Karen Al Hokayem
email: karen.al-hokayem@univ-lorraine.fr

Jaafar Ghanbaja

Stéphanie Michel

Sophie Legeai

Nicolas Stein

Insights in the two-step synthesis of single crystalline Ag 2 Te nanorods

Keywords:

Pure monoclinic single crystalline Ag2Te nanorods were synthesized by a two-step method. The first step concerns the electrodeposition of self-standing Te nanorods in an ionic liquid medium, taking advantage of its templating properties. These obtained nanorods are single crystalline with growth direction along the [001] axis in the hexagonal lattice. The average diameter of the as-deposited Te nanorods is 60 ± 13 nm with less than 300 nm long. Afterwards, the resulting nanostructured Te film is immerged in a dedicated silverbased bath in order to be transformed into Ag2Te binary compounds by a combined mechanism of chemical cementation and topotactic transformation. The chemical analysis reveals that the final nanostructures exhibit a slight excess of silver and TEM-SAED analysis shows that they are still single crystalline with an increase of the initial diameter, less than 30%. HR-TEM highlights the presence of an intermediate stützite phase and the hessite phase α-Ag2Te is obtained at the end of the synthesis process.

Introduction

Silver tellurides (AgTe, Ag2Te, Ag5Te3 and Ag7Te4…) are semi-conducting compounds that can be used in data storage devices [START_REF] Batabyal | Axialjunction nanowires of Ag2Te-Ag as a memory element[END_REF], optical materials [START_REF] Prabhune | Measurement of properties of silver telluride thin films using holography[END_REF] and magnetic measurements [START_REF] Lee | Band-gap tuning and linear Email address: karen.al-hokayem@univ-lorraine.fr magnetoresistance in the silver chalcogenides[END_REF]. Like numerous metal tellurides (PbTe, Bi2Te3, Cu1.75Te…), Ag2Te is of high interest as thermoelectric material, converting heat into electricity and vice versa [START_REF] Chang | Facile synthesis of Ag2Te nanowires and thermoelectric properties of Ag2Te polycrystals sintered by spark plasma sintering[END_REF]. In the current energy crisis context, thermoelectricity appears as a promising renewable energy route to collect heat waste thus minimizing energy losses. The best thermoelectric materials are characterized by a low thermal conductivity associated with a maximum electrical conductivity and Seebeck coefficient. At the nanoscale, thermoelectric performances can be improved by increasing the number of interfaces and the frequency of diffuse phonon reflection. Ultimately, the phonon mean free path is reduced and consequently the thermal conductivity decreases [START_REF] Lee | Thermoelectric Properties of a Single Crystalline Ag2Te Nanowire[END_REF].

Ag2Te exists in 3 different phases: monoclinic α which undergoes at 418 K a phase transition into cubic β, this latter being stable between 418 and 1075 K and the phase γ between 1075 and 1233 K. Among these phases, monoclinic Ag2Te has the best thermoelectric properties [START_REF] Fujikane | Thermoelectric properties of α-And β-Ag2Te[END_REF]. This compound behaves like a semiconductor with a narrow band gap [START_REF] Pei | Alloying to increase the band gap for improving thermoelectric properties of Ag2Te[END_REF] that can be adapted for low temperature thermoelectric applications [START_REF] Wang | Design guidelines for chalcogenide-based flexible thermoelectric materials[END_REF].

The composition can be varied in order to control the nature of the charge carriers to obtain Te-rich structures (p-type) or Ag-rich structures (n-type) [START_REF] Yang | Composition modulation of Ag2Te nanowires for tunable electrical and thermal properties[END_REF]. Ag2Te has a high power factor due to high carrier mobility (Hall mobility µH = 6000 cm²/V.s at 300 K), a low lattice thermal conductivity (K = 1 W/m.K) and a high electric conductivity (σ = 25 S/m at 300 K) [START_REF] Pei | Alloying to increase the band gap for improving thermoelectric properties of Ag2Te[END_REF]. Park et al. have measured reduced thermal conductivity (K = 0.35 W/m.K) for nanorods by comparison with bulk values [START_REF] Park | Facile fabrication of one-dimensional Te/Cu2Te nanorod composites with improved thermoelectric power factor and low thermal conductivity[END_REF]. Hence this work will focus on the synthesis of nanostructures of Ag2Te.

Ag2Te nanostructures can be obtained by different methods: solvothermal [START_REF] Chang | Facile synthesis of Ag2Te nanowires and thermoelectric properties of Ag2Te polycrystals sintered by spark plasma sintering[END_REF], [START_REF] Yang | Composition modulation of Ag2Te nanowires for tunable electrical and thermal properties[END_REF], [START_REF] Jao | A Self-Powered Temperature Sensor Based on Silver Telluride Nanowires[END_REF]-[15] or hydrothermal reduction [16]- [START_REF] Wang | One-pot fabrication and thermoelectric properties of Ag 2Tepolyaniline core-shell nanostructures[END_REF], by electrodeposition [START_REF] Prabhune | Measurement of properties of silver telluride thin films using holography[END_REF], [START_REF] Chen | Silver telluride nanowires prepared by dc electrodeposition in porous anodic alumina templates[END_REF], by high intensity ultrasonic irradiation in organic solvents [START_REF] Li | Sonochemical synthesis of nanocrystalline silver tellurides Ag2Te and Ag7Te4[END_REF], by vapor transport technique [START_REF] Lee | Thermoelectric Properties of a Single Crystalline Ag2Te Nanowire[END_REF], [START_REF] Pillai | Single Crystalline Highly Pure Long Ag2Te Nanowires for Thermoelectric Applications and its Structural Analysis[END_REF], or spark plasma sintering [START_REF] Chang | Facile synthesis of Ag2Te nanowires and thermoelectric properties of Ag2Te polycrystals sintered by spark plasma sintering[END_REF]. Common synthesis methods are hydrothermal and solvothermal routes, that can be divided in two main approaches:

-A single step synthesis [START_REF] Chang | Facile synthesis of Ag2Te nanowires and thermoelectric properties of Ag2Te polycrystals sintered by spark plasma sintering[END_REF], [START_REF] Xiao | Simple synthesis of ultra-long Ag2Te nanowires through solvothermal co-reduction method[END_REF], [16]-[18], [START_REF] Wang | One-pot fabrication and thermoelectric properties of Ag 2Tepolyaniline core-shell nanostructures[END_REF] consists in mixing Te(IV) and Ag(I) as metallic precursors in the presence of a reducing agent (ethylene glycol (EG), ascorbic acid, hydrazine, KBH4...). The precursors of Te are usually TeO2, TeCl4 and Na2TeO3, dissolved in an aqueous solution [17] or in organic solvent such as EG which can act also as a reducing agent [18]. Using these methods, the reaction kinetics is slow and takes between 24 and 48 hours. In addition, it is difficult to avoid impurities such as silver particles and Te rich nanostructures. Indeed, the reaction is very sensitive to pH that must be adjusted to basic values to avoid the formation of Ag7Te4 and Ag particles on the surface of Ag2Te nanostructures [START_REF] Xiao | Simple synthesis of ultra-long Ag2Te nanowires through solvothermal co-reduction method[END_REF]. The presence of Ag(0) particles is due to an insufficient relative quantity of reduced Te(0), leading to the reduction of silver cations into their metallic counterpart [16]. Sometimes complexing agents like disodium salt of ethylenediaminetetraacetic acid Na2(EDTA) [START_REF] Xiao | Simple synthesis of ultra-long Ag2Te nanowires through solvothermal co-reduction method[END_REF] or 3mercaptopropionic acid MPA [START_REF] Wang | One-pot fabrication and thermoelectric properties of Ag 2Tepolyaniline core-shell nanostructures[END_REF] are added to control kinetics and to avoid the apparition of silver impurities. Finally, another disadvantage of the single step synthesis is the difficulty to control the size and the crystallinity of the resulting nanostructures.

