Brahim Nabi

Anne-Laure Helbert
email: anne-laure.helbert@u-psud.fr

François Brisset

Thierry Waeckerlé

Rémy Batonnet

Thierry Baudin

Effect of the hot rolling on Goss development and magnetic induction in an advanced soft magnetic Fe-27%Co alloy

Keywords: Fe-27%Co alloy, Hot rolling, Goss texture, Magnetic induction

This paper shows a complete study about the characterization of the texture and the microstructure evolution of an industrial thermo-mechanical processing aiming to develop an advanced soft magnetic Fe-27%Co material with a sharp Goss texture. Thanks to its high magnetic performances, this alloy is able to replace the grain oriented Fe-Si alloy in transformers of electric aircrafts application.

It appears that the development of a highly Goss textured Fe-27%Co alloy could be achieved by optimizing the intensity of the Goss and rotated Cube texture during hot rolling using an adequate thermo-mechanical processing, namely finishing hot rolling in the ferritic phase.

Introduction

Due to the increase of the electric loads, the need of power and number of transformers in the aircrafts increases. This implies a growing demand for the aircraft industry to reduce the noise emitted by such equipment. This noise is due to the magnetostriction phenomenon and electromagnetic forces in the magnetic sheets. The magnetostriction, which is defined as a change of material shape or dimensions during the process of magnetization, is a consequence of magnetocrystalline anisotropy partly linked to the sheet crystallographic texture. A Fe-27%Co alloy, denominated industrially AFK1, has been existing for a long time and is devoted to magnetic circuit of aeronautic generators. This material brings high power density in airborne electrical engineering, saving precious masses during flights. Such performances of AFK1 are based on its physical advantages, which the main ones are a high saturation magnetization value (2.38 T) and rather good magnetic properties (coercive field, permeability) at the same level as FeSi N.O. (Non Oriented) electrical steel. The interest of AFK1 remains high despite its drawbacks: rather high value of magnetocrystalline constant (K1 of about 45 kJ/m3), rather high apparent saturation magnetostriction (~ 35ppm) and rather high cost. The usual AFK1 grade exhibits low texture components, then a weakly anisotropic magnetic behaviour at the macroscopic scale, well adapted to rotating electromagnetic machines. As electrical steels, it should be called AFK1 N.O.

As a continuation of Westinghous works in the 70s, Aperam Alloys Imphy develops new AFK1 grades, acting on the metallurgical route. The main one is based on the AFK1 ability to develop a Goss texture ({110}<001> orientation), at the end of an adequate thermo-mechanical processing. If Goss texture is improved and if magnetic losses are further reduces, the AFK1 sheet should be able to get a place in the market of transportation transformers, requiring high power density, such as high speed train, large-scale machines in civil engineering or manufacturing, where Fe-3%Si Grain Oriented grades are currently used.

The Goss texture offers the best magnetic properties necessary for such application [START_REF] Bernier | EBSD study of angular deviations from the Goss component in grain-oriented electrical steels[END_REF][START_REF] Sun | Effect of Single Pass Hot Rolling on Goss Texture Development in Twin-Roll Cast Silicon Steel Strip[END_REF] since high quantity of <100> crystallographic axes (easy magnetization direction) are aligned in the sheet rolling direction. Therefore, a good understanding of the origin and the development of the Goss texture in the different steps of the industrial processing of the AFK1 is necessary for the optimization of its magnetic properties.

The origin of the Goss texture obtained after recrystallization during the Goss industrial processing [3] of the grain oriented Fe-3%Si alloy was widely investigated [START_REF] Bernier | EBSD study of angular deviations from the Goss component in grain-oriented electrical steels[END_REF][START_REF] Sun | Effect of Single Pass Hot Rolling on Goss Texture Development in Twin-Roll Cast Silicon Steel Strip[END_REF][3][START_REF] Gheorghies | Evolution of Texture in Grain Oriented Silicon Steels[END_REF][START_REF] Böttcher | Influence of subsurface layers on texture and microstructure development in RGO electrical steel[END_REF][START_REF] Han | Rolled textures and magnetic properties of regular grain oriented 3% silicon steel with very low thickness[END_REF][START_REF] Mishra | On the development of the goss texture in iron-3% silicon[END_REF][START_REF] Samet-Meziou | Recovery and recrystallization study after low deformation amount by cold rolling in an IF-Ti steel[END_REF][START_REF] Inokuti | Shimanakai Transmission Kossel study of the structure during secondary recrystallization in grain oriented silicon steel[END_REF][START_REF] Sakakura | Effects of AlN on secondary recrystallization textures in cross rolled and annealed (001) (hkl) oriented single crystals of pct Si-Fe[END_REF][START_REF] Walter | An effect of impurity atoms on the energy relationship of (100) and (110) surfaces in high purity silicon iron[END_REF][START_REF] Dorner | Overview of Microstructure and Microtexture Development in Grain-oriented Silicon Steel[END_REF][START_REF] Cruz-Gandarilla | Texture and microstructure evolution in a Fe-Si CGO sheet during the processing route before the secondary recrystallization[END_REF][START_REF] Cruz-Gandarilla | A study of local microstructure and texture heterogeneities in a CGO Fe3%Si alloy from hot rolling to primary recrystallization[END_REF]. On one hand, many investigations on single or poly crystal of Fe-Si alloy showed that the Goss texture of the primary recrystallization develops by the inheritance of this texture from hot rolling [START_REF] Gheorghies | Evolution of Texture in Grain Oriented Silicon Steels[END_REF][START_REF] Walter | An effect of impurity atoms on the energy relationship of (100) and (110) surfaces in high purity silicon iron[END_REF]. In fact, the Goss orientation present after hot rolling turn into the {111}<112> component during cold rolling and then reappear during the primary recrystallization.

