The structure of Na₂MnO₃ was directly constructed by replacing the Li atoms with Na atoms in the Li₂MnO₃ structure. Then, a full geometry optimization for the structure of Na₂MnO₃ was performed. The results indicate that Na₂MnO₃ has a similar arrangement of atoms in Li₂MnO₃, where both the Na⁺ and Mn⁴⁺ ions reside in the octahedral sites of a cubic close-packed, O₃ oxygen lattice. As shown in Fig. 1(a), the manganese layers and the Na layers stack into the structure and the Na₂MnO₃ can be described as . Similar to Li⁺ in Li₂MnO₃, Na⁺ has three occupied sites in Na₂MnO₃, with the 4h and 2c sites in the Na layer and the 2b site in the Mn layer. In this case, the calculated lattice parameters and band gaps of Li₂MnO₃ (in Okamotoʼs work),^[50] Li₂MnO₃ (in this work), and Na₂MnO₃ (in this work) are shown in Table 1 . The calculated results for Li₂MnO₃ are very close to those in Ref. [50], indicating that the calculations in this work are reliable and comparable to the previous results. The results indicate that the lattice parameters of Na₂MnO₃ are slightly larger than those of Li₂MnO₃, which is reasonable given that the larger Na atom radius occupies more space in the cell. A superior electrical conductivity of Na₂MnO₃ can be expected as a result of smaller band gap of Na₂MnO₃ compared to that of Li₂MnO₃.

Fig. 1.

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	Fig. 1. (color online) (a) Top (upper left) and side (lower left) view of Na2MnO3 structure. (b) The structure of Na2MnO3 after AIMD simulation. (c) Phonon spectrum of Na2MnO3.

Table 1.

Table 1.

Table 1. Calculated lattice parameters and band gaps (Eg) of Li2MnO3 (in Ref. [50]), Li2MnO3 (in this work), and Na2MnO3 (in this work). . Space group a/Å b/Å c/Å Eg/eV Li2MnO3[50] C2/m 4.98 8.63 5.00 1.90 Li2MnO3 C2/m 4.99 8.63 5.07 1.87 Na2MnO3 C2/m 5.26 9.12 5.63 1.55

Table 1. Calculated lattice parameters and band gaps (Eg) of Li2MnO3 (in Ref. [50]), Li2MnO3 (in this work), and Na2MnO3 (in this work). .

The crystalline stability of a proposed material is of vital importance in the evaluation of its difficulty of experimental synthesis and potential application prospects. The cohesive energy of Na₂MnO₃ is calculated as 5.23 eV/atom. Such a large cohesive energy indicates that Na₂MnO₃ is energetically stable. In addition, the thermodynamic stability of Na₂MnO₃ is investigated using ab-initio molecular dynamic (AIMD) simulations that adopt NVT canonical ensembles (at a temperature of 800 K). The time step is set to be 1 fs and the total simulation time is more than 10 ps to ensure that the stability conclusions are reliable. The calculated results indicate that the structure of Na₂MnO₃ still maintains the initial state long after AIMD processing, as presented in Fig. 1(b). These results indicate that Na₂MnO₃ is also thermally stable at high temperatures. In order to further evaluate the stability of Na₂MnO₃, phonon spectrum calculations were performed. Based on the results shown in Fig. 1(c), Na₂MnO₃ is confirmed as a dynamically stable structure without any imaginary frequencies. When combined, these results confirm the crystalline stability of Na₂MnO₃.

