Computational Study of NaVOPO4 Polymorphs as Cathode Materials for Na-Ion Batteries: Diffusion, Electronic Properties, and Cation-Doping Behavior

Rechargeable sodium-ion batteries have gained considerable interest as potential alternatives to lithium-ion batteries, owing to their low cost and the wide abundance of sodium. Phosphate compounds...


INTRODUCTION
The development of lithium-ion batteries has enabled a host of new technologies, particularly portable electronic devices. In recent years, sodium-ion batteries have begun to receive interest with the aim of developing low-cost and sustainable batteries based on the high abundance and easy accessibility of sodium, especially for grid-scale storage systems for intermittent renewable energy. 1−5 Considerable effort has been expended on the exploration of new cathode materials with high reversible capacity, rapid Na insertion/extraction, and cycling durability. 3,6−8 A large variety of materials, including layered metal oxides, polyanionic compounds, and Prussian blue analogs, have been studied as cathodes for Na-ion batteries. 3,9 In particular, vanadium phosphate materials, such as Na 3 V 2 (PO 4 ) 3 , Na 3 V 2 (PO 4 ) 2 F 3 , and NaVPO 4 F, show good electrochemical performance and they possess a versatile structure. 10−12 They are considered promising cathode compounds owing to their high structural stability, long-term cycling, and low volumetric expansion during Na + insertion/extraction. 13 Song et al. explored monoclinic α-NaVOPO 4 as a cathode for Na-ion batteries. 14 More recently, Manthiram and coworkers synthesized and tested two new NaVOPO 4 polymorphs, i.e., the β and α I phases, both of which exhibited a higher potential and reversible capacity than the α phase. 15,16 The new β and α I polymorphs have both been synthesized from the lithium counterparts, via delithiation and sodiation chemical processes, but it was found that the new phases are thermodynamically less stable than the α phase.
The α I phase has the highest ionic conductivity among the three polymorphs, making it a promising cathode for Na-ion batteries, although the Na-ion transport mechanisms and electronic properties of the NaVOPO 4 polymorphs are not yet fully characterized. Li et al. designed a high-energy Na-ion battery based on structured Na 2 Ti 3 O 7 and layered α I -NaVOPO 4 compounds as the anode and cathode materials, respectively. 17 This cell battery showed excellent reversible capacity, rate capability, and cycling stability. Lin et al. 18 published a comparison of VOPO 4 polymorphs investigating their stability and electrochemistry of Li and Na intercalation.
Here, we report the diffusion, electronic properties, and cation doping of the α-, β-, and α I -NaVOPO 4 polymorphs studied by an effective combination of ab initio calculations and classical molecular dynamics (MD) simulation techniques. We were able to reproduce the experimental structural parameters of the three polymorphs. The Na-ion transport mechanisms and voltage trends from metal doping were also investigated and the doped structures show an increase in the cell voltage of up to 0.2−1.0 V vs Na/Na + . We found that the α I phase has the highest Na-ion diffusion coefficient and the lowest activation energy for Na-ion migration among the three polymorphs.

METHODS
We have employed a combination of ab initio techniques based on density functional theory (DFT) and interatomic potentialbased molecular dynamics (MD) simulations, extending our successful work on related battery materials. 19−25 All DFT calculations were performed using the Vienna ab initio simulation package (VASP). 26−29 We have used projectoraugmented wave pseudopotentials 30,31 and the Perdew− Burke−Ernzerhof exchange correlation functional revised for solids (PBEsol), 32 setting the kinetic energy cut-off at 520 eV. A k-point grid of 7 × 7 × 7 was used to converge the forces and energies of the bulk material. The DFT+U methodology was applied to account for the d orbitals of the metal atoms, with effective Hubbard values of U eff = U − J = 4.0, 3.3, 4.3, 3.9, 6.0, and 4.2 eV (J = 1 eV) for V, Co, Fe, Mn, Ni, and Ti, respectively. 11,33,34 In addition, van der Waals corrections were included, using the DFT-D3 method of Grimme, 35 as these have been shown to influence cell voltage calculations in some polyanionic systems. 19 Previous computational studies have shown that such methods are suitable to compute accurate cell voltage trends 20 as well as to determine transport and defect properties in Liand Na-ion batteries. 36−38 For all compounds, the cell voltage vs Na/Na + of the V 4+ /V 5+ redox couple was calculated using the following equation where E[NaVOPO 4 ] and E[VOPO 4 ] are the total energies of the NaVOPO 4 and VOPO 4 structures, respectively. The chemical potential of sodium, μ[Na], was calculated using metallic sodium, which is the standard practice for cell voltage calculations. We also used eq 1 to compute the cell voltage vs Na/Na + of the NaV 1−x M x OPO 4 doped compounds, where we assumed that the fully desodiated structures (V 1−x M x OPO 4 ) can be formed. The MD simulations were performed with the LAMMPS code 39 and the calculations were carried out on a large supercell made up of 7 × 7 × 7 unit cells, consisting of 10 976 atoms. The initial configuration of the supercell contained 10% Na vacancies plus corresponding V 5+ species, which were randomly distributed. Pre-equilibrium runs of 4 ps using a time step of 2 fs with the NPT ensemble (constant number of particles, constant pressure, and constant temperature) were first used to obtain stable configurations. Data collection runs were then carried out using the NVT ensemble (constant number of particles, constant volume, and constant temperature) and a time step of 1 fs for long runs of 10 ns, at temperatures in the range of 300−1400 K. The interatomic potentials used in the MD calculations are listed in Table S1. The mean squared displacement (MSD) of the Na ions was computed first to derive the diffusion coefficients (D Na ) according to where t is time and ⟨[r(t)] 2 ⟩ is the MSD.

