Jacky Even

Laurent Pedesseau

Jean-Marc Jancu

Claudine Katan

Importance of Spin-Orbit Coupling

published or not. The documents may come

associated to a spin-orbit split-off band related to a triply degenerated conduction-band of the cubic lattice without SOC. It is shown that, due to the strong SOC, the electronic states involved in the optical absorption are only slightly perturbed by local distortions of the lattice. In addition, band offset calculations confirm that CH 3 NH 3 PbX 3 /TiO 2 (X=Br, I) is a reference material for driving electrons toward the electrode in dye-sensitized solar cells. Two dimensional hybrid perovskites are also suggested to reach further flexibility for light conversion efficiency. This study is a major step towards the understanding of the optoelectronic properties of these novel class of DSSC at the level of knowledge already achieved in the field of conventional semiconductors.

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Hybrid materials, optoelectronics, titanateoxyde, density functional theory Photovoltaic (PV) solar electricity is one of the key technologies of the 21st century to reduce the world's reliance on fossil fuels for energy generation. Reduced costs and higher conversion efficiencies are of crucial importance to make PV-based technologies economically more competitive. The quest for quality and performances of future solar cells has attracted a vast research effort over the last decade in the field of semiconductor heterostructures, nanostructured materials, and thin films. Various approaches ranging from high-cost/high-performance III-V technologies, multiple junctions and concentrator systems, to low-cost thin-films technologies [1][2][3][4] have been investigated. Obviously, the design of novel and/or efficient PV devices requires a realistic modeling of underlying material's properties including chemical composition, mechanical, electrical and optical features. This can be gained with state-of-the-art ab-initio approaches. In addition, such knowledge is desirable to reach PV cells composed of earthabundant elements based materials.

Three-dimensional organic/inorganic perovskites, based on relatively small organic cations, have recently been suggested as a novel class of materials for dye-sensitized solar cells (DSSC) with improved photo-conversion efficiency. [START_REF] Kojima | Organometal Halide Perovskites as Visible-Light Sensitizers for Photovoltaic Cells[END_REF][START_REF] Im | 6.5% efficient perovskite quantum-dot-sensitized solar cell[END_REF][START_REF] Lee | Efficient Hybrid Solar Cells Based on Meso-SuperstructuredOrganometal Halide Perovskites[END_REF][START_REF] Etgar | Mesoscopic CH 3 NH 3 PbI 3 /TiO 2 Heterojunction Solar Cells[END_REF][START_REF] Kim | Lead Iodide Perovskite Sensitized All-Solid-State Submicron Thin Film Mesoscopic Solar Cell with Efficiency Exceeding 9%[END_REF][START_REF] Heo | Efficient inorganic-organic hybrid heterojunction solar cells containing perovskite compound and polymeric hole conductors[END_REF][START_REF] Ball | Low-temperature processed mesosuperstructured to thin-film perovskite solar cells[END_REF][START_REF] Baikie | Synthesis and crystal chemistry of the hybrid perovskite (CH 3 NH 3 ) PbI 3 for solid-state sensitised solar cell applications[END_REF][START_REF] Qiu | Allsolid-state hybrid solar cells based on a new organometal halide perovskite sensitizer and one-dimensional TiO 2 nanowire arrays[END_REF][START_REF] Noh | Chemical Management for Colorful, Efficient, and Stable Inorganic-Organic Hybrid Nanostructured Solar Cells[END_REF][START_REF] Mosconi | First Principles Modeling of Mixed Halide OrganometalPerovskites for Photovoltaic Applications[END_REF][START_REF] Park | OrganometalPerovskite Light Absorbers Toward a 20% EfficiencyLow-Cost Solid-State Mesoscopic Solar CellJ[END_REF][START_REF] Burschka | Sequential deposition as a route to high-performance perovskitesensitized solar cells[END_REF] Indeed, with a nanoporous TiO 2 electrode, ultra-high light power conversion efficiencies have been demonstrated. Compared to alternative strategies based on inorganic semiconductor quantum dots (QDs) or extremely thin absorbers coated upon the internal surface of a mesoporous TiO 2 electrode, hybrid perovskites offer several benefits. In particular, the ease of synthesis, tailoring of the optical absorption by chemical substitution, electronic transport, and high stability in dry air, are among the most featured properties of these systems. Interestingly, whereas conduction-band and valence-band alignments between absorbers and TiO 2 are of crucial importance in understanding charge transfer and charge transport, [START_REF] Lee | Efficient Hybrid Solar Cells Based on Meso-SuperstructuredOrganometal Halide Perovskites[END_REF][START_REF] Etgar | Mesoscopic CH 3 NH 3 PbI 3 /TiO 2 Heterojunction Solar Cells[END_REF][START_REF] Park | OrganometalPerovskite Light Absorbers Toward a 20% EfficiencyLow-Cost Solid-State Mesoscopic Solar CellJ[END_REF][START_REF] Itzhaik | Band Alignment and Internal Field Mapping in Solar Cells[END_REF] their modeling is still scanty.

Conversely, two dimensional hybrid organic/inorganic materials have attracted increasing interest over the past decade due to their potential optoelectronic applications. [START_REF] Mitzi | Conducting layered organic-inorganic halides containing (110)-oriented perovskite sheets[END_REF] Once more, the versatility of the organic part affords the possibility of fine tuning material's properties. For example, it has been shown that the optical spectra of lead halide organic/inorganic perovksites can be easily tailored by varying the organic cation, which improves the optical efficiencies and tuning of the emission wavelength. [START_REF] Mitzi | Conducting layered organic-inorganic halides containing (110)-oriented perovskite sheets[END_REF] Among them, self-assembled layered structures have recently shown enhanced non-linear optical properties in microcavities. [START_REF] Wei | Strong excitonphoton coupling in microcavities containing new fluorophenethylamine based perovskite compounds[END_REF] Interestingly, in these materials DFT calculations predict reversed ordering of band-edge states as compared to tetrahedrally-bonded semiconductors. [START_REF] Even | Electronic model for self-assembled hybrid organic/perovskite semiconductors: Reverse band edge electronic states ordering and spin-orbit coupling[END_REF] Consistently with important relativistic effects expected for lead, such calculations have also evidenced the major role of spin-orbit coupling (SOC) that significantly reduces the band-gap by inducing a large splitting of the first degenerated conduction levels. [START_REF] Even | Electronic model for self-assembled hybrid organic/perovskite semiconductors: Reverse band edge electronic states ordering and spin-orbit coupling[END_REF] Surprisingly, for the hybrid 3D materials recently proposed as efficient DSSC, [START_REF] Kojima | Organometal Halide Perovskites as Visible-Light Sensitizers for Photovoltaic Cells[END_REF][START_REF] Lee | Efficient Hybrid Solar Cells Based on Meso-SuperstructuredOrganometal Halide Perovskites[END_REF][START_REF] Etgar | Mesoscopic CH 3 NH 3 PbI 3 /TiO 2 Heterojunction Solar Cells[END_REF][START_REF] Noh | Chemical Management for Colorful, Efficient, and Stable Inorganic-Organic Hybrid Nanostructured Solar Cells[END_REF][START_REF] Mosconi | First Principles Modeling of Mixed Halide OrganometalPerovskites for Photovoltaic Applications[END_REF][START_REF] Park | OrganometalPerovskite Light Absorbers Toward a 20% EfficiencyLow-Cost Solid-State Mesoscopic Solar CellJ[END_REF] the effect of SOC has been largely overlooked, especially from the theoretical point of view. Optical absorption [START_REF] Tanaka | Comparative study on the excitons in lead-halide-based perovskite-type crystals CH 3 NH 3 PbBr 3 CH 3 NH 3 PbI 3[END_REF] and reflectivity [START_REF] Hirasawa | Excitonfeaturesin 0-dimensional, 2dimensional, and 3-dimensionalnetworksof PbI 6 4-octahedra[END_REF] measurements brought clear experimental evidence of sizeable SOC splittings in such materials more than a decade ago. Even so effects of SOC have been stressed in a recent computational work, [START_REF] Mosconi | First Principles Modeling of Mixed Halide OrganometalPerovskites for Photovoltaic Applications[END_REF] to the best of our knowledge they have not yet been accounted for in calculated band structures reported in the literature. This paper aims to investigate the role of SOC on the electronic band structure of two 3D hybrids recently investigated for PV-devices, namely MAPbX 3 compounds where X=Br,I and MA stands for methylammonium (CH 3 NH 3 ). Moreover, as the design of efficient DSSC requires a good understanding of VB alignments, we suggest a protocol to derive the band alignment between the MAPbX 3 absorbers and TiO 2 . It involves calculations on a slab of a related 2D hybrid, namely 2(C 4 H 9 NH 3 + )(PbI 4

2-

). [START_REF] Billing | Synthesis, characterization and phase transitions in the inorganic-organic layered perovskite-type hybrids (C n H 2n+1 NH 3 ) (2) PbI 4 , n=4, 5 and 6[END_REF] Such analogs also allow further comparison and offer complementary routes for band gap engineering. Actually, by mixing the composition of halogen atoms in MAPb(I 1-x Br x ) 3 structures, it has recently been shown that the band gap can be controllably tuned to cover the entire visible spectrum. [START_REF] Noh | Chemical Management for Colorful, Efficient, and Stable Inorganic-Organic Hybrid Nanostructured Solar Cells[END_REF] These mixed compounds exhibit an averaged disordered cubic phase (space group Pm3m) at room temperature. [START_REF] Poglitsch | Dynamicdisorderin methylammoniumtrihalogenoplumbates(II) observed bymillimeter-wave spectroscopy[END_REF] The structural disorder is both associated to the rotation of ammonium cations and tilt of lead halide octaedra. At low temperature, MAPbBr 3 and MAPbI 3 present an ordered orthorhombic structure of space-group Pnma, with a cell doubling when compared to the room temperature phase(Figure 1). [START_REF] Mosconi | First Principles Modeling of Mixed Halide OrganometalPerovskites for Photovoltaic Applications[END_REF][START_REF] Swainson | Phase transitions in the perovskitemethylammonium lead bromide, CH 3 ND 3 PbBr 3[END_REF] Figure 1.Overview of the crystal structures of MAPbI 3 (left) and MAPbBr 3 (right) at low temperature. The structures are both orthorhombic (space group Pnma), [START_REF] Mosconi | First Principles Modeling of Mixed Halide OrganometalPerovskites for Photovoltaic Applications[END_REF][START_REF] Swainson | Phase transitions in the perovskitemethylammonium lead bromide, CH 3 ND 3 PbBr 3[END_REF] with a cell doubling when compared to the room temperature cubic phase.

