Defect Engineering and Na-Ion Transport in NaMnPO₄: A Computational Perspective

Abstract

Rechargeable sodium-ion batteries (SIBs) have attracted considerable attention owing to the natural abundance and accessibility of sodium. Maricite NaMnPO₄, a phosphate-based cathode material with high theoretical capacity, suffers from blocked sodium-ion diffusion channels. In this study, atomistic simulations using pair potentials and density functional theory (DFT) are employed to investigate intrinsic defect mechanisms, sodium-ion migration pathways, and the role of dopant incorporation at Na, Mn, and P sites in generating Na vacancies and interstitials. Among the intrinsic defects, the Na–Mn anti-site cluster emerges as the most favorable, exhibiting a very low formation energy of 0.12 eV, while the Na Frenkel pair (1.93 eV) is the next most stable defect, indicating that sodium diffusion is primarily facilitated by vacancy formation. Nevertheless, sodium-ion mobility in NaMnPO₄ remains limited, as reflected by the relatively high migration activation energy of 1.28 eV. Among the isovalent substitutions, K is predicted to be the most favorable dopant at the Na site, whereas Ca and Cu are the most favorable at the Mn site. Thallium is identified as a promising dopant at the Mn site for generating Na vacancies that facilitate Na-ion migration, while Ge substitution at the P site is predicted to enhance the sodium content in the material.

1. Introduction

The global energy transition hinges on two fronts: replacing fossil fuel-based electricity with renewable generation and converting ground transportation from internal combustion engines (ICEs) to electric vehicles (EVs). The intermittency of renewable sources demands efficient, durable, and sustainable energy storage, yet existing technologies remain insufficient.

Lithium-ion batteries (LIBs) have enabled high energy density, rechargeable systems, with lithiated transition metal oxide cathodes (e.g., LiCoO₂ and LiFePO₄) serving as the principal lithium source [1,2,3,4,5,6,7]. The limited and uneven global distribution of lithium necessitates the development of alternative chemistries for next-generation, high-capacity energy storage devices, particularly for hybrid electric vehicles.

Owing to its natural abundance, sodium has attracted increasing attention, and rechargeable SIBs are emerging as promising candidates for large-scale energy storage [8,9,10,11,12,13,14,15]. Among the many factors influencing SIB performance, the design of high–energy-density, low-cost cathode materials is paramount. Promising cathode materials for SIBs include olivine NaMPO₄ (M = Fe, Mn, Co, Ni) [16,17,18], NASICON (Na super ionic conductor) Na₃V₂(PO₄)₃ [19,20,21], Na₂MSiO₄ (M = Fe, Mn and Co) [22,23,24] and layered NaxMO₂ (M = Fe, Mn, Co, Ni) [25,26,27,28]. The NASICON family, with its three-dimensional framework and wide interstitial channels, has been extensively investigated for sodium-ion battery cathodes [29,30,31,32].

Interest in olivine NaMPO₄ is largely motivated by the commercial success of LiFePO₄ in lithium-ion batteries. NaMnPO₄ crystallizes in two polymorphic forms: maricite NaMnPO₄ [33] and olivine NaMnPO₄ [34]. The key structural difference lies in cation site occupancy: in the maricite phase, Na⁺ ions occupy 4c Wyckoff sites and transition metals occupy 4a sites, whereas in the olivine phase this arrangement is reversed [33,35]. Electrochemically, the maricite form is largely inactive, while the olivine form is active but thermodynamically unstable and challenging to synthesize directly. The maricite structure, by contrast, is more stable but exhibits the inverted cation distribution relative to olivine [36]. Experimental studies on NaMnPO₄ synthesis have been reported. Early work by Koleva et al. [37] demonstrated the preparation of defect-free olivine and maricite-NaMnPO₄, while Venkatachalam et al. [38] improved the electrochemical performance of NaMnPO₄ via a polyol synthesis route, achieving a discharge capacity of 102.67 mAhg⁻¹. Despite these advances, significant scope remains for enhancing the electrochemical properties of NaMnPO₄-based materials. The robust crystal framework affords excellent thermal stability and strong resistance to structural degradation during cycling. Furthermore, Mn is favored as a transition metal in polyanion frameworks due to its environmental benignity, and the theoretical capacity and voltage of NaMnPO₄ make it a promising candidate for high energy density applications [33,35].

While extensive computational research has been carried out on olivine-NaMnPO₄, relatively few studies have addressed maricite NaMnPO₄. In a previous DFT simulation study [36], the electrochemical, electronic, and diffusion properties of maricite NaMnPO₄ cathodes for Na-ion batteries, as well as the effects of lattice strain on these properties, have been investigated. However, the influence of intrinsic defects on electrochemical behavior and possible doping strategies has not been explored so far. This work explores maricite NaMnPO₄ as a promising cathode material for sodium-ion batteries, with particular emphasis on its defect chemistry. Strategies to enhance electronic conductivity and ion diffusion, including elemental doping and composite formation, will be explored, as has been previously demonstrated in Li- and post-Li-ion battery materials [39,40,41,42,43,44,45,46,47,48]. By addressing these challenges, the study aims to contribute to the development of more efficient and sustainable sodium-ion batteries.

