Author Correction: Fundamental investigations on the sodium-ion transport properties of mixed polyanion solid-state battery electrolytes

one af liationdetailsfor incorrectly and of Bengaluru California, USA ’ but have been ‘ Materials Department and Materials Research Laboratory, University of Santa California, USA ’ . Also, in the original version of the Supplementary Information associated with this Article, the af ﬁ liations of Anthony K. Cheetham, Gopalakrishnan Sai Gautam and Pieremanuele Canepa were incorrectly reported and numbered. Also, this article contained mis-typing errors in Figure 2, Figure 4, Supplementary Table 1, Supplementary Table 4 and Supplementary Note 3. In Figure 2, panel b, the y-axis label was incorrectly reported as “ Na+ conductivity, log10() (S/cm) ” . In Figure4, panel a, the x-axis label was incorrectly reported as “ 2 (°) ” . In Figure 4, panel b, the y-axis label was incorrectly reported as “ ln(T) (SKcm-1) ” . In Supplementary Table 1, the numerical values “ 623.9 ” and “ 625.2 ” in the “ AE end ” column were missingthe “ - ” sign,andtheterms “ Na(1) ” and “ Na(2) ” wereswapped.InSupplementary Table4, the number “ 8 ” wasused for twodifferent “ Cluster # ” rows.In Supplementary Note 3, the terms “ Na(1) ” and “ Na(2) ” were swapped, and the term Δ E end was incorrectlyreported as “ | Δ E end | ” . The correct af ﬁ liations,axis labels,numericalvalues


Supplementary Figures
Supplementary Figure 1. Panel a Migration unit and migration events indicated as first Na(2)→Na(1) and second Na(1)→Na (2). The Na(1) site is indicated in grey, whereas (Na) sites are colored in orange. Panel b migration unit emphasizing the diffusion environment 0Si3P_3Si0P, with P and Si atoms indicated as light-blue and green, respectively. Panel c migration barriers of Na atom from (left) Na(1) site to adjacent Na(1) site via a Na(2) site for a x = 0 and (right) Na (2) to Na(1) site for a x = 3 in Na1+xZr2SixP3-xO12 structure. The naming convention αSiβP_γSiδP is used. Migration energies are adjusted so that the average energies of the initial and final images are 0. Figure 2. NEB migration barriers form the Na(2) site to an adjacent Na(2) site via a Na(1) site for x = 2 in Na1+xZr2SixP3-xO12 structure at different Si/P environment. The naming convention αSiβP_γSiδP is used. While charge neutrality is globally maintained for all barriers, local charge neutrality is not enforced. Migration energies are adjusted so that the average energies of the initial and final images are 0. Figure 3. NEB computed migration barrier form the Na(2) site to an adjacent Na(2) site via a Na(1) site for x = 2 in Na1+xZr2SixP3-xO12 structure at different Si/P environment. These configurations maintain both global and local charge neutrality by removing extra Na atoms from the migration unit of Figure 1d in the main text. The naming convention αSiβP_γSiδP is used. Migration energies are adjusted so that the average energies of the initial and final images are 0. Figure 4. Comparison between the predicted (y-axis) and calculated EKRA (a) and ΔEend (b) (xaxis) shown as blue dots. Pink ribbons indicate tolerable bounds (± 50 meV) in reproducing computed values of EKRA and ΔEend. The root mean square error was found to be ± 29.15 meV and ± 38.0 meV for EKRA and ΔEend local cluster expansion models, respectively.   Figure 7. SEM analysis of sintered pellet with composition Na2.5Zr2Si1.5P1.5O12 . Panel a shows the as recorded SEM picture. Panels b shows a filtered micrograph using an adaptive thresholding algorithm using a neighborhood area of 15x15 px 2 . Panel c shows the NASICON grains isolated using the boundary detection algorithm. Panel d shows the resulting histogram of particle size. The average grain area for Na2.5Zr2Si1.5P1.5O12 was found to be ~ 0.028 µm 2 with the maximum grain area of 9 µm 2 and minimum grain area of 4 x 10 -4 µm 2 .

