## Abstract

Solid-state ionic conduction is a key enabler of electrochemical energy storage and conversion. The mechanistic connections between material processing, defect chemistry, transport dynamics and practical performance are of considerable importance but remain incomplete. Here, inspired by studies of fluids and biophysical systems, we re-examine anomalous diffusion in the iconic two-dimensional fast-ion conductors, the β- and β″-aluminas. Using large-scale simulations, we reproduce the frequency dependence of alternating-current ionic conductivity data. We show how the distribution of charge-compensating defects, modulated by processing, drives static and dynamic disorder and leads to persistent subdiffusive ion transport at macroscopic timescales. We deconvolute the effects of repulsions between mobile ions, the attraction between the mobile ions and charge-compensating defects, and geometric crowding on ionic conductivity. Finally, our characterization of memory effects in transport connects atomistic defect chemistry to macroscopic performance with minimal assumptions and enables mechanism-driven ‘atoms-to-device’ optimization of fast-ion conductors.

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## Data availability

The scripts and templates to generate simulation structures and to run simulations and analyses are available at https://github.com/apoletayev/anomalous_ion_conduction. Source data are provided with this paper.

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## Acknowledgements

A.D.P. thanks G. McConohy, A. Sood, S. Kang and P. Muscher for helpful and invigorating discussions. A.D.P. and A.M.L. acknowledge support from the Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division (Contract DE-AC02-76SF00515). J.A.D. and M.S.I. acknowledge support from the EPSRC Programme Grant ‘Enabling next generation lithium batteries’ (grant No. EP/M009521/1 to M.S.I and J.A.D.), the EPSRC (grant No. EP/V013130/1 to J.A.D.), Research England (Newcastle University Centre for Energy QR Strategic Priorities Fund) and Newcastle University (Newcastle Academic Track (NUAcT) Fellowship). Via membership of the UK’s HEC Materials Chemistry Consortium, which is funded by the EPSRC (grant Nos. EP/L000202, EP/L000202/1, EP/R029431 and EP/T022213), this work used the ARCHER UK National Supercomputing Service.

## Author information

### Authors and Affiliations

### Contributions

A.D.P. initiated the application of anomalous-transport concepts and conceptualized the study with advice and support from A.M.L. A.D.P. carried out simulations with instruction and help from J.A.D and advice from M.S.I. A.D.P. carried out analysis. A.M.L. advised and supervised the work. All authors contributed to the writing of the manuscript.

### Corresponding authors

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*Nature Materials* thanks Ralf Metzler and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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## Extended data

### Extended Data Fig. 1 Ion diffusion in Na β-alumina.

**a**, tMSD of mobile ions. **b**, exponent of tMSD vs time lag, plotted against the time-averaged displacement. The exponent does not reach unity until displacements are ≥ distances between defects. In **b**, the horizontal guide is the Fickian limit *t*^{1}, and the vertical line is one unit cell (5.6 Å, 2 hops). **c**, Time slices of the distribution of ion displacements Δ*x* along [100] at 300 K. **d**, Distributions of ion displacements Δ*x* along [100], each rescaled by its variance σ_{Δx}. Laplace and Gaussian distributions are shown as black and grey dotted lines, respectively. **e**, The probability of an ion remaining within 1.7 Å (<1 hop, dashed) or 4.6 Å (<2 hops, solid) of an initial position. The relative change in the timescale due to varying the distance cutoff by 0.1 Å is ≤10%. **f**, Diffusion kernel correlation *C*_{D}. *C*_{D} peaks between one-hop and two-hop relaxation times at all simulated temperatures. In **b** and **f**, short-time checks of the ballistic limits (dashed), tMSD ∝ *t*^{2} and *C*_{D} → 2, respectively, used 100-ps trajectories recorded every 1 fs.

### Extended Data Fig. 2 Ion diffusion in K β”-alumina.

**a**, tMSD of mobile ions. **b**, exponent of tMSD vs. time lag, plotted against the time-averaged displacement. In **b**, the horizontal guide is the Fickian limit *t*^{1}, and the vertical line is one unit cell (5.6 Å, 2 hops). **c**, Time slices of the distribution of ion displacements Δ*x* along [100] at 300 K. **d**, Distributions of ion displacements Δ*x* along [100], each rescaled by its variance σ_{Δx}. Laplace and Gaussian distributions are shown as black and grey dotted lines, respectively. **e**, The probability of an ion remaining within 1.7 Å (<1 hop, dashed) or 4.6 Å (<2 hops, solid) of an initial position. The relative change in the timescale due to varying the distance cutoff by 0.1 Å is ≤10%. **f**, Diffusion kernel correlation *C*_{D}. Notably, unlike in Na β″-alumina (Fig. 2), *C*_{D} has multiple peaks. In **b** and **f**, short-time checks of the ballistic limits (dashed), tMSD ∝ *t*^{2} and *C*_{D} → 2, respectively, used 100-ps trajectories recorded every 1 fs.