2. Computational Methods

In this study, classical simulations were carried out using the General Utility Lattice Program (GULP) [49] to evaluate the intrinsic defect processes and Na⁺ ion migration. The total lattice energy was calculated by considering both long-range Coulombic interactions and short-range forces, including Pauli repulsion and van der Waals attraction. Short-range ionic interactions were modelled using Buckingham potentials (see

Table 1. Buckingham potential parameters [50,51,52,53] used in the classical simulations of two-body problems [Φ_ij (r_ij) = A_ij exp (−r_ij/ρ_ij) − C_ij/r_ij⁶], where A, ρ, and C are parameters that were selected carefully to reproduce the experimental data. The values of Y and K represent the shell charges and spring constants, respectively.

Interaction	A/eV	ρ/Å	C/eV·Å6	Y/e	K/eV·Å−2
Na+–O2−	1677.830	0.2934	0.00	1.0000	292.430
Mn2+–O2−	2601.394	0.2780	0.00	2.0000	148.0
O2−–O2−	22,764.0	0.149	27.88	−2.820	74.92
P5+–O2−	877.340	0.35940	0.00	5.0000	99,999

) [50,51,52,53]. Structural optimization was carried out using the BFGS (Broyden-Fletcher-Goldfarb-Shanno) algorithm [54] to relax the NaMnPO₄ crystal structure to its lowest energy configuration. For modelling point defects and ion migration, the Mott-Littleton [55] method was applied. The migration of Na⁺ ions was simulated by defining two adjacent Na vacancy sites as the starting and final points of the ionic path, with seven interstitial sites forming the migration path computed, using the midpoint between two neighboring Na vacancy sites as a reference to reduce systematic errors. Finally, the activation energy for Na⁺ ion migration was determined by calculating the energy difference between the highest local energy at the saddle point and the lowest energy required for Na vacancy formation, providing insights into the energy barrier for the migration process.

Density functional theory (DFT) calculations were performed using the Vienna Ab initio Simulation Package (VASP) [56]. The exchange–correlation functional was treated within the generalized gradient approximation (GGA), parameterized by Perdew, Burke, and Ernzerhof (PBE) [57]. A plane-wave basis set with a cutoff energy of 600 eV and standard projector augmented-wave (PAW) potentials [58], as implemented in VASP, was employed. Defect configurations were modeled using a 2 × 1 × 2 supercell consisting of 112 atoms (Na: 16, Mn: 16, P: 16, and O: 64). A 4 × 4 × 4 Monkhorst–Pack k-point mesh was used in all calculations [59]. Structural optimizations of the doped systems were carried out using the conjugate gradient algorithm [60], with atomic forces evaluated via the Hellmann–Feynman theorem, including Pulay corrections. All optimized structures converged to atomic forces below 0.001 eV/Å. To further examine the electronic effects of doping, Bader charge analysis was conducted to evaluate the charge distribution on the dopant atoms [61]. Density of states (DOS) plots were generated using the Xmgrace plotting tool. To obtain the accurate electronic structure, we employed an orbital-dependent Coulomb potential (Hubbard U) together with the exchange parameter J for Mn. The localized 3d states of Mn were treated using U = 5.00 eV and J = 0.00 eV, in accordance with values reported in previous DFT simulations [62]. The results of DFT + U calculations are highly dependent on the choice of the Hubbard U and Hund’s J parameters, as these values significantly affect key material properties such as band gaps, magnetic moments, and structural features. Experimental observations are often used to calibrate and validate these parameters by comparing calculated properties (e.g., photoemission spectra or magnetic anisotropy) with experimental measurements. The optimal selection of U and J also depends on computational factors such as the basis set and exchange–correlation functional, and different physical properties may require distinct parameter values for best agreement with experiment. In the present work, DFT simulations based on the GGA-PBE approximation employed a U value of 5.0 eV, which has been optimized for the Mn-3d states [62].

3. Results

3.1. Crystal Structure of NaMnPO₄

Maricite NaMnPO₄ crystallizes in an orthorhombic structure with space group Pmnb. The crystal structure is illustrated in Figure 1. Its lattice parameters (a = 6.892 Å, b = 9.079 Å, c = 5.108 Å, α = β = γ = 90°) were reported by Subhendhu Jana et al. [35] through Rietveld refinement of powder X-ray diffraction data. The unit cell contains 28 atoms in total (Na: 4, Mn: 4, P: 4 and O: 16), corresponding to four formula units (Z = 4) of NaMnPO₄. The atomic arrangement includes Na and P atoms occupying 4c sites, Mn atoms positioned at the 4a inversion centers, and oxygen atoms distributed over 4c and 8d Wyckoff positions. In this structure, Mn, Na, and P atoms are coordinated by six, eight, and four oxygen atoms, respectively. Each MnO₆ octahedron is linked to six PO₄³⁻ tetrahedra through shared vertices, while no connectivity occurs between adjacent PO₄³⁻ tetrahedra within the maricite framework (see Figure 1a). To assess the validity of the interatomic potentials and pseudopotentials used in this study, the bulk maricite NaMnPO₄ structure was optimized under constant-pressure conditions.

The optimised lattice parameters of bulk maricite NaMnPO₄ obtained from both classical and DFT calculations are summarized in

Table 2. Calculated and experimental lattice parameters of NaMnPO₄.

