Computational design of polaronic conductive Li-NASICON mixed ionic–electronic conductors

Abstract

Li-NASICONs featuring three-dimensional corner-sharing frameworks have been experimentally and theoretically demonstrated to be fast Li-ion conductors, with high room-temperature ionic conductivities. In this study, we conduct first-principles calculations to investigate the polaronic conduction in Li-NASICONs comprising different transition metal (TM) elements (Ti, V, Cr, Mn, Fe, Mo) and evaluate their potential as mixed electron-ion conductors. We find that the electron polaron conductivity of octahedral–octahedral (oct–oct) hopping in Li-NASICONs is strongly correlated with the Jahn–Teller activity on the TM site. While LiMn₂(PO₄)₃ and Li₃Cr₂(PO₄)₃ have a low predicted electron polaron conductivity owing to the J–T distortion resulting from the polaron formation, LiFe₂(PO₄)₃ is predicted to have the most facile polaron transport, which stems from the elimination of its intrinsic J–T distortion upon polaron formation. We also find that polaron hopping between octahedral and tetrahedral sites contributes limited electronic conductivity because of the large difference in the polaron formation energies on these sites. For polaron transport along purely tetrahedral sites, we find limited transport through the V⁵⁺/V⁴⁺ pair in LiGe₂(VO₄)₃ and LiTi₂(VO₄)₃ but better conductivity through the Mo⁶⁺/Mo⁵⁺ pair in LiTi₂(MoO₄)₃. Our research suggests that Li-NASICONs with TM elements involving no J–T distortion upon polaron formation on octahedral sites and Mo on tetrahedral sites are promising candidates for mixed ionic electronic conductors (MIECs).

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1. Introduction

Many electrochemical energy storage devices require the addition of carbon as a conductive additive to the cathode composite. While carbonaceous materials are inexpensive, they can be the source of degradation phenomena in several new technologies. For example, in solid-state batteries, carbon has been identified as the source of electrochemical breakdown of sulfide solid-state conductors.^1,2 In Li–O₂ batteries, porous carbon cathodes and organic carbonate electrolytes in non-aqueous Li–O₂ batteries are prone to degradation during cycling.³ The carbonaceous components react with the Li₂O₂ discharge product and form insulating Li₂CO₃ (ref. 4) and reduce the capacity of the cell.⁵ In addition, the incorporation of lithium as lithium carbonate at the Li₂O₂–electrolyte interface leads to an increased overpotential during cycling.⁶

To address these issues, attempts have been made to replace carbon in cathodes with other electron-conducting materials,⁷ such as noble metals and alloys,^8–11 titanium compounds (oxides,^12,13 carbides,¹⁴ and nitrides¹⁵), and perovskites in nonaqueous Li–O₂ batteries.^16,17 In addition, solid-state Li–O₂ batteries with Li-NASICON,^18,19 perovskite,²⁰ and zeolite²¹ inorganic electrolytes have also been proposed to avoid Li₂CO₃ formation.

One feasible approach to develop MIECs is to introduce electronic conductivity in crystals that already exhibit high Li-ion conductivity, such as perovskite, garnet, and Li-NASICON. For instance, Ma et al. investigated the perovskites Li_xLa_yMO_3−z with various TM elements (M = Ti, Cr, Mn, Fe, and Co) and demonstrated that Li_0.34La_0.55MnO_3−z exhibits high electronic conductivity (2.04 × 10⁻³ S cm⁻¹) and Li-ion conductivity (8.53 × 10⁻⁵ S cm⁻¹).²² This remarkable electronic conductivity was attributed to the favorable formation of oxygen vacancies, which result in a higher mobile electron concentration in the compound. Kim et al.²³ developed a MIEC cathode consisting of electron-conducting RuO₂ and Li-ion conducting La₂LiRuO_6−x to enhance the cyclability of solid-state Li–O₂ batteries. Jin et al.²⁴ introduced diverse TM elements (Mn, Fe, Co, Ni) into the Li and Zr sites of garnets and discovered that garnet with Fe in Li sites exhibits high mixed ionic–electronic conductivity ranging from 10⁻⁵ to 10⁻⁴ S cm⁻¹. The electronic conduction was attributed to polaronic conduction through the Fe²⁺–O–Fe³⁺ pathway. Recently, Li-NASICONs with compositions of LiTi₂(PO₄)_3−x(VO₄)_x and Li_1.3Al_0.3Ti_1.7(PO₄)_3−x(VO₄)_x, where x = 0.1, 0.2, 0.3, and 0.4, were shown to display mixed ionic–electronic conductivity ranging between 10⁻⁵ and 10⁻⁴ S cm⁻¹, with the enhanced electronic conductivity being linked to polaronic conduction through the V⁵⁺/V⁴⁺ pair.²⁵ Previous theoretical work has also reported low polaron hopping barriers in doped SrCeO₃ perovskite²⁶ and Li-Garnet compounds,²⁷ highlighting the possibility for electronic conduction in wide-band-gap transition metal oxides. Since Li-NASICONs can be synthesized with a diversity of TM on the octahedral and tetrahedral sites,^28–30 we evaluate in this paper the effect of each chemistry on the electronic conductivity.

Polarons are quasiparticles that involve the coupling of excess carriers with lattice vibrations in polarizable materials.³¹ Within polarons, holes or electrons are trapped in a local structural distortion. Numerous materials, including inorganic crystals (commonly oxides) and polymers, have been reported to contain polarons induced from a variety of sources, such as oxygen vacancies, photoexcitation, and extrinsic doping.³¹ Polarons contribute to electronic conductivity via a hopping mechanism, where a localized charge transfers from one atom to another assisted by the transport of the lattice distortion and the tunneling of the charge between two atomic sites.

In this study, we investigate polaronic conduction in Li-NASICONs with different transition metal (TM) elements (Ti, V, Cr, Mn, Fe, Mo) using first-principles calculation. We first compute the polaron formation energies in Li-NASICONs and find that all Li-NASICONs except Li₃Fe₂(PO₄)₃ have negative electron polaron-formation energies, indicating the favorable existence of electron polarons in the compounds. Next, we identify percolating conduction paths in Li-NASICONs and calculate their rate-determining hopping barriers for oct–oct, oct–tet, and tet–tet polaronic conduction. We use the computed hopping barriers and the electronic coupling between the initial and final polaronic state to estimate the room-temperature polaronic conductivity of Li-NASICONs. Our study suggests that Li-NASICONs with TM elements involving no J–T distortion upon polaron formation on octahedral sites or with Mo on tetrahedral sites to be promising candidates for MIECs.