On the other hand, other investigations have shown that the origin of the Goss texture of the primary recrystallization in Fe-Si alloys may derive from two types of Goss texture. By using EBSD technic, Dorner et al. [START_REF] Dorner | Overview of Microstructure and Microtexture Development in Grain-oriented Silicon Steel[END_REF] studied the evolution of the Goss texture in a single crystal of Fe-Si deformed to 89% oriented initially {110}<001>. They showed that the Goss component could come from either the inheritance of the Goss texture of the hot rolling or the strong cold deformation which involves the development of the Goss texture in the shear bands of the cold rolled microstructure. Later, in 2008, in a polycrystalline of Fe-3%Si, Cruz-Gandarilla et al. [START_REF] Cruz-Gandarilla | Texture and microstructure evolution in a Fe-Si CGO sheet during the processing route before the secondary recrystallization[END_REF] showed similar results. Besides, it has been shown that for steels, the finishing hot rolling temperature had a decisive influence on the textures produced [START_REF] Vanderchueren | Influence of transformation induced recrystallisation on hot rolling textures of low carbon steel sheet[END_REF][START_REF] Butronguillen | Effect of finishing temperature on hot band textures in an IF steel[END_REF].

In this paper, the evolution of the Goss texture from the hot rolled state to the final heat treatment of the Fe-Co electrical sheet, has been investigated in order to optimize the final Goss volume fraction of the product. The link between Goss texture and rolling process parameters will be studied all along the thermo-mechanical transformation and especially the role of the finishing hot rolling temperature and the associated texture will be discussed.

Material and experiments

The industrial Iron-Cobalt alloy with 27% (wt %) cobalt and 0.5% of Chrome, called AFK1, was provided by Aperam Alloys Imphy. The forged state was followed by a hot rolling (HR) to a final thickness of 2 mm. Six different hot rolling conditions (HR1 to HR6) were tested, they were numbered as a function of their magnetic induction. Then, the sheets underwent exactly the same thermomechanical route: a first annealing at 930°C (A1), followed by a first 70% cold rolling (CR1). An intermediate annealing at 930°C (A2) was then applied before a second 70% cold rolling (CR2). Finally, the double cold rolled sheets underwent three successive final annealing treatments at 930°C (A3, A4 and A5) enabling the primary recrystallization and the normal grain growth to improve the Goss texture development.

The local texture was carried out by EBSD and the global one was measured by X-Ray diffraction. EBSD results were collected using a scanning electron microscope Zeiss FEG SUPRA 55 VP operating at 20 kV and equipped with the OIM TM system. EBSD component fractions were determined with a 15° dispersion. The samples were firstly mechanically polished and then electropolished (Struers A2 solution) at 22 °C, with a voltage of 15 V for 12 s. X-ray data were collected from a Siemens system (K α cobalt radiation) and the {110}, {200} and {112} incomplete pole figures were recorded. The data were then analyzed using the arbitrarily defined cells (ADC) method implemented in the Labotex software to calculate the complete pole figures and the orientation distribution function (ODF).

Results

a. Relationship between the texture and the magnetic properties.

Figure 1 shows the evolution of the magnetic induction (B 10Oe ) for sheets coming from various hot rolling conditions and the associated Goss fraction for two of them available for investigation. Induction and Goss fraction were measured after the last annealing A5. It appears that induction strongly depends on hot rolling conditions. Moreover, for this Fe-Co alloy, magnetic induction is correlated to the Goss intensity, as already reported in the literature for Fe-Si alloys [START_REF] Bernier | EBSD study of angular deviations from the Goss component in grain-oriented electrical steels[END_REF][START_REF] Cruz-Gandarilla | Texture and microstructure evolution in a Fe-Si CGO sheet during the processing route before the secondary recrystallization[END_REF].

b. Texture evolution during the industrial elaboration process

As Goss texture is conditioning the magnetic properties, it seem important to understand how the Goss texture develops during sheet elaboration. The texture evolution has been determined at each step of the thermomechanical process and ODFs sections are reported in figure 2. This evolution has been characterized for samples HR1 and HR6, which correspond to extreme magnetic induction values among the samples, the lowest (HR1) and the highest (HR6). The A1 texture was measured by EBSD in the RD (Rolling Direction)-ND (Normal Direction) plan to study the texture gradient through thickness. The other states were characterized in the RD-TD (Transverse Direction) plan. Unfortunately, Aperam Alloys Imphy does not succeed to be sure about the hot rolling conditions of the samples, especially the finishing hot rolling temperatures. The latter have been sent by Aperam after the first annealing A1 for analyses. After the first annealing at 930°C (A1), the HR1 texture is quasi-isotropic. On the contrary, HR6 is mainly characterized by a strengthening around the rotated Cube (18%) and the Goss components (9%). After the first cold rolling (CR1) to 70 % thickness reduction, the texture of both samples is characterized by γ-fiber components ({111}//rolling plane) and the presence of a partial αfiber (<110>//rolling direction). The {111}<112> component is more pronounced in the case of the sample derived from HR6. The second annealing at 930°C (A2) allows the development of the Goss component, which is more pronounced in the case of HR6. After the second cold rolling (CR2), the development of the {111}<112> component is once again observed. Finally, the final annealing treatments (A3, A4 and A5) lead to the development of the Goss texture and its strengthening. Then, after the final annealing A5, the Goss fraction coming from HR6 (60%) is nearly twice that of the sample coming from HR1 (35%). The HR6 hot rolling conditions are clearly more efficient to achieve a strong Goss texture and thus good magnetic properties. A comparison of both texture filiations (Figure 2) allows to note that only the first texture after A1 is notably different from a sample to the other. More, it appears that the final Goss intensity strongly depends on this initial texture. In order to better describe this A1 texture through the sheet thickness, a cross section of both sheets has been studied by EBSD.