RESULTS AND DISCUSSION
3.1. Structural and Electronic Analysis. Lii et al. first reported the synthesis and structural characterization of NaVOPO 4 . 40 The magnetic behavior of this compound was described by O'Connor et al. as anti-ferromagnetically coupled S = 1/2 one-dimensional (1D) chains 41 with the structure of the α-VOPO 4 polymorph. Vanadyl phosphate (VOPO 4 ) crystallizes in several different structures, namely as α/α I /α II , β, γ, δ, ε, and ω-VOPO 4 . 42−48 So far, only three sodiumcontaining products have been synthesized: α (monoclinic), 40 β (orthorhombic), 15 and α I (tetragonal) 16  A comparison between the experimental and computed cell parameters and bond lengths can be found in Table 1 (all bond lengths are given in Table S2). The computed values are within 1% of the experimental values, except in the case of the α I phase. In this case, the experimental values are not available for the NaVOPO 4 stoichiometry and are referred to as the α I -Na 0.8 VOPO 4 compound; hence, the computed cell parameters and bond lengths are slightly different from the experimental data. The interlayer distance between the VOPO 4 sheets in the α I phase can vary up to 0.6 Å depending on the Na loading.
Turning to the electronic structure, the density of states (DOS) of the α phase is shown in Figure 2. The DOS of the β and α I phases, which are similar to the DOS of the α phase, can be found in Figure S1. As the NaVOPO 4 compounds are described as anti-ferromagnetically coupled chains, 40,41 the compounds have a zero total spin and the DOS are symmetric. The electronic structure corresponds to one unpaired electron per VO formula unit and the vanadyl chains show shortrange anti-ferromagnetic exchange in the crystal lattice.
The Journal of Physical Chemistry C Article Although we model the magnetic state as determined experimentally, a detailed analysis of the magnetic properties is not the focus of this study.
The valence band is described mainly by V 3d and O 2p states, whereas the conduction band is mainly made up of V 3d states. The P atoms have a relatively small contribution to both valence and conduction bands. Below the Fermi level, from −0.5 to 0.0 eV, the total DOS shows a similar contribution from the V 3d and the O 2p states, which correspond to the short VO bonds. The band gaps of the sodiated compounds are in the range of 1.5−2.5 eV, whereas the values of the desodiated compounds are all similar (∼2 eV). These values are in good agreement with previous DFT+U results of different VOPO 4 polymorphs (1.3−2.2 eV). 49 3.2. Na-Ion Diffusion Rates and Pathways. MD simulations were performed over long time scales to examine long-range Na-ion transport properties. Lin et al. computed the migration barriers of Na and Li ions in VOPO 4 polymorphs using the climbing image nudged elastic band method. 18 This practice is well established in the literature, 50,51 but it fails to capture the long-range dynamics of alkali-ion transport through the structure, which we have sought to elucidate here. From the MSD analysis, we observe that in the α phase, Na-ion diffusion takes place preferably in the a and c directions. In the case of the β phase, the b direction is preferred, in accord with experimental suggestions. 15 Finally, Na-ion diffusion in the α I phase does not show any preferred direction, with the three directions being equally probable.
The MD simulations were performed over a range of temperatures (300−1400 K), covering a wider range than the operating temperatures of a typical battery and much wider than current experiments. At 300 K, the diffusion coefficients of the α-, β-, and α I -NaVOPO 4 polymorphs were calculated as 1.3 × 10 −11 , 6.7 × 10 −12 , and 5.7 × 10 −11 cm 2 s −1 , respectively. To the best of our knowledge, there are no experimental diffusion coefficients of the NaVOPO 4 compounds for direct comparison. Nevertheless, the computed diffusion coefficients are similar to those found in other Na-ion cathode materials, such as Na x CoO 2 (10 −11 cm 2 s −1 ), 52 Na x MnO 2 (10 −11 cm 2 s −1 ), 53 and Na 2 CoSiO 4 (10 −12 cm 2 s −1 ). 54 The computed diffusion coefficients as a function of temperature can be used to derive migration activation energies (E a ) from an Arrhenius plot, as shown in Figure 3.
Ion migration activation energies of 0.35, 0.51, and 0.32 eV were derived for the α-, β-, and α I -NaVOPO 4 polymorphs, respectively. These values indicate favorable Na-ion migration, especially in the α I phase. Our findings cannot be compared with the literature as neither experimental activation energies nor conductivity studies of these compounds are available. However, these values are smaller than the activation energy in the related NaTiOPO 4 compound, which is 1.1 ± 0.1 eV in the temperature range of 480−800 K. 55 Lin et al. 18 computed the migration barrier of β-NaVOPO 4 as 0.54 eV, which is in good agreement with our computed value of 0.51 eV. However, direct comparison is not appropriate for the α I -NaVOPO 4 compound, since Lin et al. computed the migration barriers of three different 1D diffusion pathways, whereas our calculated value is for a threedimensional (3D) diffusion pathway.
The diffusion trajectories are visualized in Figure 4 by plotting accumulated Na-ion densities, indicating the lattice sites most frequently crossed by Na ions during the MD simulations. These results confirm that Na ions are highly mobile, with all of the Na sites involved in bulk diffusion. In the case of the α phase, the Na diffusion takes place in twodimensional (2D) channels, mainly in the b and c directions. In the β phase, the Na ions move in the 1D channels along the b axis. Finally, the α I phase shows the highest Na mobility, taking place in the 3D channels. Our overall results showing high diffusion rates, low migration energies, and favorable 3D diffusion pathways for α I -NaVOPO 4 makes this phase a The Journal of Physical Chemistry C Article promising Na-ion cathode material. We recognize the stability issues with the α I phase, but we consider that our predictions certainly warrant further electrochemical investigation of this promising material.
3.3. Cell Voltage Trends on Cation Doping. DFT+U was used to compute the cell voltage of the α-, β-, and α I -NaVOPO 4 polymorphs as well as the doped materials. To carry out the calculations, Na ions were removed from the optimized structure to have vacancy concentrations of 0, 25, 50, 75, and 100%, and various vacancy configurations were considered. The voltage was then calculated using eq 1 with the lowest energy configuration. The computed voltages of the α-, β-, and α I -NaVOPO 4 polymorphs are 3.3, 3.1, and 3.1 V (vs Na/Na + ), respectively. These are in good agreement with the experimental values of 3.6, 3.3−3.4, and 3.4 V (vs Na/Na + ), respectively. 14− 16 We have also investigated how doping on the vanadium site affects the cell voltage of the NaVOPO 4 polymorphs. The voltage trends of NaV 1−x M x OPO 4 (M = Al 3+ , Co 2+ , Fe 3+ , Mn 4+ , Ni 2+ , or Ti 4+ ) vs Na/Na + with x values of 0, 0.25, and 0.50 are depicted in Figure 5. These cations were chosen as they have been used as doping agents for different vanadium phosphate compounds, including Li 3 V 2 (PO 4 ) 3 and Na 3 V 2 (PO 4 ) 3 . 56−64 The doped materials show higher capacity and capacity retention than the pristine compounds. Also, the addition of the cation dopants has been shown to enhance Li-and Na-ion diffusion, and have a structural stabilization effect.
Our computed cell voltage values of some of the doped structures are above the voltage stability window for organic