2. Computational methodology

To localize an electron polaron in electronic structure methods, self-interaction needs to be fully or partially removed. Density functional theory (DFT) with a Hubbard-U correction or hybrid functionals are commonly adopted to reduce the charge self-interaction^32–37 for systems with d- or f-band electrons. The DFT+U scheme requires an appropriate selection of the Hubbard-U value for a reliable description of the d- or f-band electronic structure, which typically involves fitting theoretically predicted physical properties against their experimental counterparts. This process renders a different set of +U potentials for different physical properties and compounds, and the calculated polaron formation energy or hopping barrier is susceptible to change with +U potentials.^38,39 In contrast, hybrid functionals involve fewer empirical parameters to obtain localized polaronic solutions⁴⁰ and have been reported to be more successful in obtaining localized solutions than the DFT+U scheme.⁴¹ Hence, in this study, we adopt the HSE06 hybrid functional⁴² for all calculations, including the optimization of the polaron structure, its hopping barrier, and the projected crystal orbital Hamilton population (p-COHP) of the polaronic orbitals on the neighboring TM atoms. All the calculations are performed using the Vienna ab initio simulation package (VASP),^43–45 and the input generation is performed using the Pymatgen library.⁴⁶

To model the formation of a polaron, lattice-parameter optimizations for all neutral Li-NASICONs are first conducted. Subsequently, we add an extra electron to a Li-NASICON supercell containing 72 oxygens and perform geometrical optimization of the supercell with an excess electron with the lattice parameters of the cell fixed to those of the neutral cell. To more easily converge to a polaronic state, we introduce a structural perturbation to a TM–O₆ octahedron and a TM–O₄ tetrahedron by elongating each of the six and four TM–O bonds by 10%. This bond-distortion approach has been extensively used to break the symmetry of a crystal and facilitates electronic density localization on a polaronic site.^47–54 We note that initial bond distortions are only applied on the TM-site with an electron polaron. In addition, to promote the localization of an electron at the specified TM-site, we initialize the magnetic moment of the TM atom containing the polaron to be 1μ_B higher than those of the other TM atoms. To obtain a delocalized electronic structure, we add one excess electron to each Li-NASICON compound without bond distortion and initialization of the magnetic moments for TM atoms, which facilitates the optimization of the non-polaronic structure. For all DFT calculations, an energy cutoff of 520 eV is used for the plane-wave basis set, and a gamma-point-only K-point grid⁵⁵ is adopted. The convergence criteria for electronic and ionic steps are set to 10⁻⁵ eV for energy and 0.01 eV Å⁻¹ for force, respectively.

Polaron transport is studied using a linear-interpolation method to describe the transfer of lattice distortion^56–62 between two sites. In this approach, the intermediate images of the hopping trajectory {q_x} are obtained by linearly interpolating the initial {q_i} and final {q_f} optimized polaronic structures, i.e., {q_x} = (1 − x){q_i} + x{q_f}, where 0 < x < 1. In this study, seven intermediate images are used to locate the transition state and determine the kinetic barrier of each hop. We also initialize the magnetic moments of the two TM atoms that participate in a hop in the seven intermediate images as a linear interpolation between the magnetic moments of the two TM atoms in the initial and final polaronic structures for Fe- and Mn-containing Li-NASICONs. This magnetic moment initialization serves only to guide the convergence of the electronic structure of the intermediate images along the minimum energy pathways and does not constrain the magnetic moments during the optimization of the reaction pathway.

Although the linear interpolation approach may not always be valid for long-range polaron hops, we expect that it remains a reasonable approximation for simulating such hops in Li-NASICONs. To validate this, we perform a CI-NEB calculation for the longest hop (Hop IV) in LiTi₂(PO₄)₃, which has a hopping distance of 6.02 Å, using seven images and a force convergence criterion of 0.02 eV Å⁻¹. Notably, the hopping barrier obtained from the CI-NEB method is 0.27 eV, closely matching the 0.28 eV barrier obtained using the linear interpolation approach for the same hop. This suggests that the linearly interpolated images are already close to the minimum energy path for the hop. Therefore, we believe this approach reasonably captures the lattice distortion geometry during long-range hopping in Li-NASICON, although a more accurate hopping barrier can be obtained using the CI-NEB method at the expense of significantly higher computational cost.

We investigate the electron polaron transport for Li-NASICONs in the well-studied structure of LiTi₂(PO₄)₃, which has a trigonal lattice with space group R3̄C. The chemical space is explored by replacing different elements in octahedral or tetrahedral sites. Although Li-NASICONs can form in other structures (monoclinic,⁶³ triclinic,⁶⁴ and orthorhombic⁶⁵), the topologies of the corner-sharing framework consisting of MO₆ octahedra and PO₄ tetrahedra are similar to those of trigonal Li-NASICONs.⁶⁶

In the three-dimensional corner-sharing framework of Li-NASICONs, the octahedral sites can accommodate various TMs, including Ti, V, Cr, Mn, Fe, and Co. In contrast, the tetrahedral sites are usually occupied by main group cations (e.g. P, Si), though some fully oxidized TM such as V and Mo can occasionally also be accommodated on these sites. We study eleven different Li-NASICONs, eight of which contain one Li atom with the other three containing three Li atoms per formula unit in the R3̄C and R3̄ space group, respectively. We exclude Co-containing Li-NASICONs in this work as we are not able to localize an electron polaron to form a Co³⁺ in Co⁴⁺-containing Li-NASICON, which may be related to the ability of Co⁴⁺ to become metallic in oxides.^67,68 In this study, we only consider electron polarons as a previous work by Kaur et al.⁶⁹ found in V-oxides that electron and hole polarons behave similarly.