First of all, a heterogeneity of grain size along the thickness of both sheets HR1 and HR6 is observed (Figure 3). Grain sizes of about 35 and 45 µm at the center and 70 and 75 μm at the surfaces are obtained for HR1 and HR6, respectively. This difference between center and surface grain size may be due to the shearing introduced by the mills during the hot rolling. Indeed, more stored energy in the shearing areas induces an acceleration of recrystallization and then of grain growth, while these last processes are delayed in the sheet center. More, as it can be clearly seen in Figure 3, texture of both samples after A1 is totally different depending on the hot rolled conditions. A quasi isotropic texture is obtained throughout the thickness of the sheet HR1, while the HR6 exhibits a heterogeneous texture: the mid-thickness of the sheet is characterized by a high intensity of the rotated Cube component while the surface is characterized by a domination of the Goss one (Figure 4). As it is known from the literature, the rotated Cube texture is characteristic of hot rolling texture of bcc materials [START_REF] Mishra | On the development of the goss texture in iron-3% silicon[END_REF], which explains the development of such texture in the case of the Fe-27%Co alloy hot rolled. The Goss texture is characteristic of the shearing texture [START_REF] Bernier | EBSD study of angular deviations from the Goss component in grain-oriented electrical steels[END_REF][START_REF] Mishra | On the development of the goss texture in iron-3% silicon[END_REF]. Nevertheless, as shown in Figure 4, the texture heterogeneity through thickness can be more or less pronounced depending on the hot rolling conditions.

(a) (b)

Discussion

Figure 5 shows the evolution of the main texture component fractions as a function of the thermo-mechanical stages, for the hot rolled samples HR1 and HR6 after A1. The thermo-mechanical history of these sheets is a sequence of peaks corresponding to the {111}<112> orientation development during cold rolling and the Goss intensity increases during annealing treatments. Except after A1 for HR6 (20%), the rotated Cube fraction is quite constant and lower than 10% during the different thermo-mechanical steps. Comparing Figures 5a) and 5b), it is shown, for this Fe-27%Co alloy, that annealing Goss texture is all the more strong that the {111}<112> component is intense at the preceding cold rolling state. This is in agreement with results of the literature on single (or poly) crystal(s) of Fe-Si alloy [START_REF] Gheorghies | Evolution of Texture in Grain Oriented Silicon Steels[END_REF][START_REF] Walter | An effect of impurity atoms on the energy relationship of (100) and (110) surfaces in high purity silicon iron[END_REF], where it has been considered that the {111}<112> intensity is at the origin of the Goss development during the primary recrystallization. This point will be discussed below.

(a) (b) Since the final Goss fraction depends on the {111}<112> intensity, the origin of the {111}<112> after cold rolling has been investigated in previous works in Fe-Co alloys [START_REF] Nabi | Nouveaux alliages Fe-Co magnétiques pour l'aéronautique, à microstructure partiellement recristallisée et à texture fortement orientée[END_REF]. It has been shown that during cold rolling, Goss crystals can rotate by 35° around the transverse direction to form the {111}<112> crystals and that rotated Cube grains can rotate by 55° around TD to become {111}<112> grains. Both Goss and rotated Cube orientations have then a crucial position to promote {111}<112> formation during cold rolling. Proofs are given in Figure 5a) where a too small amount of Goss and rotated Cube (<5%) at the initial stage (A1) puts in danger the {111}<112> formation during CR1 and thus weakens the final Goss amount (A5). On the contrary, initial fractions of Goss and rotated Cube greater than 9% ensure the {111}<112> development during CR1 and lead to a high final Goss fraction (60%).

The aim of the present work is to determine which hot rolling conditions favor the Goss development during the thermo-mechanical process of Fe-27%Co sheet elaboration. The final Goss development has been associated to the non-neglecting fraction of Goss and rotated Cube at the A1 stage. Because hot rolling conditions for HR1 and HR6 were not provided, then, it has been decided to elaborate four new samples (HR2 to HR5), changing hot rolling temperatures, in order to reproduce the typical magnetic inductions of samples HR1 and HR6 (Figure 1). The goal is to deduce the unknown conditions and to find the best ones to optimize magnetic properties.

These magnetic properties have been successfully reproduced for samples HR2 (close to HR1) and HR5 (close to HR6) (Figure 1). Corresponding microstructures and textures are described in Figure 6. The HR2 hot rolling has been entirely performed in the austenitic phase (> 930°C) whereas the HR5 hot rolling has been finished at a lower temperature (around 760°C) in the ferritic domain. HR3 and HR4 correspond to intermediate conditions that will not be detailed in the present study. Figure 6 shows that the microstructure strongly depends on the hot rolling conditions. In the case of the austenitic hot rolling (HR2), microstructure is quite homogeneous and recrystallized through the thickness (Figure 6a) and texture is weak. On the contrary, when hot rolling is finished in the ferritic phase (HR5), the microstructure is still deformed. As desired, a high texture heterogeneity is obtained through the thickness (t) of HR5 with a high fraction of rotated Cube (Figure 6b) in the mid-thickness (15% (0.3% Goss)) and a noticeable Goss orientation at the sheet surface over a distance of t/4 at each side (11% (4% rotated Cube)). Because of the good magnetic induction of HR5 (Figure 1), it can be assumed that the final Goss fraction (after A5) is high. This is not surprising since there is already, at the hot rolled state, a high quantity of Goss and rotated Cube necessary to develop the {111}<112> during cold rolling and thus to guaranty an intense Goss at the end of the thermomechanical process (A5) (case of Figure 5b).

To sum up, ending the hot rolling at a temperature below the austenite/ferrite transformation seems to guaranty the Goss and rotated Cube formation during hot rolling and then ensures good magnetic induction properties for the final product. As proposed by Butron-Guillen and Jonas [START_REF] Butronguillen | Effect of finishing temperature on hot band textures in an IF steel[END_REF] in IF steels, when the finishing rolling temperature is in the austenite temperature range, the Cube texture forms in the austenite phase and is converted, during the ferritic transformation, into the rotated Cube, Goss and rotated Goss orientations. For the Fe-27%Co alloy, these three first components are observed but in low quantities (about 3%). Butron-Guillen and Jonas [START_REF] Butronguillen | Effect of finishing temperature on hot band textures in an IF steel[END_REF] showed that when the rolling is finished in the ferritic range, after the transformation starts, the texture is more intense (ODF max = 13 against 5 for HR2), and apart from marked Goss and rotated Cube, it also contains the typical  and  fibers encountered in bcc cold rolled materials. In agreement, in the present work, the  and  fibers are quite developed in HR5 (20 and 15% respectively). The role of the finishing rolling temperature is then crucial as far as texture is concerned.

As it was already shown in Figure 6, comparing to HR2 and HR5, the hot rolling performed in the ferritic domain (HR5) develops higher intensity of Goss grains in areas in contact with the rolling mills. This can be attributed to the higher shear rate induced by the low hot rolling temperature.

