Main

Linking the mechanistic features of ion transport at the atomic level to the collective descriptors of macroscopic transport within a multiscale model, or as measured in a device, yields opportunities to design new processes and applications3,9,13. Fast-ion transport in the solid state commands particular attention due to its importance in energy and information technologies such as solid-state batteries and non-volatile memory1,2,3. Solid-state ion transport is composed of rapid translations of ions between lattice sites, called hops, which constitute the fundamental steps of diffusion and conduction (Fig. 1a).

Fig. 1: Probing ionic transport in β-aluminas.
figure 1

a, Mobile metal ions (purple) hop (arrows) between energetically and crystallographically distinct lattice sites (triangles) with two- and six-fold coordination by oxygen ions (black). The same lattice sites are depicted schematically in d–f. b, If at time t0 = 0 (dark green) there is perfect knowledge of the state of the system, then at time t1 > 0, some correlations persist. By time t2 ≫ 0, all correlations and memory are lost, and the full entropy of transport is produced as for a random walk (dotted line). c, Transient birefringence experiment in β-aluminas. d, At time t0 = 0 (dark green), the terahertz pump triggers ion hops along its electric field vector. e, At time t1 > 0, the subsequent hops retain a correlation with the impulse at t0: back-hopping is predominant, which yields a birefringence signal (c, red). The anisotropic distribution of hopping directions at t1 is probed as a contribution to the anisotropy in ionic velocities (c, pink). f, The pump energy fully thermalizes by time t2 when the hopping directions reach isotropy.

Full size image

However, the correspondence between atomistic and macroscopic regimes in ion transport is often not well characterized: whereas nanoscale ionic transport is heterogeneous, dispersive and non-ergodic6,7, correlation-sensitive probes of atomistic paths analogous to single particle tracking in biophysics do not exist. Without such information, the maximum-entropy principle compels models of transport constructed from macroscopic measurements to assume that ion transport proceeds via a Markovian random walk14,15. From the standpoint of information theory16, the full entropy of transport is evolved at every step of this memory-free process. Traditionally equated to hops in solid-state transport, these steps originate randomly with an attempt frequency ν0, and succeed with a probability determined by the Gibbs free energy of a transition state6,11,12,17.

By contrast, models of correlated transport allow for the macroscopic process of conduction to consist of several interconnected steps. Examples include memory kernels in generalized master equations18,19, Burnett-order nonlinear hydrodynamics8,10 or kinetic competition6,20. In such models, the state of the material at time zero (Fig. 1b, dark green) influences transport dynamics over some non-negligible timescale (for example, t1 in Fig. 1b). The information entropy of transport is expressed as a mutual entropy between configurations a time lag t1 apart. This quantity increases (equivalently, mutual information decreases) until the full entropy of transport is produced, possibly over several successive atomistic steps or several consecutive correlated hops. Only at longer timescales (for example, at t2 in Fig. 1b) can transport parameters such as the diffusion coefficient and ionic conductivity be constant-valued, as for a random-walk process. Indeed, picosecond to nanosecond studies of ion conductors11,21,22 yield reduced activation energies, suggesting that the processes being probed at those frequencies may be incomplete with respect to overall conduction due to the persistence of such memory.

Understanding correlation effects in transport remains necessary to predict practical performance of ionic conductors from the atoms up3,13, and to exploit nonlinear nanoscale transport phenomena9 in devices. Here, we study the correspondence between the thermodynamics of ion transport and those of information23,24. Such a mechanistic investigation requires the ability to (1) impulsively trigger ion hops3 on the short timescale of 1/ν0, typically no more than 1 ps, and (2) track their outcomes over potentially much longer times7,8. Such a combination is inaccessible to current techniques11, and even the vibrational nature of ν0 remains under debate2,25,26. Furthermore, since ion hops, unlike those of electronic carriers, are stochastic and lack clear spectral resonance signatures, single-pulse reciprocal-space or Fourier-transform probes are insufficient for probing them. Instead, correlations in nanoscale transport7,8,9,10 could be probed using time-domain nonlinear optical pump–probe methods such as transient birefringence27,28,29 by means of these methods’ sensitivity to polarization. However, pump–probe studies of ionic dynamics have so far focused on coherent displacements of bound vibrational modes or order parameters30,31, in which many ions undergo coherent sub-ångström motions, rather than rare large-amplitude displacements (typically 2–3 Å) such as ionic hops.

