^{4+}$, the less-than-half-filled shell results in a g lower than $g_e$,²² so $Mn^{4+}$ with stronger net exchange interactions will have g even lower from $g_e$ than $Mn^{4+}$ centres with weaker exchange interactions to neighbouring spins. Since $Mn^{3+}$ typically has stronger exchange than $Mn^{4+}$ (as the $Mn^{3+}$ exchange interactions are mediated by $e_g$ orbitals), the more $Mn^{3+}$ centres nearby $Mn^{4+}$, the lower the g-factor is expected to be.⁷⁸ It is also expected that more $Mg^{2+}$ centres around a $Mn^{4+}$ will move g closer to $g_e$, as Mg has a smaller spin-orbit coupling constant than Mn and therefore a smaller influence on the g-factor of the unpaired electron.

Based on these observations and assumptions, resonance A (with a g-factor further from $g_e$) can be tentatively assigned to $Mn^{4+}$ with at least one $Mn^{3+}$ nearest neighbour and resonance B to $Mn^{4+}$ with only $Mn^{4+}$ and $Mg^{2+}$ nearest neighbours. The presence of JT-distorted $Mn^{3+}$

clusters below 50 K, consistent with the temperature at which the field-cooled and zero-field cooled susceptibility bifurcate²⁵ and the point below which diffuse magnetic scattering in neutron diffraction is observed [Figure S10]. These spin clusters likely develop through competing exchange interactions between domains of the lattice with different *TM* and $Na^+/vacancy$ orderings.

The effect of ordering on the *TM* sublattice has consequences for the $Na^+$ sublattice: $Na^+$ ions adopt a 'long-zig-zag' (LZZ)-like configuration, as seen in other cathode materials with a $Na^+$ content $x = 0.67$.⁹,¹⁰,⁶⁴,⁹²,⁹³ Here, $Na^+$ ions can be grouped into those $Na^+$ in the zig-zag regions of the LZZ structure, $Na_a$, or the island regions, $Na_b$. Based solely on vacancy concentration, the former is anticipated to be more mobile than the latter. This is also borne out in the observed and simulated variable-temperature ²³Na NMR spectra [Figure 6]: resonances II and III may be ascribed to $Na^+$ ions in layers with LZZ-like ordering, where $Na^+$ ion hopping occurs between P(2d) sites and P(2b) sites with two $Mg^{2+}$ nearest neighbours (and no $Mn^{4+}$ nearest neighbours) or between

<table><caption>**Table 2:** Locally electroneutral sites around $O^{2-}$ in pristine NMMO, with probabilities of each environment based on the occupancies of the Na and *TM* sublattices.</caption><thead><tr><th></th><th>No. of<br>Na<sup>+</sup><br>n.n.</th><th>No. of<br>Mg<sup>2+</sup> n.n.</th><th>No. of<br>Mn<sup>3+</sup> n.n.</th><th>No. of<br>Mn<sup>4+</sup> n.n.</th><th>Probability<br>(%)</th></tr></thead><tbody><tr><td>(1)</td><td>0</td><td>0</td><td>0</td><td>3</td><td>28.4</td></tr><tr><td>(2)</td><td>1</td><td>0</td><td>1</td><td>2</td><td>15.4</td></tr><tr><td>(3)</td><td>2</td><td>1</td><td>0</td><td>2</td><td>39.1</td></tr><tr><td>(4)</td><td>2</td><td>0</td><td>2</td><td>1</td><td>2.8</td></tr><tr><td>(5)</td><td>3</td><td>1</td><td>1</td><td>1</td><td>14.1</td></tr><tr><td>(6)</td><td>3</td><td>0</td><td>3</td><td>0</td><td>0.2</td></tr></tbody></table>

nearby the $Mn^{4+}$ will likely also increase the spin-orbit coupling contribution to the g-factor.²² This is also consistent with observations that $Mn^{4+}$ species in a $Mn^{4+}$-only lattice have g-factors of approximately 1.98, as observed [Figure 4(b)].⁶⁷,⁸¹,⁸² On the basis of empirical observations by Stoyanova *et al.*,⁸⁸,⁸⁹ in which species with stronger exchange interactions have larger g-shifts, A may be assigned to $Mn^{4+}$ with one $Mn^{3+}$ neighbour, two $Mn^{4+}$ neighbours and three $Mg^{2+}$ nearest neighbours (i.e., electroneutral environment (5)) and B assigned to $Mn^{4+}$ with three $Mn^{4+}$ neighbours and three $Mg^{2+}$ neighbours (electroneutral environment (3); Figure 8(a)). Additional high-frequency EPR results also indicated the formation of spin