To explain this trend at low temperature, it could be assumed that during the hot rolling, the high deformation rate (shearing plus compression) at the surface of the hot rolled sheet, results in the creation of numerous shear bands. Inside these bands, the local shear will induce the creation of nuclei of the Goss orientation [START_REF] Nabi | Nouveaux alliages Fe-Co magnétiques pour l'aéronautique, à microstructure partiellement recristallisée et à texture fortement orientée[END_REF][START_REF] Shuai | Development of Goss texture in Al-0.3%Cu annealed after heavy rolling[END_REF][START_REF] Jiao | Microstructural evolution and magnetic properties in strip cast non-oriented silicon steel produced by warm rolling[END_REF] and subsequently by dynamic recrystallization, these Goss grains will grow into the matrix at the expense of the γ fiber. Actually, this phenomenon of Goss nucleation inside shear bands has been observed, not during hot rolling because the dynamic kinetic makes it difficult to catch, but during static recrystallization after cold rolling. Figure 7a shows the presence of shear bands located inside the {111}<112> cold rolled grains. Goss sub-grains of 1 μm diameter are identified inside these shear bands (Figure 7b). For high rates of reduction in the Fe-Si, Dorner et al. [START_REF] Dorner | Retention of the Goss orientation between microbands during cold rolling of an Fe3%Si single crystal[END_REF] During the primary recrystallization, occurring during the subsequent annealing treatments A3 to A5, the Goss texture, present in the shear bands of the {111}<112> grains, develops at the expense of this last orientation [START_REF] Cruz-Gandarilla | Texture and microstructure evolution in a Fe-Si CGO sheet during the processing route before the secondary recrystallization[END_REF]. This development, which occurs by Strain Induced Boundary Migration (SIBM) [START_REF] Humphreys | Recrystallization and Related Annealing Phenomena[END_REF], is related to the stored energy gap, measured at the cold rolling state, between the Goss sub-grains and the surrounding {111}<112> matrix. The stored energy was estimated by ESBD using the KAM parameter (Kernel Average Misorientation) [START_REF] Wright | Electron Backscatter Diffraction in Materials Science[END_REF]. This last parameter was calculated with a kernel size of 1 micron excluding the misorientation greater than 5°. Figure 8 shows the KAM values for {111} <112> grains and for their associated Goss sub-grains inside shear bands.

The Goss sub-grains exhibit lower energy (Figure 8) than the surrounding matrix, which explains their recrystallization at the expense of the {111}<112> component, during annealing [START_REF] Etter | Influence of the Goss grain environment during secondary recrystallisation of conventional grain oriented Fe-3%Si steels[END_REF].

Conclusion

The Goss formation during thermo-mechanical processing of the advanced Fe-27%Co alloy has been studied.

The hot rolling conditions are determinant to obtain a large amount of Goss orientation in the final product. The presence of Goss and rotated Cube at the rolling stage guaranty the Goss development during the entire process.

A low temperature at the end of hot rolling allows large shear deformation, especially induced by mill rolls contact, at the origin of shear bands. Goss nuclei are formed inside these shear bands. During further annealing treatments, these nuclei grow at the expense of the surrounding matrix, since the Goss nuclei have lower stored energy and thus have an advantage to grow by strain induced boundary migration.

A hot rolling ending in the ferritic phase implies higher shearing and then higher intensity of the final Goss texture.

From an industrial point of view, applying a hot rolling in the ferritic domain will induce an enhancement of the magnetic properties of the product.

Figure 1 :

 1 Figure 1: Evolution of the magnetic induction B 10Oe and Goss fraction of samples after A5 annealing with different initial hot rolling conditions

Figure 2 :

 2 Figure 2: Texture evolution during the thermo-mechanical processing obtained by EBSD (A1) and by X-Ray diffraction of samples (a) HR1 and (b) HR6

Figure 3 :

 3 Figure 3: Microstructures of two hot rolled sheets after the first annealing A1 at 930°C. (a) HR1, (b) HR6.ND-IPF maps.

Figure 4 :

 4 Figure 4: Texture heterogeneity through thickness in the HR1 and HR6 samples after first annealing A1. Rotated Cube and Goss fractions are measured at the mid-thickness and at the surface of the sheets

Figure 5 :

 5 Figure 5: Evolution of the main texture components during the thermo-mechanical processing of the hot rolled sheet (a) HR1 and (b) HR6

Figure 6 :

 6 Figure 6: Hot rolling microstructures of (a) austenitic hot rolling HR2 and (b) ferritic hot rolling HR5. Upper map : ND-IPF map, lower map : Rotated Cube (red) and Goss (green)

  of these new Goss grains in shear bands by the effect of the local shearing which implies a sudden rotation of the {111}<112> component into the Goss one.

Figure 7 :

 7 Figure 7: Cold rolled grains having the {111} <112> orientation with shear bands containing Goss sub-grains. (a) Distribution of the image quality parameter (that describes the quality of an electron backscatter diffraction pattern) with a step of 0.1 microns and (b) pixels belonging only to Goss orientation (green). Sample HR5 after second cold rolling CR2.

Figure 8 :

 8 Figure 8: Distribution of KAM parameter for Goss sub-grains in shear bands and surrounding {111}<112> grains. Sample HR5 after the second cold rolling CR2.

Acknowledgements

The authors acknowledge 'La region Ile de France' for the funding of the 3MT project.


domain: phys chim spi

text:
Doriane Djomani

Francesco Capitani

Jean Blaise Brubach

Eugenio Calandrini

Charles Renard

Daniel Bouchier

Jean Paul Itié

Pascale Roy

Laetitia Vincent
email: laetitia.vincent@u-psud.fr

Jean Blaise Brubach

Jean Paul Itié

Atypical reversed pressure-induced phase transformation in Ge nanowires

Keywords: Phase transformation, pressure diamond anvil cell, nanowires, Raman spectroscopy, infrared absorption

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

INTRODUCTION

There has been a renewed interest in the last few years for the synthesis of Si and Ge allotrope structures offering modified band gap or innovative heterostructures for band engineering in group IV semiconductors. In Si and Ge nanowires, we have developed a process to produce hexagonal 2H phases along the 3C nanowires [START_REF] Vincent | Shear-driven phase transformation in silicon nanowires[END_REF][START_REF] Vincent | Novel Heterostructured Ge Nanowires Based on Polytype Transformation[END_REF]. The unveiled martensitic transformation occurring under shear stress suggests new mechanisms of deformation-induced phase transformation at the nanoscale.