3. Results

3.1. Polaron formation in Li-NASICONs

The localization of a polaron causes the TM–O bonds of the TM in the polaronic site to elongate and the magnetic moment on the TM to increase by approximately 1μ_B. The TM–O bond lengths and the magnetic moments of the TM atoms before and after polaron formation in Li-NASICONs are shown in Fig. S1.† The computed polaron formation energies (E_P) of each Li-NASICON, as well as the space group and the source of each crystal model are listed in Table 1. E_P is calculated as the energy difference between the polaronic and delocalized solution, i.e., E_P = E_localized − E_delocalized. This value is negative for all the Li-NASICONs considered (Table 1), indicating favorable polaron formations in all compounds, consistent with our expectation. For Li₃Fe₂(PO₄)₃, we successfully find a localized solution, indicating that the formation of a polaron in this structure is feasible, although the formation energy could not be determined as we are unable to converge a delocalized solution for Li₃Fe₂(PO₄)₃. The formation energy of an electron polaron on Ti⁴⁺ is among the least negative ones, consistent with it being an early transition metal with more extended d-orbitals and its tendency in oxides to often become metallic when partially reduced. The electrochemical stability windows of Li-NASICONs, computed with the previous defined formalism,^71–73 are listed in the Table S3.† These values are calculated without considering the possibility of Li (de)intercalation, which may occur due to the mixed ionic–electronic conductivity. Details of the methodology are provided in the ESI.†

Table 1 Polaronic site, polaron formation energy (E_P), space group (S.G.), energy above the hull, and source of the crystal model of Li-NASICONs. “mp” refers to the Materials Project id numbers.⁷⁰ Energy above the hull values marked with an asterisk are obtained from our DFT calculations using the GGA/GGA+U functional, while those without an asterisk are extracted directly from the Materials Project

	Polaronic site	Valence state	E P (eV)	S. G.	Crystal model source	Energy above the hull (eV per atom)
LiTi2(PO4)3	Octahedral	Ti^4+/Ti^3+	−0.42	R3̄C	mp – 18640	0
LiV2(PO4)3	Octahedral	V^4+/V^3+	−0.52	R3̄C	mp – 26712	0.002
LiCr2(PO4)3	Octahedral	Cr^4+/Cr^3+	−0.54	R3̄C	mp – 25838	0.044
LiMn2(PO4)3	Octahedral	Mn^4+/Mn^3+	−0.60	R3̄C	Replace Ti with Mn in LiTi2(PO4)3	0.012*
LiFe2(PO4)3	Octahedral	Fe^4+/Fe^3+	−1.71	R3̄C	mp – 26581	0.079
Li3Ti2(PO4)3	Octahedral	Ti^3+/Ti^2+	−0.16	R3̄	mp – 26883	0.064
Li3Cr2(PO4)3	Octahedral	Cr^3+/Cr^2+	−0.33	R3̄	mp – 26971	0
Li3Fe2(PO4)3	Octahedral	Fe^3+/Fe^2+	—	R3̄	mp – 19430	0.010
LiTi2(VO4)3	Octahedral	V^5+/V^4+	−0.21	R3̄C	Replace P with V in LiTi2(PO4)3	0.005*
LiTi2(VO4)3	Tetrahedral	V^5+/V^4+	−0.61	R3̄C	Replace P with V in LiTi2(PO4)3
LiGe2(VO4)3	Tetrahedral	V^5+/V^4+	−0.71	R3̄C	Replace Ti with Ge in LiTi2(VO4)3	0.008*
LiTi2(MoO4)3	Tetrahedral	Mo^6+/Mo^5+	−0.39	R3̄C	Replace P with Mo in LiTi2(PO4)3	0.132*

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3.2. Percolating paths for polaronic conduction in Li-NASICONs

Polarons must have percolating paths through the entire structure to enable overall conduction. The Li-NASICONs which contain one Li per formula unit, namely Li(TM/Ge)₂((P/V/Mo)O₄)₃, contain only one symmetrically distinct octahedral (12c) and tetrahedral site (18e) to accommodate TM atoms. In these cases, we find that an electron polaron can be localized on each TM atom. However, for Li-NASICONs with compositions Li₃TM₂(PO₄)₃, there are two symmetrically distinct octahedral TM sites in the R3̄ space group. We find that the less distorted octahedral TM site (6c1) (edge-sharing with Li–O₄ tetrahedra) is able to accommodate localized charge, whereas the more distorted octahedral TM site (6c2) (corner-sharing with Li–O₄ tetrahedra) cannot accommodate a polaron. Consequently, fewer polaron hops are available in Li₃TM₂(PO₄)₃ than in LiTM₂(PO₄)₃. The octahedral TM sites that can accommodate polaron are visualized in Fig. 1b.

Fig. 1 Electron polaron hops of (a) oct–oct hopping in LiTM₂(PO₄)₃, TM = Ti/V/Cr/Mn/Fe; (b) oct–oct hopping in Li₃TM₂(PO₄)₃, TM = Ti/Cr/Fe; (c) oct–tet hopping in LiTi₂(VO₄)₃; and (d) tet–tet hopping in LiTi₂(TMO₄)₃, TM = V/Mo and LiGe₂(VO₄)₃. Only atoms near each constitutive hop are shown. The orange spheres denote Li atoms, the blue spheres denote atoms (TMs or Ge) in octahedral sites, the purple spheres denote atoms (P, V, or Mo) in tetrahedral sites, and the red spheres denote O atoms. All constitutive electron polaron hops are represented by the arrows from the center TM atom. For Li₃TM₂(PO₄)₃ in (b), the dark blue octahedra denote 6c2 sites which cannot accommodate polarons. The hops we consider for LiTi₂(MoO₄)₃ are the hop I and hop IV in (d).

In this work, we first determine the polaron hops between two neighboring TMs with hopping distances of less than 7 Å. This includes hops between octahedral sites, octahedral-to-tetrahedral, as well as between tetrahedral sites. We exclude hops with hopping distances of more than 7 Å, as these hops are combinations of shorter hops. In addition, the tunneling probability of a hop between two TM atoms exponentially decays with respect to the hopping distance. All the hops considered are either perpendicular or parallel to [001] direction, which aligns with the z-axis of the Li-NASICON crystal structure, as shown in Fig. 1. While most ions in our compounds have reasonable valence states for the coordination they occupy, LiTi₂(MoO₄)₃ with both Mo⁶⁺ and Mo⁵⁺ on the tetrahedral site is introduced as a more hypothetical compound, as Mo⁵⁺ is unlikely to be found on a tetrahedral site.^74,75 The ground-state electronic structure of LiTi₂(MoO₄)₃ exhibits magnetic moments of 0.90μ_B on each Ti ion, suggesting a valence +3. To maintain charge neutrality, the average valence of Mo atoms is then +17/3, which contain one-third Mo⁵⁺ and two-thirds Mo⁶⁺, consistent with the magnetic moments of 0.78μ_B and 0.09μ_B found in the calculations. We find that an electron polaron in this compound preferentially localizes on Mo⁶⁺, as evidenced by the electron polaron formation energy of −0.39 eV on Mo⁶⁺ and −0.34 eV on Mo⁵⁺. Therefore, we only consider hops between Mo⁶⁺ ions.