Probing correlated hopping

We probe orientational correlations between ionic hops using a nonlinear optical method sensitive to ionic velocities (Fig. 1c). We trigger ionic hops with an impulsive single-optical-cycle pump (Fig. 1d) and measure the anisotropy in the directions of subsequent hops (Fig. 1e,f) as transient birefringence. The ionic response is pumped impulsively using single-cycle terahertz pulses with centre frequency near 0.7 THz (refs. 32,33), and transient birefringence is probed in transmission mode (Extended Data Fig. 1). The general solution for electric fields E in the sample at position z along the pulse paths at pump–probe delay t1 is34,35

$${E}_{{\rm{probe}}}\left(z,t\right)-E\left(z,t\right)\propto \frac{\partial }{\partial t}\left(P\left(z,t+{t}_{1}\right){E}_{{\rm{probe}}}\left(z,t\right)\right)$$
(1)

Here, the polarization P arises from coherent displacements of vibrational modes or bound dipoles Qi, plus an extra component from the history of hopping rates H:

$$P(t+{t}_{1})={\sum }_{i}{Q}_{i}(t+{t}_{1})+{\int }_{-\infty }^{t+{t}_{1}}H(\tau ){\rm{d}}\tau $$
(2)

The integral is zero until the pump drives ionic hops in a preferential direction (Fig. 1d). Because the emitted signal at t1 is proportional to the time derivative of the polarization, it arises from ionic velocities, to which hopping rates H(t1) contribute. In the terahertz-pumped Kerr effect (TKE) geometry, the anisotropy of velocities and hopping rates is measured specifically (Fig. 1c). This anisotropy corresponds to a time-dependent preference for hopping along the direction defined by the terahertz pump relative to an orthogonal direction. The relaxation of hopping anisotropy corresponds to the loss of memory of the pump-driven impulse by the conducting ions during the temperature-activated solid-state diffusion process. If conduction occurs via a random walk, then hopping immediately returns to isotropy and no relaxation should be observable. Furthermore, the modes Qi that couple appreciably to H contribute to attempt frequencies ν0.

Picosecond hopping dynamics in β-aluminas

We first use the fast ionic conductors β-aluminas (M1+2xAl11O17+x, where x is approximately 0.1, and the mobile ion M+ = Na+, K+, Ag+) as model systems. In β-aluminas, ion conduction occurs over two non-equivalent lattice sites (Fig. 1a), and correlations persist over timescales corresponding to several (at least two) consecutive hops7. The vibrational modes of the mobile ions fall between 0.7 and 3.0 THz, depending on the mobile ion36,37. To match the direction of ionic hopping, the pump electric field E is perpendicular to the crystalline c axis and parallel to the two-dimensional conduction planes. Figure 2a shows the time traces of the terahertz-pumped transient birefringence in β-aluminas at 300 K. All samples show both oscillatory and non-oscillatory responses at 300 and 620 K (Extended Data Fig. 1). Such incoherent, non-oscillatory relaxation has been previously observed in liquids27,29,38,39 and solids40, but was attributed to overdamped rotations or librations, which are absent in β-aluminas.

Fig. 2: TKE in β-alumina ion conductors.
figure 2

a, Time-domain traces of transient birefringence in β-alumina ion conductors: Na+ (orange), K+ (blue), Ag+ (pink) and electro-optic sampling trace of the pump field (green), smoothed with a 30 fs Gaussian window. The shaded regions correspond to ±1 s.e. of the mean signal at each time delay. The labelled feature at 8.4 ps is the rotational coherence of atmospheric nitrogen50 within a Rayleigh length (greater than 500 μm) of the samples (below 30 μm in thickness). b, Fast Fourier-transform (FFT) power spectra (points) of the signals in a, and smoothed with a Gaussian filter of 0.1 THz s.d. (lines), with peaks highlighted by arrows.

Full size image

Whereas the magnitude of the birefringence response scales with the square of the pump field (Supplementary Fig. 3), as expected for a Kerr effect signal, we do not observe any spectral changes with increasing pump fluence in either TKE or terahertz transmission experiments (Supplementary Figs. 3 and 5), indicating that we are probing a response intrinsic to the material and not a threshold-dependent response only relevant at high fields41. The non-oscillatory relaxation is substantially slower than any vibrational components. Therefore, we also rule out nonlinear phonon coupling42,43. Otherwise, the vibrational components of the TKE signals (Fig. 2b) are consistent with established far-infrared and Raman modes37 of mobile ions.