Supplementary
Supplementary Figure 8. SEM analysis of sintered pellet with composition Na3Zr2Si2P1O12. Panel a shows the as recorded SEM picture. Panels b shows a filtered micrograph using an adaptive thresholding algorithm using a neighborhood area of 15x15 px 2 . Panel c shows the NASICON grains isolated using the boundary detection algorithm. Panel d shows the resulting histogram of particle size. The average grain area for Na3Zr2Si2P1O12 was found to be ~ 0.043 µm 2 with the maximum grain area of 24.8 µm 2 and minimum grain area of 4.6 x 10 -4 µm 2 .
Supplementary Figure 9. SEM analysis of sintered pellet with composition Na3.4Zr2Si2.4P0.6O12. Panel a shows the as recorded SEM picture. Panels b shows a filtered micrograph using an adaptive thresholding algorithm using a neighborhood area of 15x15 px 2 . Panel c shows the NASICON grains isolated using the boundary detection algorithm. Panel d shows the resulting histogram of particle size. The average grain area for Na3.4Zr2Si2.4P0.6O12 was found to be ~ 0.035 µm 2 with the maximum grain area of 28.5 µm 2 and minimum grain area of 4 x 10 -4 µm 2 .
Supplementary Figure 10. Variable temperature impedance spectra of Na1+xZr2SixP3-xO12 with nominal compositions (a, d) x = 1.5, (b, e) x = 2.0 (c, f) x = 2.4, respectively. The impedance spectra are obtained for different temperatures of ~193 K, ~273 K and 313 K and high-to-mid frequency (3 GHz -1 Hz) for panel a-c, and ~373 K, ~473 K and 573 K for mid-to-low frequency (10 MHz -0.1 Hz) panels d-f. The equivalent circuits are shown above each plot. Examples of fits of the impedance spectra are shown in panels a, b and c. The bulk and apparent grain boundary resistances are qualitatively indicated by the semicircles and arrows as a guide for the eye. In high-frequency low-to-moderate temperature AC Impedance measurements, two semicircles were observed, corresponding to bulk (high frequencies) and grain boundary (low frequencies) resistance contributions (see samples   Table 6. NASICON pellet characteristics, including the mass (accuracy ±0.0001g) and thickness of the samples (accuracy ±0.01mm), the measured surface area. Thickness and surface areas are used to compute the volume of the pellets, which is then used to estimate the relative density of the sintered pellets to the NASICON bulk. The compacity of the pellets indicates the deviation of the pellet density from the NASICON theoretical density (at specific Na contents), which is assumed as 100 %. Supplementary Table 6 also reports the amount of ZrO2 impurities in wt. %. derived from the Rietveld refinements of the powder X-ray diffraction presented in the main text.

Supplementary Note 1
Quantum mechanical density functional theory (DFT) calculations were performed using the Vienna ab initio Simulation Package (VASP). 8,9 Projected augmented wave (PAW) potentials 10,11 were used with the following electrons treated explicitly: Na 3s 1 , Zr 4s 2 4p 6 4d 2 5s 2 , Si 3s 2 3p 2 , P 3s 2 3p 3 and O 2s 2 2p 4 . The exchange-correlation energy was calculated using the strongly constrained and appropriately normed (SCAN) functional. 12 A 2×2×1 Monkhorst-Pack 13 k-point mesh and a 520 eV kinetic energy cutoff were used to integrate the DFT total energy in the 1 st Brillouin zone and expand the multielectron wave-functions of the Na1+xZr2SixP3-xO12 (NASICON) structures. These settings were employed for the structure optimization of the NASICON in the hexagonal conventional unit cell (6 f.u.), whereas for the pseudo-cubic structure (8 f.u.) at low Na content (mentioned below) the total energy was only sampled at the Γ-point. Three Na concentrations were computed: x = 0, 2 and 3, where Na diffuses along Na(1)➝Na(2)➝Na(1), Na(2)➝Na(1)➝Na(2), and Na(1)➝Na(2), respectively. A charge-neutral Na vacancy was created for each structure to enable these migrations. Selected energy formation to create these vacancies were added to the energy barrier fitting (see Method section in the main text). For x = 0, a large pseudo-cubic supercell structure (8 f.u.) was used and obtained by transforming the primitive trigonal cell (2 f.u.) using a matrix: [[1, 1, -1], [-1, 1, 1], [1, -1, 1]]. For x = 2 and 3, the most stable configurations 14 were taken (6 f.u.). At x = 2, different local Si/P configurations were also calculated. Structures with charge-neutral vacancies at initial and final images in the diffusion path were fully optimized until the interatomic forces were less than 0.01 eV/Å. These images are interpolated and used for nudged elastic band simulation as described in the next section.