-A two-step synthesis [START_REF] Yang | Composition modulation of Ag2Te nanowires for tunable electrical and thermal properties[END_REF], [START_REF] Jao | A Self-Powered Temperature Sensor Based on Silver Telluride Nanowires[END_REF], [START_REF] Sutch | Electron conducting Ag 2 Te nanowire/polymer thermoelectric thin films[END_REF]-[15], [19], [20], can be alternatively conducted, where either Te or Ag (nanostructures or films) is first obtained and further converted into Ag2Te by a redox reaction [START_REF] Jao | A Self-Powered Temperature Sensor Based on Silver Telluride Nanowires[END_REF] whose mechanism is not well known until now. This two steps method allows preserving the crystallinity and the purity of the Te nanorods. Moreover, it prevents the formation of Ag particles on the surface of Ag2Te nanostructures and of by-products (Ag4.5Te3, Ag5Te3, Ag7Te4…) [START_REF] Moon | Chemical transformations in ultrathin chalcogenide nanowires[END_REF]. Different parameters influence the composition, the crystalline quality and the morphology of Ag2Te like bath temperature, time of reaction, Ag(I):Te molar ratio, addition of a Reducing Agent (RA) and the RA:Ag(I) molar ratio. In most cases EG is used, playing both roles of solvent and Ag(I) reducing agent and avoids the formation of byproducts [20]. Only Jao et al. [START_REF] Jao | A Self-Powered Temperature Sensor Based on Silver Telluride Nanowires[END_REF] and Samal et al. [19] synthesized Ag2Te nanowires without any addition of a reducing agent. In this case, Te acts both as template and Ag(I) reducing agent but polycrystalline structures are obtained even if the initial Te nanowires are single crystalline [19].

In this work, we present a two-step synthesis route of single crystalline Ag2Te nanorods of high purity. The synthesis occurs in mild conditions: room temperature, atmospheric pressure and without any addition of toxic solvents, reducing agents, surfactants or complexing agents. One-dimensional hexagonal single crystalline Te nanorods are first obtained by electrodeposition in an ionic liquid medium, taking advantage of its templating properties. Te nanorods are then transformed into pure single crystalline and defect free α Ag2Te nanorods (monoclinic phase, hessite) via a combination of cementation and topotactic reactions, with a slight increase in diameter. The influence of reaction time and Ag(I):Te molar ratio on the transformation of Te in Ag2Te is presented.

Experimental

Chemicals

TeCl4 (99%, Alfa Aesar) was used as precursor of Te(IV), whereas AgNO3 (analytical reagent, VWR) was used as precursor of Ag(I). 1-ethyl-1-octylpiperidinium bromide (EOPipBr) and 1-ethyl-1-octylpiperidinium bis(trifluoromethylsulfonyl)imide (EOPipTFSI) ionic liquids (ILs) were synthesized as described by Traore et al. [START_REF] Traore | New insight into indium electrochemistry in a Tf2N-based roomtemperature ionic liquid[END_REF] and characterized by nuclear magnetic resonance analysis (Bruker Avance 400 MHz) then stored in a glove box to minimize water/oxygen contents (Mbraun Labstar, O2/H2O levels < 1 ppm).

Synthesis of Ag2Te nanostructures

First step

The synthesis of Te nanorods was performed inside a glovebox (Mbraun Labstar, O2/H2O levels < 1 ppm). The electrochemical deposition of Te nanostructures was achieved at 100 °C in a mix of EOPipTFSI:EOPipBr 95:5 mol% ionic liquids containing 5 mM TeCl4. Bromide anion plays the role of complexing agent for the Te(IV) salt, improving its solubility in the IL [START_REF] Thiebaud | Tuning the morphology of Te onedimensional nanostructures by template-free electrochemical deposition in an ionic liquid[END_REF]. The working electrode and the counter electrode were 300 nm thick platinum-coated glass slides (25 x 10 x 1 mm) purchased from Applications Couches Minces (ACM). A Pt wire was used as a pseudo-reference electrode. The deposition surfaces were equal to 0.5 cm². All the electrodes were immerged in a 5 mL electrolyte solution. Te nanostructures were obtained by potentiostatic deposition for a fixed charge density Q (C/cm²).

Prior to Te electrodeposition, platinum-coated glass slides were cleaned by sonication in acetone (2 x 20 minutes) and then in ethanol (20 minutes). The airdried resulting surfaces were then free of organic contamination. Then the surface of the Pt working electrode undergoes an electrochemical activation by cyclic voltammetry: 50 cycles were performed in 25 mL of 0.5 M H2SO4 between -0.17 V (potential of reductive adsorption of protons) and 1.2 V (potential of formation of Pt oxides). This allows a crystal reconstruction of Pt surface by repeated adsorption and desorption of oxygen and hydrogen atoms [START_REF] Bengio | Electrochemical behavior of platinum and gold electrodes in the aprotic ionic liquid N,N-Trimethylbutylammonium Bis(trifluoromethanesulfonyl)imide[END_REF].

Second step

The second step concerns the conversion of Te to Ag2Te nanorods by chemical cementation. The Pt substrate on which the Te nanorods were previously deposited was immerged in a 10 mL aqueous solution of AgNO3 under stirring, at room temperature and without any addition of reducing agent. The cell was protected from the light with aluminum foil to avoid photochemical reduction of Ag(I). The Ag(I):Te molar ratio was varied, as well as the reaction time, in order to compare the resulting composition of nanorods.

Characterizations of the nanostructures

The morphology and chemical composition of Te and Ag2Te nanostructured films were analyzed using a Scanning Electron Microscopy SEM Quanta FEG 650 microscope equipped with an Energy Dispersive X-ray (EDX) detector.

The Te deposition yield and the molar composition of Ag2Te nanostructures were also evaluated by Inductively Coupled Plasma-Optical Emission Spectrometry (ICP-OES) using a Perkin Elmer Avio 200-OES after the mineralization of the deposits in 7 M nitric acid. Signals are collected through a cooled CCD-based detector, with simultaneous measurement of the background emission and a neon spectrum for active wavelength correction. The operating parameters were as follow: RF power: 1.5 kW; plasma gas flow 10 L min -1 , auxiliary gas flow 0.2 L min -1 , nebulizer gas flow 0.7 L min -1 and sample uptake rate 1.0 L min -1 . The following atomic line wavelengths were used in the measurements: Ag I (328.068 nm) and Te I (214.281 nm).

X-Ray Diffraction (XRD) analyses were carried out with a Bruker D8 Advance diffractometer using Cu Kα radiation (λ = 1.5406 Å) at 40 kV and 40 mA. The samples were scanned in the 2θ range from 10° to 70° but only the reduced range 20° to 60° was analyzed, where all the main peaks of the Te-based phases were expected. The crystallographic phases were analyzed on the basis of the JCPDS databases with the EVA 5 software. (cFEG TEM/STEM) operating at 200 kV and equipped with a spherical aberration (Cs) probe and image correctors (point resolution 0.12 nm in TEM mode and 0.078 nm in STEM mode). The growth direction of the nanostructures was determined by imaging and Selected Area Electron Diffraction (SAED). The chemical composition of individual nanostructures was analyzed by EDX (mapping and line profile).

Results and discussion

Synthesis of Te nanorods

The first step of the process allows to obtain a thin film of self-standing Te nanorods by applying a fixed cathodic potential. This result is a consequence of the anisotropic crystalline structure of Te [START_REF] Thiebaud | Synthesis of Te-Bi core-shell nanowires by two-step electrodeposition in ionic liquids[END_REF], the limitation of electrochemical growth by mass transport [START_REF] Thiebaud | Tuning the morphology of Te onedimensional nanostructures by template-free electrochemical deposition in an ionic liquid[END_REF] and the templating properties of the ionic liquid [START_REF] Janiak | Ionic liquids for the synthesis and stabilization of metal nanoparticles[END_REF]. The value of the applied potential is defined by the position of the peak observed in a preliminary cathodic linear sweep voltammetry (Fig. 1). This cathodic peak corresponds to the reduction of Te(IV) to Te(0) according to the following chemical equation:

TeX6 2-+ 4e - Te + 6X -
where X -may be attributed to Cl -or Br -anions [START_REF] Thiebaud | Tuning the morphology of Te onedimensional nanostructures by template-free electrochemical deposition in an ionic liquid[END_REF]. This procedure was derived from the works of J.Szymczak et al. [START_REF] Szymczak | Template-free electrodeposition of tellurium nanostructures in a room-temperature ionic liquid[END_REF] and L.Thiebaud et al. [START_REF] Thiebaud | Tuning the morphology of Te onedimensional nanostructures by template-free electrochemical deposition in an ionic liquid[END_REF] and adapted to obtain reduced and uniform dimensions (Fig. 2). It appears that the diameter and length of the 1D Te nanostructures are directly related to the coulometric charge. The tellurium concentration and the bromide content have a lesser impact. Thus, with [Te(IV)] = 5 mM and [Br -] = 0.25 mM (5% mol), the smallest nanorods were obtained by applying a charge density Q of 0.1 C/cm². The mean diameter is of 60 ± 13 nm and length of 191 ± 41 nm as shown in Fig. 3. The mass per unit area of electrodeposited Te nanorods, measured by ICP-OES, is equal to 30 µg/cm², corresponding to a faradic yield of 90%.