Lattice Parameters	Calculated		Experiment [35]	|∆|(%)
	Classical	DFT		Classical	DFT
a (Å)	6.946	6.937	6.892	0.78	0.65
b (Å)	9.099	9.195	9.079	0.22	1.28
c (Å)	5.177	5.125	5.108	1.02	0.33
α = β = γ (°)	90.0	90.0	90.0	0.00	0.00
V (Å3)	327.24	326.89	319.62	2.38	2.27

, together with the corresponding experimental values. The deviations between calculated and experimental parameters were found to be less than 2.5% in all cases, indicating good agreement. Specifically, the b, and c lattice constants showed minor variations, with the maximum error of 1.28% observed in the b parameter for the DFT calculation. The unit cell volume was slightly overestimated in both approaches, with errors of 2.38% and 2.27% for classical and DFT calculations, respectively. Overall, the close correspondence with experimental data confirms the accuracy and reliability of the chosen interatomic potentials and pseudopotentials for defect modelling in maricite NaMnPO₄.

Figure 1b presents the charge density distribution, where strong electron accumulation is observed around the oxygen atoms, confirming their highly electronegative character and indicating predominantly ionic bonding between Na–O and P–O, with some degree of covalent interaction in Mn–O bonds. This is further supported by the Bader charge analysis, which reveals effective charges of +1.00 on Na, +1.62 on Mn, +5.00 on P, and 1.90 on O, consistent with the expected oxidation states and confirming the strong charge localization on oxygen. The incorporation of the Hubbard U parameter significantly altered the electronic structure by widening the band gap from 1.40 eV to 3.20 eV (see Figure 1c,d). This increase arises from the enhanced localization of the Mn 3d states, which reduces their hybridisation with the surrounding orbitals and effectively pushes the conduction and valence band edges further apart. Such behavior is consistent with previous DFT + U studies [36], highlighting the critical role of on-site Coulomb interactions in accurately describing the electronic properties of Mn-containing systems. Its magnetic character is evidenced by the asymmetry of the DOS plot with respect to the x-axis, as also reported in earlier work [36].

3.2. Intrinsic Defect Properties

Intrinsic defects play a crucial role in governing ion transport within materials. In this section, we focus on point defects, namely vacancies, interstitials, and anti-site defects. These individual defects were systematically combined to evaluate the formation energies of Frenkel, Schottky, and anti-site defect mechanisms. In the case of anti-site defects, the energy was calculated by exchanging cations, where Na occupies a Mn site and Mn occupies a Na site. To describe these intrinsic defect processes, Kröger–Vink notation [63] was employed, and the corresponding reaction equations were derived to represent the various defect mechanisms in this material.

$$\mathrm{Na} \mathrm{Frenkel}: {\mathrm{N}\mathrm{a}}_{\mathrm{N}\mathrm{a}}^{\mathrm{X}}\to {\mathrm{V}}_{\mathrm{N}\mathrm{a}}^{\prime }+{\mathrm{N}\mathrm{a}}_{\mathrm{i}}^{•}$$

(1)

$$\mathrm{Mn} \mathrm{Frenkel}: {\mathrm{M}\mathrm{n}}_{\mathrm{M}\mathrm{n}}^{\mathrm{X}}\to {\mathrm{V}}_{\mathrm{M}\mathrm{n}}^{″}+{\mathrm{M}\mathrm{n}}_{\mathrm{i}}^{••}$$

(2)

$$\mathrm{P} \mathrm{Frenkel}: {\mathrm{P}}_{\mathrm{P}}^{\mathrm{X}}\to {\mathrm{V}}_{\mathrm{P}}^{\prime \prime \prime \prime \prime }+{\mathrm{P}}_{\mathrm{i}}^{•••••}$$

(3)

$$\mathrm{O} \mathrm{Frenkel}: {\mathrm{O}}_{\mathrm{O}}^{\mathrm{X}}\to {\mathrm{V}}_{\mathrm{O}}^{••}+{\mathrm{O}}_{\mathrm{i}}^{″}$$

(4)

$$\mathrm{Schottky}: {\mathrm{N}\mathrm{a}}_{\mathrm{N}\mathrm{a}}^{\mathrm{X}}+{\mathrm{M}\mathrm{n}}_{\mathrm{M}\mathrm{n}}^{\mathrm{X}}+{\mathrm{P}}_{\mathrm{P}}^{\mathrm{X}}+4{\mathrm{O}}_{\mathrm{O}}^{\mathrm{X}}\to {\mathrm{V}}_{\mathrm{N}\mathrm{a}}^{\prime }+{\mathrm{V}}_{\mathrm{M}\mathrm{n}}^{″}+{\mathrm{V}}_{\mathrm{P}}^{\prime \prime \prime \prime \prime }+{4\mathrm{V}}_{\mathrm{O}}^{••}+{\mathrm{NaMnPO}}_4$$

(5)

$${\mathrm{Na}}_2\mathrm{O} \mathrm{Schottky}: 2{\mathrm{N}\mathrm{a}}_{\mathrm{N}\mathrm{a}}^{\mathrm{X}}+{\mathrm{O}}_{\mathrm{O}}^{\mathrm{X}}\to 2{\mathrm{V}}_{\mathrm{N}\mathrm{a}}^{\prime }+{\mathrm{V}}_{\mathrm{O}}^{⦁⦁}+{\mathrm{Na}}_2\mathrm{O}$$

