Unexpected role of electronic coupling between host redox centers in transport kinetics of lithium ions in olivine phosphate materials

Abstract

The discrepancy between the trend in the diffusion coefficient of a lithium ion (D_Li⁺) and that in the activation energy of ion hopping signals hidden factors determining ion transport kinetics in layered olivine phosphate materials (LiMPO₄). Combining density functional theory (DFT) calculations and the Landau–Zener electron transfer theory, we unravel this hidden factor to be the electronic coupling between redox centers of the host materials. The ion transport process in LiMPO₄ is newly described as an ion-coupled electron transfer (ET) reaction, where the electronic coupling effect on D_Li⁺ is considered by incorporating the electronic transmission coefficient into the rate constant of the transfer reaction. The new model and DFT calculation results rationalize experimental values of D_Li⁺ for various LiMPO₄ (M = Fe, Mn, Co, Ni) materials, which cannot be understood solely by the calculated activation barrier of ion hopping. Interestingly, the electronic coupling between host redox centers is found to play an essential role. Particularly, the sluggish ion mobility in LiFePO₄ is due to a very weak electronic coupling. The obtained insights imply that one can improve the rate performance of intercalation materials for metal-ion batteries through modifying the electronic coupling between redox centers of host materials.

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1. Introduction

Ion transport in intercalation compounds is an elementary process underpinning rechargeable metal-ion batteries that drive the paradigm shift of storing sustainable and clean energy. LiFePO₄, a layered olivine phosphate material composed of earth-abundant and non-toxic elements, exhibits numerous merits such as low cost, high structural stability, and competitive electrochemical performance, and therefore constituting one of the mainstream cathode materials in rechargeable lithium-ion batteries.^1–4 Key challenges faced by LiFePO₄ include low electronic conductivity (10⁻¹⁰–10⁻¹⁵ S cm⁻¹) and ionic mobility, hindering the realization of high-power devices.^5–8 A practical consequence of this is that the charging time of electrical vehicles powered by Li_xMPO₄ is much longer compared to the fuel filling time of conventional fossil-fuelled cars.^9–12

Therefore, the mechanism and kinetics of ion transport in LiFePO₄ has attracted research interest for decades from both computational and experimental sides in the past few years. In experimental studies, the diffusion coefficient of lithium ions, D_Li⁺, in pure LiFePO₄ is found to vary between 10⁻¹³ and 10⁻¹⁵ cm² s⁻¹,^13–17 while theoretical calculations have arrived at numbers that are several orders of magnitude higher.^18–25 Such discrepancy is rooted in the belief that the lithium ion transport through diffusion in the lattice frame of intercalation solids has led to the use of the activation barrier of ion hopping (E_a) to estimate D_Li⁺ according to the following Arrhenius type of kinetic equation:¹⁸

where the pre-exponential factor D₀ represents the activationless (free) diffusion coefficient. First-principles-based computations mostly have given relatively small numbers for E_a. The unreasonably high D_Li⁺ values estimated by E_a thus suggest that the Li⁺ transport in LiFePO₄ is far more complicated than the simple particle diffusion process described by eqn (1).

Indeed, there are evidences showing that transport of electrons and Li⁺ ions is closely coupled in LiFePO₄. The electron-coupled Li⁺ transport^26,27 (or equivalently, Li⁺-coupled electron transfer) mechanism explains why the carbon coating technique,^28–30 aimed initially at improving the electronic conductivity, fortuitously increases the ionic diffusivity in LiFePO₄. The first experimental clue of the electron-coupled Li⁺ transport mechanism was reported by Ellis et al.²⁷ They found in Mossbauer measurements that the isomer shift corresponding to the average Fe²⁺/Fe³⁺ environment is closely correlated with the Li⁺ disorder in the lattice, and that Li⁺ transport is correlated with the vibrational modes of the phosphate lattice. First-principles calculations by Maxisch et al. revealed that the activation energy of Li⁺-coupled ET is much lower than that of the decoupled transfer process.³¹ In addition, nonadiabatic molecular dynamics simulations by Tao et al. have also revealed that the Li⁺ diffusion can be significantly enhanced upon the formation of a small polaron.²⁶