We investigate the possible connection between temperature-activated hopping conduction and the picosecond TKE relaxation by varying the temperature of the samples. In all materials, the signals from thinner (less than 30 μm) versus thicker (100–300 μm) samples match after the short-time oscillations dephase (Extended Data Fig. 2). The subsequent non-oscillatory relaxation represents a bulk material response. We analyse a thick sample of K+ β-alumina (Fig. 3a) here to highlight this long time signal. We model the response as the sum of an instantaneous response arising from intrinsic nonlinearity and a mismatch in optical constants between the pump and probe frequencies44,45 (Supplementary Notes 1 and 2 and Extended Data Figs. 3−5), and single-exponential decay, shown together as dashed lines. The slow non-oscillatory component is absent in the non-resonant optical Kerr response (Extended Data Fig. 6), and the residuals of the TKE fit (Extended Data Fig. 7a) show the same frequency, approximately 2 THz, as the non-resonant signal. Together, this suggests that both pumps (THz and optical) excite coherent ionic vibrations, but the terahertz pump excites an extra response that decays incoherently. For K+ β-alumina, this non-oscillatory relaxation accelerates from approximately 10 ps at 300 K to 3–4 ps at 620 K (Fig. 3b), with an activation energy of 40 ± 5 meV. In ambient atmosphere, the apparent time constants are slightly slower due to the overlaid signal from atmospheric water vapour27 (Supplementary Fig. 1), which is slower than the sample response and is not temperature activated.

Fig. 3: Temperature dependence of the long-lived TKE in K+ β-alumina.
figure 3

a, Time-domain traces of transient birefringence in K+ β-alumina, measured in ambient atmosphere (solid lines), normalized and offset for clarity. The shaded regions correspond to ±1 s.e.m. Coloured dashed lines are fits to the sum of single-exponent relaxation and an instantaneous polarization. The labelled feature at 8.4 ps is the rotational coherence of atmospheric nitrogen. b, Time constants of single-exponential fits to the long-lived TKE component as measured experimentally (expt, roughly 200 μm) in ambient atmosphere (filled symbols) and dry atmosphere (empty symbols). Error bars are ±1 s.e. of least-squares fitting. The dashed line at an activation energy of 40(5) meV is a fit to the purged-atmosphere measurements. c, Normalized anisotropy of hopping simulated (sim.) with molecular dynamics under applied electric field mimicking the experimental terahertz pump (Methods) at 300 K (blue) and 600 K (pink). The black dashed line shows the square of the simulated electric field.

Full size image

Eliminating several other plausible assignments, we propose that the liquid-like picosecond-timescale component of the TKE response arises from the incoherent hopping of mobile ions. We use large-scale molecular dynamics simulations with a pulsed electrical field27,46,47 mimicking the experimental THz pump in frequency and magnitude to verify this hypothesis. Following the simulated terahertz pump, the anisotropy of hopping rates is calculated from the times and crystallographic directions of all hops (Fig. 3c). The pump selectively accelerates mobile ions (Supplementary Note 4) and drives hopping along its electric field. In K+ β-alumina at 300 K, the anisotropy of hopping is distinguishable from zero until more than or equal to 6 ps after the peak applied field (Fig. 3c, blue), but relaxes to zero by 3–4 ps at 600 K, in agreement with experiment. The simulated relaxation of the hopping anisotropy is only slightly faster than the experimental TKE relaxation. We conclude that the liquid-like picosecond-timescale TKE response in β-aluminas is indeed a signature of anisotropy in the hopping of mobile ions caused by the pump, the decay of which is heat-activated with an energy of 40 ± 5 meV in K+ β-alumina (Fig. 3b).

Correlations and memory in ion hopping

We now discuss the physical meaning of the TKE signal. The contribution of hopping to the TKE signal for any time delay scales with the hopping rate H, weighed by the hop directions relative to the pump’s electric field. Hopping rates both parallel (+v||) and antiparallel (−v||) to the pump contribute positively to the TKE signal, whereas hopping orthogonal to the pump contributes negatively as ±v⊥ (Fig. 1c). The existence of the slow TKE signal signifies a mismatch between v|| and v⊥, and implies that directions of consecutive hops are correlated. This is consistent with previous simulations6,7: following pump-driven hops at t0 (Fig. 1d), the mobile ions have a preference to hop backwards (correlation factors f, fI ≪ 1 using solid-state ionics nomenclature14,15), which randomizes only slowly. TKE measures the decay of this preference and verifies that the atomistic mechanism of ion transport in β-aluminas differs from a random walk.