These ordered crystalline structures are used to perform the present theoretical study. Figure 2 shows the band structures of MAPbBr 3 and MAPbI 3 with and without the SOC interaction. Let us first analyze the results obtained without SOC. Consistently with earlier results on the room temperature cubic phase, [START_REF] Umebayashi | Electronic structures of lead iodide based low-dimensional crystals[END_REF] the ground state transitions are of direct type but at the -point instead of the R point. [START_REF] Mosconi | First Principles Modeling of Mixed Halide OrganometalPerovskites for Photovoltaic Applications[END_REF] Our study shows that they mainly involve eight active Bloch levels: two levels for the valence-band maximum (VBM1-2) and six levels for the conduction-band minimum (CBM1-6). This apparent complex sequence at -pointis associated to the low temperature symmetry breaking which leadsboth to unit cell doubling and strain. [START_REF] Swainson | Phase transitions in the perovskitemethylammonium lead bromide, CH 3 ND 3 PbBr 3[END_REF] In fact, a symmetry analysis indicates that the CBM1-6 states are related to the conduction-band minimum (CBM)at the R point in the room temperature Pm3m cubic phase which corresponds to a triplydegenerate level associated to the vectorial representation of the simple group. [START_REF] Poglitsch | Dynamicdisorderin methylammoniumtrihalogenoplumbates(II) observed bymillimeter-wave spectroscopy[END_REF] The same electronic band structure is obtained around the band gap for the related CsPbI 3 compound where the organic part of MAPbI 3 is replaced by a Cs atom in the low temperature Pnma phase (figure S1). We canfurther develop the analysis of the electronic states using the complete phase sequence ofMAPbI 3 and CsPbI 3 structures (vide infra). Similar energy dispersions occur at low temperature along the -U (1/2,0,1/2), -Y and -U' (1/2,0,-1/2) directions in figure 2, which is inherent to the small distortion of the perovskite lattice. The direct band-gap calculated at pointis in agreement with the strong absorbance observed at room temperature. [START_REF] Lee | Efficient Hybrid Solar Cells Based on Meso-SuperstructuredOrganometal Halide Perovskites[END_REF][START_REF] Tanaka | Comparative study on the excitons in lead-halide-based perovskite-type crystals CH 3 NH 3 PbBr 3 CH 3 NH 3 PbI 3[END_REF] The six CBM1-6states correspond to anti-bonding hybridizations of 6p-orbitals of lead and belong to the irreducible representations of the D 2h point group (factor group for the Pnmaspace group at  point). VBM1-2consist in anti-bonding hybridizations of the 6s-orbitals of lead and 5p-orbitals of iodine and are associated to non-polar irreducible representations of the point group. Optical activity is related to the dipolar matrix elements between the first valence and conduction-band states as defined by:

3 1 3 1 ,       CBM i VBM CBM VBM x i M  
 where i x represents the crystal axis. [START_REF] Even | Electronic model for self-assembled hybrid organic/perovskite semiconductors: Reverse band edge electronic states ordering and spin-orbit coupling[END_REF] These matrix elements take similar values for the CBM states in the MAPbX 3 family. A strong and almost isotropic optical activity is expected from symmetry and enhanced at room temperature by the disorder of the cubic phase. Similar results are obtained for MAPbI 3 , although the strain-induced conduction-band splitting is larger. We point out that a strain effect was also reported in related MASnX 3 compounds. 32 Figure 2 reveals large changes of the electronic band structure when accounting for SOC. Indeed, the fundamental transitions of MAPbI 3 and MAPbBr 3 lower to 0.5eV and 0.8eV respectively.

While the VB are nearly unaffected, the sixfirst CB undergo a dramatic splitting. The band gap remains direct and located at the center of the Brillouin zone. These findings are consistent with the value reported for a related 2D organic-inorganic hybrid, 2(pF-C 6 H 5 C 2 H 4 -NH 3 )PbI 4 , for which the correction is of about 1eV, leading to a band-gap reduction by a factor of two. [START_REF] Even | Electronic model for self-assembled hybrid organic/perovskite semiconductors: Reverse band edge electronic states ordering and spin-orbit coupling[END_REF] This 2D perovskite crystalizes similarly to the 2(AA)PbX 4 family (AA= C n H 2n+1 NH 3 ) [START_REF] Billing | Synthesis, characterization and phase transitions in the inorganic-organic layered perovskite-type hybrids (C n H 2n+1 NH 3 ) (2) PbI 4 , n=4, 5 and 6[END_REF][START_REF] Billing | Synthesis, characterization and phase transitions of the inorganic-organic layered perovskite-type hybrids (C n H 2n+1 NH 3 ) 2 PbI 4 (n=12, 14, 16 and 18)[END_REF][START_REF] Lemmerer | Synthesis, characterization and phase transitions of the inorganic-organic layered perovskite-type hybrids (C n H 2n+1 NH 3 ) 2 PbI 4 , n=7, 8, 9 and 10[END_REF] One can notice that the calculated band-gap at the GGA-PBE level is equal to 1.3eV in good agreement with previous simulations. [START_REF] Murtaza | First principle study of the structural and optoelectronic properties of cubic perovskites CsPbM 3 (M=Cl, Br, I)[END_REF] . The calculated band-gaps at the LDA level are equal to 1.2eV for CsPbI 3 and 1.3eV for CsPbBr 3 , and the spin-orbit coupling induces a large band-gap correction of about 1.1eV in both cases (figure 3(top)). All these results demonstrate that the effect of SOC on band gaps is huge and cannot be reasonably disregarded. Even if comparison with available experimental data is not straightforward, our results are in qualitative good agreement with evidence of SOC effects in the optical absorption [START_REF] Tanaka | Comparative study on the excitons in lead-halide-based perovskite-type crystals CH 3 NH 3 PbBr 3 CH 3 NH 3 PbI 3[END_REF] and reflectivity [START_REF] Hirasawa | Excitonfeaturesin 0-dimensional, 2dimensional, and 3-dimensionalnetworksof PbI 6 4-octahedra[END_REF] spectra reported for MAPbX 3 . Furthermore, the optical transition occurs at the R point in the room temperature Pm3m cubic phase. For the Pm3msymmorphic space group, it is possible to directly analyzethe irreducible representations at R point using the Oh point group. [START_REF] Bir | Symmetry and strain-induced effects in semiconductors[END_REF] The triplydegenerate (CBM1-3) (figure 3-a (top)) statesare associated to the vectorial representation of the simple group commonly described [START_REF] Even | Electronic model for self-assembled hybrid organic/perovskite semiconductors: Reverse band edge electronic states ordering and spin-orbit coupling[END_REF]32 by the X , Y , Z symbols. In the corresponding double groupincludingspinors, it is splitted by SOC into twofold degeneratestates and fourfold degenerate states (figure 3-b (top)). This situation is usually encountered in the valence band of cubic conventional semiconductors. [START_REF] Even | Electronic model for self-assembled hybrid organic/perovskite semiconductors: Reverse band edge electronic states ordering and spin-orbit coupling[END_REF][START_REF] Bir | Symmetry and strain-induced effects in semiconductors[END_REF] The conduction band minimum of CsPbX 3 and MAPbX 3 at room temperature are associated to the twofold degenerate spin-orbit split-off (SO) states

       Z iY X 3 1 3 1 2 / 1 , 2 / 1 and         Z iY X 3 1 3 1 2 / 1 , 2 / 1 (figure 3-b (top))
.The SO states lead to a strong [START_REF] Even | Electronic model for self-assembled hybrid organic/perovskite semiconductors: Reverse band edge electronic states ordering and spin-orbit coupling[END_REF] and isotropic 32 optical transition with the even S -like VBM states.To understand the influence of the cubic-tetragonal (Pm3m-I4mcm) transition of MAPbI 3 33 on its optoelectronic properties, we have also performed DFT calculations on CsPbI 3 in the tetragonal phase of MAPbI 3 including the SOC. Without SOC, the triply degenerate CBM states and the VBM state are folded back from the R point at the  pointcases (figure 3a, bottom). In addition and according to the D4h point group, the strain along the z axis inducesa small splitting of the CBM and the band gap reduces to 1.0eV. The CBM vectorialrepresentation splits into a non-degenerate Z state at lower energy and twofold degenerate X , Y states.

The SOC effect is however much larger than the effect of strain, and leads to a very small band gap (figure 3b(bottom)). The state ordering including SOC at the  point of the I4mcm phase remains similar to the one of the Pm3m cubic phase, the CBM states being again related to the

twofold degenerate SO states 2 / 1 , 2 / 1 and 2 / 1 , 2 / 1  .
Based on the band structure of CsPbI 3 , we can also investigate the cubic-orthorhombic (Pm3m-Pnma) transition. This transition is associated to a cell doubling and folding back from the R point to the  point. The triply degenerate CBM are doubled and splitted by the strain along the three directions. The six CBM states correspond to one dimensional representations of the simple D 2h point group which yield two-fold degenerate representations of the corresponding double group. The strain effect is no more negligible when compared to the SOC effect, but the CBM state of lowest energy has an electronic density similar to the one of cubic SO state. Now, one can wonder why the fundamental transitions calculated without SOC for the low temperaturePnma structures, 1.5 eV and 1.9 eV respectively for MAPbI 3 and MAPbBr 3 (figure 2) compare nicely with the values obtained experimentally, i.e. 1.5eV and 2.3eV respectively for MAPbI 3 and MAPbBr 3 . [START_REF] Kojima | Organometal Halide Perovskites as Visible-Light Sensitizers for Photovoltaic Cells[END_REF][START_REF] Noh | Chemical Management for Colorful, Efficient, and Stable Inorganic-Organic Hybrid Nanostructured Solar Cells[END_REF][START_REF] Tanaka | Comparative study on the excitons in lead-halide-based perovskite-type crystals CH 3 NH 3 PbBr 3 CH 3 NH 3 PbI 3[END_REF][START_REF] Hirasawa | Excitonfeaturesin 0-dimensional, 2dimensional, and 3-dimensionalnetworksof PbI 6 4-octahedra[END_REF] This agreement is fortuitous and stems from large error cancellations. Indeed, the band gap is known to be underestimated in the DFT ground-state computations. This can be corrected by including many-body effects by means of GW selfenergy correction for the band gap [START_REF] Hedin | New method for calculating 1-particle greens function with application to electron-gas problem[END_REF] or using the Bethe Salpeter equation resolution for the exciton. [START_REF] Salpeter | A relativistic equation for bound-state problems[END_REF] Unfortunately, such calculations are beyond available computational resources for large systems. In order to gauge the importance of the GW corrections, we have considered the cubic phase of CsPbI 3 . A one-shot GW self-energy correction on the LDA level amounts to a 0.6 eV increase of the band gap. This value, although not self-consistent, shows that GW corrections are large and in the opposite direction to SOC effects. Unfortunately, a full treatment including both SOC and many-body effects is far beyond available computational resources.

Efficient electrical power generation relies on the quality of the absorber and the band-gap energy. These conditions appear to be fulfilled and understood from the theoretical point of view.

This efficiency also relies on the ability of the PV device to drive the carriers toward the electrodes. A correct band alignment between the sensitizer and the electrode materials is thus a key parameter to predict performances of PV devices. In the case of DSSC, with a TiO 2 electrode, hybrid MAPbX 3 perovskites appear to greatly improve the electron transport. [START_REF] Etgar | Mesoscopic CH 3 NH 3 PbI 3 /TiO 2 Heterojunction Solar Cells[END_REF] To evaluate the valence-band alignment between the anatase structure of TiO 2 8 and hybrid perovskites, a slab calculation 37 was performed for the 2(AA)PbI 4 with n=4. This allows estimating the offsets between the vacuum level and its bulk potential (figure 4). It clearly demonstrates that the conduction-band offsets are favorable for the electron injection from absorber to electrode. In addition, this diagram suggests that combining 2D and 3D hybrid perovskites may provide an alternative way for the design of colorful solar cells that cover the entire visible spectrum.