Although widespread experimental and computational efforts underscore the ion-coupled ET nature of Li⁺ diffusion in LiFePO₄, there lacks a quantitative description of Li⁺ transport kinetics in this coupled charge transfer mechanism. Herein, we propose a kinetic model for ion transport in intercalation solids by describing it as an ion-coupled ET reaction. The electronic effect is incorporated into the model formula by using the Landau–Zener ET theory, which allows the ion transport kinetics to be directly correlated with the electronic coupling between the host redox centers. In the meantime, first-principles calculations based on the density functional theory (DFT) are performed to determine the key parameters in the model formula. The comparison between the model results and experimental data clearly demonstrates the determining role of electronic coupling in the ion dynamics of LiMPO₄, and thus providing guidance for rational design of ion-intercalation electrode materials.

2. Results and discussion

2.1 Microkinetic model

LiMPO₄ has an olivine structure with O forming a distorted hexagonal close-packed (hcp) framework which accommodates Li and M at the octahedral interstices and P at the tetrahedral interstices respectively (Fig. 1 for LiFePO₄). During charging (discharging) processes, Li ions are removed (added) topotactically, maintaining the topology of the MPO₄ framework. There are three possible hopping paths for Li ions, corresponding to the motion along the b direction in the hexagonal a–b planes of the oxygen hcp host, that between the b channels in the neighboring hexagonal planes along the c axis, and that between the neighboring b channels in the a–b plane.¹⁸ DFT-based nudged elastic band calculations have shown that Li hopping along the b direction has the lowest energetic barrier,¹⁸ which was later confirmed in experiments.³²

Fig. 1 Illustration of Li⁺ transport via a curved trajectory between adjacent sites along the b direction of LiMPO₄.

Considering that lithium ion transport in LiMPO₄ is mainly along the b-direction,^3–8,18 we can model it as a one-dimensional (1-D) ion hopping process. We adopt the recent idea that the ion hopping in concentrated media corresponds to an ion-vacancy coupled transfer reaction, that is, I_x − V_x ↔ I_x±a − V_x±a.^33,34 This underlying idea is that the appearance of an ion at position x is accompanied by the loss of a vacancy there. The rate of this ion-vacancy coupled transfer reaction can be formulated according to the law of mass action, that is^33,34

where c_x refers to the number density of ions at position x, k_x→y represents the rate constant of ion transfer from position x to y (y = x + a, or x − a), a is the average distance of ion hopping, and v is the average volume excluded by an ion (v = a in a 1-D system). The product vc_x thus represents the volume fraction occupied by ions at position x, and the term (1 − vc_x) is that of vacancies accordingly.

In the spirit of transition state theory and the Brønsted–Evans–Polanyi (BEP) relationship, the rate constants k_x→y can be expressed as,

where μ⁰_x refers to the standard chemical potential of ions at position x, β (≈0.5) is the BEP coefficient (also called the symmetric factor), and the pre-factor k⁰, the so-called standard rate constant, refers to the equal rate constants of the forward and backward hopping processes under thermodynamic neutral conditions, i.e., μ⁰_y = μ⁰_x. When the standard chemical potential varies very mildly so that μ⁰_y − μ⁰_x ≪ k_BT, taking the first two items of the Taylor expansion of k_x→y into eqn (2), we have the following near-equilibrium thermodynamic equation for ion transport,^33,34where D = a²k⁰ is the diffusion coefficient. Thus, the ion diffusion coefficient is linked to the standard rate constant of an ion hopping reaction. According to the transition state theory, k⁰ can be related to the activation energy of ion hopping (E_a) as k⁰ = 1/2τ₀ exp(−E_a/k_BT),³² where τ₀ refers to the average time interval of the ion hopping events. The factor 2 in the denominator of the right-hand side is the consequence of activation entropy when considering that the transition state occupies two vacancies.^33,34 It can also be understood that a transition state has the equal probability of 1/2 to go forward and backward. Thus, D can be formulated as,

In the case when the activation barrier is negligibly small, e.g., the free diffusion, one has D = a²/2τ₀, which is exactly the result of Einstein's diffusion theory for Brownian motion.