The hopping component of the TKE signal is temperature activated, but the activation is much smaller than for low-frequency conductivity (40 ± 5 meV in Fig. 3b versus more than 200 meV, ref. 48). However, any hop in isolation is not timescale-dependent. The difference between hopping at picosecond versus macroscopic timescales is not within individual hops, but in the mutual information between successive hops (Supplementary Note 5), conceptually similar to a viscoelastic effect. Small activation energies measured with TKE are consistent with the slow loss of such correlation via production of entropy (Fig. 1b), being the origin of increasing measured activation energies toward macroscopic timescales22. At picosecond timescales, as probed here, even though hopping is triggered, correlations persist throughout the measurement, and only partial activation is measurable. Conversely, the full activation for macroscopic conduction is measured only at timescales sufficient for the decay of all correlations over multiple hops.

Random-walk ion conduction in K+ β″-alumina

All β-aluminas show the picosecond ‘tail’ in the TKE response, implying that hopping remains correlated for longer than the TKE response is measurable7. This is expected because of the presence of non-equivalent lattice sites (Fig. 1). To verify the above interpretation, we next seek a control system in which ion transport could proceed via a true atomistic random walk. In such a system, any TKE tail could be expected to vanish rapidly. In the closely analogous β″-aluminas (M1.67Al10.67Li0.33O17, where the mobile ion M+ = Na+, K+, Ag+) all lattice sites for ion hopping are equivalent (Fig. 4a, inset). The β″-aluminas are expected to have true random-walk conduction at elevated temperature, but not at 300 K (ref. 7). In the TKE response of K+ β″-alumina, we observe a long-lived non-oscillatory relaxation at 300 K (Fig. 4a), as in the β-aluminas. At 620 K, this component is absent, which suggests the absence of long time correlations, and a more field-following response is observed overall.

Fig. 4: TKE in K+ β″-alumina.
figure 4

a, Normalized time-domain transient birefringence in thin polycrystalline K+ β″-alumina at 300 K (blue) and 620 K (pink). The shaded regions are ±1 s.e.m. The inset shows that in β″-aluminas, mobile ions (purple) hop (arrows) between equivalent four-coordinate sites. b, Normalized simulated (sim.) anisotropy of hopping directions in polycrystalline K+ β″-alumina under applied electric field centred at 0.7 THz mimicking the experimental pump, 300 K (blue) and 600 K (pink). The black dashed line indicates the square of the simulated pump electric field. The inset zooms in on the post-pump times and simulated hopping anisotropy at 300 K is non-zero for more than or equal to 5 ps. c, Simulated anisotropy of hopping in polycrystalline K+ β″-alumina at 300 K under applied electric fields centred at 0.7 THz (teal) and 2.5 THz (orange). The inset shows simulated in-plane optical K+ conductivity of K+ β″-alumina in the terahertz region (blue). Peaks at 1, 3, and 5 THz agree with the literature49. The spectra of simulated pumps with centre frequencies 0.7 THz (teal) mimicking the experimental pump (a,b), and 2.5 THz (orange) are shaded.

Full size image

In agreement with experiment, the simulated anisotropy of hopping following a 0.7 THz pump shows a long-lived relaxation at 300 K, but not at 600 K (Fig. 4b). Simulation reproduces the most important feature of experimental TKE: a picosecond-timescale anisotropy of hopping at 300 K that disappears by 600 K. Both experiment and simulation are consistent with a change of transport mechanism: from correlated at 300 K to a true random walk by 600 K (ref. 7). We conclude that the non-oscillatory picosecond-timescale relaxation of the TKE indeed probes the decay of orientational memory in ionic hopping at the level of individual ionic hops.