As shown in Fig. 1, the possible polaron sites lead to five distinct oct–oct hops in LiTM₂(PO₄)₃, two distinct oct–oct hops in Li₃TM₂(PO₄)₃, two distinct oct–tet and four distinct tet–tet hops in LiTi₂(VO₄)₃ and in LiGe₂(VO₄)₃. For LiTi₂(MoO₄)₃ we only considered two tet–tet hops through the Mo⁶⁺/Mo⁵⁺ pair perpendicular to [001], as the other two hops parallel to [001] create Mo⁴⁺ ions. For Li₃TM₂(PO₄)₃, we only investigate the oct–oct polaronic transport as P⁵⁺ is unlikely to accommodate a polaron.

The possible hops join to form percolating conduction paths. In this study, we classify the percolating paths into two types: perpendicular to [001] or parallel to [001]. For LiTi₂(MoO₄)₃, we only consider percolating paths perpendicular to [001]. The oct–tet hopping in LiTi₂(VO₄)₃ only leads to a percolating path perpendicular to [001]. The directions of all percolating paths, possible hops, and the polaron Wyckoff site are listed in Table 2.

Table 2 The first column gives the direction of the percolating paths. The second column gives the constitutive hops that create the percolating paths, and the last column gives the Wyckoff site involved. The //[001] and ⊥[001] direction correspond respectively to the direction parallel to and perpendicular to the z-axis in the Li-NASICON structure in Fig. 1

LiTM2(PO4)3, TM = Ti,V,Cr,Mn,Fe Oct–oct hopping			Li3TM2(PO4)3, TM = Ti,Cr,Fe Oct–oct hopping
Direction	Constitutive hop	Wyckoff site	Direction	Constitutive hop	Wyckoff site
//[001]	Hop I–Hop V	12c	//[001]	Hop I–Hop II	6c1
⊥[001]	Hop II	12c	⊥[001]	Hop I	6c1
⊥[001]	Hop III	12c
⊥[001]	Hop IV	12c

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LiTi2(VO4)3 Oct–tet hopping			LiTi2(VO4)3 and LiGe2(VO4)3 Tet–tet hopping
Direction	Constitutive hop	Wyckoff site	Direction	Constitutive hop	Wyckoff site
//[001]	Hop II–Hop III	18e	⊥[001]	Hop I–Hop II	12c (Oct) and 18e (Tet)
⊥[001]	Hop I–Hop IV	18e
⊥[001]	Hop IV	18e

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LiTi2(MoO4)3 Tet–tet hopping
Direction	Constitutive hop	Wyckoff site
⊥[001]	Hop I–Hop IV	18e

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3.3. Rate-determining polaron hopping barriers of Li-NASICONs

To determine the rate-determining polaron hopping barriers in each compound, we first identify the hop with the highest barrier for each percolating conduction path. The path with the lowest maximum barrier is the preferential conduction path to realize percolating conduction, and the lowest maximum barrier is identified as the rate-determining hopping barrier for the polaronic conduction in a Li-NASICON. The barriers for all hops in each compound are shown in Table 3. For LiTi₂(MoO₄)₃, only the barriers of hop I and IV perpendicular to [001] with the Mo⁶⁺/Mo⁵⁺ pair are calculated. The rate-determining hopping barriers are used to estimate the polaronic conductivity of Li-NASICONs.

Table 3 Electron polaron hopping barriers of the hops in Li-NASICONs. The rate-determining hopping barrier (in eV) of each Li-NASICON is highlighted by the bold number. For LiTi₂(VO₄)₃, two hopping barriers are highlighted as there are two hops with the same maximum barriers in the percolating conduction path

	Oct–oct hopping					Tet–tet hopping
	LiTi2(PO4)3	LiV2(PO4)3	LiCr2(PO4)3	LiMn2(PO4)3	LiFe2(PO4)3	LiTi2(VO4)3	LiGe2(VO4)3	LiTi2(MoO4)3
Hop I	0.16	0.28	0.26	0.43	0.14	0.40	0.51	0.26
Hop II	0.22	0.26	0.23	0.36	0.12	0.39	0.54	—
Hop III	0.27	0.27	0.19	0.43	0.13	0.49	0.58	—
Hop IV	0.28	0.33	0.26	0.35	0.09	0.40	0.52	0.29
Hop V	0.30	0.38	0.33	0.47	0.25	—	—	—

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	Oct–oct hopping			Oct–tet hopping
	Li3Ti2(PO4)3	Li3Cr2(PO4)3	Li3Fe2(PO4)3	LiTi2(VO4)3
Hop I	0.17	0.43	0.20	0.41
Hop II	0.13	0.39	0.17	0.45

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The rate-determining hopping barriers of oct–oct hopping in different Li-NASICONs range from less than 0.1 eV to more than 0.4 eV, with the lowest barrier of 0.09 eV through the Fe⁴⁺/Fe³⁺ pair and much higher hopping barriers of 0.35 and 0.43 eV through the Mn⁴⁺/Mn³⁺ and Cr³⁺/Cr²⁺ pairs, respectively. The oct–oct hopping through the Ti⁴⁺/Ti³⁺, V⁴⁺/V³⁺, Cr⁴⁺/Cr³⁺, Ti³⁺/Ti²⁺, and Fe³⁺/Fe³⁺ have barriers between 0.15 and 0.3 eV. These results indicate element-dependent oct–oct polaron transport in Li-NASICONs. The tet–tet hopping also depends on the nature of the TM present on that site. Although hopping through the V⁵⁺/V⁴⁺ pair has an energy barrier of more than 0.4 eV in LiTi₂(VO₄)₃ and LiGe₂(VO₄)₃, through the Mo⁶⁺/Mo⁵⁺ pair in LiTi₂(MoO₄)₃ a lower barrier of 0.29 eV has to be crossed. In LiTi₂(VO₄)₃, the oct–tet hopping has a high barrier of 0.45 eV through the Ti⁴⁺/V⁴⁺ pair. The rate-determining polaron hopping barriers for all Li-NASICONs are shown in Fig. 2a. All rate-determining hops occur along the percolating paths perpendicular to [001].

Fig. 2 (a) Rate-determining polaron hopping barriers of Li-NASICONs and (b) polaron formation energy in Li-NASICONs.

