Ang Gao

Qinghua Zhang

Xinyan Li

Tongtong Shang

Zhexin Tang

Xia Lu

Yanhong Luo

Jiarun Ding

Wang Hay Kan

Huaican Chen

Wen Yin

Xuefeng Wang

Dongdong Xiao

Dong Su

Hong Li

Xiaohui Rong
email: rong@iphy.ac.cn

Xiqian Yu

Qian Yu

Fanqi Meng

Cewen Nan

Claude Delmas

Liquan Chen

Yong-Sheng Hu
email: yshu@iphy.ac.cn

Lin Gu
email: l.gu@iphy.ac.cn

Topologically protected oxygen redox in a layered manganese oxide cathode for sustainable batteries

Manganese could be the element of choice for cathode materials used in large-scale energy storage systems owning to its abundance and low toxicity levels. However, both lithium and sodium ion batteries adopting this electrode chemistry suffer from rapid performance fading, suggesting a major technical barrier that must be overcome. Here we report a P3-type layered manganese oxide cathode Na 0.6 Li 0.2 Mn 0.8 O 2 (NLMO) that delivers a high capacity of 240 mAh g -1 with outstanding cycling stability in a lithium half-cell. Combined experimental and theoretical characterizations reveal a characteristic topological feature that enables the good electrochemical performance. Specifically, the -α-γ-layer stacking provides topological protection for lattice oxygen redox, whereas the reversibility is absent in P2-structured NLMO which takes a -α-β-configuration. The identified new order parameter opens an avenue towards the rational design of reversible Mn-rich cathode materials for sustainable batteries.

Sustainable energy applications in today's society, especially transportation and grid, require safe high-energy density lithium (Li) and sodium (Na) ion batteries with low-cost and/or abundant natural resources [START_REF] Lee | Reversible Mn 2+ /Mn 4+ double redox in lithium-excess cathode materials[END_REF][START_REF] Zhao | Rational design of layered oxide materials for sodium-ion batteries[END_REF] . Compared to cobalt (Co) and nickel (Ni) as the major component of commercial cathode materials, redox active manganese is the element of choice for cathode materials used in large-scale energy storage systems owning to its abundance and low toxicity levels [START_REF] Zhu | LiMnO 2 cathode stabilized by interfacial orbital ordering for sustainable lithium-ion batteries[END_REF] . In particular, Li/Na-rich Mn-based cathodes exhibit the excellent capacity and high potential (> 4 V vs. Li + /Li or Na + /Na), resulting from the extra lattice oxygen redox (LOR) [START_REF] Hu | Evolution of redox couples in Li-and Mn-rich cathode materials and mitigation of voltage fade by reducing oxygen release[END_REF][START_REF] Hua | Structural insights into the formation and voltage degradation of lithiumand manganese-rich layered oxides[END_REF][START_REF] Yabuuchi | High-capacity electrode materials for rechargeable lithium batteries: Li 3 NbO 4 -based system with cation-disordered rocksalt structure[END_REF][START_REF] Du | Exploring reversible oxidation of oxygen in a manganese oxide[END_REF] . However, the poor cycling stability of LOR-related processes, including voltage hysteresis and voltage fade, hampers its practical use because of the irreversible local structural transformation or the lattice oxygen loss [START_REF] Assat | Fundamental understanding and practical challenges of anionic redox activity in Li-ion batteries[END_REF][START_REF] Sathiya | Origin of voltage decay in high-capacity layered oxide electrodes[END_REF][START_REF] Li | Dynamic imaging of crystalline defects in lithium-manganese oxide electrodes during electrochemical activation to high voltage[END_REF] . The stability and/or reversibility of LOR is correlated significantly with the atomic structure or the local oxygen coordination environment, as elaborated by various theories, including the oxygen lone-pair states [START_REF] Maitra | Oxygen redox chemistry without excess alkali-metal ions in Na 2/3 [Mg 0.28 Mn 0.72 ]O 2[END_REF][START_REF] Seo | The structural and chemical origin of the oxygen redox activity in layered and cation-disordered Li-excess cathode materials[END_REF][START_REF] Hong | Metal-oxygen decoordination stabilizes anion redox in Li-rich oxides[END_REF] , reductive coupling mechanism [START_REF] Sathiya | Reversible anionic redox chemistry in high-capacity layered-oxide electrodes[END_REF][START_REF] Saubanère | The intriguing question of anionic redox in high-energy density cathodes for Li-ion batteries[END_REF][START_REF] Xie | Requirements for reversible extra-capacity in Li-rich layered oxides for Li-ion batteries[END_REF][START_REF] Vergnet | The structural stability of P2-layered Na-based electrodes during anionic redox[END_REF] , O-O dimer [START_REF] Mccalla | Visualization of O-O peroxo-like dimers in high-capacity layered oxides for Li-ion batteries[END_REF][START_REF] Rong | Structure-induced reversible anionic redox activity in Na layered oxide cathode[END_REF] , ligand-to-metal charge transfer (LMCT) [START_REF] Hong | Metal-oxygen decoordination stabilizes anion redox in Li-rich oxides[END_REF] , and critical oxygen hole [START_REF] Ben Yahia | Unified picture of anionic redox in Li/Na-ion batteries[END_REF] theories, as well as a number of other theories [START_REF] Chen | Unraveling oxygen evolution in Li-rich oxides: A unified modeling of the intermediate peroxo/superoxo-like dimers[END_REF][START_REF] Yabuuchi | Origin of stabilization and destabilization in solid-state redox reaction of oxide ions for lithium-ion batteries[END_REF][START_REF] Luo | Charge-compensation in 3d-transition-metal-oxide intercalation cathodes through the generation of localized electron holes on oxygen[END_REF][START_REF] Jacquet | Charge transfer band gap as an indicator of hysteresis in Li-Disordered rock salt cathodes for Li-Ion batteries[END_REF][START_REF] Okubo | Molecular orbital principles of oxygen-redox battery electrodes[END_REF][START_REF] Radin | Manganese oxidation as the origin of the anomalous capacity of Mn-containing Li-excess cathode materials[END_REF] . Thus, exploring crystal structures compatible with reversible LOR in Mn-rich oxides is desirable to meet the demand for sustainable energy storage.