For the electron-coupled ion hopping process, the value of τ₀ should be co-determined by electronic and nuclear factors. As a first approximation, we use τ₀ = (v_nκ_el)⁻¹, where v_n is the effective frequency of nuclear motion and κ_el the electronic transmission coefficient. Considering that the ion hopping event is mainly a result of the vibration of the M–O bond, we assign the typical frequency (10¹³ s⁻¹) of the normal vibration of the M–O bond to v_n.³⁵ To estimate κ_el, we use the Landau–Zener formula,^36–38 that is, , where is the electron hopping frequency at the transition state, λ is the configurational reorganization energy of electron transfer (Fig. 2), and H_AB is the electronic coupling matrix element which represents the electronic coupling strength between the redox centers,³⁹ which are the neighboring transition metal centers in LiMPO₄ (Fig. 2). Thus, we have the following formula for the ion diffusion coefficient:

Fig. 2 Schematic illustration of the potential energy surface for Li⁺ -coupled electron transfer (ET) reaction in LiMPO₄ as a function of the nuclear configuration. The solid and dashed curves represent the adiabatic and nonadiabatic cases. Q_A and Q_B are the equilibrium nuclear coordinates of the states A and B, respectively, and Q_C is the nuclear coordinate corresponding to the lowest energy on the crossing seam surface. λ is the intramolecular reorganization energy, E_a is the activation energy for the adiabatic ET, and H_AB represents the strength of electronic coupling between neighboring transition metal centers at the transition state.

Depending on the magnitude of H_AB, an ET process can be adiabatic or nonadiabatic. The adiabatic ET system has a large H_AB, which means v_el ≫ v_n, and therefore κ_el ≈ 1. In this case, τ₀ is mainly determined by the nuclei factor and the ion diffusion coefficient reads D = (1/2)a²v_n exp(−E_a/k_BT), which corresponds to a thermally activated diffusion process with an Arrhenius type of kinetics as described in eqn (1) and is negligibly effected by the electronic effect. For a nonadiabatic ET with small H²_AB and κ_el, the ion diffusion coefficient reads D = (1/2)a²v_el exp(−E_a/k_BT), and can be reformulated as,

where a proportional relationship is found between D and H²_AB. In other words, E_a and H_AB should be used together to describe the ion transport properties in LiMPO₄.

Due to the fact that the adjacent redox centers in LiMPO₄ are separated by relatively large distance, the electronic interactions should be weak, making the ET process mostly nonadiabatic. Therefore, the key parameters determining the Li⁺ diffusion coefficient include H_AB, λ, and E_a.

2.2 DFT-based parameterization

We employed the DFT calculations to determine these model parameters for Li⁺ transport in LiMPO₄ (M = Fe, Mn, Co, Ni). Specifically, minimum energy pathways (MEPs) and saddle points of Li⁺ hopping were obtained with the climbing-image nudged elastic band (CINEB) method⁴⁰ in the PWscf (Plane-Wave Self-Consistent Field) module in Quantum ESPRESSO with the ultrasoft pseudopotentials for nuclei and core electrons (see details in the Methods section).⁴¹ Usually, 5–7 images are interpolated in CINEB computations to determine the transition states and activation barriers of ion migration in solid crystals. Considering the relatively simple 1-D hopping of Li ions and small change of the solid structure in the hopping process of Li ions in LiMPO₄, five images are adequate to identify the transition states.^18,20 Therefore, we interpolated five images between the starting and end points in the ion transport process to simulate the intermediate states, with the positions of the starting point and the end point fixed. This can significantly reduce the computation cost without losing the accuracy. The transition states obtained agree with those reported in the literature.^18,20

The values of H_AB were determined by using a large-scale parallel multiconfigurational self-consistent field (MCSCF) implemented in the NWChem computational chemistry code,⁴² and the bases are set as 6-31g** for P, O and Li, and LANL2DZ ECP for Fe, respectively. In H_AB calculations, the reactant and the product had the same configurations as those of the transition state obtained in the CINEB method, but the charges were assigned differently to distinguish the reactant, transition state and product of an ET process (Fig. 2).

A 1a × 2b × 1c (S.G. Pnma) supercell was used in all calculations. The cell contains 8 Li⁺ sites in total, with 7 sites filled with Li⁺, thus having a composition of Li_0.875MPO₄, which is in consistence with the model in previous studies so that the calculated activation barriers are comparable with the reported values.

