Understanding electron behavior is crucial in secondary battery research, as electrons directly participate in electrochemical reactions. Over the past few decades, Goodenough and his co-workers developed an electron theory for secondary batteries, introducing the splitting of d-orbital energy levels in transition metals, which reconstructs the local crystal field. This theory has provided valuable insights for optimizing the structure and performance of battery materials. However, the theory is not easy to be handled and difficult to intuitively understand due to its reliance on complex atomic and molecular orbitals, as well as orbital hybridization theories. With the advancement of material science, particularly density functional theory (DFT) and first-principles high-throughput calculations in quantum crystallography, researchers can obtain more precise insights into electronic structures and key parameters, such as the Fermi energy levels, band structures near the Fermi level, electron state, and density of states (DOS). These electronic characteristics are vital for understanding the electrochemical performance of electrode materials in secondary battery. This study was to investigate the band structures of phosphate cathodes for sodium-ion batteries via the first-principles calculations in order to establish a qualitative and semi-quantitative correlation between electronic properties and electrochemical behavior.

Three typical phosphate compounds, i.e., β-NaVP₂O₇, Na₃V₂(PO₄)₂F₃ and Na₃V₂(PO₄)₃, were selected as model materials. Their crystal structures were obtained from the Inorganic Crystal Structure Database (ICSD) and optimized by the Vienna Ab initio Simulation Package (VASP) with the Perdew-Burke-Ernzerhof (PBE) functional and projector-augmented wave (PAW) pseudopotentials. The key computational parameters included a plane-wave cutoff energy of 520 eV and Monkhorst-Pack k-point grids of 4×3×3 for β-NaVP₂O₇, 3×3×2 for Na₃V₂(PO₄)₂F₃ and 2×3×3 for Na₃V₂(PO₄)₃. The convergence criteria were set at 10^-5 eV for energy and -0.01 eV/Å for forces. The band structures and density of states (DOS) were initially calculated using optimized structures for theoretically perfect crystals. However, since the actual sodium ion content in some phosphate cathodes deviates from the perfect crystal, the structural modifications were implemented. For Na₃V₂(PO₄)₃, the unit cell was doubled, and the number of sodium ions was adjusted according to symmetry rules to align with practical requirements. For Na₃V₂(PO₄)₂F₃, no unit cell expansion needed, but the number of sodium ions reduced based on symmetry considerations. For β-NaVP₂O₇, no sodium ion adjustments required. After these corrections, the band structures and DOS were recalculated. The electrochemical performance was further validated through galvanostatic charge/discharge tests at 0.2 C, with the specific capacities and cycling stability evaluated for 50 cycles.

High-throughput calculations can yield the more precise Fermi energies for the selected phosphate cathodes, with β-NaVP₂O₇ at 2.06 eV, Na₃V₂(PO₄)₂F₃ at 1.70 eV, and Na₃V₂(PO₄)₃ at 2.91 eV. Correspondingly, the bandgaps are determined to be 2.53 eV for β-NaVP₂O₇, 2.74 eV for Na₃V₂(PO₄)₂F₃, and 1.93 eV for Na₃V₂(PO₄)₃. These electronic parameters have a direct impact on the discharge voltages and conductivity of the materials. Note that β-NaVP₂O₇ with the minimum Fermi energy, achieves the maximum discharge voltage of 4.2 V, aligning with theoretical predictions that lower Fermi levels correspond to higher working voltages. Despite its relatively high Fermi level, Na₃V₂(PO₄)₃ maintains a stable 3.4 V platform due to its uniform energy band distribution near the Fermi level along the entire recommended band path, minimizing electron behavior changes and transition barriers. This uniform distribution facilitates consistent electron transport, resulting in an exceptional cycling stability with 99.4% capacity retention after 50 cycles. In contrast, β-NaVP₂O₇ and Na₃V₂(PO₄)₂F₃ exhibit non-uniform energy band distributions near the Fermi level, indicating multiple electron transition pathways that contribute to instable voltage plotforms. The discharge specific capacities of these materials are measured as 92.7 mA·h·g^-1 for β-NaVP₂O₇, 124 mA·h·g^-1 for Na₃V₂(PO₄)₂F₃ and 115.2 mA·h·g^-1 for Na₃V₂(PO₄)₃. A correlation between the integral of DOS near the Fermi level band and the discharge specific capacity of electrode materials occurs. Higher integrals of DOS for conductive electrons near the Fermi level correspond to greater discharge capacities, as demonstrated in the comparative analysis of these electrode materials.

This study established a correlation between high-throughput quantum crystallography calculations and electrochemical performance in sodium-ion battery cathodes. The Fermi levels directly affected discharge voltages, with lower Fermi levels corresponding to higher working voltages. A uniform band structure near the Fermi level over the whole recommended band path improved a cycling stability. The integral of the DOS near the Fermi level could be a critical factor determining the specific capacity of electrode materials. In addition, high-throughput calculations also revealed that structural imperfections, such as uncertain atomic occupancy in Na₃V₂(PO₄)₂F₃ and Na₃V₂(PO₄)₃, significantly could affect computational accuracy. The more precise Fermi energy values, band configurations, and DOS parameters were obtained via refining ion-occupancy corrections and implementing supercell expansions, providing deeper insights into fundamental electrochemical properties of these materials. These findings established a theoretical framework for designing next-generation battery materials via optimizing band structures for optimizing energy density and stability. This study could highlight a potential of high-throughput quantum crystallography calculations in predicting and improving electrochemical performance, thus offering a roadmap for the development of next-generation battery materials.