3.4. Polaronic conductivity in Li-NASICONs

We estimate the room-temperature polaronic conductivity of Li-NASICONs from the calculated rate-determining hopping barriers using the Marcus theory framework, which formulates the rate of charge-transfer across molecules or atoms in compounds.^76–78 In Marcus theory, a small polaron hopping is described by a parabolic free-energy curve for the polaron on each site, and their crossing point gives the hopping activation energy. When a polaron transfers from one site to another, its energy rises and reaches a maximum when the polaronic distortion is distributed between the two sites. At this crossing point, the polaron hopping rate depends on the tunneling probability. A tunneling probability close to identity indicates an adiabatic charge transfer, whereby a polaron hops to the final state once the lattice distortion reaches the transition-state configuration. In contrast, a limited tunneling probability results in a diabatic charge transfer, and the polaron may go back to the initial state even when the transition state has been reached. The tunneling probability is given by⁷⁸

P_tunnel = 1 − exp[−(V_e²/hυ_n)(π³/λk_BT)^1/2], (1)

where V_e is the electronic coupling between the initial and final state; h is Planck's constant; υ_n is the characteristic phonon vibrational frequency, which is on the order of 10¹³ Hz; λ is the reorganization energy; k_B is Boltzmann's constant; and T is the temperature in Kelvin. In a diabatic polaron transfer, the reorganization energy is approximated as 4 × E_diabatic, where E_diabatic is the diabatic hopping barrier⁷⁸ and is evaluated as the sum of the hopping barrier and the charge transfer integral. The electronic coupling, V_AB, is estimated along the lines of the Mulliken–Hush formalism within Marcus theory.⁷⁹ In this formalism, ΔE₁₂ is the energy difference between the polaronic bonding and antibonding states in the transition state during a hop and is related to the electronic coupling V_AB, as V_AB = ½ΔE₁₂.⁸⁰ Calculations of V_AB are conducted using the LOBSTER software,^81,82 which projects electron wavefunctions using local atomic orbital basis sets and reconstructs the atomic orbital-resolved bonding information between specific atomic pairs in a compound. The results for the calculated polaronic bonding and antibonding states are shown in Fig. S3.†

We estimate polaron hopping frequencies to obtain the diffusivities in Li-NASICONs. The adiabatic and diabatic polaron hopping frequencies are given by⁷⁸

f_adiabatic = υ_n exp[−E_a/k_BT] (2)

where ΔG⁰ is the energy difference between polarons in the initial and final state. Herein, Li₃Cr₂(PO₄)₃ and Li₃Fe₂(PO₄)₃ have diabatic rate-determining hops as these hops involve polaron tunneling probability deviating from identity as shown in Table 4. The ΔG⁰ is negligible as the initial and final TM site with a polaron are symmetrically equivalent. The diffusivity and polaron mobility are then expressed as:⁷⁸

D = fa²g (diffusivity) (4)

u = eD/k_BT (charge mobility). (5)

Here, f is either the adiabatic or diabatic polaron hopping frequency, a is the hopping distance between two TM atoms, g is the available TM site near the initial polaronic site for hopping and is assumed to be 1 as the rate-determining hop contains only one available final TM site for a polaron, and e is the elementary charge. Polaronic conductivity is the product of polaron concentration (n), elementary charge (e), and mobility (u):

σ = neu. (6)

Table 4 Characteristic properties of polaronic conduction in Li-NASICONs. V_e denotes the electronic coupling, n denotes the polaron concentration evaluated as one polaron per unit cell given by 1/(cell volume), a denotes the hopping distance, E_a is the hopping barrier, and P_tunnel is the tunneling probability

	V e (eV)	n (#/cm^3)	a (Å)	E a (eV)	P tunnel	Transfer type	Conductivity at 300 K (S cm^−1)
LiTi2(PO4)3	0.23	7.626 × 10^20	5.03	0.22	1	Adia; oct–oct	2.42 × 10^−2
LiV2(PO4)3	0.38	7.863 × 10^20	4.99	0.26	1	Adia; oct–oct	5.24 × 10^−3
LiCr2(PO4)3	0.30	8.067 × 10^20	5.37	0.19	1	Adia; oct–oct	9.31 × 10^−2
LiMn2(PO4)3	0.33	8.229 × 10^20	5.85	0.35	1	Adia; oct–oct	2.32 × 10^−4
LiFe2(PO4)3	0.13	8.430 × 10^20	5.85	0.09	1	Adia; oct–oct	5.51
Li3Ti2(PO4)3	0.20	7.171 × 10^20	4.89	0.17	1	Adia; oct–oct	1.49 × 10^−1
Li3Cr2(PO4)3	0.23	7.599 × 10^20	4.83	0.43	1	Adia; oct–oct	6.65 × 10^−6
Li3Fe2(PO4)3	0.18	7.405 × 10^20	4.86	0.20	1	Adia; oct–oct	4.75 × 10^−2
LiTi2(VO4)3	0.41	6.650 × 10^20	3.43	0.45	1	Adia; oct–tet	1.35 × 10^−6
LiTi2(VO4)3	0.50	6.650 × 10^20	4.51	0.40	1	Adia; tet–tet	1.62 × 10^−5
LiGe2(VO4)3	0.30	6.994 × 10^20	5.54	0.52	1	Adia; tet–tet	2.48 × 10^−7
LiTi2(MoO4)3	0.40	5.999 × 10^20	5.82	0.29	1	Adia; tet–tet	1.70 × 10^−3

---

Ultimately, for adiabatic and diabatic polaron hopping, the conductivity can be formulated as

σ_adiabatic = (ne²v_na²g/k_BT) × exp[−E_a/k_BT] (7)

All predicted characteristic properties of the rate-determining hops in Li-NASICONs are listed in Table 4.

Although the polaronic conductivity is determined by the hopping barrier and tunneling probability, for Li-NASICONs, most of the hops are adiabatic with tunneling probabilities of identity. Even the most diabatic hop, which is found in LiMn₂(PO₄)₃, has a tunneling probability of 0.41, suggesting a feasible polaron transfer when the transition state is reached, as shown in Table 5. Therefore, for Li-NASICONs the polaronic conductivity is mainly dominated by the hopping barrier and a percolating path with the lowest maximum barrier is the path with the highest polaronic conductivity. Amongst all Li-NASICONs in the current study, polaron transport via the Fe⁴⁺/Fe³⁺ pair on octahedral sites in LiFe₂(PO₄)₃ has the highest calculated room-temperature conductivity of 5.51 S cm⁻¹, whereas the transport through the Cr³⁺/Cr²⁺ pair on octahedral sites in Li₃Cr₂(PO₄)₃ and through the V⁵⁺/V⁴⁺ pair on tetrahedral sites in LiTi₂(VO₄)₃ and LiGe₂(VO₄)₃ has much lower predicted room-temperature conductivity of 10⁻⁷ to 10⁻⁵ S cm⁻¹. The oct–tet polaron transport in LiTi₂(VO₄)₃ is also predicted to be poor (near 10⁻⁶ S cm⁻¹). All rate-determining hops of Li-NASICONs are adiabatic.