(6)

$$\mathrm{MnO} \mathrm{Schottky}: {\mathrm{M}\mathrm{n}}_{\mathrm{M}\mathrm{n}}^{\mathrm{X}}+{\mathrm{O}}_{\mathrm{O}}^{\mathrm{X}} \to {\mathrm{V}}_{\mathrm{M}\mathrm{n}}^{″}+{\mathrm{V}}_{\mathrm{O}}^{⦁⦁}+\mathrm{MnO}$$

(7)

$${\mathrm{P}}_2{\mathrm{O}}_5 \mathrm{Schottky}: 2{\mathrm{P}}_{\mathrm{P}}^{\mathrm{X}}+5{\mathrm{O}}_{\mathrm{O}}^{\mathrm{X}}\to {2\mathrm{V}}_{\mathrm{P}}^{\prime \prime \prime \prime \prime }+5{\mathrm{V}}_{\mathrm{O}}^{••}+{\mathrm{P}}_2{\mathrm{O}}_5$$

(8)

$$\mathrm{Na}/\mathrm{Mn} \mathrm{Anti}\text{-}\mathrm{site} \left(\mathrm{isolated}\right): {\mathrm{N}\mathrm{a}}_{\mathrm{N}\mathrm{a}}^{\mathrm{X}}+{\mathrm{M}\mathrm{n}}_{\mathrm{M}\mathrm{n}}^{\mathrm{X}}\to {\mathrm{N}\mathrm{a}}_{\mathrm{M}\mathrm{n}}^{\prime }+{\mathrm{M}\mathrm{n}}_{\mathrm{N}\mathrm{a}}^{•}$$

(9)

$$\mathrm{Na}/\mathrm{Mn} \mathrm{Anti}\text{-}\mathrm{site} \left(\mathrm{Cluster}\right): {\mathrm{N}\mathrm{a}}_{\mathrm{N}\mathrm{a}}^{\mathrm{X}}+{\mathrm{M}\mathrm{n}}_{\mathrm{M}\mathrm{n}}^{\mathrm{X}}\to {\left\{{\mathrm{N}\mathrm{a}}_{\mathrm{M}\mathrm{n}}^{\prime }:{\mathrm{M}\mathrm{n}}_{\mathrm{N}\mathrm{a}}^{•}\right\}}^{\mathrm{X}}$$

(10)

Table 3. Defect formation energies calculated for different reaction processes.

Defect Process	Equation Number	Defect Energy/Defect (eV)
Na-Frenkel	1	1.93
Mn-Frenkel	2	2.97
P-Frenkel	3	15.98
O-Frenkel	4	4.07
Schottky	5	4.85
Na2O-Schottky	6	2.65
MnO-Schottky	7	3.13
P2O5-Schottky	8	7.48
Na-Mn anti-site (isolated)	9	0.53
Na-Mn (cluster)	10	0.12

presents the calculated defect formation energies for NaMnPO₄. Anti-site defects exhibit particularly low formation energies. The isolated Na–Mn anti-site defect requires only 0.53 eV per defect (Equation (9)), while the clustered configuration is even more favorable at 0.12 eV per defect (Equation (10)). This suggests that Na–Mn site exchanges are the most thermodynamically favorable defect processes in NaMnPO₄. They may play a dominant role in both defect chemistry and ion migration. The binding energy for forming a Na–Mn anti-site defect cluster from isolated defects is −0.41 eV, determined from the difference between 0.12 eV and 0.53 eV. The formation of anti-site defects is influenced by several factors, including the synthesis method, processing temperature, cooling rate, and chemical environment, all of which can significantly affect cation ordering and defect concentrations in the material. In olivine-NaMnPO₄, hydrothermal samples with higher concentrations of anti-site defects tend to produce more impurity phases under identical thermal treatments [64]. Precursor-based methods have been developed for the low-temperature synthesis of two structural modifications of NaMnPO₄. These approaches enable the formation of defect-free NaMnPO₄ in the 200–400 °C range. Importantly, both olivine- and maricite-type NaMnPO₄ show no signs of anti-site mixing or Na/Mn deficiencies [37]. Although the Na Frenkel defect (1.93 eV/defect, Equation (1)) is the second most favorable defect type, its formation energy is significantly higher than that reported for other phosphate materials, such as Na₃V₂(PO₄)₃ (0.46 eV) [65] and NaZr₂(PO₄)₃ (0.70 eV) [66]. This emphasizes the comparatively sluggish nature of Na-ion diffusion in NaMnPO₄. In comparison, the Mn Frenkel (2.97 eV/defect, Equation (2)) and O Frenkel (4.07 eV/defect, Equation (4)) are less favorable, while the P Frenkel (15.98 eV/defect, Equation (3)) is highly endothermic and therefore unlikely under normal conditions. For Schottky-type defects, the Na₂O Schottky defect (2.65 eV/defect, Equation (6)) is the most energetically favorable, suggesting possible Na₂O loss at elevated temperatures, whereas the complete Schottky defect (4.85 eV/defect, Equation (5)) and the P₂O₅ Schottky defect (7.48 eV/defect, Equation (8)) are less favorable.