In recent years, theoretical and experimental studies showed that mechanical properties are significantly modified at the nanoscale due to surface effects. Nanowires generally exhibit modified mechanical properties with decreasing diameter size, especially in the elastic regime -i.e. higher stiffness, larger yield stress, lower compressibility- [START_REF] Gerberich | Scale effects for strength, ductility, and toughness in "brittle[END_REF][START_REF] Ngo | Ultimate-strength germanium nanowires[END_REF]. For instance bulk modulus was reported to increase by 25% in Si nanowires (diameter of 70 nm) [START_REF] Wang | Phase Transition and Compressibility in Silicon Nanowires[END_REF]. Atomistic simulations have shown that intrinsic surface stresses are responsible for such phenomena because they induce substantial intrinsic compressive stresses that may lead to spontaneous reorientations or phase transformations. Additionally, edges on free surfaces form high stress points and they can act as preferential nucleation site for defects and/or shear failure.

Deformation mechanisms (dislocation formation, slip and twinning) can thus differ significantly from the bulk and can cause unexpected phase transformation pathway in nanowires.

Bulk crystalline germanium is known to exhibit several allotropic phases under compression and decompression [START_REF] Gogotski | High Pressure Surface Science and Engineering[END_REF][START_REF] Katzke | Theory of the high-pressure structural phase transitions in Si[END_REF]. Most of the allotrope phases obtained by static compression of a sample in a high-pressure diamond anvil cell (DAC) have been identified by means of in-situ measurements of electrical resistivity, X-ray diffraction and Raman spectroscopy in the last forty years. The cubic diamond phase 3C (Ge-I) transforms into the metallic β-tin phase (Ge-II) under hydrostatic compression conditions around 10 GPa [START_REF] Gaál-Nagy | Transition pressures and enthalpy barriers for the cubic diamond >β-tin transition in Si and Ge under nonhydrostatic conditions[END_REF][START_REF] Menoni | Germanium at high pressures[END_REF].

The pressure range where this transformation takes place is larger when hydrostatic conditions are not fulfilled and shear stress is involved: the transformation threshold is lowered to 7 GPa if compression is performed without any pressure transmitting medium [START_REF] Qadri | High pressure studies of Ge using synchrotron radiation[END_REF] and is completed around 12 GPa [START_REF] Menoni | Germanium at high pressures[END_REF][START_REF] Qadri | High pressure studies of Ge using synchrotron radiation[END_REF]. The β-tin phase of Ge is then stable up to 75 GPa.

The decompression leads to the formation of distinct metastable phases denser than the 3C form, even though they display a higher energy [START_REF] Mujica | First-principles calculations of the structural properties, stability, and band structure of complex tetrahedral phases of germanium: ST12 and BC8[END_REF]. Kinetics of transformation (i.e. enthalpy barrier) may account for this irreversibility (see in particular figure 2f in ref [START_REF] Wang | Kinetic Origin of Divergent Decompression Pathways in Silicon and Germanium[END_REF] showing that enthalpy barrier of Ge-βTin ! Ge-3C is larger).

The decompression leads mainly to a metastable tetragonal phase ST12 (referred as

Ge-III). The same phase was identified upon pressurization and decompression of amorphous Ge (a-Ge) [START_REF] Coppari | Pressure-induced phase transitions in amorphous and metastable crystalline germanium by Raman scattering, x-ray spectroscopy, and ab initio calculations[END_REF]. However, depending on the unloading conditions (hydrostaticity, kinetics, temperature) another metastable cubic phase BC8 (Ge-IV) can be obtained. At room temperature, Ge-BC8 is very unstable and transforms into the hexagonal 2H structure (Ge-V).

The phases formed after decompression, Ge-ST12 and Ge-BC8, are still under debate. In fact, both transformations Ge-βtin ! Ge-ST12 and Ge-βtin ! Ge-BC8 exhibit similar energetic barriers [START_REF] Wang | Kinetic Origin of Divergent Decompression Pathways in Silicon and Germanium[END_REF], and both structures are energetically equivalent [START_REF] Mujica | First-principles calculations of the structural properties, stability, and band structure of complex tetrahedral phases of germanium: ST12 and BC8[END_REF][START_REF] Wang | Kinetic Origin of Divergent Decompression Pathways in Silicon and Germanium[END_REF][START_REF] Malone | Electronic structure, equation of state, and lattice dynamics of low-pressure Ge polymorphs[END_REF]. Note that another intermediate phase Ge-R8 was observed solely by Haberl et al. [START_REF] Haberl | Controlled formation of metastable germanium polymorphs[END_REF] and no calculations are available regarding the barrier of Ge-βtin ! Ge-R8.

Experimentally, transition pathways depend strongly on the unloading conditions. In most cases, the ST12 phase is formed. However, BC8 is expected for very fast unloading [START_REF] Nelmes | Stability and crystal structure of BC8 germanium[END_REF] and low temperature [START_REF] Bates | High-Pressure Transitions of Germanium and a New High-Pressure Form of Germanium[END_REF][START_REF] Brazhkin | Nonequilibrium phase transitions and amorphization in Si, Si/GaAs, Ge, and Ge/GaSb at the decompression of high-pressure phases[END_REF]. According to Haberl et al. [START_REF] Haberl | Controlled formation of metastable germanium polymorphs[END_REF] the critical parameter dominating the transition pathway is the local stress environment. They showed that BC8 can be formed if very strict hydrostaticity is fulfilled. However, the total absence of shear is quite difficult to obtain in DAC. It is important to note that the control of the decompression pathway is challenging and no consensus appears in the literature still after 30 years of studies.