It is necessary to specify the process of Na⁺ de-intercalation in the simulation of electrochemical performance, which is the bottleneck step in the analysis of the mechanism of an electrochemical reaction. In the charging process, Na⁺ is continuously extracted from Na₂MnO₃. The ground states of four structures with different concentrations of sodium—i.e., Na_1.75MnO₃, Na_1.5MnO₃, Na_1.25MnO₃, and Na₁MnO₃—were considered. There are 10, 10, 9, and 9 possible configurations for Na_1.75MnO₃, Na_1.5MnO₃, Na_1.25MnO₃, and Na₁MnO₃, respectively, and these structures and their corresponding total energies are shown in Fig. S1–S4. In this case, the four ground-state configurations are selected and depicted in Fig. 2. Initially, Na⁺ was extracted from Na₂MnO₃ and subsequently, half of the ions was lost at the 2c site to become Na_1.75MnO₃, which is represented as the ground state of this compound in Fig. 2. In the case of Na_1.5MnO₃, all the Na⁺ at the 2c site was extracted, and the Na⁺ at the 4h site and the 2b site remained. Next, from Na_1.5MnO₃ to Na_1.25MnO₃, Na⁺ at the 2b site in the Mn layer were extracted and half of them will be removed. Subsequently, all the Na⁺ at the 2b site were continually extracted to the NaMnO₃.

Fig. 2.

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	Fig. 2. (color online) Ground-state configuration of Na1.75MnO3, Na1.5MnO3, Na1.25MnO3, and NaMnO3.

When the Na⁺ concentration of Na_xMnO₃ was equal to that of the Na₁MnO₃ (x=1.0), Na⁺ only remained at the 4h site. Throughout the entire charging process from Na₂MnO₃ to NaMnO₃, the framework of the layered structure was well maintained except for small changes in the lattice parameters (as shown in Table 2 ) and the Na/Mn ordering disruption of the Mn layer. Next, we also considered the further extraction of Na⁺ (4h site). The calculated results indicate that half of the Na⁺ at the 4h site can be extracted and the crystal framework maintains its stability to reach the low Na concentration structure of Na_0.5MnO₃, for which the ground state configuration is shown in Fig. S5. From Na₂MnO₃ to Na_0.5MnO₃, the Na⁺ de-intercalation capacity can achieve a value of which represents a significantly high capacity, calculated using Eq. (2). The cycle voltage between Na₂MnO₃ and Na_0.5MnO₃ was also calculated, which is in the range of 3.42–4.30 V as obtained from Eq. (3). It should be noted that all of the configurations of Na_0.75MnO₃, represented as intermediate structures between Na₁MnO₃ to Na_0.5MnO₃, will deform. Moreover, the stacking sequence of the layered structure will be transferred from O₃ to O₁ when the concentration is lower than that of Na_0.5MnO₃. This phenomenon is also observed in the Li de-intercalation process of Li₂MnO₃.^[51] Considering the potential structural instability of low Na⁺ concentrations ( ), we only investigate the Na_xMnO₃ ( ) structures, in which excellent reversible intercalation/de-intercalation of Na⁺ can be expected without phase transition.

Table 2.

Table 2.

Table 2. Lattice parameters of NaxMnO3 supercell and the average charge transfer of sodium, manganese, and oxygen, respectively. . Value x in NaxMnO3 Distribution of Na+ Lattice parameter/Å Average badercharge/e a b c Na Mn O 2.00 10.52 9.12 5.63 +0.78 +1.74 −1.10 1.75 10.32 9.13 5.70 +0.79 +1.74 −1.04 1.50 10.18 9.10 5.70 +0.79 +1.74 −0.98 1.25 10.01 9.01 5.64 +0.80 +1.76 −0.92 1.00 9.80 8.92 5.59 +0.80 +1.78 −0.86

Table 2. Lattice parameters of NaxMnO3 supercell and the average charge transfer of sodium, manganese, and oxygen, respectively. .