Table 5 Polaron hopping barriers, electronic coupling, tunneling probability and hopping types for all hops in Li-NASICONs. The rate-determining step for evaluating the polaronic conductivity of each Li-NASICON is marked by bold characters. The cutoff tunneling probability for diabatic/adiabatic hopping is arbitrarily set to 98%

		E a (eV)	V e (eV)	P tunneling	Hopping type
LiTi2(PO4)3	Hop I	0.16	0.28	1	Adiabatic
	Hop II	0.22	0.23	1	Adiabatic
	Hop III	0.27	0.13	1	Adiabatic
	Hop IV	0.28	0.13	1	Adiabatic
	Hop V	0.30	0.03	0.480	Diabatic
LiV2(PO4)3	Hop I	0.28	0.25	1	Adiabatic
	Hop II	0.26	0.38	1	Adiabatic
	Hop III	0.27	0.23	1	Adiabatic
	Hop IV	0.33	0.18	1	Adiabatic
	Hop V	0.38	0.03	0.444	Diabatic
LiCr2(PO4)3	Hop I	0.26	0.18	1	Adiabatic
	Hop II	0.23	0.40	1	Adiabatic
	Hop III	0.19	0.30	1	Adiabatic
	Hop IV	0.26	0.23	1	Adiabatic
	Hop V	0.33	0.03	0.466	Diabatic
LiMn2(PO4)3	Hop I	0.43	0.13	1	Adiabatic
	Hop II	0.36	0.28	1	Adiabatic
	Hop III	0.43	0.18	1	Adiabatic
	Hop IV	0.35	0.33	1	Adiabatic
	Hop V	0.47	0.03	0.413	Diabatic
LiFe2(PO4)3	Hop I	0.14	0.05	0.909	Diabatic
	Hop II	0.12	0.08	0.997	Adiabatic
	Hop III	0.13	0.05	0.915	Diabatic
	Hop IV	0.09	0.13	1	Adiabatic
	Hop V	0.25	0.05	0.852	Diabatic
Li3Ti2(PO4)3	Hop I	0.17	0.20	1	Adiabatic
	Hop II	0.13	0.03	0.610	Diabatic
Li3Cr2(PO4)3	Hop I	0.43	0.23	1	Adiabatic
	Hop II	0.39	0.03	0.440	Diabatic
Li3Fe2(PO4)3	Hop I	0.20	0.18	1	Adiabatic
	Hop II	0.17	0.03	0.569	Diabatic
LiTi2(VO4)3 (oct–tet)	Hop I	0.41	0.51	1	Adiabatic
	Hop II	0.45	0.41	1	Adiabatic
LiTi2(VO4)3 (tet–tet)	Hop I	0.40	0.50	1	Adiabatic
	Hop II	0.39	0.23	1	Adiabatic
	Hop III	0.49	0.18	1	Adiabatic
	Hop IV	0.40	0.48	1	Adiabatic
LiGe2(VO4)3	Hop I	0.51	0.45	1	Adiabatic
	Hop II	0.54	0.23	1	Adiabatic
	Hop III	0.58	0.18	1	Adiabatic
	Hop IV	0.52	0.30	1	Adiabatic
LiTi2(MoO4)3	Hop I	0.26	0.40	1	Adiabatic
	Hop IV	0.29	0.40	1	Adiabatic
LiTi2(VO4)1/9(PO4)8/9	Hop I	0.38	0.53	1	Adiabatic
	Hop II	0.38	0.58	1	Adiabatic

---

4. Discussion

Owing to the corner-sharing framework that alleviates the coulombic repulsion between Li ions and positively charged TM ions,⁸³ Li-NASICONs have low ionic migration barriers and high room-temperature ionic conductivities of 10⁻⁴ to 10⁻³ S cm⁻¹.^84,85 However, intrinsic wide band gaps (several eVs) and narrow bands prohibit band conduction in the structures.⁸⁵ For instance, LiTi₂(PO₄)₃ has a band gap of 2.50 eV (ref. 86) and a low room-temperature electronic conductivity of 10⁻⁸ S cm⁻¹.⁸⁷ To improve their electronic transport, attempts have been made to exploit mixed-valency in Li-NASICONs to generate localized charge carriers such that electronic conduction is activated via a polaron hopping mechanism. Specifically, Sharma and Dalvi²⁵ doped vanadium into LiTi₂(PO₄)₃ and Li_1.3Al_0.3Ti_1.7(PO₄)₃ to generate LiTi₂(VO₄)_x(PO₄)_3−x and Li_1.3Al_0.3Ti_1.7(VO₄)_x(PO₄)_3−x and improved its room-temperature electronic conductivity to 10⁻⁵ S cm⁻¹.

4.1. Polaronic conduction in LiTi₂(VO₄)_x(PO₄)_3−x

To validate our computational results, we compare the predicted hopping barrier and the room-temperature polaronic conductivity of LiTi₂(VO₄)_x(PO₄)_3−x with published experimental results.²⁵ We replace two P atoms with V atoms in LiTi₂(PO₄)₃ to approximate the experimental composition of LiTi₂(VO₄)_0.3(PO₄)_2.7 (Fig. S5†). In this V-doped Li-NASICON, the polaron formation is more favorable on V atoms than on Ti atoms, as the charge only localizes on V atoms even when a structural distortion and an initialization of the Ti magnetic moment are imposed on a Ti–O₆ octahedron. The preferential localization of an electron polaron on a V atom in a tetrahedral site is consistent with the experimental observation of V⁵⁺/V⁴⁺ pairs in the V-doped Li-NASICON. The computed tet–tet hopping barrier via the V⁵⁺/V⁴⁺ pair is 0.38 eV, comparable but somewhat higher than the experimental hopping barrier of 0.20 eV. The computed room-temperature polaronic conductivity is 3.62 × 10⁻⁵ S cm⁻¹, in reasonable agreement with the experimental conductivity of 4.46 × 10⁻⁶ S cm⁻¹. In our calculations, the polaron concentration is fixed at one polaron per supercell containing 72 oxygen atoms.