SAED and HR-TEM results (Fig. 4) show that the electrodeposited Te nanorods are single crystalline, with an hexagonal lattice and a preferential orientation along the [001] direction (JCPDS n° 00-036-1452). This observation is in perfect agreement with our previous published results for Te nanostructures with greater dimensions [START_REF] Thiebaud | Tuning the morphology of Te onedimensional nanostructures by template-free electrochemical deposition in an ionic liquid[END_REF]. In our work, the resulting nanostructured film appear to be homogeneous on the surface as shown by the SEM image (Fig. 6a). Moreover the presence of Ag and Te is confirmed by the EDX spectrum (Fig. 6b) and is uniformly distributed along the nanostructures (Fig. 6c). When a large excess of Ag(I) is used (nAg(I)/nTe = 772), SEM analysis reveals the presence of large Ag2Te 2D cross braces of micrometer size on the surface of the nanostructured film (Fig. 7). These large grains due to overgrowth appear after 30 minutes of immersion.

Synthesis of Ag2Te nanorods

The influence of immersion time was then studied for nAg(I)/nTe ratio = 2, which leads to highly crystalline nanorods and homogeneous films of nanostructures. The crystallographic transition from hexagonal Te to monoclinic hessite α-Ag2Te was followed by XRD analysis according to the following procedure: after a reaction time of 30 minutes, the Te sample was removed from Ag(I) solution, rinsed, analyzed by XRD and then immersed again in the same solution. This experimental sequence was repeated for 1h, 2h, 4h and 6h reaction time. Corresponding diffractograms are given in Fig. 8. Before reaction takes place, two peak at 27.7° and 57.07° attributed to hexagonal Te are observed. These peaks decrease as reaction time increases. Thus, the one at 27.7° disappears completely after a reaction time of 4h, while the peak at 57.07° disappears after 2h. In parallel, two small peaks at 29.9° and 31.05° that are attributed to hessite Ag2Te appear after 4h of immersion and increase with reaction time (Fig. 8). These peaks are of low intensity despite the crystallinity of the nanorods confirmed by SAED. The Pt substrate peak masks the angular domain 2theta 35°-47°. After 6h immersion time, ICP-OES analysis of the sample reveals an Ag:Te mean molar ratio of nanorods equal to 2.1 corresponding to Ag2.1Te. This ratio does not change whenever the immersion time is further increased.

However, HR-TEM analysis after a reaction time of 6h reveals the presence of an intermediate compound, the hexagonal stützite phase (Ag1.5Te JCPDS n° 04-008-4552), with [001] growth direction (Fig. 9). EDX analyses of individual nanostructures give Ag1.8Te (Fig. 10), which agrees with the deficit of silver of the stützite phase. However, this ratio is lower than the mean ratio measured by ICP-OES (Ag2.1Te), suggesting that the transformation is not fully completed on the whole surface of the sample after 6h immersion. The composition and the morphology of these stützite nanorods appear uniform (Fig. 10 and Fig. 11). Slight curvatures are observed (Fig. 9). After 24h reaction time, the complete transformation of hexagonal Te into highly pure single crystalline monoclinic α-Ag2Te (hessite Ag2Te JCPDS n° 01-081-1820), more stable than the stützite phase, is achieved, as confirmed by SAED analysis (Fig. 12). The transformation from Te to Ag2Te is thermodynamically favorable [START_REF] Moon | Chemical transformations in ultrathin chalcogenide nanowires[END_REF]. A slight increase in diameter of about 5 to 30 % ( ̴ 3-20 nm) is observed compared to initial Te nanorods (Fig. 13) along with a curvature of some nanorods (Fig. 12). By adding EG as reducing agent, the volume expansion was reported to be greater and can reach 2 times the initial nanostructures in the literature data. When the volume expansion becomes considerable, Te nanowires tend to bend and break into shorter nanowires but the single crystallinity is conserved [START_REF] Moon | Chemical transformations in ultrathin chalcogenide nanowires[END_REF]. In this work, the volume increase is avoided by sacrificing a part of the Te as reducing agent. Indeed, chemical analysis by ICP-OES revealed the presence of tellurium in the bath after the cementation process. This result is a consequence of the partial oxidation of the tellurium nanostructures in the form of dissolved tellurium (IV), avoiding the volume increase. The nanorods always remain single crystalline as a sign of the homogeneous transformation on the surface of the nanorods. The same kinetic study was performed in large excess of Ag(I) (nAg(I)/nTe = 772). Surprisingly, the kinetics of Te conversion to Ag2Te is not faster and even slightly slowed down according to XRD results (Fig. 14 From our observations, only a slight excess of Ag in the Ag2Te nanostructures is observed even if the molar ratio of precursors (n(Ag(I))/n(Te)) is superior to the stoichiometric ratio 4/3 according to the equation (3).

A time progressive incorporation of silver occurs in the tellurium hexagonal crystalline structure. After 6h, Te-rich Ag2Te nanorods are obtained with the stützite phase (hexagonal symmetry, P-62m). Then a phase transition occurs together with an increase of Ag content in the nanostructures. Finally, the synthesis is considered as complete after a reaction of 24h, where the stoichiometry reaches Ag2Te with a slight excess of Ag and the nanorods are crystallized according the hessite phase (monoclinic phase, P121/C1). The corresponding crystallographic cells of the 3 successive phases are displayed in the Fig. 15, highlighting that the hexagonal symmetry of Te atoms is kept whatever the phases. This observation of the final phase is similar to the transformation of trigonal Se nanowires into orthorhombic Ag2Se nanowires, which is justified by the topotactic lattice matching between the both phases. Similar transformation from hexagonal Te to Ag2Te was also observed and discussed by Moon at al. in terms of volume changes of the cell [START_REF] Moon | Chemical transformations in ultrathin chalcogenide nanowires[END_REF]. In our work, the partial consumption of tellurium can explain the only slight increase in diameter (5-30%) compared to the 100% volume expansion observed in literature by adding a reducing agent [START_REF] Moon | Chemical transformations in ultrathin chalcogenide nanowires[END_REF]. The final dimensions of the nanostructures being close to initial ones suggest that no breakage of the nanorods due to stress release occurs during the transformation.

Conclusions

The two-step way used in this work allows synthesizing highly pure single crystalline monoclinic Ag2Te nanorods by chemical cementation reaction, using Te nanorods both as template and reducing agent. The hexagonal single crystalline self-standing tellurium nanorods of 60 ± 13 nm in diameter and less than 200 nm long were firstly electrodeposited in an ionic liquid medium. The further cementation reaction consists in the reduction of Ag(I) to Ag(0) by Te on the surface of the nanorods, followed by the incorporation of the Ag atoms in the hexagonal lattice of Te. A phase transition occurs as the stoichiometry reaches targeted Ag2Te. The mechanism of this reaction was experimentally highlighted for the first time, thanks to the observation of an intermediate stützite phase. Single crystalline hessite Ag2Te nanorods are finally obtained with uniform size and chemical composition. This two-step synthesis, performed in mild conditions (room temperature, atmospheric pressure) and without any addition of toxic solvents, reducing agents or surfactants, appears to be attractive for the dedicated synthesis of high quality silver telluride nanostructures.

[15] E. K. Kim, D. Park, N. K. Shrestha, J.

  Fig. 1: Typical linear sweep voltammogram of the reduction of Te(IV) to Te(0) in EOPipTFSI:EOPipBr 95:5 (mol%). [TeCl4] = 5 mM. Pt-coated glass slide substrate. T = 100 °C, scan rate 5 mV s -1 .

Fig. 2 :

 2 Fig. 2: SEM image of Te nanorods electrodeposited at C1 peak potential in EOPipTFSI:EOPipBr 95:5 (mol%). [TeCl4] = 5 mM. T = 100 °C, Q = 0.1 C/cm 2 , Pt-coated glass slide.

Fig. 3 :

 3 Fig. 3: a) Diameter and b) Length of Te nanorods deposited at C1 peak potential in EOPipTFSI:EOPipBr 95:5 (mol%). [TeCl4] = 5 mM. T = 100 °C, Q = 0.1 C/cm² (number of analyzed nanowires n = 18).