Next, the charge compensation mechanism of Na_xMnO₃ ( ) was investigated during the charging process via Bader atomic charge analysis. The calculated results are listed in Table 2 and the values of the transferred charge are obtained by subtracting the values of the valence electrons from the calculated charge. The charge of Na⁺ for x=2.0 to 1.0 exhibited a negligible difference of +0.78e to +0.80e, and the charge of Mn also exhibited a small difference of +1.74e to +1.78e. These results suggest that the Mn atoms are not oxidized in the process of Na⁺ de-intercalation. However, the charge of the O atoms change from −1.10e to −0.86e with a change in x from 2.0 to 1.0. This indicates that charge compensation for Na⁺ removal is mainly contributed by the oxidation of O. A similar charge compensation mechanism is also found in the Li de-intercalation process in Li₂MnO₃.^[39] In addition, the charge of the O atom surrounding the Na₂MnO₃ has a small fluctuation in the range of −1.09e ∼−1.11e, which indicates that the MnO₆ octahedron is faultless in Na₂MnO₃. However, in Na_1.75MnO₃ the charge of the O atom has a larger range of −1.00e ∼−1.10e depending on its neighboring environment. Near the vacancy, the charge of the O atom is approximate −1.00e, which is the largest charge value compared with other O atoms. These results indicate that the O atom near the vacancy is responsible for charge compensation during charging in Na_1.75MnO₃. Similar situations of unbalanced charge distributions are also found in other configurations of Na_xMnO₃ (x=1.5, 1.25, and 1.0).

To further evaluate the phenomenon of charge compensation, the partial densities of states (PDOS) for Na_xMnO₃ (x=2.0, 1.75, 1.5, 1.25, and 1.0) were calculated. Figure 3 shows that the densities of states near the Fermi level are mainly attributable to the 2p orbits of the O atom and partially to the 3d orbits of Mn. With the removal of Na⁺, an increasing number of holes in the 2p states of O are formed, therefore, the Fermi level moves in the low energy direction as depicted in Figs. 3(a)–3(e). Meanwhile, it should be noted that the charge compensation of the 2p orbit of O is much larger than that of the 3d orbit of Mn, due to the larger density of states contributions of the O atoms near the Fermi level. In this case, the charge compensations mainly occur in the O atoms near the vacated Na during the Na⁺ de-intercalation process, which is consistent with the previous results from the Bader atomic charge analysis. This charge compensation mechanism is very similar to that presented in a previous investigation on Li₂MnO₃.^[39]

Fig. 3.

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	Fig. 3. (color online) Densities of states of (a) Na2MnO3, (b) Na1.75MnO3, (c) Na1.5MnO3, (d) Na1.25MnO3, and (e) Na1MnO3, respectively. The black, blue, and red lines represent the total densities of states of NaxMnO3, partial density of states of Mn-d, and partial density of state of O-p, respectively. The dashed line in the middle represents the Fermi level.

Since the mobility of Na-ions is closely related to the rate capability of batteries, we also performed an energy barrier analysis for the process of Na⁺ mobility in Na₂MnO₃. There are two main cases of Na⁺ migration in Na₂MnO₃: (i) Na⁺ migration within the Na layer, and (ii) Na⁺ migration between Mn layer and Na layer.

In the case of Na⁺ diffusion within the Na layer, the 4h and 2c are the two main sites. Figure 4(a) shows that within this layer, Na⁺ migration involves four paths, including two paths for 4h–4h (path I and path II) and two paths for 2c–4h (path III and path IV). As a Na⁺ ion diffuses from the 4h to an adjacent 4h site, it can migrate through the octahedral of the O atom along path I with an energy barrier of 0.85 eV, or across the bottom of the 2b site along path II with an energy barrier of 0.89 eV. In addition, as one Na⁺ diffuses between the 2c and the adjacent 4h site, it migrates through the octahedral of the O atom along path IV with an energy barrier of 0.99 eV, or across the bottom of the Mn-4g site along path III with an energy barrier of 0.82 eV. The reverse migration processes have almost equal energy barriers due to the similar formation energies of Na₂MnO₃ with a vacancy at the 2c or 4h sites. These barriers are slightly higher than those reported in a previous study on Li migration in Li₂MnO₃,^[39] and it is reasonable because the larger ion radius of Na may increase the difficulty of movement of Na compared to Li because of obstruction due to other atoms. In summary, the height of these barriers is different depending on their individual local cation landscapes for the different paths. Path IV exhibits the lowest energy barrier of the four paths within the Na layer. This path is also the lowest energy barrier path for Li migration within the Li-layer in Li₂MnO₃.^[39]

Fig. 4.

