Naohisa Okita

Kazuaki Kisu

Etsuro Iwama

Yuki Sakai

Yiyo Lim

Moulay Yusuke Takami

Tahar Sougrati

Thierry Brousse

Patrick Rozier

Patrice Simon

Kisu Kazuaki

Yuki Lim

Yiyo Takami

Yusuke Sougrati

Moulay Tahar

Yiyo Lim

Yusuke Takami

Moulay Tahar Sougrati

Wako Naoi

Katsuhiko Naoi

Stabilizing the Structure of LiCoPO4 Nanocrystals via Addition of Fe3+: Formation of Fe3+ Surface Layer, Creation of Diffusion-Enhancing Vacancies, and Enabling High-Voltage Battery Operation

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

■ INTRODUCTION

Growing global demands for electric and hybrid electric vehicles necessitate the development of 5 V-class lithium ion batteries (LIBs) in view of their potential for higher specific energy and voltage compared with the 4 V-class LIBs. From this perspective, LiCoPO 4 (LCP) is an interesting material that offers advantages complementary to those of conventional cathode materials such as LiCoO 2 and LiFePO 4 (LFP): high reaction potential of 4.8 V vs. Li/Li + , theoretical capacity of 167 mAh g -1 , and thermal stability thanks to the P-O covalent bonding. [1][2][3] Although the low electron conductivity (<10 -9 S cm -1 ) [START_REF] Liu | Unique synthesis of sandwiched graphene@(Li0.893Fe0.036)Co-(PO4) nanoparticles as high-performance cathode materials for lithium-ion batteries[END_REF] and Li + diffusivity (<10 -13 cm 2 s -1 ) [START_REF] Xie | Li-ion diffusion kinetics in LiCoPO4 thin films deposited on NASICON-type glass ceramic electrolytes by magnetron sputtering[END_REF] of LCP were initially thought to be problematic, it has since become clear that these difficulties can be remedied by controlling particle size/morphology, [START_REF] Rui | Olivine-Type Nanosheets for Lithium Ion Battery Cathodes[END_REF] doping metal ions, [START_REF] Fang | Design and synthesis of two-dimensional porous Fe-doped LiCoPO4 nano-plates as improved cathode for lithium ion batteries[END_REF][START_REF] Allen | Transport properties of LiCoPO4 and Fesubstituted LiCoPO4[END_REF] carbon coating, [START_REF] Ni | Carbon coated lithium cobalt phosphate for Li-ion batteries: Comparison of three coating techniques[END_REF] and/ or synthesizing composites with conductive nanocarbons. [START_REF] Liu | Unique synthesis of sandwiched graphene@(Li0.893Fe0.036)Co-(PO4) nanoparticles as high-performance cathode materials for lithium-ion batteries[END_REF][START_REF] Doan | Preparation of LiCoPO4/C nanocomposite cathode of lithium batteries with high rate performance[END_REF] A more serious obstruction to practical utilization of LCPand the primary focus of this studyis its poor cyclability: LIBs with LCP cathodes exhibit a dramatic decrease in capacity within just a few 10s of cycles. Mechanisms proposed to explain the plummeting capacity include instability of the ethylene carbonate (EC)-based electrolyte, [START_REF] Markevich | Fluoroethylene Carbonate as an Important Component in Electrolyte Solutions for High-Voltage Lithium Batteries: Role of Surface Chemistry on the Cathode[END_REF] cleavage of P-O bonds by nucleophilic attack of F -in LiPF 6 -containing electrolytes, [START_REF] Markevich | Reasons for capacity fading of LiCoPO4 cathodes in LiPF6 containing electrolyte solutions[END_REF] and instability of the delithiated phase (Li x CoPO 4 , x ≪ 1). [START_REF] Bramnik | Thermal Stability of LiCoPO4 Cathodes[END_REF][START_REF] Ehrenberg | Crystal and magnetic structures of electrochemically delithiated Li1-xCoPO4 phases[END_REF] Formation of a protective surface film on LCP using alternatives to EC-based solvents, such as fluorinated ethylene carbonate [START_REF] Markevich | Fluoroethylene Carbonate as an Important Component in Electrolyte Solutions for High-Voltage Lithium Batteries: Role of Surface Chemistry on the Cathode[END_REF] and other electrolyte additives, [START_REF] Sharabi | Electrolyte solution for the improved cycling performance of LiCoPO4/C composite cathodes[END_REF] has succeeded in prolonging useful life to 100 cycles. Allen et al. recently showed that Fe substitution into LCP in combination with a phosphite electrolyte additive DOI: 10.1021/acs.chemmater.8b01965 enhances electrochemical performance, extending cyclability to 500 cycles with 80% retention of initial capacity. [START_REF] Allen | Improved cycle life of Fesubstituted LiCoPO4[END_REF] Fe substitutionspecifically, aliovalent Fe 3+ substitution in Li + (M1) and transition metal M 2+ (M2) sites obtained by such simple air-annealing processhas been well-known for other phosphates such as LFP and LiMn 1-x Fe x PO 4 (LMFP) to cause changes in the structural characteristics (defective) and in the electrochemical signature (slope). [START_REF] Amisse | Nonstochiometry in LiFe0.5Mn0.5PO4: Structural and Electrochemical Properties[END_REF][START_REF] Hamelet | The effects of moderate thermal treatments under air on LiFePO4-based nano powders[END_REF][START_REF] Amisse | Singular Structural and Electrochemical Properties in Highly Defective LiFePO4 Powders[END_REF] In case of LCP, the Fe 3+ substitution yields structural stabilization at high potential such 4.5 V vs. Li/Li + and creation of vacancies in the Li + (M1) sites. [START_REF] Allen | Improved cycle life of Fesubstituted LiCoPO4[END_REF][START_REF] Hanafusa | Electrochemical and Magnetic Studies of Li-Deficient Li1-xCo1-xFexPO4 Olivine Cathode Compounds[END_REF] It is reported that these vacancies contribute to increased Li mobility, thus enhancing rate capability. [START_REF] Allen | Improved cycle life of Fesubstituted LiCoPO4[END_REF][START_REF] Allen | Cr and Si Substituted-LiCo0.9Fe0.1PO4: Structure, full and half Li-ion cell performance[END_REF] Hanafusa et al. demonstrated that Fe 3+ -substituted LCP can charge and discharge with negligible capacity loss (<5%) over 150 cycles even in conventional EC-based electrolyte. [START_REF] Hanafusa | Electrochemical and Magnetic Studies of Li-Deficient Li1-xCo1-xFexPO4 Olivine Cathode Compounds[END_REF] However, achieving such an excellent cyclability in Fe 3+ substituted LCP requires restricting the operating voltage range between 3.5 and 5.0 V vs. Li, thus sacrificing capacity contributed by the Fe 3+ /Fe 2+ redox below 3.5 V vs. Li. Moreover, the detailed mechanism responsible for improved cyclability, and the role played by the presence of Fe 3+ within the LCP crystal structure, remains unclear.

This study aims at elucidating the influence of Fe 3+ substitution on LCP cyclability by focusing on electrode/ electrolyte interface. Using LCP nanoparticles (100 nm in average diameter) highly dispersed within multiwalled carbon nanotube (MWCNT) matrix prepared via ultracentrifugation, [START_REF] Iwama | Ultracentrifugation: An effective novel route to ultrafast nanomaterials for hybrid supercapacitors[END_REF] we have synthesized different LCP-based materials with different Fe 3+ amounts. Electrochemical characterizations of the synthesized materials give clear evidence of improved electrochemical stability of LCP as well as enhanced rate capability, with Fe 3+ substitution. Surface analysis using X-ray photoelectron spectroscopy (XPS) and electron energy loss spectroscopy (EELS) suggest that Fe enrichment of the surface of LCP nanoparticles occurs through the oxidation of Fe 2+ into Fe 3+ along with the creation of vacancies at Co 2+ (M2) sites during the synthesis process. The Fe 3+ -rich phase on the LCP surface enhances the stability of the delithiated phase, preventing oxidative reactions with electrolytes during highvoltage operation. This surface protection persists over 1000 cycles at 0.2 C and 5000 cycles at 1.0 C as long as the electrochemical reduction of Fe 3+ is avoided by ensuring that the full range of operating voltages lies above the Fe 3+ /Fe 2+ redox potential. Additionally, we first demonstrated for LCP or LCFP materials that the vacancies in Co/Fe M2 sites are responsible for the Li + diffusivity enhancement within LCFP crystals and thus for the improvements of C-rate capability. Our findings may offer new approaches for stabilizing the structure of LCP and other high-voltage positive electrodes for use in 5 V-class Li-ion batteries.