Fig. 4 :

 4 Fig. 4: TEM characterization of Te nanorods: a) TEM image of a Te nanorods group, b) TEM micrograph and corresponding SAED pattern indexed to the tellurium phase along the [010] zone axis (SG: P3121), c) HRTEM micrograph with Fast Fourier Transform (FFT) and d) Zoom on the blue square of Te crystallographic structure along the [010] zone axis (JCPDS n° 00-036-1452).

Fig. 5 :

 5 Fig. 5: Influence of nAg(I)/Te molar ratio in the bath on the final composition of Ag2Te nanowires: ICP-OES analysis after chemical dissolution of the deposits.

Fig. 6 :

 6 Fig. 6: SEM characterization of Ag2Te nanorods film after 24h immersion prepared with nAg(I)/nTe = 2: a) SEM image, b) EDX spectrum, c) EDX mapping.

Fig. 7 :

 7 Fig. 7: SEM characterization of Ag2Te nanorods film after 24h immersion prepared with nAg(I)/nTe = 772: a) SEM image, b) EDX mapping, c) EDX spectrum.

Fig. 9 :

 9 Fig. 9: TEM characterization of stutzite nanorods after 6h immersion, prepared with nAg(I)/nTe = 2: a) TEM image of a stutzite slightly curved nanorods group, b) TEM image of diameter-enhanced stutzite nanorod, c) corresponding HRTEM micrograph and Fast Fourier Transform (FFT) of stutzite crystallographic structure along the [210] zone axis (SG: P-62m) (JCPDS n° 04-008-4552).

Fig. 8 :

 8 Fig. 8: XRD characterization: Crystallographic transition from hexagonal Te to monoclinic hessite Ag2Te with nAg(I)/nTe = 2.

Fig. 8 :

 8 Fig. 8: XRD characterization: Crystallographic transition from hexagonal Te to monoclinic hessite Ag2Te with nAg(I)/nTe = 2.

Fig. 10 :

 10 Fig. 10: Line profil analysis of stutzite nanorods after 6h immersion prepared with nAg(I)/nTe = 2.

Fig. 11 :

 11 Fig. 11: Elemental X maps of a stutzite nanorod after 6h immersion prepared with nAg(I)/nTe = 2.

1 )Fig. 12 :

 112 Fig. 12: TEM characterization of hessite nanorods after 24h immersion prepared with nAg(I)/nTe = 2: a) TEM image of a Ag2Te nanorods group, b) TEM micrograph and corresponding SAED pattern indexed to the hessite phase along the [001] zone axis (SG: P121/C1), c) HRTEM micrograph with Fast Fourier Transform (FFT) and d) Zoom on the blue square of Ag2Te crystallographic structure along the [001] zone axis (JCPDS n° 01-081-1820).

Fig. 13 :Fig. 14 :

 1314 Fig. 13: a) Mean diameter and b) Length of hessite nanorods after 24h immersion prepared with nAg(I)/nTe = 2 (number of analyzed nanowires n = 9).

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Acknowledgements

This work was supported by the Grand Est Region under the Nanohyb project.

Sebastien DILIBERTO (from Université de Lorraine, CNRS, IJL, F-57000 Metz, France) and Sylvie MIGOT (from Université de Lorraine, CNRS, IJL, F-54000 Nancy, France) are acknowledged for their technical support.

Author Contributions

Conceptualization K.A., S.L. and N.S Methodology K.A., S.M. and J.G. Validation K.A., S.L. and N.S Writing -original draft preparation K.A. Writing -review S.L. and N.S Editing K.A.

Conflicts of interest

There are no conflicts to declare.


Laurent Mazet
email: laurent.mazet@univ-tours.fr

Abraão Ão Mendes
email: abraao.mendes@im.ufal.br

Free boundary

published or not. The documents may come

Introduction

Over the last few years, the study of free boundary minimal hypersurfaces (FBMH for short) has occupied a prominent place in differential geometry, especially the study of FBMH in the Euclidean unit ball B ⊂ R n+1 (see e.g. [START_REF] Ambrozio | Compactness analysis for free boundary minimal hypersurfaces[END_REF][START_REF]Index estimates for free boundary minimal hypersurfaces[END_REF][START_REF] Fraser | Uniqueness theorems for free boundary minimal disks in space forms[END_REF][START_REF]Sharp eigenvalue bounds and minimal surfaces in the ball[END_REF][START_REF]Some results on higher eigenvalue optimization[END_REF][START_REF] Smith | On the Morse index of higher-dimensional free boundary minimal catenoids[END_REF][START_REF] Smith | The Morse index of the critical catenoid[END_REF] and the references therein).

In this paper, we deal with FBMH in R n+1 \ B with compact boundary in ∂B, which are called exterior free boundary minimal hypersurfaces. These hypersurfaces are critical for the n-volume functional with respect to deformations that let the boundary on the unit sphere. For such critical points, the second order derivative of the volume functional is given by the so-called stability operator which here has a contribution from the boundary. In our situation, this contribution is nonnegative due to the concavity of the unit sphere with respect to its outside. An interesting question would be to understand the geometry and the topology of these hypersurfaces in terms of their indices. In the ball, this is the study made by L. Ambrozio, A. Carlotto and B. Sharp in [START_REF]Index estimates for free boundary minimal hypersurfaces[END_REF]. A situation where non-compact FBMH have been studied is the case of Schwarzschild space: R. Montezuma [START_REF] Montezuma | On free-boundary minimal surfaces in the Riemannian Schwarzschild manifold[END_REF] and E. Barbosa and J.M. Espinar [START_REF] Barbosa | On free boundary minimal hypersurfaces in the Riemannian Schwarzschild space[END_REF] have looked at some properties of these hypersurfaces.

In R n+1 \ B, important examples of exterior FBMH are the catenoidal hypersurfaces, defined as exterior FBMH invariant by isometries fixing a straight line. In Section 4 we give a complete description of the catenoidal hypersurfaces and calculate their indices: some have index 0 and others index 1.

The aim of this paper is to prove two classification results for catenoidal hypersurfaces. The first classification theorem is the following. Definitions are given in Sections 2 and 3.

Theorem 1. Let Σ be an exterior free boundary minimal hypersurface in R n+1 \B. Let us assume that Σ is stable and has parallel embedded regular ends. Then Σ is a catenoidal hypersurface.

In order to prove Theorem 1, we first prove that saying that Σ is stable is equivalent to saying that there exists a positive Jacobi function u on Σ satisfying the Robin boundary condition ∂ ν u + u = 0 on ∂Σ. This is the content of Proposition 3. Its proof is based on the proof of a classical stability characterization due to D. Fischer-Colbrie and R. Schoen [6, Theorem 1] for manifolds without boundary, where here we make use of the Harnack inequality for positive solutions to ∆u + qu = 0 on Σ with Robin boundary condition ∂ ν u + u = 0 on ∂Σ proved in Appendix A. Second, we obtain a Bôcher type theorem for positive Jacobi functions on regular minimal ends in R n+1 which, together with Proposition 3, implies that Σ is invariant by isometries fixing a straight line, in other words, Σ is a catenoidal hypersurface.

As mentioned above, some of the catenoidal hypersurfaces have index 1, so it would be interesting to know if the index equal to 1 implies that the hypersurface is catenoidal. When n = 2, it would be a result similar to Lopez-Ros result [START_REF] López | Complete minimal surfaces with index one and stable constant mean curvature surfaces[END_REF] for boundaryless minimal surfaces. For example, it would be interesting to understand if a control on the index gives a control on the number of ends of the hypersurface. However, the positive contribution of the boundary to the stability operator seems to make this not an easy task.

The second classification theorem is the following.

Theorem 2. Let Σ be an exterior free boundary minimal hypersurface. If Σ has one regular end, then Σ is a catenoidal hypersurface.

The proof of Theorem 2 is based on a symmetrization procedure as in Schoen's paper [START_REF] Schoen | Uniqueness, symmetry, and embeddedness of minimal surfaces[END_REF].

The paper is organized as follows. In Section 2 we present some preliminaries on FBMH and prove Proposition 3. In Section 3 we state and prove an auxiliary Bôcher type result (Theorem 4) and present the proof of Theorem 1. In Section 4 we introduce the catenoidal hypersurfaces and give a complete description of them, including the calculation of their indices. In Section 5 we prove Theorem 2. Finally, in Appendix A we present a proof of the Harnack inequality for positive functions satisfying a Robin type boundary condition.