We have performed additional defect calculations. To minimise charge–charge interactions between adjacent cells, both point defects were introduced within the same supercell. The comparison between classical and DFT simulations shows a consistent trend in defect formation energies. For Na–Frenkel defects, the values are 1.93 eV (classical) and 2.12 eV (DFT), while Mn–Frenkel defects yield 2.97 eV and 3.08 eV, respectively. O–Frenkel defects are slightly higher, with 4.07 eV from classical and 4.82 eV from DFT. In the case of anti-site defects, Na–Mn isolated defects show low energies of 0.53 eV (classical) and 0.74 eV (DFT), which further decrease for the clustered configuration to 0.12 eV and 0.26 eV. The highest energies are observed for P–Frenkel defects, with 15.98 eV (classical) and 16.52 eV (DFT). Overall, the close correspondence between classical and DFT values validates the reliability of the classical approach for modeling defect energetics.

3.3. Na-Ion Diffusion

Determining the activation energy of ion migration pathways is crucial for understanding ionic conduction mechanisms. While experimental determination of ion migration pathways is challenging, computational approaches provide a reliable means to predict both the pathways and their corresponding activation energies. Using atomistic simulation techniques, two promising Na-ion migration hops were identified, labelled A and B (see Figure 2a), with Na–Na separations of 4.68 Å and 4.46 Å, respectively. Activation energies of hops A and B were calculated to be 1.28 eV (see Figure 2b) and 2.63 eV (see Figure 2c) respectively. The connection of hops A gives rise to a long-range zig-zag migration pathway in the ab plane with an activation energy of 1.28 eV. This value is considerably higher than those reported for other Na-based cathode materials with phosphate units, such as Na₃V₂(PO₄)₃ (0.70 eV), NaZr₂(PO₄)₃ (0.26 eV), olivine-NaFePO₄ (0.29 eV), and NaFeP₂O₇ (0.43 eV) [64,65,66,67]. A previous DFT study reported that maricite-NaMnPO₄ exhibits the lowest activation energy of 1.21 eV, which is consistent with the value obtained in this work [36]. In contrast, Na⁺ ions in olivine-NaMnPO₄ migrate more rapidly, with a much lower activation energy of 0.32 eV [36]. In the second hop B, Na⁺ migration occurs with a high activation energy of 2.63 eV. Since both hops are associated with large energy barriers, enhancing Na-ion transport will require strategies such as elemental doping [66], particle size reduction [68,69], carbon coating [70], and composite design [71,72,73] to enable practical applications.

3.4. Solution of Dopants

The introduction of dopants or impurities can strongly influence the properties of materials, including conductivity, optical response, and structural stability. In this study, DFT simulations were employed to calculate dopant solution energies, considering charge-compensation mechanisms such as interstitials, vacancies, and lattice energies. A range of potential dopants (both isovalent and aliovalent) was investigated to identify promising candidates for future experimental validation. Isovalent dopants, which possess the same valence state as the host cation (e.g., monovalent substitution at the Na site or divalent substitution at the Mn site), do not require charge compensation since charge neutrality is preserved. In contrast, aliovalent dopants have different valence states compared to the host ions, and their incorporation can introduce additional charge carriers or modify the conductivity of materials. For instance, tetravalent substitution at the P site necessitates charge compensation via the formation of Na interstitials.

3.4.1. Monovalent Dopants

Monovalent dopants (M = Li, K, and Rb) were first considered for substitution at the Na site. The solution energies were calculated (

Table 4. M-O bond distances, Bader charges on the dopants, magnetic moments, and changes in the volume of the monovalent doped configurations.

Dopant (M)	Ionic Radius (Å)	M-O (Å)	Bader Charge on M (|e|)	Magnetic Moment (µ)	$\frac{\mathsf{\Delta }\mathit{V}}{\mathit{V}}\times 100$ (%)
Li	0.76	2.29	+1.00	5.00	─0.26
K	1.38	2.77	+0.82	5.00	+0.79
Rb	1.52	2.81	+0.82	5.00	+1.27

) using the corresponding reaction equation, expressed in Kröger–Vink notation.

$${{\mathrm{M}}_2\mathrm{O}+2\mathrm{N}\mathrm{a}}_{\mathrm{N}\mathrm{a}}^{\mathrm{X}}\to {2\mathrm{M}}_{\mathrm{N}\mathrm{a}}^{\mathrm{X}}+{\mathrm{N}\mathrm{a}}_2\mathrm{O}$$

(11)

Figure 3 summarizes the effect of monovalent substitution at the Na site. In Figure 3a, the solution energies of Li, K, and Rb are plotted against their ionic radii, revealing a clear minimum for K. This lowest solution energy indicates that K incorporates most readily into the lattice, whereas Li (undersized) and Rb (oversized) exhibit higher energies, consistent with size mismatch and the associated elastic/structural penalty. While our default dopant reservoirs are the corresponding oxides (Li₂O, K₂O, Rb₂O), realistic conditions are captured by enforcing host-stability inequalities and considering competing dopant-bearing phases (e.g., alkali phosphates/halides/peroxides) and Na-rich/Na-poor limits; these choices may shift absolute values but do not alter the qualitative conclusion that K incorporates most readily. Figure 3b shows the charge density plot for the K-doped system, where the redistribution of electronic density around the dopant and neighboring atoms is clearly visible. Figure 3c shows the total DOS, confirming that K-doping does not introduce states at the Fermi level and preserves the insulating nature of the host material. Figure 3d illustrates the atomic DOS of K, where the p-states of K are localized deep in the valence region, far from the Fermi level, further confirming its limited electronic activity.