At present, the correlation between the specific atomic structures with reversible LOR in the electrode materials has not been well established. Kang et al. [START_REF] Eum | Voltage decay and redox asymmetry mitigation by reversible cation migration in lithium-rich layered oxide electrodes[END_REF] found that transition metal (TM) layer stacking can control the voltage decay of LOR process by comparing O2-and O3-type Li x (Li 0.2 Ni 0.2 Mn 0.6 )O 2 . Li-excess disordered (rock salt) TM oxides, such as Li 1.211 Mo 0.467 Cr 0.3 O 2 [START_REF] Lee | Unlocking the potential of cation-disordered oxides for rechargeable lithium batteries[END_REF] , Li 1.2 Mn 0.4 Ti 0.4 O 2 [START_REF] Yabuuchi | Origin of stabilization and destabilization in solid-state redox reaction of oxide ions for lithium-ion batteries[END_REF] , and Li 1.2 Ni 0.333 Ti 0.333 Mo 0.133 O 2 [START_REF] Lee | A new class of high capacity cation-disordered oxides for rechargeable lithium batteries: Li-Ni-Ti-Mo oxides[END_REF] , have been reported with high capacities of around 300 mAh g -1 , also suggesting the positive effect of mixed alkali ions and TM ions on the LOR process [START_REF] Lee | Mitigating oxygen loss to improve the cycling performance of high capacity cation-disordered cathode materials[END_REF] . Compared to Li-ion batteries, Na layered-oxide cathodes are more suitable for investigating the LOR mechanism, which can maintain the well-order layered structure because of the size mismatch between the TM and Na sites [START_REF] Zhao | Rational design of layered oxide materials for sodium-ion batteries[END_REF][START_REF] Maitra | Oxygen redox chemistry without excess alkali-metal ions in Na 2/3 [Mg 0.28 Mn 0.72 ]O 2[END_REF][START_REF] Vergnet | The structural stability of P2-layered Na-based electrodes during anionic redox[END_REF][START_REF] Rong | Structure-induced reversible anionic redox activity in Na layered oxide cathode[END_REF][START_REF] Wang | Unlocking anionic redox activity in O3-type sodium 3d layered oxides via Li substitution[END_REF] . Recent studies have demonstrated that the reversibility of LOR is related to the ordered structure or superstructure of the layered-oxide cathodes. In terms of the Na 2 RuO 3 system, Yamada et al. [START_REF] Mortemard De Boisse | Intermediate honeycomb ordering to trigger oxygen redox chemistry in layered battery electrode[END_REF] proposed that the honeycomb ordering within TM layer can encourage reversible LOR. Further, they also demonstrated a self-repairing layer stacking on desodiation guarantees the better reversibility of the electrode reaction [START_REF] Mortemard De Boisse | Coulombic self-ordering upon charging a large-capacity layered cathode material for rechargeable batteries[END_REF] . In Na 2/3 Mg 1/3 Mn 2/3 O 2 , P2 stacks can trigger a reversible collective distortion to stabilize the oxygen network, while O2 stacks suffer from voltage hysteresis accompanying the disproportionation of oxygen pairs [START_REF] Vergnet | The structural stability of P2-layered Na-based electrodes during anionic redox[END_REF] . Recently, Bruce et al. [START_REF] House | Superstructure control of first-cycle voltage hysteresis in O-redox cathodes[END_REF][START_REF] House | First-cycle voltage hysteresis in Li-rich 3d cathodes associated with molecular O 2 trapped in the bulk[END_REF] demonstrated voltage hysteresis can be suppressed in LOR cathodes through a ribbon-ordered structure in TM layers that hinders the in-plane migration of the TM. The above studies have focused largely on the structural design involving intralayer atom configurations and interlayer stacks of electrode materials, and furthermore a more comprehensive theory could be required to control the critical electrochemical process.

Over the past decade, topological structures have become an important research area in materials science and condensed-matter physics [START_REF] Blanco-Redondo | Topological protection of biphoton states[END_REF][START_REF] Peterson | Trapped fractional charges at bulk defects in topological insulators[END_REF] . More and more intriguing topological structures have been discovered, thus opening up a new world of materials science and the fundamental physics [START_REF] Seidel | Nanoelectronics based on topological structures[END_REF] , including energy storage materials [START_REF] Kanyolo | Honeycomb layered oxides: structure, energy storage, transport, topology and relevant insights[END_REF] . For instance, Saito et al. reported the topological feature of honeycomb layered oxides can enable the good electrochemical performance in rechargeable batterie [START_REF] Masese | Topological defects and unique stacking disorders in honeycomb layered oxide K 2 Ni 2 TeO 6 nanomaterials: implications for rechargeable batteries[END_REF] . Ceder et al. demonstrated that a non-topotactic reaction can improve the rate capability in a Li-rich cation-disordered rock salt cathode [START_REF] Huang | Non-topotactic reactions enable high rate capability in Li-rich cathode materials[END_REF] .

Researchers are becoming aware of the role of topological structures in improving battery performance and even LOR-related processes. Further investigations into these topological structures are urgently needed to explore the structural stability and reversible LOR in layered cathodes.

In this paper, the key role of topological protection in enhancing the reversibility of LOR 1 and2. Meanwhile, the Rietveld-refined X-ray diffraction (XRD) profiles were also employed to verify the structures of P2-and P3-NLMO samples (Supplementary Fig. 1a,b). Based on the unit cell, the diffraction peaks belonging to the superstructure were indexed as (010), (012), ( 11), (2 1), (0 2), (022), and ( 21) planes for P2-NLMO, and (0 1), ( 211), ( 20), ( 21), (223), and (2 3) planes for P3-NLMO (Supplementary Figs. 123).

Fig. 1e, f shows the first ten charge-discharge curves of P2-and P3-NLMO cycled in the voltage range of 3.5-4.5 V at 10 mA g -1 . Capacities of 106 and 73 mAh g -1 were obtained in the first charge-discharge process of P2-NLMO, corresponding to 0.36 and 0.25 mol Na + ions (Fig. 1e). After ten cycles, only 0.30 mol Na + ions were retained in P2-NLMO, indicating rapid performance degradation. Meanwhile, P3-NLMO returned capacities of 105 and 73 mAh g -1 in the first charge-discharge process, corresponding to 0.36 and 0.25 mol Na + ions (Fig. 1f). After ten cycles, however, 0.40 mol Na + ions were retained in P3-NLMO, indicating comparatively excellent performance. To further quantify the contents of Li + and Na + ions, the inductively coupled plasma-mass (ICP) analysis was employed. The results (Supplementary Table 3) show that 0.48 and 0.32 mol Na were remained after one and ten cycles, respectively, in P2-NLMO. Besides, 0.49 and 0.40 mol Na were remained after one and ten cycles, respectively, in P3-NLMO. The ICP results are consistent with those of electrochemistry.