In summary, based on DFT calculations, the electronic properties of 3D lead halide organic/inorganic perovskites used as absorbers in DSSC [START_REF] Kojima | Organometal Halide Perovskites as Visible-Light Sensitizers for Photovoltaic Cells[END_REF][START_REF] Lee | Efficient Hybrid Solar Cells Based on Meso-SuperstructuredOrganometal Halide Perovskites[END_REF][START_REF] Etgar | Mesoscopic CH 3 NH 3 PbI 3 /TiO 2 Heterojunction Solar Cells[END_REF] have been thoroughly investigated.

Their band-gap is dominated by a giant SOC effect acting mainly on the conduction band, as was already demonstrated for related 2D hybrids. [START_REF] Even | Electronic model for self-assembled hybrid organic/perovskite semiconductors: Reverse band edge electronic states ordering and spin-orbit coupling[END_REF] At room temperature, their remarkable optical properties can be related to direct and isotropic optical transitions between a triply degenerated conduction-band and a single valence-band in a simple group representation. Valence band offsets [START_REF] Kojima | Organometal Halide Perovskites as Visible-Light Sensitizers for Photovoltaic Cells[END_REF] confirm that the gathering of MAPbX 3 hybrids and TiO 2 is a relevant choice for driving the electrons toward the electrode. Our calculations on alkyl ammonium based 2D hybrids also suggest a complementary route to the chemical tuning [START_REF] Noh | Chemical Management for Colorful, Efficient, and Stable Inorganic-Organic Hybrid Nanostructured Solar Cells[END_REF] recently proposed for the design of colorful solar cells with enhanced light conversion efficiency.

The present theoretical study was performed using the DFT implementation available in the ABINIT package, [START_REF] Gonze | ABINIT: First-principles approach to material and nanosystem properties[END_REF] with the LDA or the GGA-PBE gradient correction for exchangecorrelation [START_REF] Perdew | Generalized gradient approximation made simple[END_REF] and relativistic, norm-conserving, separable, dual-space Gaussian-type pseudopotentials of Goedecker, Teter, and Hutter for all atoms [START_REF] Hartwigsen | Relativistic separable dual-space Gaussian pseudopotentials from H to Rn[END_REF] or Fritz-Haber-Institute's pseudpotentials. [START_REF] Fuchs | Ab initio pseudopotentials for electronic structure calculations of poly-atomic systems using density-functional theory[END_REF] The SIESTA code was used for the simulation of slabs at the DFT-GGA level. [START_REF] Soler | The SIESTA method for ab initio order-N materials simulation[END_REF] We have verified that the band structures near the band-gap do not depend on the inclusion of 5d-orbitals for Pb. The electronic wave-functions are expanded onto a plane-wave basis set with an energy cut-off of 950 eV. 4x4x4 and 4x4x1 Monkhorst-Pack grids are used for reciprocal space integration in 3D and 2D structures respectively. Calculations were performed with and without SOC.

Corresponding Author

*Email: claudine.katan@univ-rennes1.fr Tel: +33 (0)2 23 23 56 82

Notes

The authors declare no competing financial interests.

This work was performed using HPC resources from GENCI-CINES/IDRIS grant 2013-c2012096724. The work is supported by AgenceNationale pour la Recherche (PEROCAI project ANR-10-04).

(1) Feltrin, A.; Freundlich, A., Material considerations for terawatt level deployment of photovoltaics. Renewable Energy 2008,33 (2), 180-185.

(2) Oregan, B.; Grätzel, M., A Low-cost, high-efficiency solar-cellbasedon dye-sensitizedcolloidalTiO 2 films. Nature1991,353 (6346), 737-740.

(3) Grätzel, M., Solar energy conversion by dye-sensitized photovoltaic cells. Inorg. Chem.

Figure 2 .

 2 Figure 2.Electronic band structures of MAPbI 3 (top) and MAPbBr 3 (bottom), without (a) and

  Figure 3..Electronic band structures of CsPbI 3 in the cubic phase (top) and tetragonal phase

Figure 4 .

 4 Figure 4.Potential profile in an alkyl ammonium (AA) 2D hybrid perovskite crystal (dark line) and in a slab (red dotted line) with the same crystallographic structure 24 surrounded by

Figure 5 .

 5 Figure 5.Energy level diagram derived from the position of Pb-5d orbitals, computed VBM (figure4) and experimental band gaps of MAPbI 3 (1.5eV), MAPbBr 3 (2.3eV)[START_REF] Kojima | Organometal Halide Perovskites as Visible-Light Sensitizers for Photovoltaic Cells[END_REF][START_REF] Noh | Chemical Management for Colorful, Efficient, and Stable Inorganic-Organic Hybrid Nanostructured Solar Cells[END_REF] and an alkyl ammonium (AA) 2D hybrid perovskite (2,5 eV).[START_REF] Billing | Synthesis, characterization and phase transitions in the inorganic-organic layered perovskite-type hybrids (C n H 2n+1 NH 3 ) (2) PbI 4 , n=4, 5 and 6[END_REF] Commonly accepted values for TiO 2 electron


Jacky Even
email: jacky.even@insa-rennes.fr

Laurent Pedesseau

Jean-Marc Jancu

Claudine Katan

DFTand k.p modellingof the phase transitions of lead and tin halideperovskites for photovoltaic cells

Keywords: Perovskite, photovoltaic, first-principles calculations, k.p, spin-orbit coupling

DFT and k * p modelling of the phase transitions of lead and tin halide perovskites for photovoltaic cells

cludingspin-orbit effect.In addition, the non centrosymmetric phasesof CH 3 NH 3 MI 3 exhibit a splitting of the electronic bands away from the critical point. The present work shows that their physical properties are more similar to conventional semiconductors than to theabsorbers used in DSSC.

1 IntroductionOver the past decade, Hybrid Organic/inorganic Perovskites (HOPs) have attracted increasing interest in the field of optoelectronics [1] and more recently for solar cells [2,3], thanks to their ease of synthesis, openingnew routes for low-cost technologies. Reduced costs and higher conversion efficiencies are indeed of crucial importance to makephotovoltaic technologies a reliable choice to cover emerging energy needs [4]. The quest for higher conversion efficiencies mainly relies on semiconductor heterostructuressuch ashigh-performance III-V/silicon multiple junctions cells [5], thatrequire high level technologies. Following pioneering works [2,3], threedimensional (3D) HOPs like CH 3 NH 3 PbX 3 (where X=I, Br, Cl), haverecently emerged as a novel class of low-cost materials for solid-state mesoscopicsolar cells [6][7][8][9][10],where an efficiency of 20% is predicted [11]. Structure and internal mechanismsof these hybrid semiconductor photovoltaic cells(HSPC) maydiffer from the one of dye-sensitized solar cells (DSSC) [12]. Interestingly, the organic part provides astructural flexibility that makes these materials appealing for mesostructured HSPC, but it also yields structural disorder. The design of novel devices with enhanced responses canbenefit from predictivemodelling of these HOPs using Density Functional Theory (DFT). Surprisingly, despite the important relativistic effects expected for leador tin in HOP, spin-orbit coupling (SOC) effects have been overlooked until recently from the theoretical point of view [13,14]. Moreover, concepts from solid-state physics may be used for HOPs to reach the level of knowledge already attained for the optoelectronic properties of conventional semiconductors. In fact, it has been demonstrated that the band ordering of 3D [13] and 2D [14] HOPs close to the band gap is reversed when compared to conventional semiconductors.In addition, the critical electronic transition is dominated by a giant SOC in the conduction band (CB) [13,14]. GW many-body corrections to DFT have clearly demonstrated that the agreement previously reported between calculatedand experimental band gaps is fortuitous, and stems from error cancellation between SOC and many-body effects [13].

All-inorganic analogues of such HOPs, namely CsSnI 3 , have also been reported for hole conduction in lieu of a liquid electrolyte in DSSC [15]. It is also presumed to take part in light absorption on the red side of the solar spectrum [16].These compounds have a high temperature cubic phase (Pm3m) thatcan be considered as an ideal reference structure to understand the basic electronic properties of related HOPs and all-inorganic analogues [13,17].However, one should also consider more realistic situations such as the room temperature orthorhombic -phase of CsSnI 3 , which differs significantly from the ideal cubic -phase of CsSnX 3 and CsPbX 3 [18,19].

Until recently, it was considered that 3D HOPscrystallise in a high-temperature cubic phase with dynamical disorder of the organic cationsCH 3 NH 3 + [20,21,22]. Very recently, Single crystal X-ray diffraction analysisled to a refinement of the structures of CH 3 NH 3 MI 3 (where M=Pb, Sn)close to the room temperature [23]. Their high temperature phases correspond to anon-centrosymmetrictetragonal P4mm structure, slightly strained from the ideal cubic Pm3m phase, where carbon and nitrogen atoms can be located. Both compounds exhibit a phase transition to a tetragonal non-centrosymmetricand centered I4cm phase. For the technologically important HOP,CH 3 NH 3 PbI 3 , this transition occurs above room temperature (T c =333K). In the previously reported room temperature centrosymmetric tetragonal I4mcm phase of CH 3 NH 3 PbI 3 , carbon and nitrogen atoms were indeed not properly located on Wickoff positions [22].In addition, crystallographic data of the orthorhombic low temperature phases of CH 3 NH 3 MX 3 are also available [21,24].

In this work, thanks to DFT and k.p approaches, we investigate the properties of different phases of tin and lead based HOPs and their all-inorganic analogues.First, we start by considering the high temperature cubic -phase of all-inorganic compounds, and the phase sequence of CsSnI 3 that leads to a room temperature orthorhombic phase.Then, we investigate different phases of both CH 3 NH 3 PbI 3 and CH 3 NH 3 SnI 3 using the latest crystal data. We systematically consider SOC and propose a perturbative analysis of the electronic band structure, including strain effects. This k.p model is applied to tetragonal phases. The effect of non-centrosymmetry on the band diagram is also explored.

Methods

Calculations were performed using the DFT implementation available in the ABINIT pseudo-potential package [25] with the LDA for exchange-correlation and relativistic, norm-conserving, separable, dual-space Gaussian-type pseudopotentials for all atoms [26] in order to include the spin-orbit coupling. The electronic wave-functions are expanded onto a plane-wave basis set with an energy cut-off of 950 eV. The GW self-energy perturbation was performed within ABINIT after a LDA computation without SOC. The study of the cubic phases was also performed with the FP-LAPW Elk code [START_REF]elkfp-lapwcode[END_REF] at the GGA-level [START_REF] Perdew | [END_REF]. The Bethe-Salpeter equation (BSE) was solved for the cubic phase [29]. The dielectric constant was also determined within the Random-Phase Approximation (RPA). 4x4x4 Monkhorst-Pack grids are used for reciprocal space integration, except for the cubic phases (8x8x8). It has been previously shown that the band structure of HOPs close to the band gap may be reproduced by replacing the organic cation by Cs + located at thenitrogenposition [13].