### Extended Data Fig. 3 Ion diffusion in Na β″-alumina with a quenched distribution of \({\mathrm{Mg}}^\prime_{\mathrm{Al}}\) defects.

**a**, tMSD of mobile ions. **b**, exponent of tMSD vs. time lag, plotted against the time-averaged displacement. In **b**, the horizontal guide is the Fickian limit *t*^{1}, and the vertical line is one unit cell (5.6 Å, 2 hops). The diffusion is Fickian (**a**,**b**) by contrast with Na β″-alumina without quenching (Fig. 2). **c**, Time slices of the distribution of ion displacements Δ*x* along [100] at 300 K. **d**, Distributions of ion displacements Δ*x* along [100], each rescaled by its variance σ_{Δx}. Laplace and Gaussian distributions are shown as black and grey dotted lines, respectively. **e**, The probability of an ion remaining within 1.7 Å (<1 hop, dashed) or 4.6 Å (<2 hops, solid) of an initial position. The relative change in the timescale due to varying the distance cutoff by 0.1 Å is ≤10%. **f**, Diffusion kernel correlation *C*_{D}. Unlike the non-Gaussian parameter, *C*_{D} has multiple peaks at 300 K. In **b** and **f**, short-time checks of the ballistic limits (dashed), tMSD ∝ *t*^{2} and *C*_{D} → 2, respectively, used 100-ps trajectories recorded every 1 fs.

### Extended Data Fig. 4 Ion diffusion in Ag β-alumina.

**a**, tMSD of mobile ions. **b**, exponent of tMSD vs. time lag, plotted against the time-averaged displacement. As for Na β-alumina, the exponent does not reach unity until displacements are ≥ distances between defects. In **b**, the horizontal guide is the Fickian limit *t*^{1}, and the vertical line is one unit cell (5.6 Å, 2 hops). **c**, Time slices of the distribution of ion displacements Δ*x* along [100] at 300 K. **d**, Distributions of ion displacements Δ*x* along [100], each rescaled by its variance σ_{Δx}. Laplace and Gaussian distributions are shown as black and grey dotted lines, respectively. **e**, The probability of an ion remaining within 1.7 Å (<1 hop, dashed) or 4.6 Å (<2 hops, solid) of an initial position. The relative change in the timescale due to varying the distance cutoff by 0.1 Å is ≤10%. **f**, Diffusion kernel correlation *C*_{D}. As for Na β-alumina, *C*_{D} peaks between one-hop and two-hop relaxation times at all simulated temperatures. In **b** and **f**, short-time checks of the ballistic limits (dashed), tMSD ∝ *t*^{2} and *C*_{D} → 2, respectively, used 100-ps trajectories recorded every 1 fs.

### Extended Data Fig. 5 Ion diffusion in Ag β″-alumina.

**a**, tMSD of mobile ions. **b**, exponent of tMSD vs. time lag, plotted against the time-averaged displacement. In **b**, the horizontal guide is the Fickian limit *t*^{1}, and the vertical line is one unit cell (5.6 Å, 2 hops). **c**, Time slices of the distribution of ion displacements Δ*x* along [100] at 300 K. **d**, Distributions of ion displacements Δ*x* along [100], each rescaled by its variance σ_{Δx}. Laplace and Gaussian distributions are shown as black and grey dotted lines, respectively. **e**, The probability of an ion remaining within 1.7 Å (<1 hop, dashed) or 4.6 Å (<2 hops, solid) of an initial position. The relative change in the timescale due to varying the distance cutoff by 0.1 Å is ≤10%. **f**, Diffusion kernel correlation *C*_{D}. In **b** and **f**, short-time checks of the ballistic limits (dashed), tMSD ∝ *t*^{2} and *C*_{D} → 2, respectively, used 100-ps trajectories recorded every 1 fs.

### Extended Data Fig. 6 Ion diffusion in K β-alumina.

**a**, tMSD of mobile ions. **b**, exponent of tMSD vs. time lag, plotted against the time-averaged displacement. As for Na β-alumina, the exponent does not reach unity until displacements are ≥ distances between defects. In **b**, the horizontal guide is the Fickian limit *t*^{1}, and the vertical line is one unit cell (5.6 Å, 2 hops). **c**, Time slices of the distribution of ion displacements Δ*x* along [100] at 300 K. **d**, Distributions of ion displacements Δ*x* along [100], each rescaled by its variance σ_{Δx}. Laplace and Gaussian distributions are shown as black and grey dotted lines, respectively. **e**, The probability of an ion remaining within 1.7 Å (<1 hop, dashed) or 4.6 Å (<2 hops, solid) of an initial position. The relative change in the timescale due to varying the distance cutoff by 0.1 Å is ≤10%. **f**, Diffusion kernel correlation *C*_{D}. In **b** and **f**, short-time checks of the ballistic limits (dashed), tMSD ∝ *t*^{2} and *C*_{D} → 2, respectively, used 100-ps trajectories recorded every 1 fs.