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	Fig. 4. (color online) (a) The diffusion path of Na+ in the Na layer (path I–IV, represented by blue arrow) and between Na layer and Mn layer (path V–VI, represented by green arrow); (b) diffusion barrier of Na+ migration in the Na layer (path I–IV, represented by green line) and between Na layer and Mn layer (path V–VI, represented by blue line).

Next, we investigate how one Na⁺ ion diffuses between the Na layer and the Mn layer by drawing the two main migration paths, in Fig. 4(a), corresponding to the 4h–2b (path V) and the 2c–2b (path VI). In this case, the corresponding energy barriers from the Mn layer to the Na layer, are 1.40 and 1.33 eV for path V and path VI as shown in Fig. 4(b), respectively. However, for the reverse paths from the Na layer to the Mn layer, the energy barriers are much lower at values of 0.91 eV and 0.84 eV for path V and path VI, respectively. The difference between the opposite direction barriers is due to the remarkable difference in the formation energies of Na₂MnO₃ with a vacancy at the 4h and 2b (or 2c) sites. From these energy barrier values, it is possible to determine that Na⁺ ions migration from the Na layer to the Mn layer along path VI has the lowest barrier, which is a similar situation to Li migration from the Li layer to the Mn layer in Li₂MnO₃. Moreover, the potential migration of Na⁺ between two adjacent 2b sites in the Mn layer was also considered. However, the results indicate that this process is impossible because of the very high energy barrier associated with the MnO₆ octahedral.

Overall, these results imply that the lowest energy barrier (0.82 eV) of the four kinds of Na⁺ migration in the Na layer is associated with path IV from the 2c to 4h site, and the lowest energy barrier (0.84 eV) of Na⁺ migrations between the Na layer and the Mn layer is associated with path VI from the 2c to 2b site. These results reveal that the Na⁺ ions at the 2c site tend to migrate to adjacent 4h or 2b sites, which is consistent with the previous results of Section 3.2, which indicates that Na⁺ would be initially de-intercalation from the 2c site during charging. Furthermore, the small difference of the energy barriers of the paths , , , , and indicate that Na⁺ can easily backfill empty sites in the Mn layer and Na layer during the discharging process. This phenomenon could lead to an acceptable cycle performance of the battery, which is promising for the prospective future application.

The elastic constants of materials are closely related to their mechanical properties. In order to investigate the stiffness and stability of Na₂MnO₃ during the charging process, both the elastic constants of Na₂MnO₃ and Li₂MnO₃ were determined using first-principle calculations and the results are shown in Fig. 5(a) and Table S1. From Fig. 5(a), it can be seen that the elastic constants of Na₂MnO₃ are basically the same as Li₂MnO₃ for C₄₄, C₆₆, and only slightly lower than Li₂MnO₃ for C₁₁, C₂₂, C₃₃, and C₅₅, which proves that Na₂MnO₃ has a similar mechanical stiffness and structural stability to Li₂MnO₃.

Fig. 5.

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	Fig. 5. (color online) (a) Calculated elastic constants (GPa) of Li2MnO3 and Na2MnO3; (b) calculated elastic constants (GPa) of NaxMnO3.

Figure 5(b) shows the detailed elastic constants for Na_xMnO₃ (x=1.00, 1.25, 1.50, 1.75, and 2.00). From these results, it can be determined that the elastic constants C₁₁, C₂₂, C₄₄, and C₅₅ decrease slowly with x from 2.00 to 1.00, which is an indication of the favorable stability of Na_xMnO₃. It should be noted that small increases in the elastic constant are observed for C₃₃ and C₆₆ from Na_1.5MnO₃ to Na₁MnO₃. However, the trend of the slow decrease remained.

In conclusion, the elastic constants of Na_xMnO₃ decrease slowly during the process of Na⁺ de-intercalation, intimating that the mechanical stiffness was well-maintained the departure of Na⁺ from Na_xMnO₃. Furthermore, it should be noted that the lowest elastic constants of NaMnO₃ in Na_xMnO₃ are still high enough to maintain the structural stability.