3Results and discussion 3.1 Electronic properties of CsSnI 3 and CsPbI 3 cubic phases The electronic properties of the high temperature cubic -phase of CsSnI 3 have already been studied without SOC by DFT planewavepseudopotential approach [17]. These results showed that theoptical ground state transition is related to a direct electronic transition located at the R-point of the Brillouin zone,leading to a band gap of 0.35eV for CsSnI 3 [17]. It is worth mentioning that even though SOC was neglected, the agreement with the measured band gap at room temperature (1.3eV, [15,16,23]) is poor. Similar calculations for the room temperature orthorhombic -phase of CsSnI 3 led to a somewhat higher value (0.56eV) [17]. The inclusion of SOC, has been recently reported for CsPbX 3 (X=I, Br) [13],

showing that SOC cannot be disregarded for lead halide perovskites.

Figure 1 shows a comparison of the electronic structure of CsSnI 3 and CsPbI 3 without and with SOC using an allelectron FP-LAPW approach. In both cases, the band gap is located at the R-point and the CB critical states correspond to a triply degenerate odd states without SOC and a twice-degenerate spin-orbit split-off states with SOC: CsSnI 3 ).The CsSnI 3 band gap with (without) SOC amounts to 0.1eV(0.3eV). Similar results are obtained withinaplanewave pseudopotential method [25]. A oneshot many-body (GW) corrections to the LDA gap of CsSnI 3 (without SOC)amounts to 0.6eV. This correction is similar to that obtained for CsPbI 3 [13]. It is not sufficient to recover the measured optical band gap (1.3eV). Indeed more involved many-body techniques should be used and excitonic effects cannot be neglected.Besides it is wellknown that band gaps and effective masses are related [29].

3 2 3 0 ) ( 3 0 2 , 1 SO CB SO SO CB E E E E       1 2
Based on experimental band gaps [23], hole effective masses of both tin and lead compounds become comparable(0.17). Figure 2 shows the dielectric constant of CsPbI 3 calculated without SOC, including a scissor correction of 0.6eVfor the band gap. The first absorption peak, located at about 1.8eV, corresponds to the fundamental transition between the triply degenerate CB state and the VB state (blue line, RPA level). From a BSE computation (red line), it is found that excitonic interaction enhances the absorption with a sizeable binding energy

( eV E bX 1 . 0 
).

Phase transitions and room temperature orthorhombicphase of CsSnI 3

On cooling at 426K, the crystal converts to a P4/mbmtetragonal -phase [15], with a cell doubling perpendicular to the tetragonal axis.The correspondingstrain tensor holds  xx = yy =-0.14% and zz =0.28%. To highlight the changes of the band structure during this phase transition,figure 3compares the bands of the tetragonal phase with folded bands derived from the cubic phase. The insert illustrates the small splitting induced by the tetragonal strain on the CB states. This effect can be first analysed without SOC within a k.p approach close to the R-point. The corresponding k-dependent Hamiltonian for the triply degenerate CB reads: ,,,, 0 can be extracted either from experiment [30]or from state of the art electronic structure calculations [31].

                         2 2 2
                The n m l N L M E CB , ,
SO SO SO CB SO CB E E E E E E E                         
In the particular caseof the Pm3m-P4/mbmtransitionof CsSnI 3 , SOC is larger than strain effect

E SO    3
(figure 3) and:

3 2 , 3 0 0 SO CB SO CB E E E E E          
This shows that detailed analysis of the effect of the phase transitions on the electronic properties must involve SOC.The room temperature orthorhombic Pnma-phase is derived from the P4/mbm-phase by a cell doubling along the c-axis [15]. This leads to an additional foldingfrom the Z-point of the -phase to the point of the-phase. The SOC effect is again larger than the strain one. The splitting of the SO band with the next CB is equal to 0.4eV, consistently with earlier findings [15].

Interplay of SOC and loss of inversion symmetry inCH 3 NH 3 PbI 3 and CH 3 NH 3 SnI 3 Earlier

DFT calculations including SOC were already performed for CH 3 NH 3 PbI 3 , but using available structural data for the -phases [13]. Recent crystallographic data [23] allow to investigatefurther the high temperature -phase of CH 3 NH 3 MI 3 HOPs. It is characterized by atetragonal P4mm structure, slightly strained from the ideal cubic Pm3m phase ( zz =0.04and 0.01% for M=Pb and Sn, respectively).The band diagram is similar to that reportedfor the cubic phase of CsPbI 3 (figure 1). The SOC splitting is equal to 1.5 and 0.4eV for M=Pb and Sn, respectively. Hole effective masses evaluated from the VB dispersion are found similar to that obtained for CsPbI 3 and CsSnI 3 . The most important feature is due to the loss of the inversion symmetry that inducesa splitting of the bandsaway from the critical point. The electronic band diagram of the P4mm phase of both CH 3 NH 3 MI 3 HOPsclose to the Apoint (corresponding to the R-point of the ideal cubic phase) indeed exhibits a small splitting of the bands.

This effect is more important for the tetragonal I4cm phase in both compounds (figure 4). The bands are folded from the A-point to the -point (figure 4). The phase transition is associated to a group-subgroup relationship between P4mm and I4cm. It is similar to the previously proposed Pm3m-I4/mcm phase sequence [20][21][22]analysed without inversion asymmetry for CH 3 NH 3 PbI 3 [13]. We underline that the previous DFT calculations for the phase of CH 3 NH 3 PbI 3 [13] were performed using a refined centrosymmetric orthorhombic Pnma structure [21]. However the structure of the -phase of CH 3 NH 3 MI 3 com-pounds is still debated. Alternative monoclinic, triclinic [23] or non-centrosymmetric Pna2 1 orthorhombic [20][21][22] space groups have been suggested.Noteworthy, the k.p modelling of such phases should take into account the loss of inversion symmetry. 4 Summary and outlookIn the present work, we have thoroughly investigated different phases of tin and lead based HOP and their all-inorganic analogues, with special attention on the room temperature phases relevant for photovoltaic applications.Based on both DFT and k.p approaches, we confirm the importance of SOC for an accurate description of band edge states in this class of materials. The fundamental optical absorption of these materials is associated to transitions between a doubly degenerate spin-orbit split-off CB and a doubly degenerate VB. The spin-orbit splitting of the CB of the tin halide materials is about three times smaller than the one of lead halide compounds. The hole conduction of these materials is related to a quasi-isotropic VB dispersion and a small effective mass. Both the high temperature phase sequence of CsSnI 3 leading to the room temperature orthorhombic -phase and the and -phases of CH 3 NH 3 MI 3 HOPs, reported recently, are analysed starting from the ideal cubic phase using DFT calculations and a k.pperturbative approach. The effect of strain is systematically smaller than that induced by SOC. The loss of inversion symmetry in presence of strong SOC yields further splitting of the electronic bands away from the critical points, especially for the room temperature I4cm phase of CH 3 NH 3 PbI 3 .Except for reversed band edge states ordering and strong SOC, these compounds showmany similarities withconventional semiconductors, leading to new class of semiconductors differentfrom absorbers commonly used in DSSC.

Figure

  Figure 1Comparison of the electronic band diagram of cubic CsSnI3 (red straight lines) and CsPbI3 (black dotted lines) without (a) and with (b) SOC. The SOC splitting in CsSnI 3 is about eV SO 4 . 0   , three times smaller than the corresponding one in CsPbI 3 [13].The hole effective masses can be evaluated from the valence band (VB) hole dispersion curve with a small warping ( 12 . 0 

Figure 2 (

 2 Figure 2(a) Comparison of the dielectric constant variation of CsPbI 3 with (red dotted line, BSE calculation) and without (blue straight line, RPA calculation) the excitonic interaction. (b) Schematic representation of the two optical transitions.

Figure

  Figure 3Band structure of CsSnI3 withSOC (a) from folded bands of the cubic -phase (straight line, along R-M and dotted line along Z-) (b) for the tetragonal P4/mbm-phase. (c) Insert shows the small splitting due to tetragonal strain near the Z-point.

  Atthe R(Z)-point of CsSnI 3 , the energy splitting  befurther added perturbatively, considering spinorstates (figures 1, 3). The threeeigenvalues of the total Hamiltonian are doubly degenerated and given by:

Figure 4 (

 4 Figure 4(a)Band structure of the I4cm phaseof CH 3 NH 3 SnI 3 with SOC and zoomclose to the critical  point, (b) for CH 3 NH 3 PbI 3 and (c) for CH 3 NH 3 SnI 3 .

  The work is supported by AgenceNationale pour la Recherche (PEROCAI project ANR-10-04).

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Acknowledgements This work was performed using HPC resources from GENCI-CINES/IDRIS grant 2013-c2013096724.