Free boundary minimal hypersurfaces in R n+1 \ B

Let Σ be an n-manifold with compact boundary. We say that F : Σ → R n+1 \ B is an exterior proper immersion if F is a proper immersion and F (Σ) ∩ ∂B = F (∂Σ). In this paper, we always consider such exterior proper immersions. We also assume that Σ is orientable so that a unit normal N is well defined along F . Besides, we will often identify Σ with its image F (Σ) and just say that Σ is an exterior hypersurface.

We will consider exterior hypersurfaces that are critical for the n-volume functional with respect to any deformations keeping the boundary on ∂B. Such a hypersurface Σ has vanishing mean curvature and meets ∂B orthogonally: we call them exterior free boundary minimal hypersurfaces.

Basic examples are given by cones over minimal hypersurfaces S ⊂ S n = ∂B:

Σ = {tp ∈ R n+1 ; p ∈ S and t ≥ 1}.
One can also consider exterior FBMH that are invariant by isometries fixing a straight line: they are called catenoidal hypersurfaces. Their complete description is given in Section 4.

Let Σ be an exterior free boundary minimal hypersurface. The free boundary condition implies that, at P ∈ ∂Σ, the outgoing unit normal ν(P ) = -P is a principal direction of the second fundamental form B of Σ. Indeed, for T ∈ T ∂Σ, we have

B(ν, T ) = (D T ν, N ) = -B ∂B (T, N ) = -(T, N ) = 0, (2.1)
where D is the covariant derivative in R n+1 and B ∂B is the second fundamental form of ∂B.

Figure 1. Exterior FBMH

As in the boundaryless case, we also have a monotonicity formula for exterior free boundary minimal hypersurfaces:

|Σ ∩ B R | R n -1 - 1 R n |∂Σ| n = Σ∩B R |X ⊥ | 2 |X| n+2 , (2.2)
where B R is the Euclidean ball centered at the origin of radius R ≥ 1. In fact, let v

(r) = |Σ∩B r |, r > 1. It follows from coarea formula that d dr v(r) = Σ∩∂Br 1 |∇ Σ d| ,
where d(X) = |X| is the distance function to the origin. On the other hand, because Σ is minimal, div Σ (X ) = n. Therefore,

nv(r) = Σ∩Br div Σ (X ) = ∂Σ (X , ν) + Σ∩∂Br (X , ν) = -|∂Σ| + Σ∩∂Br |X |.
This gives

d dr v(r) r n = |∂Σ| r n+1 + Σ∩∂Br 1 r n 1 |∇ Σ d| - |X | r = |∂Σ| r n+1 + Σ∩∂Br 1 |∇ Σ d| |X ⊥ | 2 r n+2 ,
where above we have used that

∇ Σ d = X |X| and |X ⊥ | 2 = r 2 -|X | 2 .
Thus, integrating last equation from 1 to R and using coarea formula, we obtain (2.2).

2.1. The stability operator. Let {F t } be a family of exterior proper immersions of Σ such that F 0 (Σ) is free boundary minimal and ∂ t F t has compact support. Even if the volume of F t (Σ) is infinite its derivatives can be computed since the deformation has compact support. Then the first derivative of the n-volume functional vanishes at t = 0 and the second derivative at t = 0 can be computed in terms of the function u = (∂ t F t | t=0 , N ) by

d 2 dt 2 Vol(F t (Σ)) t=0 = Q(u, u) = Σ (|∇ Σ u| 2 -B 2 u 2 )dµ + ∂Σ u 2 ds,
where ∇ Σ and dµ are the gradient and the n-volume measure on Σ, and ds is the (n -1)-volume measure on ∂Σ, all with respect to the metric induced by F 0 (see [START_REF]Index estimates for free boundary minimal hypersurfaces[END_REF]). After integration by parts, one has

Q(u, u) = - Σ u(∆u + B 2 u)dµ + ∂Σ u(u + ∂ ν u)ds.
So the quadratic form Q is associated with the Jacobi operator defined by Lu = ∆u + B 2 u. Then, for any bounded domain Ω in Σ, we can consider the associated spectrum of L: a sequence of eigenvalues λ n +∞ and a L 2 -orthonormal sequence of functions u n on Ω such that

       ∆u n + B 2 u n = -λ n u n on Ω, ∂ ν u n + u n = 0 on ∂Σ ∩ Ω, u n = 0 on ∂Ω \ ∂Σ.
The number of negative eigenvalues is then called the index of Q on Ω and is denoted by Ind(Q, Ω). If (Ω n ) is an increasing sequence of domains such that ∪Ω n = Σ, then the limit of the increasing sequence (Ind(Q, Ω n )) is called the index of Σ and is denoted by Ind(Σ). When Ind(Σ) = 0, we say that Σ is stable and this is equivalent to Q(u, u) ≥ 0 for any function u with compact support on Σ. Actually, we have an alternative characterization of stability given by the following Fischer-Colbrie and Schoen type result. Proposition 3. Let Σ be an exterior free boundary minimal hypersurface. Then Σ is stable if and only if there exists a positive solution u on Σ to

(2.3) ∆u + B 2 u = 0 on Σ, ∂ ν u + u = 0 on ∂Σ.
Proof. The proof is very similar to that one of [6, Theorem 1] by Fischer-Colbrie and Schoen.

In fact, in order to adapt their proof, we just need to observe that, if u is as in (2.3), then ∂ ν ln u = -1 on ∂Σ (this is for the part (iii) =⇒ (i)) and use the Harnack inequality given in Proposition 9 in Appendix A (for the part (ii) =⇒ (iii)).

When n = 2, Fischer-Colbrie's result [START_REF] Fischer-Colbrie | On complete minimal surfaces with finite Morse index in three-manifolds[END_REF]Theorem 2] gives that an exterior free boundary minimal surface Σ has finite index if and only if it has finite total curvature. One difference is that, in our case, the quadratic form Q does not depend only on the Gauss map but also on the conformal factor along the boundary ∂Σ. A second important point is that we assume ∂Σ to be compact. For example, if Σ is stable, we have a solution u to (2.3) which can be lifted to the universal cover Σ. This implies that the associated quadratic form on Σ is nonnegative. However, the universal cover may not have finite total curvature as we are going to see below (see Example 6). Actually, the universal cover is not properly immersed and thus it is not an exterior surface according to our definition.

Regular ends.

The asymptotic of an exterior free boundary minimal hypersurface can be highly complicated. A simple asymptotic is given by regular ends introduced by Schoen in [START_REF] Schoen | Uniqueness, symmetry, and embeddedness of minimal surfaces[END_REF].

In order to describe it, we split P ∈ R n+1 as (X, z) ∈ R n × R. Then an end E of an exterior free boundary minimal hypersurface is said to be regular if, after an isometry, a representative of E is given by the graph of a function f of bounded gradient defined on {|X| ≥ R} with the following asymptotic:

f (X) = A ln |X| + B + (C, X)|X| -2 + O(|X| -2 ) if n = 2, (2.4) f (X) = B + A|X| -(n-2) + (C, X)|X| -n + O(|X| -n ) if n > 2, (2.5)
where A, B ∈ R and C ∈ R n . We notice that the above estimate on f implies similar estimates on its derivatives (see [START_REF] Schoen | Uniqueness, symmetry, and embeddedness of minimal surfaces[END_REF]). For example, one sees that ∇f (X) goes to 0 as |X| goes to +∞ and, in particular, there is a well-defined unit normal at ∞ for such an end.

If n = 2 and Σ has finite total curvature, then Σ is conformally equivalent to a compact Riemann surface with boundary minus a finite number of points. Moreover, a properly embedded annular end with finite total curvature is regular [14, Proposition 1].

In the case 3 ≤ n ≤ 6, following the arguments of J. Tysk [START_REF] Tysk | Finiteness of index and total scalar curvature for minimal hypersurfaces[END_REF], if we assume that Σ has finite index and finite volume growth in the sense that lim R→+∞ R -n |Σ ∩ B R | < +∞, then Σ has finitely many ends, all of them being regular.

Stable hypersurfaces

A Bôcher type result for the Jacobi operator.