### Extended Data Fig. 7 Ergodicity breaking parameter (EB) versus simulation length Δ.

**a**-**c**, β″-aluminas. **d**-**f**, β″-aluminas with a simulated distribution of defects corresponding to quenching. **g**-**i**, β-aluminas. The time lag used in all cases was *t* = 20 ps, but the asymptotic dependences at long simulation lengths are not sensitive to the precise value of *t* for Δ » *t*. The noted power-law relations are guides to the eye, not quantitative fits. In all simulations where EB ∝ Δ^{–1}, *C*_{D} → 0 precedes it. This is seen most clearly in Ag β-alumina at 600 K (**g**), where *C*_{D} → 0 for *t* ≈ 10 ns (Extended Data Fig. 4), and EB ∝ Δ^{–1} starting at Δ ≈ 20 ns.

### Extended Data Fig. 8 Distributions of the centre-of-mass diffusion coefficient D_{CoM} in Na β”-alumina.

**a**-**d**, Absolute values, **e**-**h**, rescaled by the standard error at each time lag. (**a**,**e**) 230 K, (**b**,**f**) 300 K, (**c**,**g**) 473 K, (**d**,**h**) 600 K. The rescaled distributions (**e**)-(**h**) are exponential. Notably, at 230 K, the distribution becomes wider rather than narrower with increasing time lag, suggesting the possibility of further glass-like collective dynamics at long timescales. At 230 K, the ordering of the mobile ions extends across the entire simulation cell within each conduction plane (Supplementary Fig. 2).

### Extended Data Fig. 9 Correlation factor *f* for hops, disaggregated by location relative to defects, versus hop residence times.

**a**-**c**, β”-aluminas. **d**-**f**, β-aluminas. For β-aluminas, hops starting in the low-energy Beevers-Ross sites are considered here. The sites with the most neighbouring defects (for β”) or those closest to defects (for β) have the lowest correlation factor for a given residence time. Yellow symbols: two-hop relaxation time from *G*_{s}. Black horizontal ranges: *C*_{D} → 0 at 600 K. At 300 K, *C*_{D} → 0 only for K β”-alumina.

## Supplementary information

### Supplementary Information

Supplementary Notes 1–9, Figs. 1–17 and refs. 1–36.

## Source data

### Source Data Fig. 2

Descriptors of ion transport in Na β”-alumina as plotted in Fig. 2.

### Source Data Fig. 3

Ionic conductivity literature and simulation data as plotted in Fig. 3.

### Source Data Fig. 4

Hopping rates in β”-aluminas as plotted in Fig. 4b–d.

### Source Data Fig. 5

Correlation factors f in Na β”-aluminas as plotted in Fig. 5.

### Source Data Extended Data Fig. 1

Descriptors of ion transport in Na β-alumina as plotted in Extended Data Fig. 1.

### Source Data Extended Data Fig. 2

Descriptors of ion transport in K β”-alumina as plotted in Extended Data Fig. 2.

### Source Data Extended Data Fig. 3

Descriptors of ion transport in Na β”-alumina with a quenched distribution of defects as plotted in Extended Data Fig. 3.

### Source Data Extended Data Fig. 4

Descriptors of ion transport in Ag β-alumina as plotted in Extended Data Fig. 4.

### Source Data Extended Data Fig. 5

Descriptors of ion transport in Ag β”-alumina as plotted in Extended Data Fig. 5.

### Source Data Extended Data Fig. 6

Descriptors of ion transport in K β-alumina as plotted in Extended Data Fig. 6.

### Source Data Extended Data Fig. 7

Ergodicity breaking parameters as plotted in Extended Data Fig. 7.

### Source Data Extended Data Fig. 8

Centre-of-mass displacements in Na β”-alumina as plotted in Extended Data Fig. 8.

### Source Data Extended Data Fig. 9

Site-resolved correlation factors as plotted in Extended Data Fig. 9.

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Poletayev, A.D., Dawson, J.A., Islam, M.S. *et al.* Defect-driven anomalous transport in fast-ion conducting solid electrolytes.
*Nat. Mater.* (2022). https://doi.org/10.1038/s41563-022-01316-z

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DOI: https://doi.org/10.1038/s41563-022-01316-z