The size-dependency of the phase transformation is another important parameter to investigate. The purpose of these experiments was to clarify the different transformation pathways of Ge-β-tin phase during decompression and evidence how nanoscale effects may affect the transformation in Ge nanowires. Therefore, size effects were examined by comparing the decompression pathways in Ge bulk and in nanowires (40 nm in diameter).

EXPERIMENTAL SECTION

Sample synthesis

N-type Ge with a resistivity on the range 1-9 Ω.cm was employed for bulk measurement. For optical measurements, round pellets with a diameter of 250 µm and a thickness of around 20 µm was shaped and mechanically polished from a <111>-oriented Ge substrate. For XRD measurements, a powder of microcrystallites was obtained by milling in ethanol a piece of the aforementioned substrate.

Ge nanowires were synthesized by VLS growth process using ultra-high vacuum chemical vapor deposition. Details of the experimental procedure were described previously [START_REF] Renard | Size effect on Ge nanowires growth kinetics by the vapor-Liquid-Solid mechanism[END_REF]. We have used nanowires of about 2 µm length and a mean diameter of 40 nm. After the growth, gold catalysts were removed by chemical etching with KI:I 2 :H 2 O solution. Ge nanowires have been scrapped off from their growth substrate with a scalpel blade and inserted in the DAC with a pressure transmission medium. The nanowires with the pressure medium (KBr, NaCl or PE) form a heap that will be compressed to a final sintered pellet.

High pressure set-up

Two membrane diamond anvil cells (DAC) with a culet surface of 500 µm diameter were used as pressure device (300 µm for XRD measurements). The samples were loaded into 150 µm hole drilled in a preindented stainless steel gasket with a thickness of 50 µm.

Different pressure transmitting media were employed depending on the performed characterizations and the transparency of the media to the wavelength range. Polyethylene (PE) and Ne were used for XRD measurements to check the effect of hydrostaticity. PE or NaCl were used for Raman measurements and KBr was used for infrared absorption measurements. For pressure monitoring, a ruby crystal or gold powder (in the case of XRD) were inserted in the DAC. The highest pressure attained was about 20 GPa.

Compression/decompression cycles were carried out at room temperature. Decompression rate was about 1 GPa/min either for bulk and nanowire measurements. After decompression the pellets can be handled for further analyses. Samples were for instance prepared by focused ion beam for transmission electron microscopy (see supporting information SI).

Diffraction set-up

Psiché beamline at synchrotron SOLEIL is partially dedicated to diffraction under extreme conditions. The x-ray diffraction pattern is recorded using a CCD detector (Marccd) or a flat panel (Perkin Elmer) detector. The high-pressure cell is mounted on a goniometer which allows to rotate the cell during acquisition. A ruby luminescence set-up is installed on the beamline in order to measure pressure. The pressure is increased using a membrane connected to a helium bottle under pressure. The x-rays are collimated down to 80 µm using two pairs of slits. The monochromatic beam is obtained using a double crystal monochromator with Si-(111) crystals and the beam is focused vertically with a large vertical mirror down to 100 µm. To obtain powder diffraction data the sample used in that case was beforehand milled as described previously.

Raman set-up

Raman spectra were collected using a Horiba iHR320 spectrometer equipped with a 1800 grooves/mm grating, a Peltier cooled CCD, a red solid state laser (632 nm) as excitation source, a fiber coupled Horiba Raman Head with a 50x objective and an edge filter to reject the Rayleigh line in the backscattering geometry. Raman spectra were calibrated using the emission lines of a Ne lamp.

Optical set-up

Infrared absorption spectra were recorded by means of a Bruker 125 spectrometer coupled to a custom high pressure setup [START_REF] Voute | New high-pressure/low-temperature set-up available at the AILES beamline[END_REF]

RESULTS and DISCUSSION

Structure and pathway determined by XRD

In situ compression/decompression XRD measurements have been performed to monitor the evolution of the lattice parameters as a function of the pressure and to reveal the formation of new phases through the appearance and disappearance of Bragg peaks (figure 1).

In bulk, when increasing the pressure, the expected Ge-3C to β-tin phase transformation occurred around 8.5 GPa as indicated by the emergence of the (200) diffraction peak of Ge-βtin. The transition onset was the same for Ne and PE transmitting media indicating similar (non)-hydrostatic conditions. In the 8.5-12 GPa pressure range, 3C and β-tin phases coexisted (figure 1). Increasing pressure up to 15 GPa led to the total disappearance of the main (111) peak of Ge-3C and thus confirmed the purity of the Ge β-tin phase. The pressure was then stabilized over 12 hours.

Shortly after the decompression that occurred in a couple of seconds (ultra-fast decompression only in that case), we observed the presence of a mixture of Ge-ST12 and a small quantity of Ge-BC8 as shown in figure 2. Similar spectrum was obtained 24 hours later, although the amount of Ge-BC8 seemed to decrease with time according to the reduction of the intensity of the peak corresponding to the Bragg reflection on the (211) planes of Ge-BC8 at 7.57°. This suggested the transformation to the expected 2H phase, nonetheless we did not identify any 2H diffraction peak in the spectrum probably due to the superimposition of the expected low intensity 2H peaks with the Ge-ST12 peaks and from the gasket. For lower rate decompression (1GPa/min), performed with PE transmitted medium, only Ge-ST12 was found. It is worth noticing, the absence of Ge-3C on all the spectra after the decompression.

Ge nanowires inserted in the DAC with Ne were loaded up to 17.5 GPa. From the diffraction patterns (Figure 3) the lattice parameter and the unit-cell volume of Ge-3C in nanowires were obtained. In supporting information, figure SI.1 illustrates the evolution of the crystal cell volume as a function of pressure enabling to obtain respectively the isothermal bulk modulus B 0 and its first pressure derivative B' 0 . We found B' 0 = 4 and B 0 = 73.8 GPa which are in well agreement with the experimental values reported for bulk [START_REF] Menoni | Germanium at high pressures[END_REF][START_REF] Raffy | Properties of hexagonal polytypes of group-IV elements from first-principles calculations[END_REF][START_REF] Cicco | Highpressure and high-temperature study of phase transitions in solid germanium[END_REF]. Thus, contrarily to results obtained from Ge nanocrystallines [START_REF] Wang | High-pressure structural behaviour of nanocrystalline[END_REF], no size effect is observed here on the compressibility of the Ge nanowires down to a diameter of 40 nm. Above 12 GPa, the phase transformation is evidenced in figure 3 by the emergence of the ( 200) and (101) Bragg diffraction peaks of Ge-β-tin phase and the concomitant intensity reduction of the Ge-3C peaks (see the indexed peaks). A coexistence of Ge-3C and Ge-β-tin phases was reported up to 15 GPa. The transformation range for nanowires (12-15 GPa) is therefore shifted towards higher pressure compared to their bulk counterpart under the same loading conditions although bulk modulus derived from P-V measurements is comparable to the reference value of bulk. This result suggests a higher stability for nanometric size.