In this section, we analyze the asymptotic behavior of positive Jacobi functions (i.e. solutions to Lu = 0) on regular ends. Theorem 4. Let E be a regular minimal end in R n+1 (let X ∈ R n be a coordinate associated to the end as in (2.4) and (2.5)) and consider a positive Jacobi function u on E. Then u has the following asymptotic: there exist

A, B ∈ R such that u(X) = A ln |X| + B + v(X) if n = 2, u(X) = A + B|X| -(n-2) + v(X) if n > 2,
where v is such that the function

|X| n-1 v is C 2 -bounded on R n \ B R . Moreover, either A > 0 or A = 0 and B > 0.
Proof. Writing X = e t p with t ∈ R and p ∈ S n-1 , a regular end can be parametrized by [t 0 , +∞) × S n-1 with a metric g having the asymptotic g = e 2t (δ + O(e -2(n-1)t )), where δ is the product metric on R × S n-1 . Moreover, the second fundamental form can be estimated by

B 2 = O(e -2nt
). Thus the Jacobi operator can be computed as

∆u + B 2 u = e -2t (u tt + (n -2)u t + ∆ σ u + M (u)) ,
where ∆ σ is the Laplacian on S n-1 and M (u) is a second order linear operator whose coefficients have C 0,α -norm bounded by Ce -2(n-1)t for some constant C > 0.

Therefore a Jacobi function u satisfies (3.1)

u tt + (n -2)u t + ∆ σ u + M (u) = 0,
which is a uniformly elliptic equation on [t 0 , +∞) × S n-1 . As a consequence, by Harnack inequality ([10, Corollary 8.21]), there is a constant C > 0 such that, for any p, q ∈ S n-1 and t, s ≥ t 0 + 1 with |t -s| ≤ 1, and any positive Jacobi function u, we have u(t, p) ≤ Cu(s, q). By Schauder's elliptic estimates ([10, Corollary 6.3]), we also have

(3.2) u C 2,α ([t-1 2 ,t+ 1 2 ]×S n-1 ) ≤ C u C 0 ([t-1,t+1]×S n-1 ) for t ≥ t 0 + 2. Let us define ū(t) = 1 |S n-1 | S n-1 u(t, p)dσ. By Harnack inequality, we obtain (3.3) u C 0 ([t-1,t+1]×S n-1 ) ≤ C min p∈S n-1 u(t, p) ≤ C ū(t).
Then, combining with (3.2), there is a constant C > 0 such that

(3.4) M (u) C 0,α ([t-1 2 ,t+ 1 2 ]×S n-1 ) ≤ Ce -2(n-1)t ū(t) and M (u)(t) = 1 |S n-1 | S n-1 M (u)(t, p)dσ satisfies (3.5) M (u) C 0,α ([t-1 2 ,t+ 1 2 ]) ≤ Ce -2(n-1)t ū(t)
. By integrating (3.1) over S n-1 , we obtain that ū solves ūtt +(n-2)ū t +M (u) = 0. Considering first the case n > 2, let a and b be two functions such that

ū ū = a 1 0 + b 1 2 -n .
Then we have the system

a = -1 n-2 M (u)(t), b = -(n -2)b + 1 n-2 M (u)(t).
Using the above equations, we obtain

∂ t a 2 + b 2 ≤ C|M (u)(t)| ≤ Ce -2(n-1)t ū(t) ≤ Ce -2(n-1)t a 2 + b 2 .

Thus

√ a 2 + b 2 and ū stay bounded on [t 0 , +∞). In particular, |M (u)(t)| ≤ Ce -2(n-1)t . Also,

∂ t e nt a 2 + b 2 ≥ ne nt a 2 + b 2 -(n -2)e nt b 2 √ a 2 + b 2 -Ce -(n-2)t a 2 + b 2 ≥ (2 -Ce -2(n-1)t )e nt a 2 + b 2 ≥ 0
for t sufficiently large. Therefore e nt √ a 2 + b 2 cannot converge to 0 at t goes to +∞. We can also solve the system to obtain

a = A + +∞ t 1 n-2 M (u)(s)ds, b = Be -(n-2)t -e -(n-2)t +∞ t 1 n-2 e (n-2)s M (u)(s)ds.
We notice that if A = B = 0 then lim t→+∞ e nt a = lim t→+∞ e nt b = 0, which is not possible. Then we can be sure that either A or B is nonzero. As ū = a + b is positive and A = lim t→+∞ (a + b), then either A > 0 or A = 0 and B > 0. Observe that ū -A -Be -(n-2)t = O(e -2(n-1)t ).

If n = 2, we notice that

∂ t ū2 + ū2 t ≤ ūū t ū2 + ū2 t + C|M (u)| ≤ 1 2 ū2 + ū2 t + Ce -2t ū ≤ 1 2 + Ce -2t ū2 + ū2 t .
Thus ū = O(e

ū(t) = At + B - +∞ t +∞ s M (u)(r)dr ds.
If A and B vanish, then ū, ūt = O(e -3 2 t ), which is not possible. Then either A > 0 or A = 0 and B > 0. Notice that ū -At -B = O(te -2t ). In fact, last equation gives that ū = O(t). Then, from (3.5) we have M (u) = O(te -2t ). Therefore, using last equation again, we obtain ū -At -B = O(te -2t ).

In both cases, we have M (u) = O(te -2(n-1)t ). Now, to conclude, we need to estimate u -ū. Let v i be a L 2 -unit eigenfunction for the Laplace operator on S n-1 associated to a nonzero eigenvalue λ (in particular, λ ≥ n -1). Let

u i = S n-1 uv i dσ. Equation (3.1) implies u itt + (n -2)u it -λu i = - S n-1 M (u)v i dσ = f i = O(te -2(n-1)t ).
Observe that µ 2 + (n -2)µ -λ = 0 has two roots: µ + ≥ 1 and µ -≤ -(n -1). Then, solving the above equation, we obtain

(3.6) u i (t) = e µ + t a i - +∞ t e -µ + s f i (s) µ + -µ - ds + e µ -t b i - t t 0 e -µ -s f i (s) µ + -µ - ds for some a i , b i ∈ R. Using (3.
3) and the fact that ū = O(t), we have u i = O(t) and thus a i = 0. We also have

b i = e -µ -t 0 u i (t 0 ) + e (µ + -µ -)t 0 +∞ t 0 e -µ + s f i (s) µ + -µ - ds.
Now, by Cauchy-Schwarz,

+∞ t e -µ + s f i (s)ds 2 ≤ 1 µ + e -µ + t +∞ t e -µ + s f 2 i (s)ds, t t 0 e -µ -s f i (s)ds 2 ≤ 1 -µ - (e -µ -t -e -µ -t 0 ) t t 0 e -µ -s f 2 i (s)ds.
Thus, by squaring (3.6), we obtain

u 2 i (t) ≤ 16 e µ + t µ + (µ + -µ -) 2 +∞ t e -µ + s f 2 i (s)ds + e 2µ -(t-t 0 ) u 2 i (t 0 ) + e 2µ -(t-t 0 )+µ + t 0 µ + (µ + -µ -) 2 +∞ t 0 e -µ + s f 2 i (s)ds + e µ -t -µ -(µ + -µ -) 2 t t 0 e -µ -s f 2 i (s)ds .
Let us define

U (t) = S n-1 (u(t, p) -ū(t)) 2 dσ, M (t) = S n-1 (M (u)(t, p) -M (u)(t)) 2 dσ.
Using that µ + ≥ 1 and µ -≤ -(n -1), we can sum the above inequalities with respect to i to obtain

U (t) ≤ 16 e µ + t +∞ t e -µ + s M (s)ds + e 2µ -(t-t 0 ) U (t 0 ) + e 2µ -(t-t 0 )+µ + t 0 +∞ t 0 e -µ + s M (s)ds + e µ -t t t 0 e -µ -s M (s)ds .
Since M (t) = O(t 2 e -4(n-1)t ), it follows that

u -ū L 2 ([t-1,t+2]×S n-1 ) = O(e -(n-1)t ).
Actually, u -ū solves the equation

z tt + (n -2)z t + ∆ σ z + M (z) = M (u) -M (ū).
Then, combining the above L 2 -estimate with (3.4) and (3.5), Schauder's estimates give

u -ū C 2 ([t,t+1]×S n-1 ) = O(e -(n-1)t ).
We have then proved that

u -A -Be -(n-2)t C 2 ([t,t+1]×S n-1 ) = O(e -(n-1)t ) if n > 2, u -At -B C 2 ([t,t+1]×S n-1 ) = O(e -t ) if n = 2.
This gives the expected result after going back to the original coordinate system.