■ EXPERIMENTAL SECTION

Materials. Co(CH 3 COO) 2 •4H 2 O (>99%, Wako Pure Chemicals), Fe(CH 3 COO) 2 (>99%, Sigma-Aldrich Corp.), CH 3 COOLi (>98.0%, Wako Pure Chemicals), and H 3 PO 4 (>85.0%, Wako Pure Chemicals) were used to prepare LiCo 0.8 Fe 0.2 PO 4 (LCFP). Citric acid (C 6 H 8 O 7 ) (>99.5%, Sigma-Aldrich Corp.) was used as a chelating agent. Multiwalled carbon nanotubes (MWCNTs) with a specific surface area of 240 m 2 g -123 were used for the preparation of carbon composite. Distilled water (17 MΩ cm) was used as a medium for the entire preparation scheme.

Preparation of LiCo 0.8 Fe 0.2 PO 4 /MWCNT Composite under UC Treatment. First, two kinds of solution (solution A and B) were prepared. Solution A was composed of 0.4000 g of MWCNT and 0.6714 g of H 3 PO 4 (1.0 equiv) mixed in 50 mL of ultrapure water under ultrasonication. Solution B was composed of 1.1725 g of Co(CH 3 COO) 2 •4H 2 O (0.8 equiv), 0.2027 g of Fe(CH 3 COO) 2 (0.2 equiv), 0.3922 g of CH 3 COOLi (1.0 equiv), and 1.1247 g of citric acid (1.0 equiv) dissolved in 20 mL of ultrapure water. Solution A was subjected to the ultracentrifugation (uc) treatment [START_REF] Iwama | Ultracentrifugation: An effective novel route to ultrafast nanomaterials for hybrid supercapacitors[END_REF] at 80°C for 5 min before and after the addition of solution B. The uc-treated sol was further dried at 130°C under vacuum for 12 h. The obtained powder made of LCFP precursor and MWCNT was preannealed at 350°C in air for 1 h to remove absorbed water and citric-acid-derived amorphous carbon, then left to cool to room temperature (RT). The preannealed powder was then fast annealed at 650°C under N 2 flow (RT to 650°C in 3 min, dwell for 5 min, 20 min to cooling to RT) to form the LCFP/MWCNT composite (pristine LCFP/ MWCNT). The pristine LCFP/MWCNT composite was further annealed at 350°C for 3 h under air in order to oxidize Fe 2+ into Fe 3+ without incinerating MWCNTs (air-annealed LCFP/MWCNT).

Physicochemical Characterization of LiCo 0.8 Fe 0.2 PO 4 / MWCNT Composite. Structure analysis on pristine and air-annealed LCFP samples was performed by X-ray diffraction [XRD, Smart-lab (Rigaku), Cu Kα radiation (λ = 1.54 Å), operating at 45 kV, 200 mA]. XRD patterns were recorded in the 10°-90°2θ range at a scan speed of 0.0033°s -1 . Time-of-flight neutron powder diffraction (NPD) measurements were performed at the iMATERIA (BL20) facility of the Japan Photon Accelerator Research Complex (J-PARC, Ibaraki, Japan) to obtain NPD patterns corresponding to 0.5-4.8 Å in dspacing. 57 Fe Mossbauer spectra were collected at room temperature in transmission geometry on a constant acceleration spectrometer using a 57 Co source in a Rh matrix. Isomer shift (IS) calibrations were performed using α-Fe as a standard at room temperature. The obtained spectrum was fitted by two Lorentzian doublets (Fe 2+ and Fe 3+ ). The LCFP/MWCNT nanostructure and the particle size were characterized by high-resolution transmission electron microscopy (HRTEM, Hitachi model H9500). Thermogravimetric analysis was performed under a synthetic air (O 2 : 20%, N 2 : 80%) using a thermogravimetric/differential thermal analyzer (TG/DTA, Seiko Instruments TG/DTA6300). X-ray photoelectron spectroscopy (XPS JEOL Ltd. JPS-9200) was carried out using Mg X-ray source. Prior to the XPS characterization, the sample electrodes were thoroughly washed by dimethyl carbonate (DMC) and dried overnight. X-ray adsorption fine structure (XAFS) measurements at the Co and Fe Kedges for the composite were performed in transmission mode at the beamline BL01 of the synchrotron radiation facility SPring-8 (Hyogo, Japan). Laminate-type two-electrode cells (pouch cells) were assembled using lithium metal foil as a negative electrode and LiCo 0.8 Fe 0.2 PO 4 /MWCNT composite as a positive electrode. The obtained XAFS spectra were analyzed using the spectral fitting software REX2000 (Rigaku Corp.) to evaluate the ratio of Co and Fe species using references selected to exhibit different oxidation states, such as Co 2+ (LiCoPO 4 bulk sample), Co 3+ (Cobalt acetylacetonate), Fe 2+ (LiFePO 4 bulk sample), and Fe 3+ (FePO 4 bulk sample oxidized by the chemical method using NO 2 BF 4 ).