4.2. Correlation between J–T distortion and rate-determining hopping barrier of oct–oct polaron transport in Li-NASICONs

We classify the oct–oct polaron transport into three classes depending on the presence of any Jahn–Teller (J–T) activity with or without the electron polaron present: only the state with polaron is J–T active (Class 1: Mn⁴⁺/Mn³⁺ and Cr³⁺/Cr²⁺); both the polaronic and pristine state of the TMs are J–T inactive (Class 2: Ti⁴⁺/Ti³⁺, V⁴⁺/V³⁺, Cr⁴⁺/Cr³⁺, Ti³⁺/Ti²⁺ and Fe³⁺/Fe²⁺); and only the state without the polaron is J–T active (Class 3: Fe⁴⁺/Fe³⁺). We note that only cations with a doublet degeneracy in the e_g band (Exe J–T coupling) involving a static J–T distortion are classified as J–T active, whereas cations with either a triplet degeneracy in the t_2g band (T_x(t + e) coupling) involving a dynamic J–T effect, or cations with no degeneracy, are classified as J–T inactive. For Class 1, the J–T distortion is stabilized by the electron polaron formation and results in a relatively high hopping barrier as the lattice distortion that must be migrated is largely caused by the J–T coupling with the polaron. This leads to low polaronic conduction in LiMn₂(PO₄)₃ and Li₃Cr₂(PO₄)₃. For Class 2, no J–T distortion is involved during polaron hopping, resulting in lower hopping barriers than those of class one. In Class 3, the J–T distortion vanishes upon the formation of an electron polaron leading to a lower lattice distortion, while the TM sites without the extra electron are “pre-distorted” and possess multiple degrees of freedom by which they can distort at low cost in energy (i.e., the J–T modes).⁸⁸ Hence, the polaronic conductivity in LiFe₂(PO₄)₃ is the highest with a very low barrier of 0.09 eV. This obvious correlation between hopping barriers and J–T distortion in Li-NASICONs reflects the adiabatic nature of polaron transport, which is phonon-assisted and dominated by the transport of the lattice distortion. The TM–O bond lengths of the J–T active species are shown in Table S1.†

With the assumption that polaron transport can be approximated as adiabatic hopping at room temperature in most oxides,^69,89,90 other materials involving TM–O₆ octahedra might also exhibit a correlation between high conductivity and J–T distortion in the non-polaronic TM ion. For instance, in olivine LiMPO₄ structured phosphates, the hole polaron hopping barrier via the Mn²⁺/Mn³⁺ pair is more than 0.3 eV, higher than those through the Fe²⁺/Fe³⁺ and Co²⁺/Co³⁺ pairs with no J–T distortion during hopping,⁵⁹ consistent with the fact that Mn³⁺ is J–T active. In addition, both the hole polaron hopping through the Co³⁺/Co⁴⁺ pair and the electron polaron hopping through the Co³⁺/Co²⁺ pair involve no J–T distortion in LiCoO₂ (ref. 60) and have energy barriers roughly between 0.1 and 0.3 eV, which are similar to the hopping barriers in class two. Furthermore, the electron polaron transport through the Mo⁶⁺/Mo⁵⁺ J–T inactive pair in MoO₃ ranges from 0.11 to 0.35 eV,⁶¹ similar to the hopping barriers in class two. Notably, the electron polaron hopping barrier through the Mn³⁺/Mn²⁺ pair in MnPO₄ is only two-thirds of the hole polaron hopping barrier through the Mn²⁺/Mn³⁺ pair in LiMnPO₄.⁶² The applicability of J–T distortion-correlated polaron hopping barriers to other oxides of TMs underpins the 3d electronic configuration as an indicator for polaronic conduction. Despite the hopping barriers of the oct–oct polaronic transport being quite different among various mixed-valency pairs, a majority of Li-NASICONs have predicted room-temperature polaronic conductivity above 10⁻⁴ S cm⁻¹, including those with the Ti⁴⁺/Ti³⁺, Ti³⁺/Ti²⁺, V⁴⁺/V³⁺, Cr⁴⁺/Cr³⁺, Fe⁴⁺/Fe³⁺, and Fe³⁺/Fe²⁺ mixed-valency pairs. Polaronic conduction through the oct–oct pathway in Li-NASICONs may open up the possibility of Li-NASICON mixed ionic electronic conductors.

Previous study on polaron hopping in AgCl compound found that the hopping barrier depends on the orientation of the J–T axis in the octahedra.⁹¹ To investigate whether the orientation of the J–T axis influences the polaron hopping barrier in Li-NASICONs, we examine whether polaronic states can be stabilized along all three principal axes. We reorient the elongated axis of the original extrinsic J–T distorted octahedron of Mn³⁺ and Cr²⁺ along the other two principal axes and then perform geometry optimizations. Our results show that, in both Mn³⁺ and Cr²⁺ octahedra, the elongated axis reverts back to the original principal axis, indicating that J–T distortion is only stable along a single principal axis in Li-NASICONs. This finding rules out the possibility that the polaron hops by first changing direction of the JT axis and then hopping.

4.3. The effect of the Li charge state on polaron hopping in Li-NASICONs

Another factor that may contribute to the oct–oct polaron transport in the Ti-based Li-NASICONs is the amount of Li present in the NASICON. The additional Li ions in Li₃Ti₂(PO₄)₃ introduce attractive coulombic interactions with the polaron during a hop, potentially leading to a smoother hopping energy landscape and a lower hopping barrier. This effect is reflected in the two hopping barriers of 0.13 eV and 0.17 eV in Li₃Ti₂(PO₄)₃ compared to the hopping barriers in LiTi₂(PO₄)₃ which are mostly above 0.2 eV as shown in the Table 3. In contrast, for Cr-based NASICONs, although the additional Li⁺ ions in Li₃Cr₂(PO₄)₃ could facilitate polaron hopping, their effect may be overpowered by the J–T active Cr²⁺ upon polaron formation resulting in a higher barrier. Hence, unlike in Ti-based NASICONs where increasing the Li content lowers the hopping barriers, Li₃Cr₂(PO₄)₃ has higher hopping barriers than in LiCr₂(PO₄)₃, as shown in the Table 3.