Meanwhile, the Li contents (0.2 mol Li) were proved to be essentially unchanged within ten cycles according to the ICP results. With the increase in cycles, the reversible Na + ions decreased from 0.25 to 0.16 mol in P3-NLMO, and from 0.25 to 0.08 mol in P2-NLMO (Supplementary Fig. 4) according to the electrochemical results. Thus, it can be concluded that the capacity of P3-NLMO after ten cycles was almost twice that of P2-NLMO. It is worth noting that the reversible capacity (3.5-4.5 V) of both materials was mainly provided by LOR [START_REF] Rong | Structure-induced reversible anionic redox activity in Na layered oxide cathode[END_REF][START_REF] House | Superstructure control of first-cycle voltage hysteresis in O-redox cathodes[END_REF][START_REF] Wu | Dissociate lattice oxygen redox reactions from capacity and voltage drops of battery electrodes[END_REF] (for details see 'Oxygen redox analysis' in Methods). Hence, the inferior capacity retention of P2-NLMO indicated the irreversibility of LOR process during cycling. Moreover, compared to P3-NLMO, the voltage polarization of P2-NLMO gradually increased with cycling, implying the irreversibility of LOR process. Topological characteristics of P2-and P3-NLMO. P2-NLMO exhibits inferior cycling stability compared with P3-NLMO, although both have a ribbon-ordered structure in the TM layer. Here, the stacking characteristics of both structures were analyzed to explain the origin of the above discrepancy, which can affect battery performance significantly [START_REF] Li | Dynamic imaging of crystalline defects in lithium-manganese oxide electrodes during electrochemical activation to high voltage[END_REF][START_REF] Eum | Voltage decay and redox asymmetry mitigation by reversible cation migration in lithium-rich layered oxide electrodes[END_REF][START_REF] Han | Stacking faults hinder lithium insertion in Li 2 RuO 3[END_REF][START_REF] Yu | Crystalline grain interior configuration affects lithium migration kinetics in Li-rich layered oxide[END_REF] . In addition to the intrinsic stacking differences between P2 and P3 structures (Fig. 1b The contrast in the high-angle annular dark-field (HAADF) images varied according to the atomic number in terms of a Z 1.7 dependency [START_REF] Zhang | Atomic-resolution imaging of electrically induced oxygen vacancy migration and phase transformation in SrCoO 2.5-σ[END_REF] , which is more sensitive to heavy atoms (e.g., Mn). The HAADF-STEM image of pristine P2-NLMO (Fig. 2d and Supplementary Fig. 12) revealed the ribbon-ordered structure of a TM layer with four-atom dumbbells (Mn-Mn-Mn-Mn). More importantly, the ---stacks (Fig. 2a) were revealed clearly in Fig. 2d (the brown area), which was consistent with the results of the DFT calculations.

Furthermore, various Na sites were detected by annular bright-field (ABF)-STEM analysis, which is more sensitive to lighter atoms [START_REF] Okunishi | Visualization of light elements at ultrahigh resolution by STEM annular bright field microscopy[END_REF] (Fig. 2e). The Na-ion distribution (black spots in the Na layer) was in good agreement with the theoretical predictions, as shown in Fig. 2c and the insert of Fig. 2e. The HAADF-STEM image of pristine P3-NLMO (Fig. 2g and Optimized structure (side and top views). g,h, HAADF-and ABF-STEM images (the inset shows the optimized P3 structure). The cyan ball is the Li ion, the brown ball the Mn ion, the yellow ball the Na ion, and the red ball the O ion. Scale bar = 2 nm in (d,g) and Scale bar = 1 nm in (e,h).

Evolution of ODT structures on cycling. Then, we carefully probed the evolution of ODT structures upon cycling using STEM analysis (Fig. 3 and Supplementary Fig. 14). Fig. 3a-c presents the HAADF images of the P2-NLMO samples in the first-charged (4.5 V vs. Na + /Na), first-discharged (3.5 V vs. Na + /Na), and tenth-discharged states, respectively. In the HAADF-STEM image of pristine P2-NLMO (Fig. 2d), the ---stacks of the TM layers were clearly visible (the brown area), which was also verified by DFT calculations. In contrast, the HAADF image of the charged samples (4.5 V vs. Na + /Na) (Fig. 3a) revealed clear --stacks of the TM layers (the cyan area). In the STEM image of the first-discharged P2-NLMO (3.5 V vs. Na + /Na) (Fig. 3b), the stacks of the TM layers partially returned to the pristine state (---) and partially preserved the stacking form of the charged state (---). Surprisingly, the stacks of the TM layer were converted completely to ---sequences after ten cycles (Fig. 3c).