P(2d) and P(2b) with two $Mn^{4+}$ nearest neighbours (and no $Mg^{2+}$ nearest neighbours, Figure 8(b)), respectively. We see only three peaks because exchange is a local, rather than a long-range, process (on the time scale probed by the NMR experiments) in which $Na^+$ ions explore only two local environments (the initial and final states), such that local electroneutrality before and after the hop is preserved. The tendency of NMMO to order limits the total number of local environments, resulting in a spectrum with many fewer lines than expected.

Resonance II is assigned to $Na^+$ ions hopping between P(2d) and P(2b) with two $Mg^{2+}$ nearest
neighbours. Of all these local exchanges, one third arises from $Na_a \leftrightarrow Na_a$ hopping, one third from $Na_b \leftrightarrow Na_b$ hopping and one third from $Na_a \leftrightarrow Na_b$ hopping. Here, the difference in resonant frequency of the exchanging $Na^+$ sites is small; since resonance II is sharp and high in intensity at most temperatures but rapidly decreases in intensity at lower temperatures, we anticipate the barriers to hopping to be small, such that these ions are (on the NMR timescale) in the fast (high $T$) or intermediate (lower $T$) regime of motion. The low barriers likely stem from the low Coulomb repulsion experienced in the P(2b) site coordinated by two $Mg^{2+}$ neighbours, in contrast to the higher repulsion expected when either one or two Mn centres coordinate the P(2b) site. The relative energies of these pathways are also consistent with the bond valence energy landscapes obtained from the PXRD Rietveld-derived structures [Figure S12]. We note, however, the difference in the barriers to hopping to a P(2b) site with only $Mg^{2+}$ neighbours and to hopping to a P(2b) with Mn neighbours (according to bond valence energy landscapes) is approximately 100 meV, so not significantly larger.

When $Na^+$ ions hop between P(2d) sites and P(2b) sites with two $Mn^{4+}$ nearest neighbours, the difference in resonant frequency is large: approximately 1840 ppm at 318 K. We therefore suggest that, due to the large difference in site frequencies, these $Na^+$ centres only enter the fast-hopping regime at the highest temperatures. On this basis, we assign these local hops to $^{23}$Na resonance III, consistent with simulations. These local environments are present in both $Na_a$ and $Na_b$ sites seen in PXRD [Figure 2(f)]; approximately 40% of these local exchanges are from $Na_a \leftrightarrow Na_a$ hopping, 40% is from $Na_b \leftrightarrow Na_b$ exchanges and 20% from $Na_a \leftrightarrow Na_b$ exchange. Again, the barrier to this hopping is not expected to be significantly larger than that for the exchange in resonance II, as these hops were those seen most clearly in variable-temperature PXRD and PDF [Figure 3(d)].

Resonance I is assigned to $Na^+$ ions in the parent-like layers, where the adjacent $TMO_2$ layers have a honeycomb arrangement of Mg and Mn, but the relative lateral positions of Mg and Mn from one layer to the next are random. Such local environments may also be achieved in anti-phase boundaries between different domains of the crystallites. The loss of order in

these regions result in a loss of LZZ-like ordering of these layers, with $Na^+$ ions instead randomly occupying sites in these parent material layers. It is expected that the energy barrier for hops between these sites lies between the barriers for the $Na^+$ ions corresponding to resonances II and III, again based on local vacancy concentration. Since the difference in the $^{23}$Na NMR frequencies of the island sites (either P(2d) and P(2b) with two $Mg^{2+}$ nearest neighbours or P(2d) and P(2b) with two $Mn^{4+}$ nearest neighbours) is, on average, lower than the difference for the honeycomb border sites [Figure 7, Figure S29] but the rate of hopping is likely slower, it is expected that the island sites will behave in a similar fashion to the honeycomb border $Na^+$ ions—i.e., resonances from these $Na^+$ ions will only sharpen at high temperatures, where $Na^+$ is in the fast hopping regime; at lower temperatures, the resonance will remain broad and low in intensity, as observed.