The pressure was maintained at 18 GPa for 12 hours to ensure the total transformation and stabilization of the Ge-β-tin compressed structure. After unloading, the pattern (figure 3) evidenced the restoration of diffraction peaks that can be indexed to Ge-3C. Thus Ge-β-tin phase in nanowires go back to 3C structure after decompression as opposed to the bulk. This may be attributed to size effects on the transition kinetics.

Raman pathway determination

As a complementary characterization, we have used in-situ Raman spectroscopy to probe vibrational dynamics under pressure in order to confirm the differences in decompression pathways between bulk and nanowires. Samples were loaded in the DACs with NaCl pressure medium. Raman spectra were recorded up to 14 GPa (figure 4a). The first order Raman peak of the Ge-3C structure, corresponding to the F 2g mode, was found at 304 cm -1 indicating a small pressure induced by the DAC setting. Indeed, this band shifted from 300 cm -1 value at 0 GPa [START_REF] Olego | Pressure dependence of Raman phonons of Ge and 3C-SiC[END_REF] to higher frequency with pressure. At P=10 GPa, the peak intensity increased and the band edge opens up to about 2 eV [START_REF] Welber | Effect of hydrostatic pressure on the direct absorption edge of germanium[END_REF], corresponding to the energy of the excitation laser (632 nm), thus giving rise to a resonant Raman effect.

Compression led ultimately to the disappearance of the F 2g band indicating that the Ge-3C has been totally transformed. Above 12 GPa, the spectra were dominated by a new broad band at low frequency 220 cm -1 corresponding to the TO phonon mode of the compressed Ge-β-tin [START_REF] Olijnyk | Raman scattering in metallic Si and Ge up to 50 GPa[END_REF]. This peak also presented a blue shift with increasing pressure.

During decompression, spectra compatible with the Ge-β-tin phase remained down to 8 GPa. There was clear evidence for the transformation toward Ge-ST12 with the appearance of the six characteristic peaks in the 150 to 350 cm -1 range [START_REF] Huston | Thermal stability of simple tetragonal and hexagonal diamond germanium[END_REF][START_REF] Zhao | Properties of the exotic metastable ST12 germanium allotrope[END_REF] as shown in figure 4b.

For comparison, figure 5 shows the first order Raman spectra on compressed and decompressed Ge nanowires. Under compression, the main F 2g peak was blue shifted with a progressive increase of FWHM from 10.5 to 12.5 cm -1 . The TO peak of the β-tin structure appeared above 9.5 GPa and a mixture of both Ge-3C and Geβ-tin was observed up to 13 GPa. The complete phase transition was unveiled by the disappearance of the F 2g mode at 14 GPa. The transformation onset was thus similar for both nanowires and bulk but the coexistence pressure range of Ge-3C and Geβ-tin was larger for nanowires, although this range was reduced using NaCl instead of Ne (used for XRD measurements -suggesting that shear stresses reduce the pressure range of transformation). Following the decompression, the β-tin phase appeared stable down to 9 GPa. At 4.7 GPa, the F 2g mode of the Ge-3C phase reappears at 321.4 cm -1 . The broad band around 270 cm -1 is characteristic to an amorphouslike phase during decompression. After complete unloading, the F 2g peak is finally located at 307 cm -1 indicating a compressive strain in the final state. From this decompressed bulk pellet, XTEM sample was prepared using focused ion beam milling. Ge-ST12 is clearly evidenced by the SAED pattern in the <110> zone axis in figure SI.2 of supporting information. The sample remains monocrystalline with the appearance of some fractures.

Optical results in the near infrared

Figure 6 shows the absorption coefficient spectra recorded in bulk before compression and after decompression unveiling clearly the irreversibility. There is no ambiguity on the difference of optical inter-band absorption between both structures Ge-3C and Ge-ST12 (Fig. 6a). For Ge-3C, the allowed direct transition obtained from Tauc plot analysis is measured at 0.9 eV at an initial pressure of 0.03 GPa in somewhat moderate agreement with the generally accepted value of 0.82 eV [START_REF] Dash | Intrinsic Optical Absorption in Single-Crystal Germanium and Silicon at 77°K and 300[END_REF][START_REF] Goi A R, Syassen | Direct-band-gap absorption in germanium under pressure[END_REF]. For Ge-ST12 after release, Tauc plot analyses illustrated in figure 6 (b andc) provide an indirect allowed and a direct allowed transitions with an energy of 0.53 eV and 0.73 eV respectively. These measurements agree with recently reported results of 0.59 eV and 0.74 eV from Zhao et al. [START_REF] Zhao | Properties of the exotic metastable ST12 germanium allotrope[END_REF]. Note that a pressure between 2 or 3 GPa is generally retained in the DAC after a cycle of compression and decompression present for all the experiments carried out, such residual stress may therefore lead to an overestimation of the gap values in the final state.

Absorption

coefficients measured in NWs before and after the compression/decompression cycle are shown in figure 7. Before compression, the direct energy transition was calculated by Tauc plot analysis at 0.8 eV. After decompression, Tauc analysis allows to estimate a direct gap value of 0.9 eV which is in agreement with Ge-3C with some residual pressure [START_REF] Goi A R, Syassen | Direct-band-gap absorption in germanium under pressure[END_REF] (note that a value of 2 or 3 GPa was estimated with the ruby fluorescence after decompression). This confirms again the reversible transition in nanowires.