Supplementary Note 2
Climbing image nudged elastic band (CI-NEB) calculations were performed using the method by Henkelman et al. 15 Table 1. The NEB calculations were carried out using the SCAN functional and following the computational settings indicated in the main manuscript. 12 The different local environments were created by replacing the Si/P from the two triangles shown in Supplementary Figure 1a. Our NEB plots follow the convention of αSiβP_γSiδP. In this nomenclature, the values of α, β, γ and δ are derived by counting the occupation of Si and P in the two Si and P triangles (Supplementary Figure 1a). The atomic species (Si or P) for sites numbered 1-2-3 are set α and β indices, and sites 4-5-6 are set γ and δ indices, as shown in Supplementary Figure 1b. For example, 0Si3P_3Si0P correspond to a migration unit, where sites 1-2-3 are occupied by Si only and sites 4-5-6 by P only.
As per Supplementary Figure 1a, the calculated barrier follows two distinct migration events indicated in Supplementary Table 1 as first Na(2)→Na(1) and second Na(1)→Na (2), which are captured by the NEB barriers. Thus, we compute the complete migration barrier Na(2)→Na(1)→Na(2) by averaging these two events (i.e., Na(2)→Na(1) and Na(1)→Na (2)). The only exception is Na4Zr2(SiO4)3 where all Na(2) sites are occupied, in this case we have only explored half path Na(1)→Na (2). Supplementary Figure 1a shows the two migration paths and an example of nomenclature of migration environment.

Supplementary Note 3
As mentioned in Figure 1 in the main text, a local cluster expansion (LCE) model is constructed by searching the nearest-neighbour Na and Si/P sites around a Na(1) site with a cut-off of 5 Å. The local structural motif used in the model is listed in Supplementary Table 1. Within the local structure, cutoffs of 10 Å and 6 Å were used for searching the pairs and triplets, respectively, which results in the generation of 24 distinct orbits. These orbits are used to generate the Hamiltonian to determine the EKRA and the endpoint energy as shown in Supplementary Equation 1: where V0 and Vorbit are called empty cluster interaction and kinetic effective cluster interactions (KECI), which are the coefficients to be determined. The information of the points, pairs and triplets terms are shown in Supplementary Supplementary Table 3. E is either kinetically resolved activation barrier 16 (EKRA) or energy difference between Na located at Na(1) and Na(2) sites (ΔEend = E[Na@Na (2)] -E[Na@Na(1)]). ΔEend and EKRA are fitted separately using 2 sets of V0 and Vorbit (Supplementary Table  and Supplementary Table ) and the diffusion barrier between Na(1) and Na(2) sites can be calculated as below: σ is the site occupation (σ[Na/Si] = -1 and σ[Va/P] = 1). σNa(1) and σNa(2) are the Na site occupations at the Na(1) and Na(2) sites for each diffusion event. An orbit is a collection of all symmetrically equivalent clusters (points, pairs, and triplets), and orbit is called the correlation vector of an orbit. This can be evaluated by summing up the products of the occupation of each site within each cluster using Supplementary Equation 3:

Supplementary Equation 3
Then a lasso fitting is performed with an L1 regularization parameter of α = 1.8. 17 Supplementary Table  shows

Supplementary Note 4
Rejection-free kinetic Monte Carlo (kMC) simulations were performed using the method described in van der Ven et al., 16,18 using the LCE Hamiltonian (V0 and Vorbit in Supplementary Equation 1). Only Na migration is considered in this study and Si/P/O are kept fixed. The migration probability of each Na diffusion event can be determined by: ; ;

Supplementary Equation 4
where the Ebarrier for this event can be calculated using Supplementary Equation 2, and * is related to the vibrational entropy of the image at the energy maximum and minimum along the migration path, which is taken as 5×10

Supplementary Equation 5
where Γtot is the summation of the probability for all events and m is the index of the event to be chosen. The time Δt for this Na migration event can be computed using Supplementary Equation 6:
All Na + were tracked during kMC simulations and Na + diffusivity D, Na + tracer diffusivity D * , Haven's ratio HR, and averaged correlation factor f, can be evaluated from Na + trajectories. All computed quantities (D, D * , HR and f), as well as Na + occupancy are arithmetically averaged for 50 structures at each composition for each temperature. D is computed using Supplementary Equation 7: where is the displacement vector of a Na + with an index i at after time t and N is the total number of Na + in the simulation cell. The correlation between different Na + can be measured by Haven's ratio calculated as Supplementary Equation 8. Averaged correlation factor f measures the correlation between successive Na + jumps for a single Na + , then averaged by the total number of Na + in a structure, as in Supplementary Equation 10.