F Ben Cheikhlarbi

Jacky Even

Jean-Marc Jancu
email: jean-marc.jancu@insa-rennes.fr

C Hajlaoui

L Pedesseau

F Raouafi

F Ben Cheikhlarbi

J.-M Jancu

First

published or not. The documents may come

Semiconductor core/shellnanowires (NWs) 1,2 hold lot of promises for applications in the field in electronics 3,4 ,photonics 5,6 , and biochemical sensors. 7 These systems, compared to pure semiconductor wires, demonstrate areductionof surface scattering and ionized impurity scattering 8 with improved electronic confinement properties. The synthesis processis mainly based on the gold-catalyzed vapor liquid-solid (VLS) growth mechanisms, which allows for a precise controlof geometrical dimensionsand strain accommodation. 9 The VLS method is particularly suited in interfacing highly strained III-V materials onto Si substrates. 10 Interestingly, it favors the hexagonal phase at the nucleation stage leading toa crystallization of III-V semiconductorsinahexagonal wurtzite (WZ) structure whereasthe bulk counterpart for most of them is cubic zinc-blende (ZB). 11 Recently, much effort has been devoted to the growth and fabrication of strained InP/InAs/InPNW devices for the realizationof high field-effect electron mobility transistors and circuits. Advantages of thesesystems include the high electron mobility (30 000 cm 2 /V s at 300 K) and velocity (43000 cm/s) of InAsand a large conduction band offset between InAs and InP(0.5 eV). 12 Strain engineering is a well-established technique to tune the electronic band gap of nanostructures. For instance,the lattice mismatch ofInAs onInPis about -3 %and it is expected to tailor the band offsets and carrier mobility by lifting the degeneracy in the electronic bands. In addition, strain effects cangeneratea piezoelectric polarization charge at interfaces leading to a band bending in the active region of the wire.Determining thematerials properties ofInAs/InP NWs iskey to improving current generating efficiencies of underlying devices.The WZ and ZB band structuresfor bulk InAs and InPhave been extensively studied both theoretically and experimentally. [12][13][14] Moreover, electronic deformation potentials (DPs) of cubic semiconductors are well documented in literature. 12 However, to the best of our knowledge,no workon DPs has dealt withWZ III-V materials so far. This first step is mandatory for realistic modeling of InAs/InP NWs in the WZ structure 15 , including the effects of linear and non-linear electro-elastic coupling. 16 In this work , we performab initio calculationsofdeformation potentials in wurtziteInAs and InP. The structural relaxations with strain perturbations are performed using the plane-wave projector augmented wave method (PAW) as implemented in the VASP code. 17 We used the Heyd-Scuseria-Ernzerhof (HSE06) DFT hybrid exchange correlation functional 18,19 with a screening parameter μ=0.2 and a mixing parameter of α = 0.25. This approach has been foundto give reliable resultsin group III-nitrides 20 and group II-oxides. 21 Generalized-Gradient Approximation (GGA) in the Perdew-Burke-Ernzerhof (PBE) parameterization 22 and single shot GW (G 0 W 0 ) calculations implemented in VASP are also performed for comparison. We used GGA eigenvalues as input to the single shot G 0 W 0 calculations leading to more reliable parameters such as band gap and deformation potentials.The semi-core d-electrons of Inaretreated as valence electrons.The procedure is carried out with a plane-wave cutoff of 600 eV and a 6x6x4 Monkhorst-Pack mesh for k-point sampling. This allows for a full relaxationof the structural parameters.The accuracy of the predicted equilibrium lattice constants for both InP and InAscorrelates directly with the level of approximation in the functional as shown in Table I. HSE gives the most accurate resultsandthe well-known overcorrection ofGGA-PBE 23 become apparent. We also note in Tab. Ithat the WZstructure is not perfectly tetrahedral with non-ideal c/a and u(an ideal WZ lattice consists ina tetragonal ratio c/a=1.633 of the lattice constants and an ideal internal cell parameter u=3/8)as found for bothsimulations and experiment. Fig. 1 shows the InAs and InPband structures at equilibrium near the -point computed with HSE. Calculations are performed without spin-orbit coupling. The direct band gap is formed by an s-like  1c (BC)conduction-band state and doubly degenerate p-like  6v (HH, LH) and  1v (CH)valence-band states. Two energy gaps can be defined according to the optical transitions: 6v  1c (A, B) and  1v  1c (C). In our simulations, E A/B is the fundamental transition energyforboth InAs and InP. As evidenced in Fig. 1, away from the Brillouin zone-center,the valence bands are strongly admixed and their dispersion is non-parabolic.Therefore the valenceband effective massescannot be easily derived for application in k.p models. On the other hand, the conduction band energy dispersionis parabolic near the -point and ourHSEcalculation of the electron masses perpendicular and parallel to the c-axis gives: 0.055 and 0.032 (in units of free electron mass).These results are consistent with thoseof De and Prior obtained within an empirical pseudopotentialapproach. 24 As seen in Tab. I, the band-gap energy issensitive to the details of the exchange and correlation functionaland both the hybrid functionaland single G 0 W 0 reproducevalues close tothe experimental data.Conversely, an equivalent crystal field splitting  cr =HH-CHis obtainedfor PBE and HSEandthe value of InAscompares well with the self-consistent GW calculationof Zanoliet al. 12,13 A similar behavior between PBE and HSE for  cr under deformation is evidenced both for biaxial strain   and for uniaxial strain  zz as shown in Fig. 2. As definition,  = ( xx+  yy )and  zz stand for the strain components in and perpendicular of the c-plane. In the calculations, the internal parameter uwas fully relaxed for each strain. Anonlinear behavior is observed for the two semiconductorswhichcan be interpolatedwitha polynomial regression. It should be noted that (Fig. 2b) HSE exhibits a strong nonlinearity for the crystal field splitting of InAs even for moderate biaxial strain.

Theelectronic deformation potentials are defined as the linear coefficients in the expansion of the k.p band energies with respect to strain. Weinterpolatedthe present first-principle calculationsfor small deformationswith a derivation ofDPs from a 3x3 band k.p model 25 which gives the linear dependence of E A/B and E C on strain as:

/ = / 0 + -1 + -2 ⊥ -( 3 + 4 ⊥ ) = 0 + -1 + -2 ⊥
(a cz , a ct ) and (D 1 , D 2 , D 3 , D 4 ) characterize thedeformation potentialsof  1c and 1v and  6v respectively.These relations are compatible with hexagonal symmetry C 6v . On the other hand, the breaking of symmetryC 6v to orthorhombic C 2v with in-plane anisotropic strain lifts the degeneracy ofHH and LH levels as:

= -= 2 5 ( -)
The electronic deformation potentials of WZ InAs and InPare listed in table II.Differences are evidenced between PBE, G 0 W 0 , and HSE. However, the calculations ensure an overall uniform behavior for the DPsincreasingdistinctly from InAs to InP.This trend with the band-gap has beenalso observed in group III-nitrides 20 and group II-oxides. 21 As a conclusion, a systematic study of the electronic band structure in wurtziteInAsand InP is presented. Hybrid-functional calculations allow for a realistic description of the strained band energies. From accurate calculations of the electronic deformation potentials, we found the widely used QC approximation is not fulfilledfor both InAs and InP. We demonstrated that the dependence of electronic bands on strain is strongly nonlinear and it should be taken into account for a proper modeling of InAs/InP nanowire-based devices.

This work was performed using HPC resources from GENCI CINES and IDRIS 2013-2013096724.

The work is supported through the participation of the INSCOOP ANR project Tables.

  Moreover, the band-gap-related deformation potentialsofa cz -D 1 and a ct -D 2 are found more sensitive to exchange correlation functionalthan D 3 , D 4 and D 5 . Because there is currently no available experimental information and in order to gain insight into the quality of our simulations, we performed DP simulationsin ZB InAs and InP as a test. By comparison to our previous DFT-LDA calculations26 ,HSE-based calculations lead to a better agreement with experiment.The same conclusion is drawn from our present GGA computation. This gives us confidence in the ability of the hybrid functional to model deformation potentialsin wurtzite III-V semiconductors.Finally,the QC approximation wastested for InAs and InP from HSE results. The QC approximation is extensively used in k.pHamiltonians and based on a similar local atomic coordination between WZ and ZB crystals leading to a correlation of their physical properties along the [111] direction of the zinc-blende. Herein, we evaluated the deviation from the QC approximation of deformation potentials.Indeed, the same approximation in k.p theorylead to: 3 + 2 4 = 0and 1 + 3 = 2 . Actually, neither for InAs nor InPsatisfiedthese equations.For instance the first one yields-0.93 eV forInP and the failure must be related to the nonideal c/a ratio.

Figure 1 :

 1 Figure 1: Blow-up of the WZ-InAs (a) and WZ-InP (b) band structurescalculated with HSEclose to the  point. anddenote the in-plane and parallel directions with respect toc-axis. The origin of the energy scale is taken at the top of the valence band (VBM).

Figure 2 :

 2 Figure 2: Left Panel:GGA-PBE dependence of the crystal-field splitting  cr ()of InAs and InPon (a) in-plane and(c) perpendicular strain with respect to the c-plane. Right Panel: Dependence of  cr ()- cr (0) on (b) in-planeand(d) perpendicular straincalculated with HSE for InAs and InP. Quantization axis z is along the[0001] WZ direction.

Table I :

 I Results of GGA-PBE, HSE,and G 0 W 0 calculations forwurtziteInAs and InP at equilibrium for the lattice constants (a, c), internal parameter u, direct band gapE A/B , crystal-field splitting  cr , and compared with the experimental values 2,14,27 (Expt.).Single G 0 W 0 is performed on top of GGA calculations with identical structural parameters of GGA-PBE.

		Method	a(Å)	c(Å)	u	E A/B (eV)	 cr (eV)
	InP	PBE	4.229	6.881	0.3776	0.67	0.086
		G 0 W 0				1.43	0.090
		HSE	4.174	6.824	0.3742	1.47	0.093
		Expt.	4.150 a	6.777 a	-	1.49 a	-
	InAs	PBE	4.426	7.164	0.3817	-0.31	0.070
		G 0 W 0				0.42	0.088
		HSE	4.330	7.066	0.3756	0.40	0.091
		Expt.	4.283 b	6.995 b	0.3750 b	0.52 c	-
	a reference 27
	b reference 14
	c reference 2

Table II :

 II Deformation potentials (eV) of wurtziteInAs and InP obtained with GGA-PBE, G 0 W 0 , and HSE.

		Method	-1	-2	3	4	5
	InAs	PBE	-3.67	-5.31	4.70	-2.18	-2.21
		G 0 W 0	-3.06	-6.12	5.62	-2.77	-2.63
		HSE	-2.69	-8.33	5.85	-3.17	-2.78
	InP	PBE	-3.13	-7.20	5.55	-3.09	-2.84
		G 0 W 0	-3.98	-7.43	5.67	-3.15	-2.88
		HSE	-4.05	-7.64	5.92	-3.28


Laurent Pedesseau
email: laurent.pedesseau@insa-rennes.fr

Jean-Marc Jancu

Alain Rolland

Emmanuelle Deleporte

Claudine Katan

Jacky Even

Electronic properties of 2D and 3D hybrid organic/inorganic perovskites for optoelectronic and photovoltaic applications

Keywords: Organic-inorganic perovskite semiconductors, Density functional theory, optoelectronic properties, photovoltaic, exciton

We herein investigate theoretically both 2D and 3D Hybrid Organic/inorganic Perovskite (HOP) crystal structures based on Density Functional Theory (DFT) calculations and symmetry analyses. Our findings reveal the universal features of the electronic band structure for the class of lead-halide hybrids (R-NH 3 ) n PbX m , where (n,m)=(2,4) and (1,3) respectively for 2D and 3D structures. Among those, the large spin-orbit coupling acting on the conduction band is shown to play a major role on the band gap of these materials. Moreover, this approach can easily be generalized to related layered and 3D hybrids, thus providing a clear-sighted inside in their electronic and optical properties.

Introduction

Over the past decade, Hybrid Organic/inorganic Perovskites (HOP) have attracted increasing interest in the field of optoelectronics [START_REF] Mitzi | Conducting Layered Organic-inorganic Halides Containing <110>-Oriented Perovskite Sheets[END_REF] and solar cells [START_REF] Lee | Efficient Hybrid Solar Cells Based on Meso-Superstructured Organometal Halide Perovskites[END_REF], thanks to their ease of synthesis that opens the way to low-cost technologies. Moreover, the organic part allows tailoring their optical properties by chemical substitution, leading for example to higher luminescence efficiencies and a fine tuning of emission wavelength [START_REF] Zhang | Synthesis and optical properties of novel organicinorganic hybrid nanolayer structure semiconductors[END_REF]. Most of the recent work has focused on lead-halide based materials of general formulae (R-NH 3 ) n PbX m , where R is an organic group and X a halogen atom. Control of the stoichiometry (n,m) affords crystal packing of various dimensionalities, ranging from three-dimensional (3D) corner-shared perovskite lattices to 0D structures with isolated inorganic octahedra [START_REF] Mercier | Structural diversity and retro-crystal engineering analysis of iodometalate hybrids[END_REF]. The quantum and dielectric confinement expected in the 2D layered HOPs (n=2,m=4) prompt active research to develop optoelectronic devices with enhanced performances [START_REF] Mitzi | Conducting Layered Organic-inorganic Halides Containing <110>-Oriented Perovskite Sheets[END_REF]. Indeed, such layered structures have been demonstrated to show enhanced non-linear optical properties in microcavities [START_REF] Wei | Strong exciton-photon coupling in microcavities containing new fluorophenethylamine based perovskite compounds[END_REF]. More recently, 3D HOPs (n=1, m=3), with relatively small organic cations, have been suggested as a novel class of low-cost materials for high efficiency hybrid semiconductor photovoltaic cells (HSPC) [START_REF] Kojima | Organometal Halide Perovskites as Visible-Light Sensitizers for Photovoltaic Cells[END_REF][START_REF] Lee | Efficient Hybrid Solar Cells Based on Meso-Superstructured Organometal Halide Perovskites[END_REF][START_REF] Park | Organometal Perovskite Light Absorbers Toward a 20% Efficiency Low-Cost Solid-State Mesoscopic Solar Cell[END_REF][START_REF] Burschka | Sequential deposition as a route to high-performance perovskite-sensitized solar cells[END_REF].