The configuration reorganization energies (λ) were calculated using the procedure shown in Fig. 3. First, structural optimization was performed to obtain the stable configurations of LiMPO₄ (Fig. 3(a)) and MPO₄ (Fig. 3(d)), which have energies of E₁ and E₂ respectively. Then, the LiMPO₄ structure was obtained by filling Li atoms in the stable configuration of MPO₄ without altering the host structure (Fig. 3(c)), giving a total energy of E₃; and the structure of MPO₄ was obtained by simply removing all Li atoms from the stable configuration of MPO₄ while leaving other constituents unaltered (Fig. 3(b)), which gives an energy of E₄. Thus, λ₁ = (E₃ − E₁) is assigned to the reorganization energy of a LiMPO₄ site in an ET process, and λ₂ = (E₄ − E₂) to that of a MPO₄ site. The total configuration reorganization energy for the ET process between a pair of LiMPO₄ and MPO₄ sites is the sum of λ₁ and λ₂, namely, λ = λ₁ + λ₂.

Fig. 3 Schematic diagram for computation of the configuration reorganization energies of the reaction in LiMPO₄.

Fig. 4 shows the energy profiles of MEPs of Li⁺ hopping between two adjacent sites along the b axis in Li_0.875MPO₄ (M = Fe, Mn, Ni, and Co) calculated using the CINEB method, with the inset numbers indicating the corresponding E_a values. These values (0.40 eV for LiFePO₄, 0.51 eV for LiMnPO₄, 0.64 eV for LiCoPO₄, and 0.36 eV for LiNiPO₄) are close to that obtained from the DFT+U calculations by Dathar et al. (0.47 eV for LiFePO₄),²⁰ but differ from those obtained from the Born model of solids (0.55 eV for LiFePO₄, 0.62 eV for LiMnPO₄, 0.49 eV for LiCoPO₄, and 0.44 eV for LiNiPO₄)²³ and those obtained from the DFT calculations without Hubbard correction for strongly correlated d electrons (0.27 eV for LiFePO₄, 0.25 eV for LiMnPO₄, 0.36 eV for LiCoPO₄, and 0.13 eV for LiNiPO₄).¹⁸

Fig. 4 DFT-calculated energy profiles of Li⁺ hopping along the MEP between adjacent Li sites in Li_0.875MPO₄.

2.3 Electronic coupling

Given the DFT-calculated parameters of a², H_AB, λ, and E_a, we estimated the diffusion coefficients of Li⁺ using eqn (6). The DFT-calculated parameters and the corresponding theoretical diffusion coefficients are listed in Table 1, where the experimental diffusion coefficients taken from the literature are also given for comparison. It is revealing to note that LiFePO₄ and LiNiPO₄ show a difference of three orders in experimentally-measured diffusion coefficients, although they have very close E_a values. Besides, compared with LiMnPO₄, LiCoPO₄ has a higher experimental diffusion coefficient in spite of a larger E_a. Such anomalies indicate that the activation energy inadequately predicts the ion transport kinetics in olivine phosphate materials as described in eqn (1).

Table 1 Calculated values of the activation barriers, the strength of electronic coupling matrix elements, the configuration reorganization energies, and the diffusion coefficients for Li⁺ transport in Li_0.875MPO₄

M	a (Å)	E a (eV)	λ/4 (eV)	H AB (eV)	v el (s^−1)	(cm^2 s^−1)	(cm^2 s^−1)
Fe	3.634	0.398	0.390	1.39 × 10^−3	2.59 × 10^10	4.43 × 10^−13	10^−15–10^−13 (ref. 21–25)
Mn	3.684	0.506	0.815	0.24 × 10^−3	5.35 × 10^8	2.08 × 10^−15	10^−15–10^−14 (ref. 8 and 43–45)
Co	3.572	0.639	0.179	4.91 × 10^−3	4.78 × 10^11	6.06 × 10^−15	10^−15–10^−12 (ref. 46 and 47)
Ni	3.439	0.356	0.197	3.25 × 10^−2	1.99 × 10^13	1.24 × 10^−9	10^−11–10^−9 (ref. 48 and 49)

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The inconsistency between diffusion coefficients and hopping activation barriers can be understood when the electronic coupling factors are considered. One can see that the H_AB is much higher for LiNiPO₄ compared with LiFePO₄, and higher for LiCoPO₄ compared with LiMnPO₄. The values estimated by considering the electronic factor through the Landau–Zener ET theory reasonably agree with the experimental data. This further confirms the electron-coupled ion transfer mechanism of Li⁺ transport in these materials and the essential role of the electronic coupling between transition metal centers.