The introduction of alkali-metal dopants leads to distinct structural and electronic modifications depending on the ionic radius of the substituent. Li, with the smallest ionic radius (0.76 Å), forms the shortest M–O bond length of 2.29 Å and carries a Bader charge of +1.00 |e|, while maintaining a magnetic moment of 5.00 µB (see Table 4). This substitution results in a slight lattice contraction, as reflected by a negative volume change (ΔV/V = −0.26%). In contrast, the larger dopants K (1.38 Å) and Rb (1.52 Å) exhibit elongated M–O bond distances of 2.77 Å and 2.81 Å, respectively, and slightly reduced Bader charges of +0.82 |e|. However, their magnetic moments remain unchanged at 5.00 µB. The incorporation of these larger cations induces a lattice expansion, with ΔV/V values of +0.79% and +1.27% for K and Rb, respectively.

3.4.2. Divalent Dopants

We categorize the divalent dopants into two groups: alkali earth elements (Mg, Ca, Sr, and Ba) and transition metals (Fe, Co, Ni, Cu, and Zn), since they exhibit distinct behaviors. Figure 4 shows the results of alkali earth dopant substitution at the Mn site in NaMnPO₄. The following equation describes the doping process.

$${\mathrm{M}\mathrm{O}+\mathrm{M}\mathrm{n}}_{\mathrm{M}\mathrm{n}}^{\mathrm{X}}\to {\mathrm{M}}_{\mathrm{M}\mathrm{n}}^{\mathrm{X}}+\mathrm{M}\mathrm{n}\mathrm{O}$$

(12)

Figure 4a shows the solution energies of Mg, Ca, Sr, and Ba as a function of their ionic radii. Among these, Ca exhibits the lowest solution energy, suggesting the most favorable incorporation, while Mg and Ba appear less stable. Importantly, all four dopants have exothermic solution energies, indicating that they are thermodynamically favorable for incorporation into the host lattice under equilibrium conditions, at least in the dilute limit. Figure 4b presents the charge density distribution for the Ca-doped structure, which reveals localized electron accumulation around oxygen atoms and confirms the strong ionic interaction between Ca and its neighboring oxygen atoms. The DOS analysis (Figure 4c) shows that Ca-doping does not introduce states at the Fermi level, thereby preserving the insulating character of the host material. Furthermore, the atomic DOS of Ca (Figure 4d) demonstrates that its orbitals lie far from the Fermi level, indicating minimal direct contribution to the electronic states near the band edges.

For alkaline earth doping, clear trends emerge with increasing ionic radius. Mg (0.72 Å) forms the shortest M–O bond (2.17 Å) and shows the highest Bader charge transfer (+2.00 |e|), leading to a slight lattice contraction (ΔV/V = −0.43%) (see

Table 5. M-O bond distances, Bader charges on the dopants, magnetic moments, and changes in the volume of the alkali earth doped configurations.

Dopant (M)	Ionic Radius (Å)	M-O (Å)	Bader Charge on M (|e|)	Magnetic Moment (µ)	$\frac{\mathsf{\Delta }\mathit{V}}{\mathit{V}}\times 100$ (%)
Mg	0.72	2.17	+2.00	4.69	−0.43
Ca	1.00	2.33	+1.60	4.69	+0.77
Sr	1.18	2.42	+1.63	4.69	+1.38
Ba	1.35	2.59	+1.62	4.69	+2.36

). In contrast, Ca, Sr, and Ba exhibit progressively longer M–O bond lengths (2.33–2.59 Å) and reduced charge transfer (~+1.6 |e|), resulting in lattice expansion that scales with size, from +0.77% for Ca to +2.36% for Ba. All alkaline earth-doped systems maintain a magnetic moment of 4.69 µB. These observations suggest that smaller dopants strengthen M–O interactions and contract the lattice, while larger cations induce expansion with weaker charge localization.

Although metastable mixed-metal olivine phases Na[Mn_1−XM_X]PO₄ (M = Fe, Ca, Mg; 0 ≤ x ≤ 0.5) have been accessed at low temperature via a topotactic solid-state route, analogous substitution in maricite-NaMnPO₄ has not been explored [74]. Future experimental studies can verify the feasibility and limits of Fe-, Ca-, and Mg-incorporation into the maricite framework by preparing doped compositions under comparable topotactic conditions and establishing phase purity, dopant uptake, site occupancy, and oxidation states using synchrotron/neutron diffraction, as well as X-ray absorption spectroscopy.

The incorporation of transition-metal dopants (Fe, Co, Ni, Cu, and Zn) at the Mn site in NaMnPO₄ was examined by plotting their solution energies against ionic radii. Among these, Cu emerges as the most favorable substitution, with a negative solution energy, while Fe, Co, Ni, and Zn exhibit higher positive values, indicating less stable incorporation (Figure 5a). The charge density distribution of the Cu-doped system (Figure 5b) shows pronounced charge localization around oxygen atoms, consistent with the ionic nature of Cu–O bonding. Density of states analysis (Figure 5c) confirms that Cu-doping preserves the insulating character of the host lattice, as no states are introduced at the Fermi level. The atomic DOS of Cu (Figure 5d) reveals that Cu 3d orbitals dominate near the valence band edge, providing states close to the Fermi level that may influence the electronic structure and enhance local conductivity.