Given the versatility offered by the organic part, the design of novel devices with enhanced responses may greatly benefit from a realistic modeling of these HOPs. Such knowledge can be gained from Density Functional Theory (DFT) based simulations. Surprisingly, despite the important relativistic effects expected for lead, the effect of spin-orbit coupling (SOC) has been overlooked [START_REF] Burschka | Sequential deposition as a route to high-performance perovskite-sensitized solar cells[END_REF][START_REF] Sourisseau | Reduced Band Gap Hybrid Perovskites Resulting from Combined Hydrogen and Halogen Bonding at the Organic-Inorganic Interface[END_REF] until recently. In fact, in two recent papers by some of us, we have demonstrated the importance of SOC both for 2D [START_REF] Even | Electronic model for self-assembled hybrid organic/perovskite semiconductors: Reverse band edge electronic states ordering and spin-orbit coupling[END_REF]) and 3D [START_REF] Even | Importance of Spin-Orbit Coupling in Hybrid Organic/Inorganic Perovskites for Photovoltaic Applications[END_REF]) lead-halide based HOPs. The present work aims to generalize our findings using related 2D and 3D structures and compare their properties, especially in terms of optical activity. To this end, we consider two structures for 2D HOPs, namely the orthorhombic and monoclinic phases of (C 5 H 14 N + ) 2 (PbI 4 2-) [START_REF] Billing | Synthesis, characterization and phase transitions in the inorganic-organic layered perovskite-type hybrids [(C n H 2n + 1 NH 3 ) 2 PbI 4 ], n = 4, 5 and 6[END_REF] and(C 2 H 7 IN + ) 2 (PbI 4 2-) [START_REF] Sourisseau | Reduced Band Gap Hybrid Perovskites Resulting from Combined Hydrogen and Halogen Bonding at the Organic-Inorganic Interface[END_REF]) respectively (Fig. 1), and one for the 3D HOPs, namely the low temperature orthorhombic phase of (CH 3 -NH 3 )PbI 3 [START_REF] Baikie | Synthesis and crystal chemistry of the hybrid perovskite (CH 3 NH 3 )PbI 3 for solid-state sensitised solar cell applications[END_REF], hereafter abbreviated to MAPbI 3 . We mention that while numerous selfassembled HOP (SAHOP) systems with lead-halides have been studied, only a few related crystallographic structures are known precisely. In fact, growth of monocrystals for X-ray diffraction is difficult because of the lattice disorder and strain induced by the organic molecule that however plays a fundamental role in the dielectric confinement of 2D HOPs. Nevertheless, we believe that this study will contribute to reach the level of knowledge already attained for conventional semiconductors that will be used as a reference.

Computational details

All DFT calculations were performed with the ABINIT package [START_REF] Gonze | First-principles computation of material properties: the ABINIT software project[END_REF] or VASP code (G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996).). For both 2D HOPs, we employed the LDA or the PBE gradient correction for exchange-correlation [START_REF] Perdew | Generalized Gradient Approximation Made Simple[END_REF] to ensure appropriate comparison to earlier results on (pFC 8 H 12 N + ) 2 (PbI 4 2-

) [START_REF] Even | Electronic model for self-assembled hybrid organic/perovskite semiconductors: Reverse band edge electronic states ordering and spin-orbit coupling[END_REF]) and MAPbI 3 [START_REF] Even | Importance of Spin-Orbit Coupling in Hybrid Organic/Inorganic Perovskites for Photovoltaic Applications[END_REF]. For the latter compound, additional band structure calculations were carried out within the Local Density Approximation (LDA) on the low-temperature orthorhombic phase. A plane-wave basis set with an energy cutoff of 950 eV was used to expand the electronic wave-functions. The reciprocal space integration was performed over 4 × 4 × 4 and 4 × 4 × 1 Monkhorst-Pack grids [START_REF] Monkhorst | Special points for Brillouin-zone integrations[END_REF] for MAPbI 3 , (C 5 H 14 N + ) 2 (PbI 4 2-

) and

(C 2 H 7 IN + ) 2 (PbI 4 2-
), respectively. Energies were computed from the linear response method and convergence was accurately checked with tolerance on the residual potential that stems from differences between the input and output potentials. To reveal the importance of SOC, calculations were carried out both with and without SOC and relativistic norm conserving separable dual-space Gaussian type pseudopotentials [START_REF] Hartwigsen | Relativistic separable dual-space Gaussian pseudopotentials from H to Rn[END_REF] were employed. This approach was already checked for HOP crystals against all-electron computations [START_REF] Even | Importance of Spin-Orbit Coupling in Hybrid Organic/Inorganic Perovskites for Photovoltaic Applications[END_REF].

2D lead halide HOPs

The electronic band structure calculated without SOC of (C 5 H 14 N + ) 2 (PbI 4 ), the optical absorption near the band-edge involves three active Bloch states at the -point: a non-degenerate level for the VB maximum (VBM) and two nearly doubly-degenerate levels for the CB minimum (CBM). This is consistent with the band structure reported for the monoclinic phase of (pFC 8 H 12 N + ) 2 (PbI 4 2-) [START_REF] Even | Electronic model for self-assembled hybrid organic/perovskite semiconductors: Reverse band edge electronic states ordering and spin-orbit coupling[END_REF]. For the orthorhombic phase of (C 5 H 14 N + ) 2 (PbI 4 2-), the number of electronic states involved at the CBM and VBM is twice larger but still degenerated, due to the unit cell doubling along the stacking axis. Nevertheless, this cell doubling has no direct consequences on properties since inorganic layers are electronically decoupled. Indeed, as shown Fig 2(a), no dispersion can be seen along the -Z direction (corresponding to the stacking axis in real space) even though (C 5 H 14 N + ) 2 (PbI 4 2-

) is one of the HOPs with the shortest aliphatic chain (Fig. 1 (a)) [START_REF] Lemmerer | Synthesis, characterization and phase transitions of the inorganic-organic layered perovskite-type hybrids [(C n H 2n+1 NH 3 ) 2 PbI 4 ], n = 7, 8, 9[END_REF]. This 2D confinement is a direct consequence of the dielectric mismatch between PbI 4 anionic layers and the organic cationic sheets [START_REF] Even | Electronic model for self-assembled hybrid organic/perovskite semiconductors: Reverse band edge electronic states ordering and spin-orbit coupling[END_REF]). As a result, the density of states close to the band gap exhibits a reduced 2D dimensionality in connection with the 2D character of excitons reported for related SAHOPs [START_REF] Hong | Dielectric confinement effect on excitons in PbI 4 -based layered semiconductors[END_REF]. Interestingly, the band structure of (C 2 H 7 IN + ) 2 (PbI 4 2-) (Fig. 2(b)) reveals small dispersions of the band edge states along the -Y direction (corresponding to the stacking axis in real space). This effect is rarely observed in the simulations of the electronic properties of 2D HOPs. In fact, the

(C 2 H 7 IN + ) 2 (PbI 4 2-
) crystal structure exhibits a particular feature (Fig. 1 (b)): the iodoethylammonium cations are organized so as to form a sheet of iodine atoms intercalated between the layers. The small dispersion observed along -Y may thus be the consequence of a reduced dielectric mismatch between the sheets and a weak coupling between inorganic layers. Moreover, the DFT band structures reveal a direct band-gap that is preserved when taking SOC into account (Fig. 3). The direct character of the fundamental transition is consistent with the luminescence reported at room temperature for related SAHOPs [2]. For (C 5 H 14 N + ) 2 (PbI 4 2-

), the calculated band gaps amounts to 2.0 and 1.2 eV, respectively without (Fig. 2(a)) and with SOC (Fig. 3). Similarly to earlier findings on (pFC 8 H 12 N + ) 2 (PbI 4 2-

), it evidences a large spin-orbit split-off of the first CB levels (1.2eV) that cannot be reasonably disregarded. Direct comparison to experimental band gaps is hampered due to the well-known underestimation afforded by ground state DFT computations. Including many-body effects, by means of GW self-energy correction for the band gap or resolution of the Bethe Salpeter equation for the excitons, may cure this problem. Unfortunately, such calculations are beyond available computational resources for these large crystal structures. However, the one-shot GW self-energy correction of +0,6eV reported for the cubic phase of CsPbI 3 [START_REF] Even | Importance of Spin-Orbit Coupling in Hybrid Organic/Inorganic Perovskites for Photovoltaic Applications[END_REF]) is a clear indication that GW corrections should be large and act in the opposite direction to the SOC effect that reduces the band gaps red. Thus, we underline that a good agreement between experimental and DFT band gaps calculated without SOC is fortuitous and a consequence of error compensations. Despite this shortcoming, the overall conclusions related to the energy band dispersions and symmetries are reliable and can be used to build semi-empirical Hamiltonians, such as a k.p model, where detailed information of Bloch states and selection rules are required, or inspect the band gap changes related to phase transitions. In fact, the calculated band gap change when going from the monoclinic low temperature phase (293K) to the orthorhombic high temperature phase (334K) of (C 5 H 14 N + ) 2 (PbI 4 2-

) amounts to 0.15eV both with and without SOC. This band gap reduction is consistent both with the color change from orange to red reported by Billing and Lemmerer and with that reported on a C 12 -alkylammonium.

Overall, the present results on (C 5 H 14 N + ) 2 (PbI 4 ) [START_REF] Even | Electronic model for self-assembled hybrid organic/perovskite semiconductors: Reverse band edge electronic states ordering and spin-orbit coupling[END_REF]. The fundamental transition of these 2D lead halide HOPs displays a nearly-perfect transverse electric character (TE). This TE character might be further enhanced in SAHOP thanks to orientational disorder introduced by the organic part. It is similar to that reported for conventional zinc-blende quantum wells with D 2d point symmetry or anisotropic würtzite crystals having C 6v point symmetry. However, symmetry and ordering of the Bloch states are reversed between the HOPs and the tetrahedrally bonded conventional semiconductors. This is also the case for the SOC splitting that acts on the CB in HOPs while it splits VB in conventional semiconductors. Pushing further the analogy with conventional semiconductors, one can define Kane matrix elements (P ij = ħ M VBMi,CBMj /m e , where M VBMi,CBMj is the dipolar matrix elements between states VBMi and CBMj) and derive corresponding Kane energies. For III-V semiconductors, the Kane energy E 11 is about 20eV and excitons have low binding energies of a few meV, typical for extended Wannier-Mott excitons. In contrast, the Kane energy of the present 2D HOPs is four times smaller and excitons are strongly bound, given the large binding energies of a few hundreds of meV reported in related SAHOP [START_REF] Hong | Dielectric confinement effect on excitons in PbI 4 -based layered semiconductors[END_REF]. We believe that both the exciton's binding energy and the Kane energy are at the origin of the peculiar optical properties of this class of materials.