The STEM analysis unequivocally confirmed the irreversible interlayer stacking transformation from ---to ---sequences during the successive charge and discharge of P2-NLMO. Fig. 3d-f presents the HAADF images of the P3-NLMO samples in the first-charged (4.5 V vs. Na + /Na), first-discharged (3.5 V vs. Na + /Na), and tenth-discharged states, respectively. In contrast to P2-NLMO, the stacks of TM layers in P3-NLMO maintained ---sequences consistently during successive charge and discharge (Fig. 3d,e), even after ten cycles (Fig. 3f). The evolution of ODT structures further was detected by ex-situ XRD (Supplementary Figs. 15-20 and Supplementary Note 1), which was consistent with the STEM analysis. ,2d,e and Supplementary Fig. 7f[i],). Meanwhile, according to the electrochemistry results, during the charging process, approximately 0.4 Na + ions were removed from P2-NLMO (Fig. 1e), where the phase transition from P2 to O2 was verified through desodiation [START_REF] House | Superstructure control of first-cycle voltage hysteresis in O-redox cathodes[END_REF] . The automatic conversion from a P2-Na 12 Li 4 Mn 16 O 40 to a O2-Na 12 Li 4 Mn 16 O 40 model was also reproduced in the relaxation process using DFT calculations (Supplementary Fig. 24). The phase transition from P2 to O2 further was verified by ex-situ XRD and STEM (Supplementary Figs. 15-18 and Supplementary Note 2). Fig. 4e shows six possible glide paths from P2 to O2 in the TM layer, which form ---, ---, or ---stacks in the O2 structure. As the Li + ions in the TM layer migrate into the Na layer in charged states [START_REF] Wang | Unlocking anionic redox activity in O3-type sodium 3d layered oxides via Li substitution[END_REF][START_REF] House | Superstructure control of first-cycle voltage hysteresis in O-redox cathodes[END_REF] , the ---and ---stacks will tend to be prohibited because of the Coulomb repulsion between Li + and Na + ions (Supplementary Fig. 24). Meanwhile, the paths can be classified as edge-shared (e) or face-shared (f) between the NaO 6 and the TMO 6 octahedron in the O2 structure [START_REF] Kubota | Electrochemistry and solid-state chemistry of NaMeO 2 (Me = 3d transition metals)[END_REF] . Here, the remaining Na + ions preferred to occupy Na1 sites rather than Na2 sites in the O2 structure (Supplementary Fig. 25), leading to the prohibitive face-shared paths (Fig. 4e). Thus, only one glide path was feasible (represented by the green arrow in Fig. 4e), and the resulting O2 structure presented ---stacks and edge-shared between NaO 6 and under the TMO 6 octahedron (Fig. 4b), which explains the ---stacks observed in the charged states (4.5 V vs. Na + /Na), as shown in Fig. 3a. When the Na + ions were reinserted into O2-Na 0.2 Li 0.2 Mn 0.8 O 2 , there were three paths to return to the P2 structure (Fig. 4f), which formed ---, ---, or --(-)-stacks in the P2 structure, as shown in Fig. 4c, d. As noted above, partial Na2 sites were inhibited by the Coulomb repulsion among the Na + ions in the ---model (Fig. 4d and Supplementary Fig. 7e), which led to a higher energy of +204 meV/cell, implying a labile structure. Thus, compared with the ---model (Fig. 4c and Supplementary 7f[i]), the ---model with a higher energy corresponds to the lower redox potential. Within the voltage range of 3.5~4.5 V, the ---model accommodated less than 0.4

Na + ions (Fig. 4d), even though both stacking models presented a P2-type structure (for details see 'Oxygen redox analysis' in Methods). This implies that the ODT structure of pristine P2-NLMO evolved gradually from ---stacks to ---stacks on cycling (Fig. 3b,c), which revealed the origin of the rapid capacity decay during cycling in the case of P2-NLMO (Fig. 1e).

Discussion

Manganese is a promising element due to its resource-abundant, non-toxic, low-cost, and compatible characteristics with LOR, which can well satisfy the requirements of the sustainable development of large-scale energy storage. Substituting proper TM elements [START_REF] Xu | Elucidating anionic oxygen activity in lithium-rich layered oxides[END_REF][START_REF] Nayak | Al doping for mitigating the capacity fading and voltage decay of layered Li and Mn-rich cathodes for Li-ion batteries[END_REF] and designing stable crystal structures [START_REF] House | Superstructure control of first-cycle voltage hysteresis in O-redox cathodes[END_REF] have been considered as effective strategies to enhance LOR reversibility, but still could not meet the practical demands at present. Thus, a new understanding of LOR is required. Here, we demonstrated that the topological protection mechanism controls the reversible Na storage with LOR by comparing P2-and P3-NLMO.

The ordered arrangement of Li/Mn in the TM layer constituted the ribbon-ordered structure in the intralayer (Fig. 1a). As a result of the incorporation of Li + ions into the TM layers, the stacks between the layers generated ODT structures, including ---, ---, and --sequences (Fig. 2a). In high-charged states, the LOR process is coupled with Li + ion migration from the TM layers to the Na layers [START_REF] Wang | Unlocking anionic redox activity in O3-type sodium 3d layered oxides via Li substitution[END_REF][START_REF] House | Superstructure control of first-cycle voltage hysteresis in O-redox cathodes[END_REF] , which is feasible in ---stacks (Fig. 5c), but is suppressed in ---or ---stacks (Fig. 5a,b) because of the Coulomb repulsion between the alkali metal ions. Further, the accessibility of Li + ion migration in different ODT structures was also validated by Li migration barriers via CINEB method (Supplementary Fig. 26). In short, the topological feature of ---stacks provides topological protection for reversible LOR, whereas that of other stacks does not. The structure with ---or --stacks will be transformed into ---stacks on charging, as is the case with, for example, P2-NLMO (a → b in Fig. 4). In the discharging process, the structure cannot be fully restored to ---stacks (b → d in Fig. 4), and the ODT structure of pristine P2-NLMO will change.

However, the ---model accommodates less Na + ions than the ---model in the case of P2-NLMO, thus accelerating the capacity decay during cycling. In contrast, the structure always maintains ---stacks in P3-NLMO, thus guaranteeing cycle stability. In general, the protected topological state is revealed in P3-NLMO through the evolution of P2-and P3-type structures.

Further, the same ODT structure can be either in the same phase or in different phases, which is independent of the specific phase. For example, in our study, both ---and --models could exist in P2-NLMO, whereas an ---model also existed in P2-, O2-, and P3-NLMO. In fact, this ODT structure can be present in any layered material, including O3, O1, P3, O2, O2', and P2. We defined a new order parameter of layered cathodes, namely, topological order (R), which was used to describe the interaction between heterogeneous TM layers, independent of the conventional phase definition (O-or P-type). Taking the ODT structure of the P3-NLMO as an example, the position of the Li + ion string within the layer is denoted by , where n is the serial number of layers (Fig. 5d). Then, the topological order ( , transposed matrix) of P3-NLMO can be described as the following matrix: as shown in Fig. 5f, exhibiting 74% capacity retention after 20 cycles. As a comparison, P2-NLMO cathode in Na half-cell showed a discharge capacity of 171 mAh g -1 with 52% capacity retention after 20 cycles (Fig. 5g). Surprisingly, P3-NLMO cathode in Li half-cell delivered a reversible capacity of ~ 240 mAh g -1 at the second cycle within the voltage range of 2.0-4.8 V at 10 mA g -1 , exhibiting 98% capacity retention after 30 cycles as shown in Fig. 5h. On the contrary, the capacity of ~ 183 mAh g -1 was obtained in P2-NLMO with only 60% capacity retention after 30 cycles as shown in Fig. 5i. The excellent cycling performance in Na and Li half-cell once again proves the importance of topological protection for LOR.