Electrochemical Characterization of LiCo 0.8 Fe 0.2 PO 4 / MWCNT Composite. Half-cells were assembled using a negative Li metal electrode and a positive LiCo 0.8 Fe 0.2 PO 4 /MWCNT electrode in 2032 coin-type cells. The electrolyte was a mixture of ethylene carbonate, propyl carbonate, and dimethyl carbonate (EC: PC: DMC = 1:1:3) containing 1.0 M of lithium hexafluorophosphate (LiPF 6 ). LiCo 0.8 Fe 0.2 PO 4 /MWCNT electrode was prepared by mixing 90 wt % of the composite and 10 wt % of polyvinylidene difluoride (PVdF) in N-methyl pyrrolidone (NMP). The mixture was coated on an etched-Al foil (current collector) and dried at 80°C in vacuum for 12 h. The loading mass of the composite was 1.4 mg cm -2 on etched Al current collector (1.54 cm 2 ). Charge-discharge tests were performed in constant-current constant-voltage mode between 2.5 and 5.0 V or 4.3 and 5.0 V vs. Li/Li + at current density of 0.2 C-rate except the 1 st where 0.1 C-rate current was applied as pre-conditioning, assuming that 1C-rate equals 167 mA g -1 . The cutoff current density at the constant-voltage mode was 8.35 mA g -1 (0.05 C). 1. These values of the a and b lattice parameters and the cell volume are smaller than those of LiCoPO 4 (LCP, see Table S1), in good agreement with previous reports. [START_REF] Hanafusa | Electrochemical and Magnetic Studies of Li-Deficient Li1-xCo1-xFexPO4 Olivine Cathode Compounds[END_REF] The obtained stoichiometry of t h e a i ra n n e a l e d L C F P c r y s t a l s t r u c t u r e i s (Li 0.98 Fe 0.02 ) M1 (Co 0.76 Fe 0.13 □ 0.11 ) M2 PO 4 , where M1, M2, and □, respectively, represent Li, Co, and vacancy sites. Fe 3+ is present primarily at transition metal sites (M2), whereupon vacancies are present at the same M2 site to compensate the valence, unlike previous reports which suggests the creation of vacancies in Li (M1) sites. [START_REF] Allen | Improved cycle life of Fesubstituted LiCoPO4[END_REF][START_REF] Allen | Cr and Si Substituted-LiCo0.9Fe0.1PO4: Structure, full and half Li-ion cell performance[END_REF] One has to note that vacancies appear overestimated as the composition (Li 0.98 Fe 0.02 ) M1 (Co 0.785 Fe 0.13 ⧖ 0.085 ) M2 PO 4 , is expected to ensure charge compensation. However, this result is related to the evidence of the inhomogeneous composition of the particles as described latter. Results of Mossbauer spectroscopy for air-annealed LCFP support the NPD analysis. The obtained spectrum was best fitted by two doublets: one with an isomer shift (IS) = 0.43 mm s -1 and a quadrupole splitting (QS) = 0.83 mm s -1 , and the other with IS = 1.2 mm s -1 and QS = 2.9 mm s -1 . The former, accounting for 94% of the absorption area, corresponds to the Fe 3+ present on M1 and disordered M2 sites, [START_REF] Amisse | Nonstochiometry in LiFe0.5Mn0.5PO4: Structural and Electrochemical Properties[END_REF][START_REF] Hamelet | The effects of moderate thermal treatments under air on LiFePO4-based nano powders[END_REF] and the latter, accounting for 6% of the spectra, corresponds to Fe 2+ on M2 sites. The high ratio of Fe 3+ to Fe 2+ is due primarily to the oxidation of Fe 2+ in pristine LCFP through the air-annealing process, as pristine LCFP contains only 12% Fe 3+ (88% Fe 2+ ), as confirmed by the additional Mossbauer spectrum shown in Figure S2b. Most of the Fe 3+ oxidized from Fe 2+ was found to remain on the M2 site including also all created vacancies thus leading to highly distorted octahedral surrounding. HRTEM observations of airannealed LCFP indicate the presence of nanocrystals of ca. 100 nm highly dispersed within the MWCNT matrix (Figure 1d). The high magnification HRTEM image (Figure 1e) shows a highly crystalline bulk part of LCFP nanocrystal surrounded by an amorphous phase. The crystalline bulk part shows clear lattice fringes with interplanar lattice distances of 3.86, 5.10, and 5.92 Å, corresponding respectively to the (210), (200), and (010) planes of the olivine structure. X-ray photoelectron spectroscopy (XPS) analysis of Co 2p 3/2 and Fe 2p 3/2 energy levels was performed on air-annealed LCFP (Figure 2a,b) subjected to different Ar ion etching duration. The Co/Fe ratio was determined quantitatively by computing the areas under the Co 2p 3/2 and Fe 2p 3/2 peaks (Figure 2c). The examination of the results shows that the distribution of Fe and Co varies from the surface to the bulk of the particles. While an average Co/Fe ratio of 83/17 is calculated from NDP refinement, the XPS experiments show that at the surface (no Ar ion etching) this ratio is equal to 57.4/42.6 and progressively increases when moving toward the center of the particle (increasing Ar ion etching duration). In contrast, the Co/Fe ratio of pristine LCFP/MWCNT composite remains fixed at approximately 80/20 independent of location between surface and center of the particle (independent of Ar ion etching duration) (Figure S2). This suggests that the air-annealing process induces, beside the oxidation of the Fe and creation of Co vacancies to balance the charge, the migration of Fe 3+ ions toward the surface of the particles in the amorphous domain. Such phenomenon is confirmed by the analysis of air-annealed samples of a composite with higher Fe concentration, namely, Li-Co 0.5 Fe 0.5 PO 4 /MWCNT. Elemental mapping (Figure S4a) and line analysis (Figure S4c) of this sample confirms the existence of a Fe-rich surface layer on the nanoparticle surface, while the elemental mapping of the pristine LiCo 0.5 Fe 0.5 PO 4 shows uniform distribution of Fe, Co, O, and P within the particle (Figure S4b). This inhomogeneous distribution of Fe allows, in addition, interpreting the apparently overestimated vacancy amount as deduced from the NDP analysis. The migration of Fe 3+ is associated with equivalent migration of vacancies in the opposite ways leading to an excess of vacancies in the bulk part of the particle and an excess of Fe at the surface. As this surface is amorphous, as revealed by HRTEM, the refinement of NDP take to account only the vacancy-rich part of the particle thus explaining the apparent overestimation.

The electrochemical behavior of LCFP/MWCNT composites was investigated in half-cells versus Li anode at a rate of 0.2C (1C = 167 mA g -1 ) (Figure 3). During the charge/ discharge processes, a voltage shoulder around 3.5 V and two plateaus were observed at 4.76 and 4.83 V (Figure 3a). The shoulder corresponds to the redox reaction of Fe 2+ /Fe 3+ in LiCo 0.8 Fe 0.2 PO 4 . The two plateaus at high potentials correspond to a two-step redox reaction of Co 2+ /Co 3+ in Li 0.8 Co 0.8 Fe 0.2 PO 4 , which well agrees with previous reports. [START_REF] Hanafusa | Electrochemical and Magnetic Studies of Li-Deficient Li1-xCo1-xFexPO4 Olivine Cathode Compounds[END_REF][START_REF] Bramnik | Phase Transitions Occurring upon Lithium Insertion-Extraction of LiCoPO4[END_REF][START_REF] Kaus | Electrochemical Delithiation/Relithiation of LiCoPO4: A Two-Step Reaction Mechanism Investigated by in Situ X-ray Diffraction, in Situ X-ray Absorption Spectroscopy, and ex Situ 7Li/ 31P NMR Spectroscopy[END_REF] The air-annealed LCFP/MWCNT composites showed a high initial discharge capacity of 130 mAh g -1 (Figure 3a) and increased stability of the cycle performance with 86% capacity retention after 50 cycles compared to LCP/ MWCNT and pristine LCFP/MWCNT composites (Figure 3d). During cycling between 2.5 and 5.0 V, dQ/dV plots show an increase near the Fe 2+ /Fe 3+ redox peak at 3.5 V and a decrease near the broad redox peak at 3.4 V (Figure 3c). Such change in redox peaks suggests that Fe 3+ in the air-annealed LCFP is gradually reduced into Fe 2+ at low potentials and that the energy level of resultant Fe 3+ produced by reoxidation of Fe 2+ is different from that for Fe 3+ originally existed in the airannealed LCFP. To avoid this phenomenon, we perform charge and discharge tests of the composites with operating voltages confined to the range 4.3-5.0 V. This results in excellent cycle performance, with 99% capacity retention at 0.2C after 1000 cycles, and 85% at 1C after 5000 cycles (Figure 3e) largely exceeding the best reported one by Hanafusa et al. [START_REF] Hanafusa | Electrochemical and Magnetic Studies of Li-Deficient Li1-xCo1-xFexPO4 Olivine Cathode Compounds[END_REF] Additionally, air-annealed LCFP/MWCNT composite enabled discharge at a 100C rate with capacity of 45 mAh g -1 , larger than observed for LCP/MWCNT and pristine LCFP/ MWCNT composites (Figure 3f). Comparison of Li + diffusion coefficients (D Li+ ) between the pristine (Fe 2+ ) and airannealed LCFP (Fe 3+ ) composites shown in Figure S5 supports such improvement in C-rate capability. The obtained D Li+ for the air-annealed LCFP is higher in both charge and discharge compared to the pristine. Such results indicate that the creation of vacancies in M2 sites resulting from the oxidation/migration of Fe ions enhances the Li + diffusivity within LCFP crystal, unlike previous reports which suggest the enhancement of Li + mobility due to the vacancies in Li (M1) sites. [START_REF] Allen | Improved cycle life of Fesubstituted LiCoPO4[END_REF][START_REF] Allen | Cr and Si Substituted-LiCo0.9Fe0.1PO4: Structure, full and half Li-ion cell performance[END_REF] EIS analysis using symmetric cells [START_REF] Kisu | Internal resistance mapping preparation to optimize electrode thickness and density using symmetric cell for high-performance lithium-ion batteries and capacitors[END_REF] (see Figure S6) shows that the pristine LCFP shows higher impedance behavior than air-annealed LCFP (with Fe 3+ -rich surface). Both the charge transfer and mass transport processes were improved for Fe 3+ -rich LCFP: the relevant absolute impedance becomes roughly 1/10 for the charge transfer process and 1/5 for the diffusion process when moving from pristine LCFP to air-annealed LCFP.