Transition metal dopants display distinct structural and electronic features compared to alkali and alkaline earth substitutions. In general, their smaller ionic radii (0.61–0.74 Å) yield shorter M–O bond lengths (2.05–2.21 Å) than those observed for larger alkali or alkaline earth cations. Among them, Ni (0.69 Å) forms the shortest bond (2.05 Å), consistent with stronger hybridization with oxygen, while Zn (0.74 Å) produces the longest bond (2.21 Å). The Bader charge transfer varies significantly across the series, from +0.98 |e| in Cu to +1.52 |e| in Fe, reflecting differences in d-electron configurations and bonding characteristics (see

Table 6. M-O bond distances, Bader charges on the dopants, magnetic moments, and changes in the volume of the transition metal-doped NaMnPO₄.

Dopant (M)	Ionic Radius (Å)	M-O (Å)	Bader Charge on M (|e|)	Magnetic Moment (µ)	$\frac{\mathsf{\Delta }\mathit{V}}{\mathit{V}}\times 100$ (%)
Fe	0.61	2.12	+1.52	4.94	−0.20
Co	0.65	2.11	+1.34	4.86	−0.38
Ni	0.69	2.05	+1.11	4.64	−0.34
Cu	0.73	2.15	+0.98	4.63	−0.18
Zn	0.74	2.21	+1.42	4.69	−0.20

). All transition metal dopants reduce the magnetic moment slightly compared to the undoped system, with values ranging from 4.63 µB (Cu) to 4.94 µB (Fe), indicating partial quenching of magnetization due to d-state participation. Unlike alkali and alkaline earth elements, transition metal substitution consistently leads to lattice contraction, with ΔV/V values between −0.18% and −0.38%, highlighting the stronger electronic interactions between the transition metal d-orbitals and the oxygen sublattice.

3.4.3. Trivalent Dopants

The doping of trivalent dopants (M = B, Al, Ga, In, and Tl) on the Mn site can promote Na vacancies for enhancing vacancy-mediated Na-ion migration according to the following equation.

$${\displaystyle \frac{1}{2}}{{\mathrm{M}}_2{\mathrm{O}}_3+\mathrm{M}\mathrm{n}}_{\mathrm{M}\mathrm{n}}^{\mathrm{X}}+{\mathrm{N}\mathrm{a}}_{\mathrm{N}\mathrm{a}}^{\mathrm{X}}\to {\mathrm{M}}_{\mathrm{M}\mathrm{n}}^{⦁}+{\mathrm{V}}_{\mathrm{N}\mathrm{a}}^{\prime }+\mathrm{M}\mathrm{n}\mathrm{O}+{\displaystyle \frac{1}{2}}{\mathrm{N}\mathrm{a}}_2\mathrm{O}$$

(13)

Figure 6 highlights a strong size–solubility correlation for trivalent dopants in NaMnPO₄. The solution energy decreases monotonically with the ionic radius of M³⁺ (B ≫ Al > Ga > In > Tl) (see Figure 6a), indicating that lattice strain and coordination mismatch dominate the thermodynamics—very small, hard cations such as B³⁺/Al³⁺ are disfavored, whereas larger ions (In³⁺, Tl³⁺) are comparatively compatible with the Mn site. The solution energies are prohibitively high for B³⁺ (>9.00 eV) and Al³⁺ (>4.50 eV), clearly indicating that these dopants are highly unlikely to be incorporated into the lattice under typical synthesis conditions. The charge-density isosurface for Tl (see Figure 6b) shows modest, fairly isotropic Tl–O bonding, consistent with Tl acting mainly as a structural/size-matched substituent while the host bears charge compensation. Although the electronic-structure results (Figure 6c) largely preserve a wide band gap with the Fermi level positioned inside it, a small number of mid-gap states are observed. The Tl-projected DOS (Figure 6d) indicates that these states predominantly originate from Tl-s orbitals, with negligible weight near the Fermi energy, confirming that Tl contributes minimally to the electronic states at the Fermi level. Together, these results suggest the highest trivalent solubility for Tl (and likely In), limited solubility for Ga, and essentially none for Al/B; compensation is expected via Mn²⁺→Mn³⁺ redox and/or Na vacancies rather than free carriers.

The local structure and charge analysis in

Table 7. M-O bond distances, Bader charges on the dopants, magnetic moments, and changes in the volume of the trivalent-doped NaMnPO₄.

Dopant (M)	Ionic Radius (Å)	M-O (Å)	Bader Charge on M (|e|)	Magnetic Moment (µ)	$\frac{\mathsf{\Delta }\mathit{V}}{\mathit{V}}\times 100$ (%)
B	0.27	2.39	+0.60	4.63	+0.59
Al	0.54	1.95	+3.00	4.63	−0.29
Ga	0.62	2.21	+2.66	4.63	+0.99
In	0.80	2.36	+3.00	4.63	+2.19
Tl	0.89	2.64	+0.92	4.63	+2.84