Classification of stable hypersurfaces.

If Σ is an exterior free boundary minimal hypersurface with regular ends, the unit normal to Σ has a well-defined limit at each end. Then we say that such a hypersurface has parallel ends if these limits coincide up to a sign. We notice that, if Σ is embedded, then its ends are always parallel. Now, we are going to use the above Bôcher type theorem in order to give a classification of stable exterior FBMH with parallel regular ends.

Proof of Theorem 1. Consider the (X, z) coordinate system on R n+1 . After an isometry, we can assume that the unit normal to the ends of Σ are given by ±e z . Now, let M ∈ M n (R) be a skew-symmetric matrix and consider the Killing vector field K(X, z) = M X. Notice that K generates isometries fixing the z-axis. Then the scalar product u = (K, N ) is a solution to ∆u + B 2 u = 0 on Σ. Moreover, since K is tangent to ∂B, u satisfies ∂ ν u + u = 0 on ∂Σ.

Each end of Σ can be parametrized by the graph of a function f with the asymptotic given by (2.4) or (2.5) (depending on n). In particular,

N (X, f (X)) = ± 1 1 + |∇f (X)| 2 (-∇f (X) + e z ).
So the asymptotic of f gives that u = O(|X| -(n-1) ).

On the other hand, since Σ is stable, there is a positive solution v to (2.3). The asymptotic of v is given by Theorem 4. As a consequence, we see that u(X)/v(X) goes to 0 as |X| goes to +∞. Also, the function w = u/v satisfies ∆w + 2(∇ ln v, ∇w) = 0 on Σ, ∂ ν w = 0 on ∂Σ.

Therefore the maximum principle gives that w = u/v is constant and thus equals zero. This implies that u = 0 and then Σ is invariant by the isometries generated by K. So Σ is a catenoidal hypersurface.

Catenoidal hypersurfaces

Theorem 1 gives that stable hypersurfaces are invariant by isometries fixing an axis. In this section, we describe this kind of exterior free boundary minimal hypersurfaces Σ. We fix the axis to be Re z .

Let φ be a primitive of the function r → (r 2(n-1) -1) -1/2 defined on [1, +∞). The hypersurface

Σ = {(X, z) ∈ R n × R; |X| ≥ 1 and z 2 = φ 2 (|X|)}
is a minimal hypersurface invariant by isometries fixing Re z . Actually, any connected piece of a minimal hypersurface invariant by isometries fixing Re z is a subset of λΣ+µe z for some λ, µ ∈ R.

Half of Σ can be parametrized by the map

F : [1, +∞) × S n-1 → R n+1 (r, p) → (rp, φ(r)) .
Given α ∈ (0, π 2 ), we look for a rotational exterior free boundary minimal hypersurface with boundary in {z = sin α}. Let R α = (sin α) -1/(n-1) be such that φ (R α ) = tan α. Notice that R α decreases with α from +∞ to 1. Let C α be the hypersurface parametrized by

F α : [R α , +∞) × S n-1 → R n+1 (r, p) → λ α (rp, φ(r)) + µ α e z ,
where λ α and µ α are chosen such that C α has the expected boundary:

(4.1) λ α = R -1 α cos α = (sin α) 1 n-1 cos α, µ α = sin α -λ α φ(R α ).
The hypersurface C α has free boundary because of the choice of R α . Thus, for any α ∈ [0, π 2 ), there is exactly one rotational exterior free boundary minimal hypersurface: C α for α = 0 and

C 0 = {z = 0} \ B for α = 0.
Therefore C α is an exterior free boundary minimal catenoidal hypersurface and any connected exterior free boundary minimal catenoidal hypersurface is the image of some C α by a linear isometry of R n+1 .

∂ α λ α = (sin α) -n-2 n-1 1 n -1 cos 2 α -sin 2 α . Let α n = arctan( 1 √ n-1
). It follows that, on [0, α n ], λ α is increasing from 0 to λ αn and, on [α n , π 2 ), λ α is decreasing up to 0. Moreover, C α converges to the hyperplane {z = 1} as α goes to π 2 .

The stability properties of C α is described by the following result, whose proof is inspired by the computation of the index of catenoids in R n+1 by L.-F. Tam and D. Zhou [START_REF] Tam | Stability properties for the higher dimensional catenoid in R n+1[END_REF].

Proposition 5. There exists ᾱn ∈ [α n , π

2 ) such that C α is stable for α ∈ [0, ᾱn ), and C α has index 1 for α ∈ (ᾱ n , π 2 ). Actually, ᾱ2 = α 2 = π 4 and ᾱn > α n for n > 2. Proof. We first study some preliminary stability properties of C α . Notice that C α is a graph over part of R n . Therefore C α is stable as a graph with fixed boundary: the stability operator is nonnegative for any test functions that vanish on ∂C α .

Let us consider on R n coordinates (x, Y ) ∈ R × R n-1 . Let K(x, Y, z) = (-z, 0, x) be the Killing vector field generating rotations around {x = 0, z = 0} in R n+1 . Then the scalar product k = (K, N ) defines on C α a solution to (2.3). The boundary condition comes from the fact that K is tangent to ∂B. Actually, one can compute k in the (r, p) coordinates. By (4.1), we have

k = 1 1 + (φ (r)) 2 (λ α φ(r) + µ α )φ (r) + λ α r p x , where p x is the x coordinate of p ∈ S n-1 ⊂ R × R n-1 . Observe that, by (4.1), (λ α φ(r) + µ α )φ (r) + λ α r ≥ (λ α φ(R α ) + µ α )φ (r) + λ α R α = sin α φ (r) + cos α > 0.
Hence k has constant sign when p x has constant sign. This implies that the half catenoidal hypersurfaces C α ∩ {±x ≥ 0} are stable.

Let us now study the global stability of C α . We have a one-parameter family {C α } of catenoidal hypersurfaces. Therefore the derivative with respect to α gives a deformation field whose scalar product with the unit normal to C α is a function u which solves (2.3). In the F α parametrization and for the upward pointing unit normal, u can be computed as

u = 1 1 + φ (r) 2 (-∂ α λ α rφ (r) + ∂ α λ α φ(r) + ∂ α µ α ).
So u depends only on r and is equal to 1 on ∂C α , i.e. at r = R α . In order to study the sign of u close to r = +∞, let us have a look on λ α . By (4.2), we have

∂ 2 α λ α = - cos α (sin α) -2n-3 n-1 (n 2 + n -2) sin 2 α + (n -2) cos 2 α (n -1) 2 ≤ 0.
Thus ∂ α λ α is decreasing.

If n = 2, we have lim r→+∞ φ(r) = +∞. Then, for α = α 2 = π 4 , we have lim r→+∞ u = ±∞ depending on sign(∂ α λ α ): close to r = +∞, u is positive for α < α 2 and negative for α > α 2 .

If n > 2, we have lim r→+∞ φ(r) < +∞ and, by (4.1),

lim r→+∞ u = ∂ α µ α + ∂ α λ α lim r→+∞ φ(r) = n n -1 cos α + ∂ α λ α lim r→+∞ φ(r) -φ(R α ) .
This limit is positive when α ≤ α n . Moreover, when α ≥ α n , the limit is decreasing with α and negative for α close to π 2 . Then there exists ᾱn > α n such that the limit is positive for α < ᾱn and negative for α > ᾱn .

Thus, for α < ᾱn , u is positive on ∂C α and close to the infinity. Therefore, if u changes sign on C α , {u < 0} would be a precompact subdomain of C α with u = 0 on its boundary, but this would contradict the stability of C α as a graph. Hence, for α < ᾱn , u is positive and then, by Proposition 3, C α is stable. C ᾱn is also stable as limit of stable minimal hypersurfaces.

When α > ᾱn , u changes sign on C α . Thus there is A > R α such that u is nonnegative on [R α , A] × S n-1 and vanishes on {A} × S n-1 . This implies that C α has index at least 1. We notice that there is no value B > A such that u vanishes on {B} × S n-1 . Indeed, this would contradict that [A, +∞) × S n-1 is stable as a graph.