HRTEM observation and X-ray photoelectron spectroscopy (XPS) analysis of the C 1s energy level were performed on airannealed LCP/MWCNT and LCFP/MWCNT composites during cycling to confirm the electrochemical behavior (Figure 4). HRTEM images of air-annealed iron-free LCP/MWCNT composites after 30 cycles between 2.5 and 5.0 V showed amorphization of LCP nanocrystals and SEI production on the surface of LCP nanocrystals and MWCNTs (Figure 4a). XPS measurements show an increase of the peak intensity for carboxyl, carbonyl and alkoxy groups (Figure 4 e) which evidence the decomposition of the electrolyte. This suggests that these degradations were caused by unstable CoPO 4 , which reacts readily with electrolyte. Normally, a stable SEI produced during charge (oxidation) should protect such CoPO 4 from reacting with electrolyte. However, SEI decomposition might occur during discharge from 5.0 to 2.5 V. Then, exposed CoPO 4 may degrade by reaction with electrolyte, yielding new SEI upon subsequent charging back to 5.0 V. This new SEI obstructs the Li insertion/deinsertion and increases the resistivity. To prevent this, we restrict the voltage range to 4.3-5.0 V to ensure protection of the SEI. The clear difference can be seen both in TEM images and XPS spectra, as neither SEI production nor LCP amorphization are observed after several cycles in the [4.3 V-5.0 V] voltage range (Figure 4b,f). In contrast, air-annealed LCFP/MWCNT composite, exhibits no electrolyte decomposition even upon cycling between 2.5 and 5.0 V (Figure 4c, d, g andh) which confirms that the Ferich surface layer contributes to stabilize the delithiated (Co 0.8 Fe 0.2 PO 4 ) structure and prevent for electrolyte decomposition. Despite that, cycling the air-annealed LCFP/ MWCNT composite between 2.5 and 5.0 V still results in decreasing capacity indicating the existence of another process. This phenomenon is similar to reported one by Paolella et al., who reported a progressive capacity decay for LiFePO 4 due to the mixing of Fe and Li cations as deduced from cell parameters evolution. [START_REF] Paolella | Cation exchange mediated elimination of the Fe-antisites in the hydrothermal synthesis of LiFePO4[END_REF] Also, the cell volume deduced from lattice parameter refinement (Figure S7) increases with increasing number of cycles in agreement with a progressive cation mixing in LCFP crystals. Differently, the refinement of the lattice parameters shows that the cell volume remains constant for the samples cycled in the high potential range to avoid the Fe 3+ /Fe 2+ redox couple (Figure S7). These results confirm that in the case of LCFP, the reduction of Fe 3+ to Fe 2+ at low voltage is also responsible for the cation mixing.

The evolution of Co and Fe valence state along cycling has been determined by X-ray absorption measurements. The Xray absorption near edge structure (XANES) spectra of Co and Fe K-edge for air-annealed LCFP/MWCNT composite before cycling and at cut off voltages of the two investigated voltage ranges (2.5-5.0 V and 4.3-5.0 V) are shown in Figure S8, while calculated average oxidation states for Co and Fe are reported in Table S3.

In the air-annealed LCFP/MWCNT composite material before cycling, valence states of Co and Fe are +2.1 and +2.9, respectively, in good agreement with previous Mossbauer spectroscopy results (Figure 1c). During cycling in the (2.5-5.0 V) voltage range, the Fe valence state changes between +2.3 and +3.0 at lower and upper cutoff voltages, respectively, and these values remain stable along cycles. The Co valence state at low cut off voltage is equal to +2.1 and remain stable along cycles while at 5.0 V at the end of the first charge it is equal to +2.8 and progressively decreases along cycles to reach +2.55 at the 30 th charged state. During cycling in the (4.3-5.0 V) voltage range, Fe valence state, as expected, increases during the first charge from +2.9 in the sample before cycling up to +3. The valence state of +3 is maintained then whatever the voltage along cycles. For the Co, the valence state is +2.1 and +2.8 at 4.3 and 5.0 V, respectively. These values are stable along cycles. This stable cycle performance confirms the specific role of the Fe 3+ which together with vacancies at M2 sites of LCFP effectively prevents irreversible changes in crystal structure. This stabilizing effect added to the previously detailed prevention of decomposition of electrolyte during charge and discharge processes, result in stable cycle performance with improved rate performance for Fesubstituted samples.

Based on the results of XRD, neutron diffraction analysis, Mossbauer, XANES, and XPS studies, we propose the schematic picture illustrated by Figure 5 for the structural changes occurring during cycling. Upon cycling, LCP as a reference material suffers both structural degradation (Li-Co cation mixing) and SEI accumulation on the particle surface, while only the latter can be restrained by a high cutoff voltage operation within the range of 4.3 and 5.0 V (Figure 5a). Fe 3+ was introduced as a partial substitute for Co in our material. It was observed that, upon air annealing, some of the introduced Fe 3+ cations migrated to the nanoparticle surface, producing vacancies at M2 sites. However, the vacancies disappeared during cycling between 2.5 and 5.0 V, due to the reduction of Fe 3+ into Fe 2+ at 3.5 V, resulting in a destabilization of the charged Co 0.8 Fe 0.2 PO 4 structure. In contrast, the Fe 3+ surfacerich structure with vacancies at M2 sites and intact SEI remains stable as long as the charge/discharge potential is confined between 4.3 and 5.0 V to avoid redox reactions of Fe 2+ /Fe 3+ . This structure results in stable cycle performance.

■ CONCLUSIONS

In conclusion, a detailed understanding of the mechanism of LCP structure stabilization by Fe 3+ substitution suggests that the Fe 3+ -rich phase formed on the LCP surface plays a significant role in yielding stable charge-discharge performance. The oxidation of Fe 2+ into Fe 3+ during air annealing of the pristine LCFP results in diffusion of Fe ions to the surface due to the increased repulsion between Fe 3+ and Co 2+ . The Fe 3+ -rich phase on the surface stabilizes the unstable delithiated phase of (Li x Fe 0.02 ) M1 (Co 0.76 Fe 0.13 ⧖ 0.11 ) M2 PO 4 (x ≪ 1), preventing both the crystal structure degradation and continuous SEI formation on the LCFP nanoparticle surface. The phase stabilization yields excellent cyclability, with 85% capacity retention over 5000 cycles at 1C and 96% capacity retention over 1000 cycles even at a slow C-rate of 0.2C. The importance of Fe 3+ retention in the crystal structure, which requires to prevent Fe 3+ reduction to Fe 2+ around 3.5 V vs. Li/ Li + , was demonstrated by a combination of XRD and XAFS analysis on samples with and without operation-voltage extending below 3.5 V vs. Li. Meanwhile, differently from previous reports, it was found that the vacancies on Co/Fe M2 sites improve the overall electrochemical performances such as the capacity and C-rate capability, thanks to the enhancement of Li + diffusivity within LCFP crystals. By demonstrating the existence of the Fe-rich phase on LCFP nanoparticles and its critical impact on electrochemical performance, this study may open the way to the design of stable 5 V cathode materials via fine-tuning of surface phenomena.

Notes

The authors declare no competing financial interest.

■ ACKNOWLEDGMENTS

This study was supported by the Global Innovation Research Organization in TUAT. This work was supported by JSPS KAKENHI grant numbers JP17K14920. Time-of-flight neutron powder diffraction measurements were performed at the iMATERIA (BL20) facility of the J-PARC (Proposal No. 2017AM0006). The synchrotron radiation experiments were performed at the BL01 of SPring-8 with the approval of the Japan Synchrotron Radiation Research Institute (JASRI) (Proposal No. 2017A1516). We are grateful to Dr. McMahon Thomas Homer Reid for helpful discussion and proofreading.

■ NOTE ADDED AFTER ASAP PUBLICATION

This article was published ASAP on September 21, 2018, with errors in Figures 1 and3-5, Table 1, and in the Supporting Information. The corrected version was reposted on September 27, 2018. attributed to MWCNT, no other peaks due to possible impurities such as Co3O4, Fe2O3, LiFePO4 and FePO4 were detected, indicating successful substitution of Fe into the LiCoPO4 (LCP) crystalline structure. Lattice parameters for the pristine LCFP were determined through Rietveld refinements on the XRD data, as shown in Table S2. Mössbauer spectroscopy for the pristine LCFP is shown in Figure S2(b) and compared with that for the air-annealing LCFP. 57 Fe Mössbauer spectra were collected in transmission geometry on a constant acceleration spectrometer using a 57 Co source in a Rh matrix operated at room temperature. Isomer shift (IS) calibrations were performed using a foil -Fe as a standard at room temperature. The obtained spectrum for the pristine LCFP was best fitted by two Lorentzian doublets: one with isomer shift (IS) = 0.51 mm s -1 and quadrupole splitting (QS) = 0.73 mm s -1 , and the other with IS = 1.2 mm s -1 and QS = 3.0 mm s -1 . The former, accounting for 12% of the full spectra, corresponds to the Fe 3+ present on M1 and disordered M2 sites 1,2 , and the latter, accounting for 88% of the spectra, corresponds to Fe 2+ on M2 sites. The drastic increase in Fe 3+ ratio from 12% to 88% indicates that the air-annealing is the main responsible process for the Fe 3+ creation in the LCFP crystal structure. Table S2 Rietveld refinement results of X-ray powder diffraction patterns from Figure S2(a) for pristine and the air-annealed LCFP / MWCNT composite.