Finally, it is noted that $Na^+$ ion motion in NMMO (both in the parent phase and superstructure) is expected to be highly correlated, based on CI-NEB and AIMD results. This cooperativity is likely a direct consequence of retaining local electroneutrality and explains the difficulty in obtaining a good Rietveld fit of the long-range structure to the observed PXRD and neutron diffraction [Figure 2(b)] whilst faithfully capturing the average local structure in both NMR and PDF. In other words, the cooperative $Na^+$ ion motion results in several local ordering
<figure><img src="image_16.png" /><figcaption>**Figure 8:** Identifying the superstructure with (a) EPR and (b) NMR observations. In (a), the two most likely local electroneutral environments around O are depicted and assigned to the two resonance classes seen in EPR, A and B. In (b), a view down the c axis of the Na⁺/vacancy ordering obtained from PXRD refinements is shown, with loops drawn around pairs of Na⁺ ions which exchange between the local environments shown below, labelled as I, II and III, corresponding to the three resonances seen in <sup>23</sup>Na NMR, shown to the right.</figcaption></figure>

schemes, each of which could be described by a superposition of several symmetry modes describing Na⁺ displacement.

The superstructure adopted by layered NIB cathodes such as NMMO has important consequences on the electrochemical performance of the cathode. In general, the presence of ordering on the Na sublattice makes Na⁺ ion diffusion sluggish compared to disordered materials, owing to the increased (enthalpically-driven) energy barrier to Na⁺ ion hopping;⁷ this will likely result in large activation barrier to Na⁺ extraction at the early stages of

desodiation to overcome the energy barrier to disrupting the Na/vacancy ordering.<sup>6,8–10</sup> Indeed, the Na<sub>0.67</sub>Mg<sub>y</sub>Mn<sub>1-y</sub>O<sub>2</sub> family (of which NMMO is a member) adopts multiple polymorphs depending on the stoichiometry of the material and synthetic conditions, with each member of the family having different electrochemical performances depending on the Mg content.<sup>94,95</sup> The best performance was observed for Na<sub>0.67</sub>Mg<sub>0.10</sub>Mn<sub>0.90</sub>O<sub>2</sub>, where the random distribution of Mg in the *TM* layer disrupted Na⁺ ion ordering, leading to large capacities even at fast cycling rates. In NMMO, the superstructure is expected to enable fast
Na⁺ ion motion for the Na⁺ in the hexagonal borders, but slower motion in the island sites, as seen from NMR simulations. The ordering on the *TM* site is anticipated to have implications for the electronic structure of NMMO: it has been proposed by several authors that the superstructure in layered NIB (and lithium-ion battery) cathodes can enable more reversible oxygen redox reactions by developing more delocalised states or by facilitating *TM* migration and pairing of oxidised O centres to form O₂.¹⁵,⁹⁶,⁹⁷ Therefore, the superstructure present in the pristine material may be the cause of the high capacities at relatively fast cycling rates seen in NMMO.

## Conclusion

In conclusion, an ordering scheme accounting for the additional superstructure reflections in Na₀.67[Mg₀.28Mn₀.72]O₂ has been proposed using synchrotron X-ray and neutron diffraction, pair distribution function analysis, variable-temperature ²³Na NMR and variable-temperature and -frequency EPR. This ordering arises from a tendency for local electroneutrality.