This concept of topological order could be extended further to the three-dimensional case, as shown in Fig. 5e (for details see '3D topological order' in Methods). As popular layered materials, ternary cathodes benefit from the synergistic effect of Ni, Co, and Mn (Al) elements, which results in superior performance [START_REF] Wei | Kinetics tuning of Li-Ion diffusion in layered Li(Ni x Mn y Co z )O 2[END_REF][START_REF] Zheng | Tuning of thermal stability in layered Li(Ni x Mn y Co z )O 2[END_REF] . And numerous elements tend to be used as dopants to improve the performance of electrodes [START_REF] Lee | A new class of high capacity cation-disordered oxides for rechargeable lithium batteries: Li-Ni-Ti-Mo oxides[END_REF][START_REF] Wang | A disordered rock-salt Li-excess cathode material with high capacity and substantial oxygen redox activity: Li 1.25 Nb 0.25 Mn 0.5 O 2[END_REF][START_REF] Ma | Synthesis and electrochemical properties of layered LiNi 2/3 Sb 1/3 O 2[END_REF] . For these materials with LOR, the incorporation of Li, Na, Mg, vacancy, etc. into TM layer [START_REF] Assat | Fundamental understanding and practical challenges of anionic redox activity in Li-ion batteries[END_REF][START_REF] House | The role of O 2 in O-redox cathodes for Li-ion batteries[END_REF] triggers and/or stabilizes the LOR process. In different systems, it is of great scientific significance to design and control the variable units or topological order (R) to improve the materials' performance. There are many reasons affecting the topological order (R), not only related to the type and contents of component elements, but also closely related to the synthesis conditions. For example, although having the same composition and the Li/Mn ordering within the layers for both P2and P3-NLMO materials, the synthesis temperature affects the stacking sequences (topological order (R)). In addition, the coordination environment of Li + and Na + ions and their migration, as well as the main structural network (honeycomb, ribbon, ...), all have an impact on the topological structures. The design concept of topology is based on long-range interactions between heterogeneous units and distinguishes itself from traditional methods of performance improvement (doping, coating, interfacial design, etc.). It is undeniable that the topology needs to be designed in terms of the overall structure. In general, as a new degree of freedom, the topological order (R) will have an effective, almost "magical" impact on preserving the cycle reversibility for layered materials and even more so for electrode materials, thus presenting a compelling strategy in the search for high-energy-density cathode materials.

In summary, the key role of topological protection in enhancing the reversibility of LOR is revealed by investigating P2-and P3-Na 0.6 Li 0.2 Mn 0.8 O 2 . The topological feature of --stacks provides topological protection for reversible LOR on Na-ion deintercalation in P3-NLMO, whereas the topological feature of ---stacks in pristine P2-NLMO is broken and evolves gradually from ---stacks to ---stacks on cycling, which accommodates less Na + ions and leads to capacity decay. Specially, P3-NLMO cathode delivers a high capacity of 240 mAh g -1 with outstanding cycling stability in a lithium half-cell. The protected topological state is identified in terms of the topological order , a new order parameter of layered cathodes, describing the interaction between heterogeneous TM layers, which differs from conventional phase definition (O-or P-type). This work provides robust guidance for designing reversible Mn-rich cathode materials for sustainable batteries. In a broader context, our findings define a clear strategy to design materials for applications beyond battery where the topological order (R) is effective, such as electrocatalysts [START_REF] Yang | Atomically dispersed Ni(I) as the active site for electrochemical CO 2 reduction[END_REF] , photovoltaics [START_REF] Domanski | Migration of cations induces reversible performance losses over day/night cycling in perovskite solar cells[END_REF] , and fuel cells [START_REF] Ott | Ionomer distribution control in porous carbon-supported catalyst layers for high-power and low Pt-loaded proton exchange membrane fuel cells[END_REF] . pseudopotential [START_REF] Perdew | Generalized gradient approximation made simple[END_REF] was dealt with Projector augmented wave 65 potentials. A plane wave representation for the wave function with a cutoff energy of 500 eV was applied. Geometry optimizations were performed using a conjugate gradient minimization until all the forces acting on the ions were less than 0.01 eV Å -1 per atom. The K-point mesh with a spacing of ca. 0.03 Å -1 was adopted. The Li migration barrier energy was calculated by CINEB method with 5 images as intermediate states. The crystal structures were built using VESTA software [START_REF] Momma | VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data[END_REF] . The crystal orbital Hamilton population (COHP) was computed with the DFT calculations by the Lobster program [START_REF] Deringer | Crystal orbital Hamilton population (COHP) analysis as projected from plane-wave basis sets[END_REF] .

Methods

Oxygen redox analysis. The charge compensation of the two cathodes in the voltage range of 3.5-4.5 V is mainly attributed to the reversible LOR, which has been adequately demonstrated by many advanced characterizations (hard and soft X-ray absorption spectroscopy, resonant inelastic X-ray scattering, scanning transmission X-ray microscopy, etc.) [START_REF] Rong | Structure-induced reversible anionic redox activity in Na layered oxide cathode[END_REF][START_REF] House | Superstructure control of first-cycle voltage hysteresis in O-redox cathodes[END_REF][START_REF] Wu | Dissociate lattice oxygen redox reactions from capacity and voltage drops of battery electrodes[END_REF] , making it possible to assess the reversibility of LOR with the voltage-plateaus capacity as the measure.