MWCNT composite shows an Fe-rich surface layer on the nanoparticle surface which is supported by its line analysis [Figure S4(c)], where as that of the pristine in Figure S4 As shown from GITT measurement (Fig. S5), the apparent DLi+ was enhanced for the airannealed LCFP (with Fe 3+ rich surface) compared with pristine LCFP. However, GITT analysis generally represents overall diffusion kinetics of active materials, thus, it is hard to distinguish between effects of charge transfer and diffusional properties. We made additional EIS experiments on (a) pristine LCFP and (b) Fe 3+ -rich-surface LCFP at two different SOCs, namely 0 and 70% (middle range of second plateau as shown in Figure 2b). As shown in the Figure S6, the pristine LCFP shows higher impedance behavior than the air-annealed LCFP (with Fe 3+ rich surface). Both the charge transfer and mass transport processes have been improved for Fe 3+ -rich LCFP: the relevant absolute impedance becomes roughly 1/10 for the charge transfer process and 1/5 for the diffusion process when moving from pristine LCFP to air-annealed LCFP. In order to compare the cycling stability of crystal structure depending on the operation voltage ranges (2.5-5.0 V or 4.3-5.0 V), unit cell volumes for the air-annealed LCFP/MWCNT composites after 1, 10, and 30 cycles were plotted along with those for the air-annealed LCP/MWCNT. Unit cell volumes for the air-annealed LCFP operated within the high voltage range (4.3-5.0 V) remain constant, compared to those for other three samples which show clear increase because of Li-Co mixing during cycling. Such results strongly support the hypothesis that Fe 3+ and vacancies at M2 sites serve to stabilize the LCFP crystal structure. The air-annealed LCFP operated within 2.5-5.0 V, where its Fe 3+ undergoes Fe 3+ /Fe 2+ redox [see Figure 3 (c) in the main text], cannot resist its initial volume increase. Meanwhile, unit cell volume for the air-annealed LCP continuously increase regardless of its operation voltage range.

Figure 1 .

 1 Figure 1. Structural characterization of air-annealed LiCo 0.8 Fe 0.2 PO 4 (LCFP)/ MWCNT composite. (a) XRD patterns and (b) neutron powder diffraction patterns for air-annealed LCFP/MWCNT composite. (c) Mossbauer spectrum fitted with two Fe environments (Fe 2+ and Fe 3+ ). (d) Lower-magnification (20 000×) higher-resolution TEM image for highly dispersed LCFP nanocrystalline within the MWCNT matrix. (e) Magnified (400 000×) HRTEM image of the LCFP nanocrystalline. Clear lattice fringes of d = 3.86, 5.10, and 5.92 Å appear, corresponding to the (210), (200), and (010) planes of LCFP, respectively.

Figure 3 .

 3 Figure 3. (a) Constant-current charge/discharge curves of a half-cell consisting of Li/1 M LiPF 6 EC:PC:DMC(vol. 1:1:3)/(air-annealed LCFP/ MWCNT composite) at a 0.2C rate between 2.5 and 5.0 V. (b) Constant-current charge/discharge curves of a half-cell consisting of Li/1 M LiPF 6 EC:PC:DMC(vol. 1:1:3)/(air-annealed LCFP/MWCNT composite) at a 0.2C rate between 4.3 and 5.0 V. (c) dQ/dV plot for air-annealed LCFP/MWCNT composite from panel a. (d) Cycle performance for the composites at a 0.2C rate. (e) Cycle performances for the air-annealed LCFP/MWCNT composites of panel b. (f) Rate performance for the composites over the range 2.5-5.0 V.

Figure 4 .

 4 Figure 4. Higher-resolution TEM images of air annealed (a), (b) LCP/MWCNT and (c), (d) LCFP/MWCNT composites after 30 cycles. XPS measurements of air annealed (e)(f) LCP/MWCNT and (g)(h) LCFP/MWCNT composites after 1, 10, and 30 cycles. (a), (c), (e), and (g) are conducted in potential range between 2.5 and 5.0 V. (b), (d), (f), and (h) are conducted in potential range between 4.3 and 5.0 V.

Figure 5 .

 5 Figure 5. Structural changes of synthesized (a) LCP and (b) LCFP during charge and discharge.

■

  ASSOCIATED CONTENT * S Supporting Information The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.chemmater.8b01965. Additional experimental data: Neutron powder diffraction, XRD, Mossbauer spectroscopy, XPS, XAFS and elemental mapping line analysis for air-annealed LCP/ MWCNT, LCFP/MWCNT and pristine LCFP/ MWCNT composites (PDF) ■ AUTHOR INFORMATION Corresponding Authors *E-mail: iwama@cc.tuat.ac.jp. *E-mail: k-naoi@cc.tuat.ac.jp. ORCID Patrice Simon: 0000-0002-0461-8268 Katsuhiko Naoi: 0000-0002-0265-2235

Figure

  Figure S2 Structural characterization of pristine and the air-annealed LCFP / MWCNT composites. (a) XRD patterns for LCFP /MWCNT composites. (b) Mössbauer spectrum fitted with two Fe environments (Fe 2+ and Fe 3+ ).

  (b) shows uniform distribution of Fe, Co, O, and P within the particle.

Figure

  Figure S4 Elemental mapping line analysis of (a)air annealed LiCo 0.5 Fe 0.5 PO 4 / MWCNT and (b) pristine LiCo 0.5 Fe 0.5 PO 4 / MWCNT composites, (c)line analysis of the air annealed LiCo 0.5 Fe 0.5 PO 4 / MWCNT composite [shown in white frame in Figure S4(a)] obtained from electron energy loss spectroscopy.

Figure S5 .

 S5 Figure S5. Plots of Li diffusion coefficients vs. state of charge/discharge calculated from GITT curves during (a) charge and (b) discharge for the pristine and air-annealed LCFP. GITT was conducted with 0.1C (1C = 167 mAh g -1 per active material) for 20 min. and rest time for 1h in the potential range of between 4.3-5.0 V vs. Li/Li + . Before GITT measurements, a pre-cycling was conducted at 0.1C in the same potential range as GITT.

Figure

  Figure S6 Impedance spectroscopy on (a) pristine LCFP and (b) air-annealed LCFP with Fe 3+rich-surface at two different SOCs, namely, 0 and 70% (middle range of second plateau). The measurement was made on symmetric cells 3 comprising of both LCFP or LCFP (Fe 3+ -rich) electrodes. Perturbation amplitude was 10 mV (peak to peak). Frequency ranges was 10 mHz to 65 kHz.

Table 1 .

 1 Rietveld Refinement Results of Neutron Powder Diffraction Patterns for Air-Annealed LCFP/MWCNT Composite a

	space group		a/Å		b/Å	c/Å		v/Å 3
	Pnma		10.1592(7)		5.9114(4)	4.7091(3)		282.80(3)
		atom	Wyckoff	x/a	y/b	z/c	B ISO	occupancy
	M1 site	Li +	4a	0	0	0	2.6(2)	0.984(7)
	(Li site)	Fe 3+	4a	0	0	0	2.6(2)	0.024(1)
	M2 site	Co 2+	4c	0.2763(2)	0.25	0.9892(4)	1.19(5)	0.762(2)
	(Co site)	Fe 3+	4c	0.2763(2)	0.25	0.9892(4)	1.19(5)	0.129(2)
		P	4c	0.0925(2)	0.25	0.4200(2)	0.89(2)	1
		O -	4c	0.10209(9)	0.25	0.7443(2)	0.84(2)	1
		O -	4c	0.45393(9)	0.25	0.1982(2)	1.13(2)	1
		O -	8d	0.16820(8)	0.0466(2)	0.2817(2)	1.09(2)	1

a R wp = 5.76, R p = 4.49, R e = 2.34, X 2 = 6.07

Figure 2. Consecutive XPS measurements with data acquisition at 5 s intervals during 0-30 s Ar ion etching, yielding a depth profile of airannealed LCFP/MWCNT composite: (a) Co 2p 3/2 peak and Co satellite peak, (b) Fe 2p 3/2 peak and Co LMM Auger electron peak, (c) atomic ratio of Co and Fe calculated from Co 2p 3/2 and Fe 2p 3/2 peaks.