3D versus 2D lead halide HOPs

Contrarily to the 2D hybrids, 3D HOPs show sizeable dispersion in the whole Brillouin zone as illustrated Fig. 4. Therefore, theory predicts an almost isotropic absorption for these materials. The direct band gap at the center of the Brillouin zone is consistent with the strong absorbance reported at room temperature [START_REF] Tanaka | Comparative Study on the Excitons in Lead-Halide-Based Perovskite-Type Crystals CH 3 NH 3 PbBr 3 CH 3 NH 3 PbI 3[END_REF][START_REF] Lee | Efficient Hybrid Solar Cells Based on Meso-Superstructured Organometal Halide Perovskites[END_REF]. The fundamental transitions for the low temperature phase of MAPbBr 3 calculated at the GGA level of theory amount to 2.0 eV (Fig. 4 (a)) without SOC with VASP code. Identically, the fundamental transitions for the low temperature phase (T<161K) [START_REF] Baikie | Synthesis and crystal chemistry of the hybrid perovskite (CH 3 NH 3 )PbI 3 for solid-state sensitised solar cell applications[END_REF] of MAPbI 3 calculated at the GGA level of theory amount to 1.6 eV (Fig. 4 (b)) and 0.7 (Fig. 4 (c)) respectively without and with SOC with VASP code. These values confirmed the large effect of SOC [START_REF] Even | Importance of Spin-Orbit Coupling in Hybrid Organic/Inorganic Perovskites for Photovoltaic Applications[END_REF]. Moreover, the splitting is large and comparable to that calculated for the 2D lead halide HOPs. These theoretical results are consistent with the SOC effects revealed in optical reflectivity [START_REF] Hirasawa | Exciton Features in 0-Dimensional, 2-Dimensional, and 3-Dimensional Networks of PbI 6 4-Octahedra[END_REF] and absorption [START_REF] Tanaka | Comparative Study on the Excitons in Lead-Halide-Based Perovskite-Type Crystals CH 3 NH 3 PbBr 3 CH 3 NH 3 PbI 3[END_REF] measurements on MAPbI 3 . Thus, the same conclusion holds regarding SOC inclusion for relevant band structure calculations on such materials. In addition, as for the 2D HOPs, accurate prediction of absolute band gaps of these 3D analogues requires also many-body technics whereas both dispersions and symmetry considerations stay reliable on the DFT level of theory. Among the noticeable differences between 2D and 3D lead-halide HOPS considered in this work, strain effects induce a splitting of the VBM in the latter (Fig. 4). Strain effects act also on the CBM but the splitting introduced by SOC effects is much larger leading to comparable band gap reductions in both 2D and 3D HOPs. The high temperature (T>330K) phase of MAPbI 3 is cubic and the room temperature phase is a slightly distorted tetragonal phase [START_REF] Baikie | Synthesis and crystal chemistry of the hybrid perovskite (CH 3 NH 3 )PbI 3 for solid-state sensitised solar cell applications[END_REF]. The analysis of critical electronic states most relevant for applications can indeed be performed using the cubic phase as a reference [START_REF] Even | Importance of Spin-Orbit Coupling in Hybrid Organic/Inorganic Perovskites for Photovoltaic Applications[END_REF]). When SOC is not taken into account, the cubic phase has a triply-degenerated CB associated to the vectorial representation of a cubic simple group, commonly described by the X , Y , Z symbols. For the low temperature orthorhombic phase considered here, the number of CB states is doubled (Fig. 4 (b)). In the corresponding double group including spinors, the CB of the high temperature cubic phase gets split by SOC into twofold degenerate states and fourfold degenerate states [START_REF] Even | Importance of Spin-Orbit Coupling in Hybrid Organic/Inorganic Perovskites for Photovoltaic Applications[END_REF]. A similar effect is predicted for the low-temperature phase although strain effects must be taken into account for a detailed analysis (Fig. 4 (c)). The high (and room) temperature CBM is associated to the twofold degenerate spin-orbit split-off states and These spin-orbit split-off states lead to strong and isotropic optical transitions with the even S -like VBM states. We underline that for both 2D and 3D HOPs, one obtains reversed ordering of band-edge states as compared to tetrahedrally bonded conventional semiconductors. In 3D crystals, strain effects induce an additional contribution to splitting on the conduction bands. For both types of HOPs, the calculated band gap is direct in agreement with strong absorbance or luminescence observed at room temperature. The electronic band structure of 2D hybrids reveals effectively the 2D character with none or little dispersion in the direction of reciprocal space corresponding to the layers staking axis, depending on the nature of the cationic layer that may affect the dielectric contrast between organic and inorganic sheets. The fundamental transitions have a nearly perfect transverse electric character, similarly to zinc-blende quantum wells or würtzite bulk compounds with respectively D 2d or C 6v point symmetry. In contrast, for 3D hybrids almost isotropic optical activity is expected from symmetry. Indeed, for the high temperature cubic phase, this activity stems from transitions between a triply degenerated conduction-band and a single valence-band, in a simple group representation. In the double group representation, the resulting twofold degenerate spin-orbit split-off states are build from a weighted mixture of these conduction band states that retain the isotropic nature of the optical activity.

Summary

Based on DFT calculations and symmetry analyses of the band-edge states, the electronic and optical properties of both 2D and 3D lead-halide hybrid perovskites have been investigated. In these materials, spin orbit effects induce a dramatic splitting of the conduction bands and appear to dominate their band gap. These results are consistent with evidence of SOC in reflectivity and absorption spectra reported earlier on MAPbX 3 . Both types of hybrids exhibit a reversed band ordering when compared to conventional

       Z 3 1 iY X 3 1 2 / 1 , 2 / 1         Z 3 1 iY X 3 1 2 / 1 , 2 / 1
semiconductor quantum wells and bulk materials.

Fig. 1 .

 1 Fig. 1. Orthorhombic and monoclinic crystal structures of (C 5 H 14 N + ) 2 (PbI 4 2-) (a) and (C 2 H 7 IN + ) 2 (PbI 4 2-) (b), respectively.

  a) and (b) respectively. For the monoclinic phase of (C 2 H 7 IN + ) 2 (PbI 4 2-

Fig. 2 .

 2 Fig. 2. Electronic band structures of the (C 5 H 14 N + ) 2 (PbI 4 2-) (a) and (C 2 H 7 IN + ) 2 (PbI 4 2-) (b) compounds without SOC calculated at the GGA PBE level. The origin of the energy scale is taken at the top of the VB.

Fig. 3 .

 3 Fig. 3. Electronic band structures of (C 5 H 14 N + ) 2 (PbI 4 2-) with SOC calculated at the GGA PBE level. The origin of the energy scale is taken at the top of the VB.

Fig. 4 .

 4 Fig. 4. Electronic band structures of MAPbBr 3 without (a) SOC, MAPbI 3 without (b) and with (c) SOC calculated at the GGA level. The origin of the energy scale is taken at the top of the VB.

Acknowledgment

This work was performed using HPC resources from GENCI CINES and IDRIS 2013-2013096724. The work is supported through the participation of the PEROCAI ANR project.


Résumé :

La détermination de l'ensemble des 6 actions de coupe présente un grand intérêt lors de la modélisation des procédés de coupe tels que le fraisage, le tournage et le perçage dans le but d'optimiser les paramètres géométriques et cinématiques et ainsi améliorer la qualité des surfaces obtenues et la durée de vie des outils. Les modélisations analytiques ou semi-analytiques de coupe donnent souvent un ordre de grandeur de ces actions mécaniques de coupe à la pointe de l'outil. L'intérêt de ces modélisations est lié à la rapidité de l'obtention des résultats et au faible coût par rapport à la mise en oeuvre de dispositifs expérimentaux. Quatre modèles analytiques présents dans la littérature (modèle de [START_REF] Merchant | Mechanics of the metal cutting process, I: orthogonal cutting[END_REF], modèles thermomécaniques d' [START_REF] Hastings | A machining theory for predicting chip géometry, cutting forces etc. from work material proprieties and cutting conditions[END_REF] de [START_REF] Gilormini | Contribution à la modélisation de la formation du copeau en usinage des métaux[END_REF] de [START_REF] Dargnat | Modélisation semi-analytique par approche énergétique du procédé de perçage des métaux monolithiques[END_REF]) sont analysés dans ce travail, en portant une attention particulière à leurs domaines de validité. Ainsi, les effets de la variation de chaque paramètre géométrique et cinématique sur les résultats obtenus par chacune des modélisations sont présentés et comparés, en tenant compte des hypothèses propres à chaque modélisation. De ce fait, cette étude est réalisée en configuration de coupe orthogonale, configuration la plus simple à modéliser, qui suppose une arête de coupe perpendiculaire à la vitesse de coupe et à l'avance.

Abstract:

The determination of all the 6 mechanical actions of cutting processes is very interesting for modeling of milling, turning and drilling in order to optimize the geometric parameters, kinematic, improve the quality of the obtained surface and the tool life. Analytical or semi-analytical models of cutting often give an order of the mechanical action magnitude in the cutting tool tip. The interest of these models is linked to the rapidity of the results obtaining and to the moderate cost compared with experimental approach In the literature there are several main analytical models (Merchant model -1945, thermomechanical Oxley models -1980, [START_REF] Gilormini | Contribution à la modélisation de la formation du copeau en usinage des métaux[END_REF][START_REF] Dargnat | Modélisation semi-analytique par approche énergétique du procédé de perçage des métaux monolithiques[END_REF]. They are analyzed in this work, paying particular attention to their validity domains. Thus, the effects of variation of each parameter on the geometric and kinematic results obtained by each of the models are presented and compared, taking into account the specific modeling assumptions. This study will be conducted in orthogonal cutting configuration, because of its simplicity for modeling the cutting process, this configuration assumes that the cutting edge is perpendicular to the cutting speed and feed directions.

Présentation des modèles

La détermination des efforts de coupe par modélisation analytique est abordée dans nombreux travaux de recherche ayant pour but l'amélioration du procédé d'usinage. Ces modélisations peuvent intégrer plusieurs paramètres, ce qui permet de quantifier l'ensemble des grandeurs physiques apparaissant lors de l'opération d'usinage. Compte tenu de la complexité des phénomènes physiques présents, la coupe orthogonale a été la configuration la plus exploitée dans la modélisation analytique de coupe. Dans cette configuration, la vitesse de coupe est perpendiculaire à l'arête de coupe, le rapport entre l'avance et la largeur de coupe est d'au moins de 10 et l'état de déformation est bidimensionnel (cas de déformation plane). Les trois modèles seront appliqués à l'acier 42CrMo4.

Modèle de Merchant (1945)

Dans la modélisation de la coupe orthogonale de Merchant [START_REF] Merchant | Mechanics of the metal cutting process, I: orthogonal cutting[END_REF], la formation du copeau est supposée être due à un cisaillement intense dans un plan de cisaillement incliné d'un angle 𝛷 par rapport à la direction de la vitesse de coupe (V c ) comme indiqué sur la figure 1.