4.4. Oct–tet and tet–tet polaron transport in Li-NASICONs

Oct–tet polaron transport is kinetically less favorable than oct–oct polaron transport in the only compound for which we evaluate this because of the difference in the polaron formation energy on a Ti and V atom in LiTi₂(VO₄)₃. Although a polaron on Ti has a formation energy of −0.21 eV, it is more stable on V with a formation energy of −0.61 eV, which leads to an asymmetric hopping energy landscape that creates a high barrier of 0.45 eV, as shown in Fig. 3. Oct–tet polaron hopping may also have high barriers in other systems due to the difference in the polaron formation energy in the different coordinated environments. Similar results were reported in a previous study on oct–tet electron polaron hopping in Co₃O₄.⁹²

Fig. 3 Asymmetric polaron hopping energy curve of oct–tet polaron transport in LiTi₂(VO₄)_3.

As fewer TM elements (V and Mo) have been reported to occupy tetrahedral sites in Li-NASICONs, tet–tet polaron hopping may be of less practical importance for application in MIECs. Nonetheless, the room-temperature polaronic conductivity via the Mo⁶⁺/Mo⁵⁺ pair on tetrahedral sites in LiTi₂(MoO₄)₃ is predicted to be on the order of 10⁻³ S cm⁻¹, comparable to the room-temperature ionic conductivities in Li-NASICONs reported in previous studies.^93–95 This result indicates that molybdenum-based Li-NASICONs may also exhibit efficient electronic conductivity via polaronic conduction.

4.5. The effect of lattice volume and polaronic orbital overlap on polaronic conduction in Li-NASICONs

Aside from the J–T distortion, other factors have also been proposed to account for the different hopping barriers in a compound. For instance, Kweon et al.⁵² and Moradabadi et al.⁹⁰ independently pointed at an increase in the polaron hopping barrier as a result of lattice expansion for BiVO₄ and LiCoO₂, respectively. In our study, we also find a similar trend in the hopping barriers of LiTi₂(PO₄)₃ and LiGe₂(VO₄)₃, as shown in Fig. 4. Lattice expansions elongate the hopping distance between two TM atoms, which is accompanied by the increase in the hopping barrier, consistent with the previous studies. However, the volume changes arising from the chemistry differences in Li-NASICONs cannot explain the barrier variation seen with chemistry. Fig. 5 shows the oct–oct hopping barrier as a function of the volume of each LiTM₂(PO₄)₃ unit cell. Although the lattice volume of Li-NASICONs monotonically increases from Ti to Fe, the hopping barriers of oct–oct polaronic transport via their +4/+3 mixed-valency TMs show no such monotonic change.

Fig. 4 Polaron hopping barrier versus lattice parameter variation of (a) LiGe₂(VO₄)₃ and (b) LiTi₂(PO₄)₃. The lattice parameter variation is the percentage of the homogeneous compression or elongation of the lattice parameters in all three dimensions, i.e., −1% denotes compressing all three lattice parameters by 1%; 1% denotes elongating all three lattice parameters by 1%. Numbers near squares denote the polaron hopping distance.

Fig. 5 Oct–oct polaron hopping barrier as a function of the lattice volume of LiTM₂(PO₄)₃, TM = Ti, V, Cr, Mn, and Fe.

Another possible factor reported by Garcia-Lastra et al.⁹⁶ to affect polaron hopping barrier is the extent of polaronic orbital overlap. These authors found that both hole polaron and electron polaron hopping barriers exhibit differences along different hopping directions in Li₂O₂ stemming from the different extent of polaronic orbital overlap in the transition states. Nevertheless, these differences in hopping barriers were found to be less than 0.1 eV in Li₂O₂ and may have limited effect on the polaronic transport. The polaronic orbital overlap is quantified by the integration of the projected crystal orbital Hamilton population (pCOHP) of the polaronic state with respect to energy up to the highest occupied state (the Fermi level);⁹⁷ namely, the integrated projected crystal orbital Hamilton population (IpCOHP). We compute the IpCOHP of the polaronic 3d orbitals of two TM atoms participating in the rate-determining hop in the transition states of various Li-NASICONs, and plot them in Fig. 6. There appears to be no correlation between the overlaps and the hopping barriers.

Fig. 6 Polaron hopping barrier as a function of the polaronic orbital overlap between the 3d orbitals of two TMs participating in the rate-determining hop in the transition state of Li-NASICONs. The blue data points denote oct–oct hopping, the pink data point denotes oct–tet hopping, and the purple data points denote tet–tet hopping.

5. Conclusion

We examine polaronic conduction in Li-NASICONs using first-principles calculations and demonstrate that significantly enhanced electronic conductivity may be achieved from polaron hopping if a high enough carrier concentration can be generated. Polaronic conductivities through the Ti⁴⁺/Ti³⁺, Ti³⁺/Ti²⁺, V⁴⁺/V³⁺, Cr⁴⁺/Cr³⁺, Fe⁴⁺/Fe³⁺, Fe³⁺/Fe²⁺, and Mo⁶⁺/Mo⁵⁺ pairs are predicted to be over 10⁻⁴ S cm⁻¹ at room temperature, with the highest conductivity of 5.51 S cm⁻¹ achieved through the Fe⁴⁺/Fe³⁺ pair in LiFe₂(PO₄)₃. Our calculations indicate that in Li-NASICONs, the hopping barrier of oct–oct polaron transport is correlated with the J–T distortion associated with polaron formation, whereas the lattice volume and the polaronic orbital overlap have minor effects on polaronic conduction. Our understanding of polaronic conduction in Li-NASICONs may assist in identifying effective strategies to develop Li-NASICON MIECs for electrochemical energy storage devices.

Data availability

The authors confirm that the data supporting the findings of this study are available within the article and its ESI.†

Conflicts of interest

The authors declare no conflict of interest.

Acknowledgements

This work is supported by the Samsung Advanced Institute of Technology (SAIT). The computational analysis is performed using computational resources sponsored by the Department of Energy's Office of Energy Efficiency and Renewable Energy located at the National Renewable Energy Laboratory (NREL) under the saepssic and ahlssic allocations. Computational resources are also provided by the Advanced Cyberinfrastructure Coordination Ecosystem: Services & Support (ACCESS) program, which is supported by National Science Foundation grants #2138259, #2138286, #2138307, #2137603, and #2138296. K. Jun gratefully acknowledges support from the Kwanjeong Educational Foundation scholarship.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d5ta00367a

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