Attempt frequencies for ion hopping

Finally, we use the TKE measurements and simulations of K+ β″-alumina to identify the vibrational origin of ionic hops: the attempt frequency ν0. To cause an anisotropy in hopping, a pump field must excite the vibrational modes Qi that couple directly to hopping and trigger hopping aligned with the applied field. This must happen before thermalization, or else pump-driven heating enhances hopping rates in all directions isotropically. Having established the correspondence of the experimental TKE and simulated anisotropy of ionic hopping, we use further molecular dynamics simulations as in silico TKE at pump frequencies presently inaccessible to us experimentally. We simulate the anisotropy of hopping in K+ β″-alumina following a 2.5-THz pulse that overlaps with the known strong infrared-active vibration of the K+ ions in the conduction plane at 3.0 THz (ref. 49) (Fig. 4c, orange). Despite the material absorbing nearly the same amount of energy from simulated 0.7 and 2.5 THz pulses (Supplementary Fig. 6), and despite the 3.0 THz vibration being coherently driven by the 2.5 THz pulse (Supplementary Fig. 7), the 0.7 THz pulse creates an approximately 20-times stronger anisotropy of hopping than the 2.5 THz pulse (Fig. 4c). This suggests that the pulse centred close to 1 THz couples directly to hops, whereas the 2.5 THz pulse heats the material isotropically. We conclude that the attempt frequency in K+ β″-alumina is roughly 1 THz. The excitation of this vibration is evident in the experimental TKE response (Fig. 4a and Extended Data Fig. 7b).

The attempt frequencies in β-aluminas can be similarly identified once the distinct dynamics of bound defect clusters and non-equivalent lattice sites are disaggregated (Supplementary Note 6). In Na β-alumina, the simulated attempt frequencies are 1.4 ± 0.2 THz for the dominant ion-pair hopping mechanism and approximately 2.3 THz for the more rare single-ion hopping. The latter corresponds to the known vibration37 commonly taken as the attempt frequency41 and is verified here with THz transmission (2.1 THz, Supplementary Fig. 5 and Supplementary Note 6). Instead, the lower frequency vibration, highlighted by neutron scattering at 5–6 meV (ref. 36), drives most hopping at 300 K, despite constituting a minor part of the vibrational density of states for the mobile ion.

Conclusions

In summary, we have used impulsive near-resonant terahertz excitation to trigger ionic hopping in solid-state ionic conductors. Picosecond transient birefringence arises from correlations in the hopping of mobile ions. Therefore, we establish TKE, a nonlinear optical measurement, as a direct probe of path dependence in ionic transport7,8. TKE highlights both the vibrational origination of ionic conduction, and the slow loss of memory during diffusion. Ionic conductivity and its activation reach their low-frequency limits only at timescales sufficiently long to scramble all memory within the system. The transport of ions can be characterized by a random walk only at timescales longer than the persistence of correlations, but such a phenomenological random walk may not correspond to a true atomistic one. In other words, macroscopic measurements cannot be interpreted in terms of atomistic quantities without accounting for such correlations. By distinguishing rapid ion hops from the persistent correlations connecting them, our work demonstrates the correspondence between thermodynamics and information for thermally activated mass transport.

In addition to probing the mechanisms of ionic conduction, the correlation effects we highlight are of importance for transport under strong driving forces, at short timescales and in confined dimensions9,10, such as in switching applications3. This study provides a framework for the use of nonlinear optical techniques to probe the atomistic mechanisms of non-equilibrium transport phenomena, which will be highly valuable in the development of energy technologies such as solid-state batteries, and in the related fields of nanofluidics, engineering phase transformations and neuromorphic computing.

Methods

Sample preparation

Single crystals of Na β-alumina were graciously shared with us by O. Kamishima. They were ion-exchanged to Ag and K in molten nitrates until no mass change was detectable, at least 3 days. For Ag, the process had to be repeated. For K, mixed nitrate compositions were used first to avoid mechanical damage from the thermodynamically favourable ion exchange. The c lattice constants for the Na, Ag and K β-alumina samples were 22.53, 22.49 and 22.73 Å, respectively. Polycrystalline K β″-alumina was purchased from Ionotec Ltd as a pellet. To produce thin samples (roughly 5–30 µm), single crystals and the polycrystalline material were hand-polished using a T-tool and then dried. Double-side polished sapphire (0001) was purchased from MTI Corp.