It is also worthwhile mentioning that the potential energy profiles shown in Fig. 4 are in general not parabolic, as assumed in the standard ET theory by Marcus.⁵⁰ We note that the parabolic potential energy curves are expected only for outer-sphere ET processes involving no molecular structural change other than harmonic relaxation. Nevertheless, the ET process here is accompanied by the nuclear motion of Li atoms, which changes the electronic environment during the course as well. Consequently, the deviation of the real energy profiles from the parabolic curves is unsurprising. As a direct consequence, E_a and λ does not satisfy the relationship of E_a = λ/4 as predicted by the classical Marcus theory (Table 1). More specifically, the deviation is much more pronounced for LiNiPO₄ and LiCoPO₄ than that for other compounds considered here, suggesting a unusual charge transfer behavior for LiNiPO₄ and LiCoPO₄.

It is now clear that, owing to the ion-coupled ET nature, neither the thermal diffusion nor the classical ET theory is insufficient for a quantitative description of ion transport kinetics in olivine phosphate materials of LiMPO₄. A comprehensive model for the ion-coupled ET process in condensed matters should include the detailed electronic ion–host interactions and their effects on the structural relaxation of the host metal–ligand systems, which are beyond the scope of this work. What we show here is that modelling the ion transport in terms of the hoping reaction kinetics provides a route to access this coupled transfer mechanism. Despite the great approximation, the electron coupling effect can be included in the ion transport kinetics by simply incorporating the electronic transmission coefficient given by Landau–Zener theory into the pre-exponential frequency factor of the rate constant of the hopping reaction. This allows the apparent diffusion coefficient of Li⁺ in LiMPO₄ to be reasonably estimated using the parameters obtained from first-principles computations.

3. Conclusions

We have developed a new theoretical model for Li⁺ transport in olivine phosphate materials of LiMPO₄, which is described as an ion-coupled electron transfer process. In particular, the effect of electronic coupling between host redox centers on the ion transport kinetics is emphasized and can be described using the Landau–Zener theory. In this model, the diffusion coefficient is co-determined by the electronic coupling matrix element between the neighboring transition metal centers, the configuration reorganization energy of electron transfer, and the activation energy of ion hopping. The values of these parameters can be obtained from DFT calculations. The model can rationalize the anomalous trend of diffusion coefficients of Li⁺ in LiMPO₄ (M = Fe, Mn, Co, Ni), which cannot be understood solely by the activation barrier of ion hopping. It is revealed that the electron coupling plays a vital role. The electron-transfer perspective of ion transport in LiMPO₄ opens new pathways to design high-performance ion intercalation materials for metal-ion batteries.

4. Computations details

Spin-polarized DFT calculations were performed using Quantum Espresso with the ultrasoft pseudopotentials for nuclei and core electrons.⁴¹ Generalized gradient approximation (GGA) in the parameterization of the Perdew–Burke–Ernzerhof (PBE) functional was used to describe the exchange–correlation and Hubbard-type correction U was taken into account due to the strongly correlated nature of the Fe 3d electrons.⁵¹ As recommended in a previous study, the effective U values for LiMPO4 are set to 4.3 eV, 4.5 eV, 5.7 eV, and 5.1 eV for M = Fe, Mn, Co, and Ni, respectively.⁵² A kinetic energy cutoff of 30 Ry and a charge-density cutoff of 300 Ry were used in all calculations. Geometry optimization was performed by using BFGS minimization until all the forces acting on each atom were below 10⁻³ Ry per Bohr and the total energy was converged to within 10⁻⁶ Ry. The Brillouin-zones were sampled with a 12 × 5 × 6 k-point mesh for a 1a × 2b × 1c (S.G. Pnma) supercell containing 8 Li sites.

Data availability

The data that support the findings of this study are available on request from the corresponding author.

Author contributions

SC conceived the project. YG, JH and SC derived the model equations. YG performed the DFT calculations. YL discussed the derivation and results. YG, JH, and SC wrote the paper. All authors have given approval to the final version of the manuscript.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (21673163, 21802170).

Notes and references

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Footnote

† Present address: Institute of Theoretical Chemistry, Ulm University, 89069 Ulm, Germany.

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