Different from the materials with long-slope discharge voltage profiles, such as ODT structures. To predict the stacking models, P2-and P3-type ribbon-ordered structures were first constructed, as shown in Supplementary Figs. 5 and6, with only the adjacent stacks considered to simplify the calculations. For example, the P2-NLMO model with ---stacks contained ---and --(-)-sequences (Supplementary Fig. 7b), forming a Z-type array rather than a linear array of the Li + ion string along the out-of-plane. Other P2-and P3-NLMO models are shown in Supplementary Figs. 7a-c and 8a-c. The stacking behavior directly determines the coordination environment of Na + ions (e.g., octahedron in O-type structures and trigonal prismatic in P-type structures 69 ); in turn, the Na + ions configurations also influence the stacking of the TMO 2 layers. In conventional P2 structures (Supplementary Fig. 9), two types of Na sites can be identified: an edge-shared trigonal prismatic cavity (V-Na-V,

3) and a face-shared trigonal prismatic cavity (Mn-Na-Mn, 4) 2 . As a result of the incorporation of Li + ions into the TM layers, the trigonal prism can be coplanar with LiO 6 in addition to MnO 6 , giving rise to two new Na sites, namely, Li-Na-Li (1) and Li-Na-Mn (2).

In conventional P3 structures (Supplementary Fig. 9), only one Na site exists, that is, an edge/face-shared trigonal prismatic cavity (Mn-Na-V, 2) [START_REF] Bianchini | The interplay between thermodynamics and kinetics in the solid-state synthesis of layered oxides[END_REF] . Similarly, Mn can be replaced by Li to generate a Li-Na-V site (1). Before considering the Na-ion configurations in the Na layer, it should be noted that all Na + ions present the most uniform distribution to reduce the Coulomb repulsion [START_REF] Lee | An advanced cathode for Na-ion batteries with high rate and excellent structural stability[END_REF][START_REF] Gao | K-Birnessite electrode obtained by ion exchange for potassium-ion batteries: Insight into the concerted ionic diffusion and K storage mechanism[END_REF] . Based on this, the priority of Na-ion position is assumed to be Na1 > Na2 > Na3 > Na4 in P2-NLMO and Na1 > Na2 in P3-NLMO, as shown in Supplementary Fig. 9. Based on the above principles, a series of structures with different Na-ion distributions were designed to verify these assumptions and to determine the stacking structures in P2-and P3-NLMO, as shown in Supplementary Figs. 7d-f and 8d-f. The energy of the most stable P2-NLMO structure was 0.19 meV/atom (Supplementary Fig. 10), with the structure characterized by four Na2 sites and two Na3 sites in the ---stacks (Supplementary Fig. 7f[i]). The energy increased to 0.88 meV/atom for the structure with two Na2 sites and four Na3 sites in the ---stacks, suggesting that Na + ions prefer to occupy Na2 sites rather than Na3 sites (Supplementary Fig. 7f[ii]). This result was further verified by the structure with three Na2 sites and three Na3 sites in the ---stack, the energy value of which (0.40 meV/atom) was between those of the previous two structures (Supplementary Fig. 7f[iii]).

Compared with the ---(i) model (Supplementary Fig. 7f[i]), the ---model was more labile with a higher energy of 3.17 meV/atom, which was attributed to the prohibitive Na2 sites limited by the ---stacks and the Coulomb repulsion. In terms of the P3-NLMO structure, the most stable Na + ions distribution (0.00 meV/atom) was characterized by four Na1 sites and two Na2 sites in the ---stacks (Supplementary Figs. 10 and8f[iii]). In such a model, the Na2 sites can be set to two alternative distributions, as shown in Supplementary Fig. 8f[i,ii], which both present an energy of 0.50 meV/atom, thus implying a precisely equivalent position. However, the energy increase from 0.00 to 0.50 meV/atom suggested that a cross-distribution (Supplementary Fig. 8f[iii]) was preferred. Like the P2-NLMO structure, partial Na1 sites were inhibited by the Coulomb repulsion between the Na + ions in the --stacks (Supplementary Fig. 8e). Thus, the structure with ---stacks had two Na1 sites and four Na2 sites with a higher energy of 4.25 meV/atom. Based on Density functional theory (DFT) calculations, the most stable ODT structures of pristine P2-and P3-NLMO were predicted to ---and ---stacks, respectively, but only adjacent stacks were considered. To ensure alignment with the linear model of the Li + ion string along the inter-plane (Fig. 2b), the optimized structure of P2-and P3-NLMO were transformed from Z-type models into linear-type models (Supplementary Fig. 11), with the side and top views of the optimized P2and P3-NLMO structures shown in Fig. 2c,f. 3D topological order. The variable unit in the heterogeneous layer is located at , where n

  Fig. 1a, can inhibit first-cycle voltage hysteresis by suppressing the manganese disorder and O 2 formation in Na 0.6 [Li 0.2 Mn 0.8 ]O 2 cathode 34 . Here, ribbon-ordered P2-and

Fig. 1|

 1| Fig. 1| Ribbon-ordered structure and electrochemistry of P2-and P3-Na 0.6 Li 0.2 Mn 0.8 O 2 cathode materials. a, Ribbon-ordered structure (-Li-4Mn-Li-) of the TM layer. b, Illustrations of P2-and P3-type structures. c,d, Refinement results of the neutron diffraction data of P2-and P3-NLMO samples using the structural model from DFT calculations, in which the red dots represent the experimental pattern; black line the calculated pattern; purple line the background curve; blue line the difference curve; green bars the Bragg reflections. e,f, Capacity-voltage curves of ten cycles for P2-and P3-NLMO in a Na half-cell, with the Na content (x) near the green or red circle at various states of charge or discharge in Na x Li 0.2 Mn 0.8 O 2 .

  ), there are three possible stacking models based on the incorporation of Li + ions in the TM layer, namely, ---, ---, and ---sequences (Fig. 2a), with the three stacking models superimposed to form one-dimensional topological (ODT) structures, as shown in Fig. 2b. In terms of the P2 structure, an ---model can exist in a P2-type unit cell (1 × 1 × 1) with two TMO 6 layers, whereas ---and ---models generate the topological structure (1 × 1 × 5) along the out-of-plane with ten TMO 6 layers, as shown in the left panel of Fig. 2b. Similarly, in the P3 structure, an ---model can exist in a P3-type unit cell (1 × 1 × 1) with three TMO 6 layers, whereas ---and ---models generate the topological structure (1 × 1 × 5) along the out-of-plane with 15 TMO 6 layers, as shown in the right panel of Fig. 2b. The ODT structures of pristine P2-and P3-NLMO were predicted to ---and ---stacks, respectively, by density functional theory (DFT) calculations (for details see 'ODT structures' in Methods) as shown in Supplementary Figs. 5-11. Fig. 2c, f shows the ODT structures and Na sites of the optimized P2-and P3-NLMO from the side and top views.