CONCLUSIONS
NaVOPO 4 vanadyl phosphates are among the most promising cathode materials for sodium-ion batteries because of their good electrochemical performance. We have studied the diffusion, electronic properties, and cation-doping behavior of the α-, β-, and α I -NaVOPO 4 polymorphs. Our fundamental study shows good reproduction of the observed experimental structures. From large-scale MD calculations, we have derived low Na-ion activation energies of 0.3−0.5 eV, and diffusion coefficients (D Na ) at 300 K of 10 −11 −10 −12 cm 2 s −1 for the three NaVOPO 4 polymorphs. From analysis of the migration pathways, the α, β, and α I phases show Na-ion diffusion along 2D, 1D, and 3D channels, respectively. Among the three  The Journal of Physical Chemistry C Article polymorphs, the α I phase has the most favorable Na-ion conductivity, activation energy, and 3D pathways, making it a promising cathode material.
The computed cell voltages of 3.3, 3.1, and 3.1 V vs Na/Na + for the α-, β-, and α I -NaVOPO 4 compounds, respectively, agree well with the reported electrochemical data. Cation doping (Al 3+ , Co 2+ , Fe 3+ , Mn 4+ , Ni 2+ , and Ti 4+ ) at the vanadium site is predicted to increase the cell voltage. The Mnand Ti-doped structures of the α and β phases show an increase in the cell voltage within the electrochemical stability window of current liquid electrolytes. In the case of the α I phase, all of the doped structures show cell voltages within the electrochemical stability range. Overall, the fundamental insights presented in this study will aid the improvement of phosphate cathodes for sodium-ion batteries.