By carefully modelling the PXRD, an average bulk structure was generated using distortion modes on the Na and *TM* sublattices to describe ordering on both the *TM* and Na sublattices. It is acknowledged that the true superstructure may be a combination of two different *k*-vectors (supercells), one for each sublattice, or that it contains more subtle distortions, with more modes active within the supercell. We note, however, that the peaks which are not well fit are small relative to the main peaks in the pattern.

At a local level, the superstructure presented in this work is consistent with neutron PDF, variable-temperature, variable-frequency EPR and variable-temperature NMR. Analysis of the EPR data revealed at least two Mn⁴⁺ environments which differ in the number of Mn³⁺ and Mn⁴⁺ nearest neighbours, rationalised by considering the magnetic exchange interactions for each unique resonance. A thorough analysis of the observed ²³Na NMR spectra using Bloch-McConnell simulations enabled the mobilities of three distinct honeycomb Na⁺ environments to be identified, rationalised and explained within the superstructure ordering model. It was

determined that those Na⁺ closest to Mg²⁺ centres hop fastest, whilst those closer to Mn⁴⁺ centres) hop at a much slower rate, a consequence of lower local vacancy concentration.

Our analysis shows that the relatively simple ²³Na NMR spectrum at room temperature stems from the presence of short-range, rather than long-range, hopping of Na⁺ ions between sites. In other words, Na⁺ ions only explore a relatively small region of space. Furthermore, hops encourage nearby Na⁺ ions to move to preserve local electroneutrality, resulting in little change to the short- and long-range structure before and after hopping.

The methodology presented in this work— studying local and bulk structures using non-invasive scattering and magnetic resonance techniques in combination with *ab initio* calculations and simulations—is critical to understanding the local structures adopted by cathode materials and the consequence of superstructure on their electrochemical performance.

## AUTHOR INFORMATION

Corresponding Author

*E-mail: cpg27@cam.ac.uk

ORCID

Euan N. Bassey: 0000-0001-8827-7175

Ieuan D. Seymour: 0000-0002-9550-9971

Joshua D. Bocarsly: 0000-0002-7523-152X

David A. Keen: 0000-0003-0376-2767

Guido Pintacuda: 0000-0001-7757-2144

Clare P. Grey: 0000-0001-5572-192X

<footnote>aPresent address: Materials Research Laboratory, University of California, Santa Barbara, Santa Barbara, CA 93106-5121, United States of America</footnote>

<footnote>bAdvanced Centre for Energy and Sustainability (ACES), Department of Chemistry, School of Natural and Computing Sciences, University of Aberdeen, King's College, AB24 3UE Aberdeen, Scotland, UK</footnote>
Present address: University of Houston, Department of Chemistry, Lamar Fleming Jr. Building, 3585 Cullen Blvd, Room 112, Houston, Texas 77204-5003, United States of America

## Notes

There are no conflicts of interest to declare.

## Supporting Information

The Supporting Information, containing a detailed experimental, additional variable-temperature <sup>23</sup>Na NMR, NMR simulations, additional Rietveld refinements, details on calculations and assignments and a discussion of the theory behind the Bloch-McConnell equations is available free of charge on ... at DOI:

## ACKNOWLEDGEMENTS

E.N.B. acknowledges funding from the Engineering Physical Sciences Research Council (EPSRC) via the National Productivity Interest Fund (NPIF) 2018 (EP/S515334/1). J.D.B. acknowledges funding from the Faraday Institution (EP/S003053/1, FIRG016). We also thank the Science and Technology Facilities Council (STFC) and ISIS Neutron and Muon source for neutron data (experiment no.: RB2010350). Additional thanks are given to the staff scientists at beamline I11 of the Diamond Light Source for synchrotron data using block allocation group time under proposal CY34243. This work also utilised the ARCHER UK National Supercomputing Service via our membership in the UK's HEC Materials Chemistry Consortium, funded by the EPSRC (EP/L000202). Research was also carried out at the Center for Functional Nanomaterials, Brookhaven National Laboratory, through the U.S. Department of Energy, Office of Basic Energy Sciences, Contract DE-AC02-98CH10866. E.N.B. would also like to thank A. Van der Ven and M.A. Jones for illuminating discussions.

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