Supporting Information

Stabilizing the structure of LiCoPO 4 nanocrystals via addition of Fe 3+ : Formation of Fe 3+ surface layer, creation of diffusion-enhancing vacancies, and enabling high-voltage battery operation Naohisa Okita, 1 Kazuaki Kisu, 1, 2 Etsuro Iwama, *, 1, 2 Yuki Sakai, 1 Yiyo Lim, 1 Yusuke Takami, 1 Moulay Tahar Sougrati, [START_REF] Liu | Unique synthesis of sandwiched graphene@(Li0.893Fe0.036)Co-(PO4) nanoparticles as high-performance cathode materials for lithium-ion batteries[END_REF][START_REF] Xie | Li-ion diffusion kinetics in LiCoPO4 thin films deposited on NASICON-type glass ceramic electrolytes by magnetron sputtering[END_REF] Thierry Brousse, 2,[START_REF] Liu | Unique synthesis of sandwiched graphene@(Li0.893Fe0.036)Co-(PO4) nanoparticles as high-performance cathode materials for lithium-ion batteries[END_REF][START_REF] Rui | Olivine-Type Nanosheets for Lithium Ion Battery Cathodes[END_REF] Patrick Rozier, 2,[START_REF] Liu | Unique synthesis of sandwiched graphene@(Li0.893Fe0.036)Co-(PO4) nanoparticles as high-performance cathode materials for lithium-ion batteries[END_REF][START_REF] Fang | Design and synthesis of two-dimensional porous Fe-doped LiCoPO4 nano-plates as improved cathode for lithium ion batteries[END_REF] Patrice Simon, 2,[START_REF] Liu | Unique synthesis of sandwiched graphene@(Li0.893Fe0.036)Co-(PO4) nanoparticles as high-performance cathode materials for lithium-ion batteries[END_REF][START_REF] Fang | Design and synthesis of two-dimensional porous Fe-doped LiCoPO4 nano-plates as improved cathode for lithium ion batteries[END_REF] Wako Naoi, [START_REF] Allen | Transport properties of LiCoPO4 and Fesubstituted LiCoPO4[END_REF] and Katsuhiko Naoi *, 1, 2, 3 1 Department of Applied Chemistry, 2 Global Innovation Research Organization, and 3

0002, Japan

* E-mail: iwama@cc.tuat.ac.jp / k-naoi@cc.tuat.ac.jp

Neutron powder diffraction analysis on the air-annealed LiCoPO 4 (LCP) / Multiwalled carbon nanotube (MWCNT) composite.

To support the structure analysis on the air-annealed LiCo0.8Fe0.2PO4 (LCFP), neutron powder diffraction (NPD) of the air-annealed LiCoPO4 (LCP) was recorded. The synthesis procedure of the air-annealed LCP was as same as described in the main text without an use of Fe species (Fe(CH3COO)2). Time-of-flight neutron powder diffraction (NPD) measurements were performed at the iMATERIA (BL20) facility of the Japan Photon Accelerator Research Complex (J-PARC, Ibaraki, Japan) to obtain NPD patterns corresponding to 0. S1. These values of the a and b lattice parameters and the cell volume are larger than those of LCFP (see Table 1 in the main text). The atomic-position analysis on LCP can be well fitted without vacancies in the synthesized crystal structure unlike the air-annealed LCFP.

Figure S1 Neutron powder diffraction patterns for the air-annealed LCP / MWCNT composite.

Table S1 Rietveld refinement results of neutron powder diffraction patterns from Figure S1 for the air-annealed LCP / MWCNT composite.

X-ray diffraction and room temperature Mössbauer spectroscopy on the LCFP/MWCNT with and without the air-annealing process.

Mössbauer spectroscopy was conducted on the pristine LCFP/MWCNT, in order to verify the idea that the air-annealing of the pristine LCFP/MWCNT is the responsible process to create Fe 3+ in the structure. To begin with, X-ray diffraction (XRD) pattern of the pristine LCFP crystal was recorded.

Structure analysis on pristine and air-annealed LCFP samples was performed by X-ray diffraction


Marcella Bonazzoli
email: marcella.bonazzoli@unice.fr

Victorita Dolean
email: victorita.dolean@strath.ac.uk

Ivan G Graham
email: i.g.graham@bath.ac.uk

Euan A Spence
email: e.a.spence@bath.ac.uk

Pierre-Henri Tournier
email: tournier@ljll.upmc.fr

A two-level domain-decomposition preconditioner for the time-harmonic Maxwell's equations

Introduction

The construction of fast iterative solvers for the indefinite time-harmonic Maxwell's system at mid-to high-frequency is a problem of great current interest. Some of the difficulties that arise are similar to those encountered in the case of the mid-to high-frequency Helmholtz equation. Here we investigate how domain-decomposition (DD) solvers recently proposed for the Helmholtz equation work in the Maxwell case.

The idea of preconditioning discretisations of the Helmholtz equation with discretisations of the corresponding problem with absorption was introduced in [START_REF] Erlangga | On a class of preconditioners for solving the Helmholtz equation[END_REF]. In Graham et al. [2017a], a two-level domaindecomposition method was proposed that uses absorption, along with a wavenumber dependent coarse space correction. Note that, in this method, the choice of absorption is motivated by the analysis in both Graham et al. [2017a] and the earlier work [START_REF] Gander | Applying GMRES to the Helmholtz equation with shifted Laplacian preconditioning: what is the largest shift for which wavenumber-independent convergence is guaranteed?[END_REF].

Our aim is to extend these ideas to the time-harmonic Maxwell's equations, both from the theoretical and numerical points of view. These results will appear in full in the forthcoming paper [START_REF] Bonazzoli | Domain Decomposition preconditioning for the high-frequency time-harmonic Maxwell equations with absorption[END_REF].

Our theory will apply to the boundary value problem (BVP)

∇ × (∇ × E) -(k 2 + iκ)E = J in Ω E × n = 0 on Γ := ∂Ω (1)
where Ω is a bounded Lipschitz polyhedron in R 3 with boundary Γ and outward-pointing unit normal vector n, k is the wave number, and J is the source term. The PDE in (1) is obtained from Maxwell's equations by as-suming that the electric field E is of the form E(x, t) = ℜ(E(x)e -iωt ), where ω > 0 is the angular frequency. The boundary condition in ( 1) is called Perfect Electric Conductor (PEC) boundary condition. The parameter κ dictates the absorption/damping in the problem; in the case of a conductive medium, κ = kσZ, where σ is the electrical conductivity of the medium and Z the impedance. If σ = 0, the solution is not unique for all k > 0 but a sufficient condition for existence of a solution is ∇ • J = 0. We will also give numerical experiments for the BVP (1) where the PEC boundary condition is replaced by an impedance boundary condition, i.e. the BVP

∇ × (∇ × E) -(k 2 + iκ)E = J in Ω (∇ × E) × n -i k n × (E × n) = 0 on Γ := ∂Ω (2)
In contrast to the PEC problem, the solution of the impedance problem is unique for every k > 0. There is large interest in solving (1) and ( 2) both when κ = 0 and when κ = 0. We will consider both these cases, in each case constructing preconditioners by using larger values of κ. Indeed, a higher level of absorption makes the problems involved in the preconditioner definition more "elliptic" (in a sense more precisely explained in [START_REF] Bonazzoli | Domain Decomposition preconditioning for the high-frequency time-harmonic Maxwell equations with absorption[END_REF]), thus easier to solve. Note that the absorption cannot increase too much, otherwise the problem in the preconditioner is "too far away" from the initial problem.