The transformation cycle has formed a polycrystalline Ge-3C pellet (as shown in SI.3).

Discussion

Under compression in DAC with identical conditions, Ge bulk and bundles of Ge nanowires (φ= 40 nm) experience similar behavior although a higher stability is observed in NWs. This higher stability may be explained by a possible lower shear stress or size effect.

As expected, the loading/unloading transformation cycle at room temperature is irreversible in bulk. In most cases the decompression from the pressurized β-tin phase is the metastable Ge-ST12 phase. The studies reported in bulk showed the importance of decompression rate and shear stresses for the phase transformation during release. The shear strain in DAC generally enhances the formation of ST12-Ge as described in [START_REF] Haberl | Controlled formation of metastable germanium polymorphs[END_REF], and also in the high-pressure torsion experiments [START_REF] Ikoma | Allotropic phase transformation and photoluminescence of germanium nanograins processed by highpressure torsion[END_REF]. The high rate enables the formation of BC8 [START_REF] Haberl | Controlled formation of metastable germanium polymorphs[END_REF][START_REF] Nelmes | Stability and crystal structure of BC8 germanium[END_REF]. With our experimental conditions, BC8 was only observed after very fast decompression

(1 GPa/s) within a mixture with Ge-ST12. No trace of 3C phase was detected in bulk whatever the decompression conditions. This irreversible behavior may be controlled by kinetics of the transformation pathways as demonstrated by Wang et al. [START_REF] Wang | Kinetic Origin of Divergent Decompression Pathways in Silicon and Germanium[END_REF].

In contrast, XRD, Raman and infrared spectroscopy clearly evidence a different pathway in Ge nanowires; the structure goes back to Ge-3C and no trace of Ge-ST12 was observed. For these measurements, same decompression rate (1 GPa/s) was used for bulk and nanowires. Indeed, in the case of a nanowire heap filling the DAC, shear stresses are also introduced as a consequence of bridging the diamond culets as well as direct interactions between nanowires. For the case of Ge bulk, decompression rate and shear stress had no effect on the "reversible" formation of Ge-3C. Then the major difference in phase transformation kinetics is ascribed here to size effects. As described in ref. [START_REF] Wang | Kinetic Origin of Divergent Decompression Pathways in Silicon and Germanium[END_REF] Ge-3C may be realized when the system has a chance to overcome the kinetic barrier. Our observations suggest that interface energies may increase the total energy of the system and in turn modifies kinetics of transition under decompression. To model this effect, further experiments are required to gain a comprehensive knowledge as a function of nanowire size.

Conclusion

We have evidenced a size dependence on the phase transformation pathways under Figure 2: XRD patterns of Ge bulk powder after decompression at room temperature from Ge-β-tin structure. The X-ray wavelength was 0.3738 Å. After a fast decompression (GPa/s) Ge-ST12 and Ge-BC8 are present.

Asterisk indicates an unresolved peak that could be composed of two peaks one arising from the gasket.

Figure 3: XRD patterns of Ge nanowires under compression up to 17.5 GPa and after decompression at room temperature in Ne transmitting pressure medium. The X-ray wavelength was 0.3344 Å. After unload, the pressure was around 0.5 GPa. Asterisks in yellow, blue and grey identify peaks from resp. Au, Ne and stainless-steel gasket. The values of the gap energies are obtained from the linear fits and correspond to the energy value at zero absorption. (In a sake of clarity of the curve, the origin of the y-axis in plot b is not zero, and the gap value has to be extrapolated the exact value is 0.53 eV).

  . The setup features a couple x15 Cassegraintype objectives to focus the radiation onto the sample and a system of motorized translations and flip mirrors allow the continuous control of the alignment under vacuum. The pressure applied on the sample was determined by in situ measuring the ruby fluorescence. To match the spectral range of Ge gap, the experimental setup exploited the tungsten internal source, a Quartz beamsplitter and the InSb detector. An aperture of 5 mm was focused by the Cassegrain to create a light spot onto the 250 microns area of the sample inside the hole of the gasket. Absorption spectra were determined by the following relation A = -log(I s /I 0 ), where I s and I 0 are the light intensities transmitted by the sample and the empty DAC, respectively.

  decompression of Ge bulk and nanowires. The decompression of bulk leads to the formation of ST-12 structure while in nanowires the structure returns back to the initial standard cubic diamond phase 3C. Phases were fully identified by XRD, Raman and near-infrared spectroscopy. Moreover, NIR absorption provided the optical absorption edge in Ge-ST12 phase. The reversal transformation to Ge-3C in NWs may be accounted for an effect of interface energy on the transformation kinetic. Further experiments coupled with theoretical simulations could help in understanding the size effect on the phase transitions. Acknowledgement This work was supported by PhD grant and funding from the LIDEX Nanodesign plateform and in the framework of the ANR HEXSIGE project (ANR-17-CE030-0014-01) funded by the French Agence Nationale de la Recherche. Samples were performed in micro/nano fabrication facility of C2N, which is part of the national network Renatech of French clean rooms centers.

Figure caption Figure 1 :

 caption1 Figure caption

Figure 4 :

 4 Figure 4: Raman spectra in the range 200-530 cm -1 recorded on bulk Ge for different values of pressure up to 14 GPa in (a) and after decompression in (b). Ge-ST12 peaks are indexed in the upper part of (b).

Figure 5 :

 5 Figure 5: Raman spectra on Ge NWs (a) under compression and (b) after decompression.

Figure 6 :

 6 Figure 6: (a) absorption coefficients extracted from transmittance measurements for Ge-3C bulk before compression and for Ge-ST12 after decompression (2 GPa). Derived Tauc's plot of Ge-ST12 (b) for indirect allowed optical transition (c) for direct allowed optical transition.The values of the gap energies are obtained from the linear fits and correspond to the energy value at zero absorption. (In a sake of clarity of the curve, the origin of the y-axis in plot b is not zero, and the gap value has to be extrapolated the exact value is 0.53 eV).

Figure 7 :

 7 Figure 7: (a) absorption coefficients extracted from transmittance measurements for Ge-3C NWs before compression and after decompression.

Figure 4 Figure 7

 47 Figure 4