Fig. 3|

 3| Fig. 3| Evolution of the ODT structures in P2-and P3-Na x Li 0.2 Mn 0.8 O 2 cathodes on cycling. P2-type: The stacking model of TM layers transformed from ---to ---sequences on Na-ion deintercalation. HAADF-STEM images of the P2-NLMO at (a) first-charged (4.5 V), (b) first-discharged (3.5 V), and (c) tenth-discharged (3.5 V) states. P3 type: the stacking model of TM layers maintained ---sequences on Na-ion deintercalation. HAADF-STEM images of P3-NLMO at (d) first-charged (4.5 V), (e) first-discharged (3.5 V), and (f) tenth-discharged (3.5 V) states. These samples were collected ex-situ at different states indicated in Fig. 1e, f by red circles. Scale bar = 1 nm.

Fig. 4|

 4| Fig. 4| Topological protection mechanism during Na-ion deintercalation. P2-type: a, Pristine P2-NLMO structure with the ---sequence. b, O2-Na 0.2 Li 0.2 Mn 0.8 O 2 structure with the ---sequence at the state of charge of 4.5 V. c, P2-NLMO structure with the ---sequence at the state of discharge of 3.5 V (the top view is shown in the right panel). d, P2-NLMO structure with the ---sequence at the state of discharge of 3.5 V (the top view is shown in the right panel). e, glide paths from the P2 structure to O2 structure (a  b). f, glide paths from the O2 structure to P2 structure (b  c or d). P3-type: g, pristine P3-NLMO structure with the ---sequence. h, P3-Na 0.2 Li 0.2 Mn 0.8 O 2 structure with the ---sequence at the state of charge of 4.5 V. i, P3-NLMO structure with the ---sequence at the state of discharge of 3.5 V (the top view is shown in the right panel). The cyan ball is the Li ion, the brown ball the Mn ion, the yellow ball the Na ion, and the red ball the O ion. The number of Na ions represents the type of Na ions, as shown in Supplementary Fig.9.

Fig. 5|

 5| Fig. 5| Topological order. The structures at the charged states with ---(a), ---(b), and ---(c) sequences. d, one-dimensional topological order; and e, three-dimensional topological order. f,g, Capacity-voltage curves of twenty cycles for P3-and P2-NLMO in the voltage range of 1.5-4.5 V at 10 mA g -1 in a Na half-cell. h,i, Capacity-voltage curves of thirty cycles for P3-and P2-NLMO in the voltage range of 2.0-4.8 V at 10 mA g -1 in a Li half-cell. Inset shows the capacity retention over cycles.

.

  ..................................(1) For the P3-NLMO structure with an ---stack, is equal to , where q is the integer. The ODT structure of P3-NLMO with the topological order is protected on cycling, thus guaranteeing cycle stability. In contrast, the ODT structure of P2-NLMO with is broken, thus diminishing the performance. To further verify the role of topological order in enhancing the reversibility of LOR, the electrochemical properties of P3-and P2-NLMO were examined versus metallic Na or Li in Na or Li half-cell. P3-NLMO cathode in Na half-cell delivered a high discharge capacity of ~ 177 mAh g -1 within the voltage range of 1.5-4.5 V at 10 mA g -1

P2-Na 2 / 3 [

 23 Mg 0.28 Mn 0.72 ]O 2[START_REF] Maitra | Oxygen redox chemistry without excess alkali-metal ions in Na 2/3 [Mg 0.28 Mn 0.72 ]O 2[END_REF] and P2-Na 0.72[Li 0.24 Mn 0.76 ]O 2[START_REF] Rong | Anionic redox reaction-induced high-capacity and low-strain cathode with suppressed phase transition[END_REF] working in a wide voltage range (1.5-4.5 V or 2.0-4.5 V) with mixed oxygen and manganese electrochemistry, P2-NLMO and P3-NLMO with separated redox voltage are almost the perfect objects of study to investigate the LOR mechanism solely. The charge compensation mechanism in P2-and P3-NLMO was further revealed by DFT calculations. As shown in Supplementary Fig. 27a, the interlayer O-O distance was approximately 2.55 Å for the oxygen ions coordinated with two Mn 4+ ions and one Li + ion in pristine P2-NLMO. An increased partial density of states can be found close to the Fermi level for O ions because of the specific Na-O-Li configuration 12 . Here, when charged to 4.5 V, the P2 structure was transformed into a O2 structure with the decrease in the O-O distance (2.48 Å). Meanwhile, electron holes emerged at the O 2p orbital, implying oxidated O 2-ions.

  Material synthesis. The solid-state reaction method was used to obtain the P3-type and P2-type Na 0.6 [Li 0.2 Mn 0.8 ]O 2 powder samples. Na 2 CO 3 (99.9%, Alfa, 2% excess), LiOH (98%, Alfa, 2% excess), and MnO 2 (99.9%, Alfa) were ground to homogeneous mixtures, which were then calcined in an air atmosphere. The holding temperature was set to be 700 o C for P3-type Na 0.6[Li 0.2 Mn 0.8 ]O 2 , and 900 o C for P2-type Na 0.6 [Li 0.2 Mn 0.8 ]O 2 , with the heating (12 h) and cooling rate of 2 o C min -1 .

	Electrochemical measurements. The electrodes were fabricated via roller pressing the
	mixture of active material (80 wt.%), carbon nanotube (15 wt.%), and polytetrafluoroethylene
	(5 wt.%), which were then cut into isometric square pieces. CR2032 coin cells were
	assembled in Ar-filled glove box to evaluate the charge and discharge behaviors on a Land
	CT2001A battery test system (Land, Wuhan, China), with Na or Li metal foil as the counter
	electrode and glass fiber as the separator. 1 M NaClO 4 in ethylene, dimethyl carbonate,
	propylene carbonate (1:1:1 in vol.) with fluoroethylene carbonate (2 vol.%), and 1 M LiPF 6 in
	ethylene, dimethyl carbonate (1:1 in vol.) were used as the electrolyte for Na half cells and Li
	half cells, respectively.