Variational formulation and discretisation

Let H 0 (curl;

Ω) := {v ∈ L 2 (Ω), ∇ × v ∈ L 2 (Ω), v × n = 0}.
We introduce the k-weighted inner product on H 0 (curl; Ω):

(v, w) curl,k = (∇ × v, ∇ × w) L 2 (Ω) + k 2 (v, w) L 2 (Ω) .
The standard variational formulation of (1) is:

Given J ∈ L 2 (Ω), κ ∈ R and k > 0, find E ∈ H 0 (curl; Ω) such that a κ (E, v) = F (v) for all v ∈ H 0 (curl; Ω), (3)
where

a κ (E, v) := Ω ∇ × E • ∇ × v -(k 2 + iκ) Ω E • v (4)
and F (v) := Ω J • v. When κ > 0, it is well-known that the sesquilinear form is coercive (see, e.g., [START_REF] Bonazzoli | Domain Decomposition preconditioning for the high-frequency time-harmonic Maxwell equations with absorption[END_REF] and the references therein) and so existence and uniqueness follow from the Lax-Milgram theorem. Nédélec edge elements are particularly suited for the approximation of electromagnetic fields. They provide a conformal discretisation of H(curl, Ω), since their tangential component across faces shared by adjacent tetrahedra of a simplicial mesh T h is continuous. We therefore define our approximation space V h ⊂ H 0 (curl; Ω) as the lowest-order edge finite element space on the mesh T h with functions whose tangential trace is zero on Γ . More precisely, over each tetrahedron τ , we write the discretised field as E h = e∈τ c e w e , a linear combination with coefficients c e of the basis functions w e associated with the edges e of τ , and the coefficients c e will be the unknowns of the resulting linear system. The Galerkin method applied to the variational problem (3) is

find E h ∈ V h such that a κ (E h , v h ) = F (v h ) for all v h ∈ V h .
(5)

The Galerkin matrix A κ is defined by (A κ ) ij := a κ (w ei , w ej ) and the Galerkin method is then equivalent to solving the linear system A κ U = F, where F i := F (w ei ) and U j := c ej .

Domain decomposition

To define appropriate subspaces of V h , we start with a collection of open subsets { Ω ℓ : ℓ = 1, . . . , N } of R d of maximum diameter H sub that form an overlapping cover of Ω, and we set

Ω ℓ = Ω ℓ ∩ Ω.
Each Ω ℓ is assumed to be non-empty and is assumed to consist of a union of elements of the mesh T h . Then, for each ℓ = 1, . . . , N , we set

V ℓ := V h ∩ H 0 (curl, Ω ℓ ),
where H 0 (curl, Ω ℓ ) is considered as a subset of H 0 (curl; Ω) by extending functions in H 0 (curl, Ω ℓ ) by zero, thus the tangential traces of elements of V ℓ vanish on the internal boundary ∂Ω ℓ \Γ (as well as on ∂Ω ℓ ∩ Γ ). Thus a solve of the Maxwell problem (3) in the space V ℓ involves a PEC boundary condition on ∂Ω ℓ (including any external parts of ∂Ω ℓ ). When κ = 0, such solves are always well-defined by uniqueness of the solution of the BVP (1). Let I h be the set of interior edges of elements of the triangulation; this set can be identified with the degrees of freedom of V h . Similarly, let I h (Ω ℓ ) be the set of edges of elements contained in (the interior of) Ω ℓ (corresponding to degrees of freedom on those edges). We then have that

I h = ∪ N ℓ=1 I h (Ω ℓ )
. For e ∈ I h (Ω ℓ ) and e ′ ∈ I h , we define the restriction matrices (R ℓ ) e,e ′ := δ e,e ′ . We will assume that we have matrices (D ℓ ) N ℓ=1 satisfying

N ℓ=1 R T ℓ D ℓ R ℓ = I; (6)
such matrices (D ℓ ) N ℓ=1 are called a partition of unity.

For two-level methods we need to define a coarse space. Let {T H } be a sequence of shape-regular, tetrahedral meshes on Ω, with mesh diameter H. We assume that each element of T H consists of the union of a set of fine grid elements. Let I H be an index set for the coarse mesh edges. The coarse basis functions {w H e } are taken to be Nédélec edge elements on T H with zero tangential traces on Γ . From these functions we define the coarse space V 0 := span{w H ep : p ∈ I H }, and we define the "restriction matrix"

(R 0 ) pj := ψ ej (w H ep )= ej w H ep • t, j ∈ I h , p ∈ I H , (7)
where ψ e are the degrees of freedom on the fine mesh.

With the restriction matrices (R ℓ ) N ℓ=0 defined above, we define

A κ,ℓ := R ℓ A κ R T ℓ , ℓ = 0, . . . , N
For ℓ = 1, . . . , N , the matrix A κ,ℓ is then just the minor of A κ corresponding to rows and columns taken from I h (Ω ℓ ). That is A κ,ℓ corresponds to the Maxwell problem on Ω ℓ with homogeneous PEC boundary condition on ∂Ω ℓ \Γ . The matrix A κ,0 is the Galerkin matrix for the problem (1) discretised in V 0 . In a similar way as for the global problem it can be proven that matrices A κ,ℓ , ℓ = 0, . . . , N , are invertible for all mesh sizes h and all choices of κ = 0.

In this paper we consider two-level preconditioners, i.e. those involving both local and coarse solves, except if '1-level' is specified in the numerical experiments. The classical two-level Additive Schwarz (AS) and Restricted Additive Schwarz (RAS) preconditioners for A κ are defined by

M -1 κ,AS := N ℓ=0 R T ℓ A -1 κ,ℓ R ℓ M -1 κ,RAS := N ℓ=0 R T ℓ D ℓ A -1 κ,ℓ R ℓ . (8)
In the numerical experiments we will also consider two other preconditioners: (i) M -1 κ,ImpRAS , which is similar to M -1 κ,RAS , but the solves with A κ,ℓ are replaced by solves with matrices corresponding to the Maxwell problem on Ω ℓ with homogeneous impedance boundary condition on ∂Ω ℓ \Γ , and (ii) the hybrid version of RAS

M -1 κ,HRAS := (I -ΞA κ ) N ℓ=1 R T ℓ D ℓ A -1 κ,ℓ R ℓ (I -A κ Ξ) + Ξ, Ξ = R T 0 A -1 κ,0 R 0 .
(9) In a similar manner we can define M -1 κ,HAS , M -1 κ,ImpHRAS , the hybrid versions of AS and ImpRAS.

Theoretical results

The following result is the Maxwell-analogue of the Helmholtz-result in [Graham et al., 2017b, Theorem 5.6] and appears in [START_REF] Bonazzoli | Domain Decomposition preconditioning for the high-frequency time-harmonic Maxwell equations with absorption[END_REF]. We state a version of this result for κ ∼ k 2 , but note that [START_REF] Bonazzoli | Domain Decomposition preconditioning for the high-frequency time-harmonic Maxwell equations with absorption[END_REF] contains a more general result that, in particular, allows for smaller values of the absorption κ.

Theorem 1 (GMRES convergence for left preconditioning with κ ∼ k 2 ). Assume that Ω is a convex polyhedron. Let C k be the matrix representing the (•, •) curl,k inner product on the finite element space V h in the sense that if v h , w h ∈ V h with coefficient vectors V, W then

(v h , w h ) curl,k = V, W C k . ( 10
)
Consider the weighted GMRES method where the residual is minimised in the norm induced by C k . Let r m denote the mth residual of GMRES applied to the system A κ , left preconditioned with M -1 κ,AS . Then

r m C k r 0 C k 1 -1 + H δ 2 -2 m/2 , (11)
provided the following condition holds:

max {kH sub , kH} ≤ C 1 1 + H δ 2 -1 . ( 12
)
where H sub and H are the typical diameters of a subdomain and of the coarse grid, δ denotes the size of the overlap, and C 1 is a constant independent of all parameters.

As a particular example we see that, provided κ ∼ k 2 , H ∼ H sub ∼ k -1 and δ ∼ H ("generous overlap"), then GMRES will converge with a number of iterations independent of all parameters. This property is illustrated in the numerical experiments in the next section. A result analogous to Theorem 1 for right-preconditioning appears in [START_REF] Bonazzoli | Domain Decomposition preconditioning for the high-frequency time-harmonic Maxwell equations with absorption[END_REF].