Supplementary Equation 10
where indices i and j refer to different Na + and different jumps for a single Na + , respectively. Mi is the total number of hopping for each Na + and is the averaged hopping distance (3.4778 Å). f is taken as 0 when there is no diffusion after simulation ends.

Supplementary Note 5
The pellet dimensions are shown in Supplementary Table 6. Supplementary Table 7, Supplementary  Table 8 and Supplementary Table 9 list the structural parameters obtained from the Rietveld refinements of the PXRD diffractograms of Na2.5Zr2Si1.5P1.5O12, Na3Zr2Si2PO12 and Na3.4Zr2Si2.4P0.6O12 presented in the main text. Supplementary Figure 10 shows the AC impedance spectra measured at high-frequency (3 GHz -1 Hz) and low temperature (panels a-c), as well as mid-to-low frequency (10 MHz -0.1 Hz) and high temperature (panels d-f). Details on the setup of these measurements are given in the methodology section of the main text. Appropriate equivalent circuits are also shown. Dashed blue lines in Supplementary Figure 10 indicate an example of such fit. The equivalent circuits used to fit the low-tomoderate temperature, high-frequency AC impedance are shown in Supplementary Figure 10a-c, whereas the high-temperature measurements are shown in Supplementary Figure 10d-e.
The equivalent circuits used to extract the total conductivity (resistivity), bulk and apparent grain boundary conductivities are shown in of each Na1+xZr2SixP3-xO12 sample are shown in Supplementary Figure 10. Supplementary Table 10 shows the resistance, capacitance and inductance values of resistors, constant phase elements and inductors of each equivalent circuit as obtained from fitting selected impedance data of Supplementary Figure 10a-c.The tail phenomenon observed in sample x = 2.0 is common in solid electrolytes, e.g. Li7La3Zr2O12. 19 At low temperatures, in sample x = 2.0, the apparent grain boundary resistance is smaller than the bulk resistance (Supplementary Figure  10b), which signifies that grain boundaries do not impede Na-ion transport.
Supplementary Figure 11 shows Arrhenius plots derived from the high-frequency (3 GHz to 0.1 Hz) AC impedance data in the temperature range of 433 to 173 K of Na1+xZr2SixP3-xO12 compositions: x = 1.5 (panel a), x = 2.0 (b) and x = 2.4 of the sintered pellets in Supplementary Table  5. These measurements can resolve the bulk, Ea(bulk) and grain boundary Ea(GB) activation energies from the total activation energy Ea(total), which quantify the resistive effects of grain boundaries on the overall Na-ion transport in Na1+xZr2SixP3-xO12. As temperature increases in Supplementary Figure 11a and Supplementary Figure 11b, the grain boundaries conductivities approach monotonically the values of bulk conductivities, and their differences appears negligible for any measurement above 333 K (~60 °C). Therefore, it is safe to say that at temperature above 333 K (~60 °C) the total ionic conductivity is only dominated by Na-ion transport in the bulk of the NaSICON particles. In Na3Zr2Si2P1O12 grain boundaries appear to help Na + transport, so the limiting factor appears the bulk resistance.
The highest regime of ion transport in NaSICONs with compositions for 1.8 < x < 2.5 is achieved at temperature above 150 °C, and beyond monoclinic-to-rhombohedral phase transition. Indeed, separating bulk from grain boundary phenomena at high temperatures is not a trivial task, hitting the capabilities of high-frequency impedance spectrometers. Because for NaSICON compositions with 1.8 < x < 2.5 show a phase transition from monoclinic-to-rhombohedral, 1,20 a change in slopes in the Arrhenius curves at high temperature is observed at high temperatures (see Figure 4b of the main text). In contrast, at composition x < 1.8 and x > 2.5, Na1+xZr2SixP3-xO12 remains always in the rhombohedral phase. 14,20 To this end, we can only extrapolate grain boundaries effects at temperatures higher than ~300 K (Supplementary Figure 12) using the impedance data of Na2.5Zr2Si1.5P1.5O12 (x = 1.5), which is guaranteed to remain rhombohedral in the full temperature range (-100 to 300 °C). Supplementary  Figure 12 demonstrates that the grain boundary contributions to the total conductivity is insignificant.
The available body of facts presented in Supplementary Figures 10-12 suggest that grain boundary contributions to the total conductivity will be only important at low temperatures, but not at temperatures above 60 °C (100 °C, 200 °C and 300 °C) central to this manuscript. An exception to this is trend is Na3Zr2Si2P1O12, where grain boundary resistance helps Na + transport.