TKE

The output of a Ti:sapphire oscillator (Coherent Micra) is amplified (Coherent Spitfire) to roughly 4.2 mJ at 1 kHz, and pulse width optimized to maximize the peak terahertz field at the sample, roughly 150 fs full-width at half-maximum. Of the output, 99% is used to generate terahertz pulses (roughly 6 μJ) via optical rectification in lithium niobate using the tilted pulse front method32,51. The terahertz pulse is focused on the sample using a pair of off-axis parabolic mirrors. Peak field amplitudes at the sample position were about 700 kV cm−1 in ambient atmosphere, and about 600 kV cm−1 in purged (at or below 0.1% RH) atmosphere. For the probe pulse, 1% of the amplifier output is used and polarized at 45° from the pump pulse, overlapped with the pump at the sample position and passed through a quarter-wave plate and a Wollaston prism. The birefringence of the transmitted probe is measured with two photodiodes in a balanced detection scheme. The terahertz waveform at the sample position is measured with a free-standing uniaxially poled mixture of an electro-optic dye and amorphous polycarbonate polymer52,53, and the peak field strength measured by electro-optic sampling in GaP(110). The time delay between pump and probe pulses is varied with a mechanical delay stage. For temperature control and measurements at elevated temperature, a transmission-mode heating stage (Linkam) was used without windows to eliminate their contributions to transient birefringence. The transient birefringence of ambient and purged air (Supplementary Note 2) was measured by aligning to a thin sample and removing the sample. Most experimental data are adapted from A.D.P.’s doctoral dissertation54.

Terahertz transmission

Thin (30 µm or below in thickness) samples are mounted on thin metallic pinholes of diameter 1 mm, fully covering the pinhole. The terahertz pulse transmitted through the sample is focused with a second pair of off-axis parabolic mirrors and sampled with the free-standing film of an electro-optic dye and amorphous polycarbonate polymer52,53. A time delay sweep of an empty pinhole is measured following every time delay sweep measuring the sample. Optical conductivity was fit to the time-domain spectra assuming a slab geometry55,56 using a nonlinear fitting procedure initialized with ñ = 2 + 0.5j as the complex refractive index. This yielded stable fitting without the single-pass assumption.

Optical Kerr effect

The output of a Ti:sapphire oscillator (Coherent Micra) was amplified (Coherent RegA) at 100 kHz to about 1 µJ and compressed to about 50 fs full-width at half-maximum (the width of the coherent artefact in the optical Kerr effect). The pump and probe pulses were overlapped on the sample so that probe E ⊥ c for single crystals, and pump roughly 15° off. The transmitted probe was detected using a pair of Si photodetectors in a balanced detection scheme. Frequencies of oscillatory components in the optical Kerr effect signals were fit using linear prediction fitting57 optimizing for least-squares error over the start and end points, which serve as hyperparameters.

Post-measurement characterization

The thin K β″-alumina sample was subjected to further high-intensity kHz-pulsed 800 nm illumination, including hitting the Inconel support. The heating of the Inconel support during this procedure created a small hole in the sample where it was thinnest. The local phase composition of the polycrystal K β″-alumina sample was subsequently analysed with X-ray microdiffraction (Bruker D8 Venture). All samples were imaged with an optical microscope to check for damage.

Steady-state molecular dynamics

Steady-state classical miolecular dynamics simulations were carried out in LAMMPS58 using Buckingham pairwise potentials with Coulombic interactions as described previously7. The vibrational density of states was calculated from the velocity autocorrelation functions for the mobile ions, and the optical conductivity from the velocity autocorrelation function for the centre of mass of the mobile ions59.

Molecular dynamics with terahertz pumping

An impulsive electrical field was included in the molecular dynamics simulations to mimic the experimental terahertz pulse27,28,46,47. Peak field E = 300 kV cm−1 polarized parallel to the conduction plane was used to account for a front-surface reflection of roughly 50%. Charge-compensating defects were placed in the scaled-up simulations using the same procedure as described previously7. To sample hopping events, 800 randomized iterations per temperature point, with 4,896 mobile ions each, were used for β-aluminas. For K β″-alumina, the simulations contained 5,940 mobile ions each. Each iteration started with an anneal to 1,000 K to randomize starting positions of the mobile ions, and a short equilibration at the simulation temperature in the constant-volume (NVT) ensemble. The electric field was applied in the microcanonical (NVE) ensemble and the simulations propagated for 15 ps, enough for mobile ions and the host lattice to come to the same temperature. For K β″-alumina, polycrystal orientations to match the experimentally available sample were averaged by changing the polarization of the simulated field over 12 angles in increments of 30°. No notable dependence on the pump polarization within the conduction plane was observed in simulations of β-aluminas. The hopping statistics were extracted from all simulations and analysed as described previously7. The direction of each hopping event was taken as the vector connecting the start and end crystallographic sites of the hop. The anisotropy of hopping was calculated by weighing the hopping directions by (cos θ)2 − ⟨(cos θ)2⟩, where θ is the angle between the simulated pump field and the hop direction and ⟨(cos θ)2⟩ = 0.5 in two dimensions.