Numerical results

In this section we will perform several numerical experiments in a cube domain with PEC boundary conditions (Experiments 1-2) or impedance boundary conditions (Experiments 3-4). The right-hand side is given by J = [f, f, f ], where f = -exp(-400((x -0.5) 2 + (y -0.5) 2 + (z -0.5) 2 )).

We solve the linear system with GMRES with right preconditioning, starting with a random initial guess, which ensures, unlike a zero initial guess, that all frequencies are present in the error; the stopping criterion, with a tolerance of 10 -6 , is based on the relative residual. The maximum number of iterations allowed is 200. We consider a regular decomposition into subdomains (cubes), the overlap for each subdomain is of size O(2h) (except in Experiment 1, where we take generous overlap) in all directions. All the computations are done in FreeFem++, an open source domain specific language (DSL) specialised for solving BVPs with variational methods (http://www.freefem.org/ff++/). The code is parallelised and run on the TGCC Curie supercomputer and the CINES Occigen supercomputer. We assign each subdomain to one processor. Thus in our experiments the number of processors increases if the number of subdomains increases. To apply the preconditioner, the local problems in each subdomain and the coarse space problem are solved with a direct solver (MUMPS on one processor). In all the experiments the fine mesh diameter is h ∼ k -3/2 , which is believed to remove the pollution effect.

In our experiments we will often choose H sub ∼ H and our preconditioners are thus determined by choices of H and κ, which we denote by H prec and κ prec . The absorption parameter of the problem to be solved is denoted κ prob . The coarse grid problem is of size ∼ H -2 prec and there are ∼ H -2 prec local problems of size (H prec /h) 2 (case H sub ∼ H). In the tables of results, n denotes the size of the system being solved, n CS the size of the coarse space, the figures in the tables denote the GMRES iterations corresponding to a given method (e.g. #AS is the number of iterations for the AS preconditioner), whereas Time denotes the total time (in seconds) including both setup and GMRES solve times. For some of the experiments we compute (by linear least squares) the approximate value of γ so that the entries of this column grow with k γ . We also compute ξ so that the entries of the column grow with n ξ (here ξ = γ • 2/9, because n ∼ (h 3/2 ) 3 = k 9/2 ).

Experiment 1. The purpose of this experiment is to test the theoretical result which says that even with AS (i.e. when solving PEC local problems), provided H ∼ H sub ∼ k -1 , δ ∼ H (generous overlap), κ prob = κ prec = k 2 , the number of GMRES iterations should be bounded as k increases. In Table 1 we compare three two-level preconditioners: additive Schwarz, restricted additive Schwarz, and the hybrid version of restricted additive Schwarz. Note that in theory we would expect AS to be eventually robust, although its inferiority compared to the other methods is to be expected Graham et al. [2017a].

Experiment 2. In this experiment (Table 2) we set κ prob = κ prec = k 2 and H ∼ H sub ∼ k -0.8 and the overlap is O(2h) in all directions. As we are not in the case H prec ∼ k -1 and we do not have generous overlap, we do not expect a bounded number of iterations here. Nevertheless, the method still performs well. Not surprisingly, the best method is ImpHRAS, as better transmission conditions at the interfaces between subdomains are used in the preconditioner. It is important to note that the time is growing very much slower than the dimension of the problem being solved.

Experiment 3 In this case we take κ prob = k. Moreover, we take impedance boundary conditions on ∂Ω. We take H ∼ H sub ∼ k -α , κ prec = k β , and we use ImpHRAS as a preconditioner.

In Table 3 on the bottom we see that the dimension of the coarse space is

n CS = (k -0.8 ) -3 = k 2.4 = O(n 0.5 ).
This is reflected in the γ and ξ figures in the n CS column. For this method the reduction factor n CS /n is substantial (about 3.9 × 10 -4 when k = 40).

The computation time grows only slightly faster than the dimension of the coarse space, showing (a) weak scaling and (b) MUMPS is still performing close to optimally for Maxwell systems of size 5 × 10 4 . Iteration numbers are growing with about n 0.3 at worst. Note that the iteration numbers may be improved by separating the coarse grid size from the subdomain size, making the coarse grid finer and the subdomains bigger.

Experiment 4. Here we solve the pure Maxwell problem without absorption, i.e. κ prob = 0, with impedance boundary conditions on ∂Ω. In the preconditioner we take κ prec = k. Results are given in Table 4, where H sub ∼ k -α , H ∼ k -α ′ . These methods are close to being load balanced in the sense that the coarse grid and subdomain problem size are very similar when α + α ′ = 3/2.

Out of the methods tested, the 2-level method (ImpHRAS) with (α, α ′ ) = (0.6, 0.9) gives the best iteration count, but is more expensive. The method (α, α ′ ) = (0.7, 0.8) is faster but its iteration count grows more quickly, so its advantage will diminish as k increases further. For (α, α ′ ) = (0.6, 0.9) the coarse grid size grows with O(n 0.64 ) while the time grows with O(n 0.65 ). For (α, α ′ ) = (0.7, 0.8) the rates are O(n 0.54 ) and O(n 0.69 ). The subdomain problems are solved on individual processors so the number of processors used grows as k increases. In the current implementation a sequential direct solver on one processor is used to factorize the coarse problem matrix, which is clearly a limiting factor for the scalability of the algorithm. The timings could be significantly improved by using a distributed direct solver, or by adding a further level of domain decomposition for the coarse problem solve.

Table 1 δ

 1 ∼ H (generous overlap), H ∼ H sub ∼ k -1 , κ prob = κprec = k 2 .

			k	n	N sub n CS	#AS #RAS #HRAS
			10 4.6 ×10 5 1000 7.9×10 3 53	26	12
			15 1.5 ×10 6 3375 2.6×10 4 59	28	12
			20 1.2 ×10 7 8000 6.0×10 4 76	29	17
	k	n	N sub n CS		#RAS (#HRAS) #ImpRAS (#ImpHRAS) Time ImpHRAS
	10 3.4 × 10 5 216 1.9×10 3	34 (23)	27 (20)	11.0
	20 7.1 × 10 6 1000 7.9×10 3	43 (31)	35 (28)	42.6
	30 4.1 × 10 7 3375 2.6×10 4	47 (34)	39 (32)	100.9
	40 1.3 × 10 8 6859 5.1×10 4	49 (36)	42 (35)	264.5
	γ	4.5			2.23

Table 2 δ

 2 ∼ 2h, H ∼ H sub ∼ k -0.8 , κ prob = κprec = k 2 .

			α = 0.6					α = 0.8
	k	n	N sub	n CS	#2-level	n	N sub	n CS	#2-level
	10 2.6 × 10 5 27 2.8×10 2	31		3.4 × 10 5 216 1.8 × 10 3	29
	20 6.3 × 10 6 216 1.9 × 10 3	87		7.1 × 10 6 1000 7.9 × 10	60
	30 3.3 × 10 7 343 2.9 × 10 3	148		4.1 × 10 7 3375 2.5 × 10 4	90
	40 1.1 × 10 8 729 5.9 × 10 3	200		1.3 × 10 8 6859 5.1 × 10 4	154
								β = 1		β = 2
		k	n	N sub	n CS		#2-level(Time) #2-level(Time)
		10 3.4 × 10 5 216 1.8 × 10 3		29 (12.9)		37 (13.1)
		20 7.1 × 10 6 1000 7.9 × 10 3		60 (63.7)		70 (69.8)
		30 4.1 × 10 7 3375 2.5 × 10 4		90 (200.4)	101 (221.2)
		40 1.3 × 10 8 6859 5.1 × 10 4	154 (771.7)	137 (707.6)
		γ	4.5		2.4			1.2 (2.9)		0.94 (2.8)
		ξ	1.0		0.5			0.3 (0.6)		0.2 (0.6)

Table 3 κ

 3

prob = k, δ ∼ 2h, H ∼ H sub ∼ k -α , κprec = k β ; Top: β = 2, α = 0.6, 0.8; Bottom: α = 0.8, β = 1, 2.

Acknowledgement This work has been supported in part by the French National Research Agency (ANR), project MEDIMAX, ANR-13-MONU-0012.