mirror the thermodynamics of Figure 6. The average M–O distance increases with the dopant ionic radius (Al 1.95 Å < Ga 2.21 Å < In 2.36 Å < Tl 2.64 Å), and the cell responds accordingly: Al causes a slight contraction (ΔV/V ≈ −0.29%), whereas Ga, In, and Tl expand the lattice (+0.99 to +2.84%). B is anomalous—its very small radius (0.27 Å) paired with an unusually long M–O (2.39 Å) indicates severe mismatch and local distortion, consistent with its very unfavorable solution energy. Bader charges track ionicity: Al and In are ~+3, Ga is slightly reduced (+2.66), while Tl (+0.92) and B (+0.60) transfer far less charge, implying much more covalent, weakly ionic bonding to O and a largely structural (rather than electronic) role for Tl. Across all cases the Mn moments remain ~4.63 μB, indicating the Mn sublattice retains high-spin Mn²⁺ character; thus, charge compensation in these dilute models is unlikely to proceed via substantial Mn²⁺→Mn³⁺ oxidation and is more consistent with compensating Na defects and/or ligand charge redistribution. Overall, larger, softer cations (In, Tl) are most easily accommodated structurally, whereas small hard cations (B, Al) impose strain and are disfavored.

3.4.4. Tetravalent Dopants

The doping of tetravalent dopants (M = Si, Ge, Ti, Sn, Zr and Pb) on the P site can generate Na interstitials for enhancing the capacity of NaMnPO₄. The following equation describes this process.

$${\mathrm{M}\mathrm{O}}_2+{\mathrm{P}}_{\mathrm{P}}^{\mathrm{X}}+{\displaystyle \frac{1}{2}}{\mathrm{N}\mathrm{a}}_2\mathrm{O}\to {\mathrm{M}}_{\mathrm{P}}^{\prime }+{\mathrm{N}\mathrm{a}}_{\mathrm{i}}^{⦁}+\mathrm{M}\mathrm{n}\mathrm{O}+{\displaystyle \frac{1}{2}}{\mathrm{P}}_2{\mathrm{O}}_5$$

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The impact of tetravalent dopants substituting at the P site in NaMnPO₄ was investigated to assess their stability and electronic effects. As shown in Figure 7a, solution energies plotted against ionic radius reveal that Si and Ge have the lowest values (~3.7–3.8 eV), appearing relatively more favorable than Ti, Sn, Zr, and Pb. Nevertheless, even these “most favorable” dopants (Si⁴⁺ and Ge⁴⁺) still exhibit prohibitively high positive solution energies, indicating poor solubility and making their incorporation into the lattice under typical synthesis conditions highly unlikely. The charge density distribution for the Ge-doped system (Figure 7b) shows strong electron localization around oxygen atoms, consistent with ionic Ge–O interactions. The total DOS (Figure 7c) further confirms that Ge-doping preserves the insulating nature of the host lattice, with no states introduced at the Fermi level. The atomic DOS of Ge (Figure 7d) indicates that Ge orbitals lie deep in the valence region and slightly near the conduction band edge, suggesting only a minimal contribution to electronic states near the Fermi level.

Under Na-interstitial compensation for M⁴⁺→P⁵⁺ substitution,

Table 8. M-O bond distances, Bader charges on the dopants, magnetic moments, and changes in the volume of the tetravalent-doped NaMnPO₄.

Dopant (M)	Ionic Radius (Å)	M-O (Å)	Bader Charge on M (|e|)	Magnetic Moment (µ)	$\frac{\mathsf{\Delta }\mathit{V}}{\mathit{V}}\times 100$
Si	0.26	1.65	+4.00	4.94	−0.38
Ge	0.39	1.78	+4.00	4.94	+0.30
Ti	0.42	1.84	+2.52	4.94	+0.55
Sn	0.55	1.96	+3.82	4.94	+1.03
Zr	0.59	1.98	+3.08	4.94	+1.37
Pb	0.65	2.22	+2.42	4.94	+1.86

shows a clear size–structure–bonding trend that supports the mechanism. As ionic radius increases from Si to Pb, the M–O distance lengthens (1.65→2.22 Å) and the unit cell expands (ΔV/V from −0.38% to +1.86%). Si and Ge are essentially +4 (more ionic), Sn and Zr are slightly lower (+3.82, +3.08), and Ti and especially Pb are more covalent (+2.52, +2.42), consistent with longer M–O and softer bonding. The magnetic moment remains ~4.94 μB across the series, indicating that Mn stays high-spin Mn²⁺ and that charge neutrality is furnished by Na interstitial rather than Mn oxidation or electronic carriers. Practically, larger M⁴⁺ (Sn, Zr, Pb) are structurally accommodated but introduce greater lattice expansion (likely perturbing Na-ion pathways), whereas smaller M⁴⁺ (Si, Ge) maintain tighter frameworks with minimal electronic impact.

4. Conclusions

Atomistic simulations were carried out to investigate point defects (vacancies, interstitials, Frenkel, Schottky, and anti-site defects), Na-ion migration pathways, and the solution of dopants in maricite NaMnPO₄. Among the intrinsic defects, the Na–Mn anti-site was identified as the most favorable process, indicating that a small fraction of Na and Mn ions readily exchange positions. The Na Frenkel defect was the second most favorable, suggesting that Na vacancies likely facilitate Na diffusion in this material. Sodium-ion migration is sluggish, with a calculated activation energy of 1.28 eV. For isovalent substitution, K is predicted to be most favorable at the Na site, and Ca/Cu at the Mn site. Thallium at the Mn site is expected to generate Na vacancies that enhance Na-ion migration, whereas Ge substitution at the P site is predicted to raise the sodium content