FIG.1 -Modèle géométrique de Merchant

Dans cette modélisation, la zone de dépouille n'est pas prise en compte (pas de rayon d'arête) et le matériau usiné est supposé parfaitement plastique. Une contrainte d'écoulement en cisaillement τ est définie. La contrainte de cisaillement dans la zone de cisaillement primaire est définie par la relation de Merchant suivante:

τ ab = ( ) ( ) ( ) (1)
La contrainte de compression dans la zone de cisaillement primaire est définie par la relation [START_REF] Hastings | A machining theory for predicting chip géometry, cutting forces etc. from work material proprieties and cutting conditions[END_REF]. Le comportement du matériau dans la zone primaire est traduit par la loi de Caquot-Bridgman. σ = τ ab .

(𝛷 ) = ( ) ( ) ( ) avec 𝛷 = .(C 1 +γ-τ) (2)
L'effort de coupe et l'effort tangentiel s'expriment par : coefficient de frottement cohérent. Le modèle de Merchant permet une variation plus importante des angles de coupe positifs jusqu'à 30 degrés, alors que cette variation est plus réduite pour les angles de coupe négatifs (inférieur à 20 degrés).

F c = ( ) ( ) ( ) ( ) , F t = ( ) ( ) ( ) ( ) (3)

Modèle d'Oxley (1980)

Le modèle purement mécanique de Merchant repose sur un comportement parfaitement plastique du matériau. Il est donc incomplet dans la mesure où il ne prend pas en compte la sensibilité de la contrainte d'écoulement à l'écrouissage, à la température et à la vitesse de déformation. Dans le but de mettre en place une approche plus réaliste, Oxley [P.L.B, Oxley-1967] a considéré un comportement thermo-viscoplastique du matériau. La contrainte d'écoulement le long de la zone de cisaillement primaire est de la forme :

σ ab = ( ) ( ) (5)
La déformation et la vitesse de déformation subie par un élément de matière ayant traversé la zone de cisaillement primaire sont exprimées par :

ε ab = ( ) √ ( ) ( ) , ȧb = ( ) ( ) √ ( ) (6)
L'effort de coupe et l'effort tangentiel s'expriment par :

F c = R.cos (λ-α), F t = R.sin (λ-α) avec R : résultante d'efforts= ( ) = √ .b.h. ( ) ( ) (9)

Effet de la variation de l'angle de coupe et d'avance sur le coefficient de frottement apparent du modèle d'Oxley

Pour une vitesse de coupe de 100 m/min et une largeur de coupe de 4 mm, les résultats du modèle d'Oxley sur un acier 42CrMo4 sont représentés par la courbe figure3 :

FIG. 3-Effet de l'angle de coupe et de l'avance sur le coefficient de frottement apparent A partir des courbes, la variation du coefficient de frottement en fonction de l'angle de coupe est d'avance est similaire au modèle de Merchant.

Modèle de Gilormini (1982)

Le modèle de Gilormini [P. Gilormini-1982] Dans la zone de cisaillement secondaire, Le modèle de Gilormini propose une nouvelle description de l'échauffement pour les zones de déformation. Le champ de vitesse est proposé avec une température uniforme dans l'épaisseur du copeau à la sortie de cisaillement primaire. Le profil de température dans l'épaisseur du copeau est constitué de deux paraboles de concavité inverse.

Copeau

FIG.4-Température et vitesse dans le copeau

La valeur moyenne de la température moyenne à l'interface outil copeau T bc est donnée par: T bc = T ab+ g(Ca). . . √

L'effort de coupe et l'effort tangentiel sont alors exprimés par :

F c = ̇ = , F t = ( ) ( √ ( )) (14), (15)

Effet de la variation de l'angle de coupe sur le coefficient de frottement apparent du modèle de Gilormini

FIG. 5: Effet de l'angle de coupe et de l'avance sur le coefficient de frottement apparent pour V c =100 m/min, b=4 mm Contrairement aux résultats du modèle de Merchant et d'Oxley, le domaine de validité du modèle de Gilormini par rapport à une variation de l'angle de coupe est plus important (figure5 (a)) et le coefficient de frottement apparent décroit en fonction de l'avance (figure5 (b)). Les valeurs de ces derniers sont de -30 ° à plus de 30 °. Pour l'avance de 0.075 mm le coefficient de frottement apparent varie brusquement pour un angle de coupe de 5° puis il revient à sa variation normale, cette constatation est de plus en plus remarquable pour les avances les plus faibles. Cette instabilité est due à des divergences de calcul pour quelques valeurs d'angle de coupe (pour une avance de 0.075 mm et au voisinage de l'angle de coupe 5° l'effort de coupe passe de 577 N à 1890 N puis il revient à 612 N).

Comparaison de la variation de coefficient de frottement apparent en fonction de la vitesse de coupe entre les modèles d'Oxley et Gilormini

Pour une avance élevée de 0.3 mm et deux angles de coupe proches des valeurs expérimentales (0 ° et -10 °) les résultats de la variation du frottement apparent, μ app en fonction de la vitesse de coupe pour les modèles d'Oxley et de Gilormini sont présentés sur la figure 6. Nomenclature : τ ab : Contrainte de cisaillement dans la zone de cisaillement primaire (MPa) A s , A 0 : Section cisaillée, Section coupée (mm²) 𝛷, : angle de cisaillement primaire, Angle de coupe (°) Fs : Effort tangentiel dans la zone de cisaillement primaire (N) Fc : Effort de coupe (N) λ : Angle de frottement dans la zone de cisaillement secondaire (°) b , h , L c : largeur de coupe, Avance, longueur de contact (mm) τ 0 : Paramètre du matériau (MPa), C 1 : Paramètre du matériau (°) ̅ ̇ ̅ ̇ : Vitesse de déformation et vitesse de déformation de référence (s -1 ) V c , V: Vitesse de coupe et vitesse de copeau (m.s -1 ) σ ab , σ bc : Contrainte d'écoulement dans la zone de cisaillement primaire et secondaire (MPa) T ab , T bc : Température dans la zone de cisaillement primaire et secondaire C, Ca : paramètre d'Oxley, nombre de Caremon ρ, S, K : masse volumique (kg/m³) , chaleur spécifique (J/kg.°C), conductivité thermique (W/m.°C) b t :effusivité thermique (J/m².√ .°C) , ̇ : Puissance (W)

1

  Analyse de la sensibilité et des domaines de validité des modélisations analytiques de la coupe W. Yousfi a , R. Laheurte a , M. Calamaz b , P. Darnis a , O. Cahuc a a. Université de Bordeaux, I2M, UMR 5295, F-33400 Talence, France b. Arts et Métiers Paris Tech, I2M, UMR 5295, F-33400 Talence, France

  Mots clefs : Usinage, Efforts de coupe, Modélisation thermomécanique, Coupe orthogonale, Domaine de validité.

, ( 4 ) 1 . 1 . 1 FIG. 2 -

 41112 FIG.2-Variation du coefficient de frottement apparent en fonction de l'angle de coupe et l'avanceLes valeurs du coefficient de frottement (μ app =F t /F c ) à considérer sont celles comprises entre 0 et 1, qui ont une signification physique. Les angles de coupe varient entre -15 et 30 degrés pour un

FIG. 6 :

 6 FIG.6: Effet de la vitesse de coupe sur le coefficient de frottement apparent ( =0 ° et =-10 °) La variation du coefficient de frottement apparent est nulle en fonction de la vitesse de coupe pour le modèle d'Oxley alors que cette variation est faiblement décroissante et de valeur plus faible pour le modèle de Gilormini. En augmentant la vitesse de coupe l'effort tangentiel diminue sensiblement pour le modèle Gilormini alors que cette variation est proportionnelle à l'effort de coupe pour le modèle d'Oxley de manière à maintenir un coefficient de frottement constant. Cette constatation peut mettre en évidence l'effet de l'adoucissement thermique du matériau en augmentant la vitesse de coupe sur le coefficient de frottement pour le modèle de Gilormini.1.4 Modèle de Dargnat : déformations et vitesses de déformation dans le plan de cisaillement primaire:Un élément de matière passant de la pièce au copeau suit une trajectoire hyperbolique et selon le modèle de Dargnat[Dargnat-2006]. Les variations de la déformation et de la vitesse de déformation en fonction de l'angle de cisaillement et de coupe sont présentées sur la figure 7 :

  tient compte d'une zone de cisaillement secondaire supposée rectangulaire. Le contact entre le copeau et l'outil est collant et le profil de vitesse dans le copeau croit linéairement dans l'épaisseur du cisaillement secondaire, et est uniforme sur le reste du copeau. Dans la zone de cisaillement primaire, la contrainte d'écoulement est définie par : σ ab =

	̅ =	√	( )	( ( )	)	, ̅̅̅̅ ̇=	( )	(	) avec 𝛷=artg (	( ) ( )	)	(11), (12)
	apparent				1 1,4 1,8			a	apparent	0,8			b
	coefficient	-40	-0,2 0,2 0,6 Angle de coupe en ° 0 20 µapp,h=0.05 mm -20 µapp,h=0.025 mm 40 µapp,h=0.075 mm µapp,h=0.1 mm µapp,h=0.125 mm µapp,h=0.25 mm µapp,h=0.35 mm µapp,h=0.5 mm	0,3 -0,2 Coefficient	0	0,1 µapp=f(h);γ=30 ° 0,2 avances (mm) 0,3 µapp=f(h);γ=20 ° 0,4 0,5 µapp=f(h);γ=10 ° µapp=f(h);γ=30 ° µapp=f(h);γ=0 °	0,6

̅ .(T ab -T 0 ) (10) La déformation et la vitesse de déformation subies par un élément de matière ayant traversé la zone de cisaillement primaire ont les expressions suivantes:

  Gilormini et de Dargnat avec des valeurs plus faibles pour ce dernier. Cette vitesse est très sensible au changement d'angle de coupe, avec une variation croissante pour le modèle de Dargnat et décroissante pour celle de Gilormini. La déformation dépend de l'avance dans le modèle d'Oxley alors que cette sensibilité à l'avance est moins remarquable dans les deux autres modèles. Au-delà d'une certaine valeur de l'angle de coupe, le coefficient de frottement est supérieur à 1. Les résultats de modèle d'Oxley ne sont alors plus stables et les déformations et vitesses de déformation varient aléatoirement rendant les résultats inexploitables. 'amélioration des modèles analytiques de coupe fait partie des objectifs de tous les travaux de recherche en modélisation dans le but d'avoir un domaine de validité plus important avec des résultats les plus fiables possible. Ce travail consiste à présenter pour deux modèles de base (modèle d'Oxley et modèle de Gilormini) et un modèle récent(Dargnat-2006) le domaine de validité de chacun ainsi que leurs sensibilités à quelques paramètres de coupe tels que la vitesse de coupe (V c ) et l'avance (h). A partir de ces résultats on a constaté que le domaine de validité s'améliore pour chaque nouveau modèle et les résultats deviennent plus stables et moins perturbés par les variations extrêmes des paramètres tels que les angles de coupe négatifs. Le modèle de Dargnat est le modèle le plus réaliste physiquement, avec une modélisation plus fine de chaque zone de déformation basée sur la mécanique des milieux continus permettant une détermination des résultats (température, déformation …) en chaque point discrédité. "Ce travail a été conduit dans le cadre du groupe de travail Manufacturing 21 qui regroupe 11 laboratoires de recherche français. Les thèmes étudiés sont : la modélisation du processus de fabrication, l'usinage virtuel, le développement de nouvelles méthodes de fabrication."

	1.5 Conclusion :

L