Acknowledgements

This work was supported by Beijing Natural Science Foundation (Z190010), National Key R&D Program of China (No. 2019YFA0308500), the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDB07030200, XDA21070500), the National Natural Science Foundation of China (Grant Nos. 51672307, 51991344, 52025025, 52072400, 52002394, 51725206, 11805034, 21704105, and U1930102), Beijing Natural Science Fund-Haidian Original Innovation Joint Fund (L182056), the Basic Science Centre Program of NSFC (grant no. 51788104), and the Natural Science Foundation of Guangdong Province (2017A030313021).

Data availability

The data that support the findings detailed in this study are available in the Article and its

Supplementary Fig. 13) also revealed the ribbon-ordered structure of a TM layer with largely ---and --(-)-stacks (cyan area in Fig. 2g) and a small number of ---stacks, which was also consistent with the results of the DFT calculations. The slight stacking fault was confirmed in our previous work [START_REF] Rong | Structure-induced reversible anionic redox activity in Na layered oxide cathode[END_REF] . The Na-ion distribution (black spot in the Na layer) was also in good agreement with the theoretically predicted models, as shown in Fig. 2f and the insert of Fig. 2h. Moreover, the Na-ion locations in both materials were also verified by the refinement of ND as displayed in Supplementary Tables 1 and2. ,2g, h and Supplementary Fig. 8f[iii]). Our previous study [START_REF] Rong | Structure-induced reversible anionic redox activity in Na layered oxide cathode[END_REF] demonstrated that the characters of P3 layered structure could maintain after the initial charge for P3-NLMO, indicating that the framework of Na containing layered oxides was stable for LOR. The stable P3 phase further was verified by ex-situ XRD and STEM (Supplementary Figs. 1920212223and Supplementary Note 2). According to the electrochemistry results, during the charging process, approximately 0.4 Na + ions were removed from P3-NLMO (Fig. 1f). Fig. 4h shows the optimized desodiated phase of P3-Na 0.2 Li 0.2 Mn 0.8 O 2 with the successive migration of Li + ions into the Na layer, which retained the ---stacks. When Na-ion reinsertion occurs, the structure can return to a pristine structure with ---stacks, which guarantees the reversible Na + ion storage. The above evolution process agrees with the STEM results in that the TM layer stacks in P3-NLMO always maintained ---sequences on cycling (Fig. 3d-f), that is, the ODT structure was maintained during the entire cycling process. On comparing the evolution of the ODT structures in P2-and P3-NLMO, the topological features of the --stacks were found to provide topological protection for reversible LOR on Na-ion deintercalation in P3-NLMO, whereas the ---stacks in P2-NLMO did not.

Material characterization. The neutron data was measured at the Multiple Physics

Instrument (MPI) at China Spallation Neutron Source. Roughly 2g of sample was measured for 6 h at ambient condition. GSAS2 was used to refine the neutron data. The structure was characterized using a D8 Bruker X-ray diffractometer with Cu-K  radiation (= 1.5405 Å) in the scan range (2) of 10-80. The Na, Li and Mn ratio of samples was measured using an inductively coupled plasma mass spectrometry (ICP-MS, 7700, Agilent Technologies Inc.,USA). An ARM-200F (JEOL, Tokyo, Japan) transmission electron microscope (operated at 200 keV and equipped with double spherical aberration (Cs) correctors) was used for the spherical aberration-corrected STEM test, with both HAADF/ABF techniques. The attainable resolution of the probe defined by the objective pre-field was 78 picometers, and the ABF and HAADF images were acquired at acceptance angles of 11-22 and 90-370 mrad, respectively.

All the ABF and HAADF images in this paper were Fourier-filtered to minimize the effect of the contrast noise, which did not have effect on our measurement results. Ex-situ hXAS and ex-situ sXAS experiments were performed at beamline BL14W and BL02B02 at Shanghai Synchrotron Radiation Facility (SSRF, Shanghai, China). The ex-situ hXAS test were carried out in transmission mode with a Si(111) double-crystal monochromator, which was detuned to the 35% value of its original maximum intensity to eliminate the higher-order harmonics. [START_REF] Rong | Structure-induced reversible anionic redox activity in Na layered oxide cathode[END_REF] . Meanwhile, the valence state of Mn in P3-NLMO was proved to remain unchanged upon charge/discharge via various spectroscopic techniques [START_REF] Rong | Structure-induced reversible anionic redox activity in Na layered oxide cathode[END_REF] . Supplementary Fig. 30 shows the Mn K-edge X-ray absorption near-edge spectroscopy (XANES) results of P2-NLMO samples in different states of charge/discharge. There is no obvious absorption edge shift during charge and discharge, which indicates valence states of Mn remains unchanged. The same conclusion could also be demonstrated by Mn L-edge X-ray absorption spectroscopy as shown in Supplementary Fig. 31, showing consistent L3 and L2 edge positions. These results indicate that in the voltage range of 3.5-4.5 V (vs. Na + /Na), the charge compensation is not related with Mn redox.

The specific capacity from LOR in P3-NLMO also declined gradually with the electrochemical cycles, which has been verified in previous reports [START_REF] Du | Exploring reversible oxidation of oxygen in a manganese oxide[END_REF][START_REF] Rong | Structure-induced reversible anionic redox activity in Na layered oxide cathode[END_REF][START_REF] Wu | Dissociate lattice oxygen redox reactions from capacity and voltage drops of battery electrodes[END_REF] . The STEM images suggested that no clear TM migration or cracking occurred in the P2-and P3-NLMO samples after ten cycles (Supplementary Fig. 32). Raman spectrum of Mn-O bond in P3-NLMO (Supplementary Fig. 33) broadened and weakened at the state of charge (4.5 V), resulting from the lattice distortion induced by LOR. And the peak becomes sharp again when discharge to 3.5 V (Supplementary Fig. 33), corresponding to the reversible structure evolution. In fact, Yang et al. [START_REF] Wu | Dissociate lattice oxygen redox reactions from capacity and voltage drops of battery electrodes[END_REF] proposed that the substantial capacity loss stems from non-lattice oxygen oxidation and that the voltage decay can be attributed to an increasing Mn redox contribution on cycling in P3-NLMO. and m are the serial number of layers and variable units, respectively (Fig. 5e). The 3D topological order (R) can be described as the following matrix: ... (2) Here, the variable units within the layer can be doped element, vacancy, lattice distortion, charge, electron spin states, and so on. Supplementary Information or from the corresponding authors upon reasonable request.

Author contributions

Competing interests

The authors declare no competing interests.