Let us now prove that, for α > ᾱn , C α has index 1. If it is not the case, then there is B > R α such that the Jacobi operator has index at least 2 on [R α , B] × S n-1 . Let us consider u 2 the eigenfunction associated to the second eigenvalue

λ 2 < 0 on C α (B) = F α ([R α , B] × S n-1 ): u 2 is a solution to        ∆u 2 + B 2 u 2 = -λ 2 u 2 on C α (B), ∂ ν u 2 + u 2 = 0 on ∂C α , u 2 = 0 on r = B.
We are going to prove that u 2 depends only on the r variable. As above, let us consider (x, Y ) coordinates on R n and let S be the symmetry of R n+1 with respect to {x = 0}. C α (B) is invariant by S and then we can consider on it the function v defined by v(p) = u 2 (p) -u 2 (S(p)). v is then a solution to

       ∆v + B 2 v = -λ 2 v on C α (B), ∂ ν v + v = 0 on ∂C α , v = 0 on r = B. Moreover, we have v = 0 on C α (B) ∩ {x = 0}. If v = 0, this implies that C α (B) ∩ {x ≥ 0}
is unstable since λ 2 < 0, which contradicts the stability of C α ∩ {x ≥ 0}. So v ≡ 0 and u 2 is invariant by S. Changing the choice of the x coordinate, we obtain that u 2 is invariant by isometries fixing the z-axis and then depends only on r.

As u 2 is associated to the second eigenvalue, u 2 must change sign. Then there is C ∈ (R α , B) such that u 2 = 0 on {r = C}. As λ 2 < 0, this implies that {C ≤ r ≤ B} is unstable, which contradicts the stability of C α as a graph. Hence C α has index 1 for α > ᾱn .

Example 6. Consider the universal cover of C α for n = 2:

F α : [R α , +∞) × R → C α ⊂ R 3 (r, θ) → λ α (r cos θ, r sin θ, φ(r)) + µ α e z .
Straightforward computations give that the area element and the Gaussian curvature of F α are given by

dµ α = λ 2 α r(1 + (φ ) 2 ) 1 2 drdθ, K α = φ φ λ 2 α r(1 + (φ ) 2 ) 2 . Therefore, K α dµ α = φ φ (1 + (φ ) 2 ) 3 2 drdθ = - drdθ r 2 √ r 2 -1 .
Thus, the total curvature of C α is given by

2 2π 0 +∞ Rα dr r 2 √ r 2 -1 dθ = 4π +∞ Rα dr r 2 √ r 2 -1 = 4π(1 -cos α),
while the total curvature of its universal cover is infinite. For α ≤ π 4 , this example shows that when the boundary is not compact, even if the stability operator is nonnegative, the total curvature can be infinite.

Classification of one-ended examples

This section is devoted to the proof of Theorem 2. The idea of the proof is based on a symmetrization procedure as in Schoen's paper [START_REF] Schoen | Uniqueness, symmetry, and embeddedness of minimal surfaces[END_REF].

After a rotation, we can assume that the end of Σ is the graph of a function f over the outside of a compact set with the following asymptotic:

f (X) = A ln |X| + B + O(|X| -1 ) if n = 2,
with A ≥ 0 and, if A = 0, B ≥ 0, and

f (X) = B + A|X| -(n-2) + O(|X| -(n-1) ) if n > 2, with B ≥ 0.
The first step consists in proving that either Σ = {z = 0} \ B = C 0 or A > 0 and Σ ⊂ {z > ε} for some ε > 0.

Observe that ∂Σ ⊂ {z ≥ -2} and Σ \ K ⊂ {z ≥ -2} for some compact set K ⊂ R n+1 . Then, by the maximum principle, Σ ⊂ {z ≥ -2}. In fact, for each t < 0, we have Σ \ K ⊂ {z ≥ t} (for a possibly different compact set K). Therefore, if Σ∩{z < 0} = ∅, we can start from t = -2 and let t < 0 increase up to finding a first contact point in Σ ∩ {z = t 0 } for some t 0 < 0. We notice that, since Σ is free boundary, the first contact point cannot be at ∂Σ. Then the maximum principle can be applied at the first contact point in order to guarantee that Σ = {z = t 0 } \ B, which is not free boundary. This shows that Σ ⊂ {z ≥ 0}. Then either ∂Σ ⊂ {z > 0} or ∂Σ has a point in {z = 0} and the boundary maximum principle can be applied so that Σ = {z = 0} \ B.

If ∂Σ ⊂ {z > 0}, then we see that

- ∂Σ ∂ ν z = ∂Σ (P, e z ) > 0.
Since z is harmonic on Σ, using the asymptotic of f we obtain that 0

< Σ∩{|X|=R} ∂ ν z = Σ∩{|X|=R} (ν, e z ) = S n-1 -(n -2)A 1 R n-1 R n-1 + O(R -1 ) dσ = -(n -2)|S n-1 |A + o(1)
for n > 2. The same estimate gives that 0 < 2πA + o [START_REF] Ambrozio | Compactness analysis for free boundary minimal hypersurfaces[END_REF] for n = 2. Therefore, if n = 2, we have A > 0 and this implies that f (X) > 1 for |X| sufficiently large. If n > 2, we have A < 0 and, since f ≥ 0, this implies that B > 0 and f (X) > B 2 for |X| sufficiently large. In any case, we obtain that Σ ⊂ {z ≥ ε} for small positive ε since there cannot be any first contact point with {z = t} for 0 ≤ t ≤ ε. This ends the first step.

We fix a (x, Y ) coordinate system in R n = R × R n-1 . We want to prove that Σ is symmetric with respect to {x = 0}. In order to do this, we are going to follow a symmetrization procedure.

Given θ ∈ [0, π 2 ], let Π θ be the hyperplane of equation -x sin θ + z cos θ = 0, S θ be the symmetry with respect to Π θ , and Σ - θ = Σ ∩ {-x sin θ + z cos θ ≤ 0}. If B ρ is the ball centered at the origin of radius ρ > 0, we notice that S θ (B ρ ) = B ρ .

Lemma 7. Given θ ∈ [0, π
2 ), there exists ρ θ > 0 such that, outside B ρ θ , S θ (Σ - θ ) is above Σ. Moreover, ρ θ can be chosen as an increasing function of θ.

Proof. In R 2 , let p = (a cos α, a sin α) be a point with a > 0 and 0 ≤ α ≤ θ and R t be the rotation of angle t. If 0 ≤ t ≤ 2(θ -α), then the angle between ----→ pR t (p) and the vertical z-axis is α -t 2 and then at most θ (see Figure 3). Because of the asymptotic of Σ, the intersection Σ∩∂B ρ can be parametrized by ∂B ρ ∩{z = 0} in the following way: there is a function g such that

Σ ∩ ∂B ρ = (1 - g 2 (X) |X| 2 ) 1/2 X, g(X) ; X ∈ ∂B ρ ∩ {z = 0} ,
where g satisfies

g(X) = A ln |X| + B + O(|X| -1 ) if n = 2, g(X) = B + A|X| -(n-2) + O(|X| -(n-1) ) if n > 2,
and

(5.1) dg(X)(V ) = O(|X| -2 )|V |
for any vector V tangent to ∂B |X| .

In the (x, Y, z) coordinates, we can extend the rotation R t to R n+1 by fixing the Y coordinates. Let ρ be large and P ∈ Σ - θ ∩∂B ρ . We can write P = (1-g 2 (X) |X| 2 ) 1/2 X, g(X) for some X = (x, Y ). Then S θ (P ) is above Σ if R t (P ) does not meet Σ for 0 < t ≤ 2(θ -α), where 0 ≤ α ≤ θ is such that

(1 - g 2 (X) |X| 2 ) 1/2
x, g(X) = a(cos α, sin α) for some a > 0 (see Figure 4). In particular, S θ (P ) = R 2(θ-α) (P ). We notice that R t (P ) belongs to ∂B ρ . Then, if R t (P ) ∈ Σ, we must have

R t (P ) = (1 - g 2 (X ) |X | 2 ) 1/2 X , g(X )
for some X ∈ ∂B ρ ∩ {z = 0}.

|g(X) -g(X )| ≤ Cρ -2 |X -X | ≤ C ρ -2 (1 - g 2 (X) |X| 2 ) 1/2 X -(1 - g 2 (X ) |X | 2 ) 1/2 X .
This implies that -----→ P R t (P ) makes an angle less than C ρ -2 with the horizontal plane {z = 0}. Therefore, if ρ θ is chosen such that this angle is less than π 2 -θ, we obtain a contradiction with R t (P ) ∈ Σ and the lemma is proved.

We are now ready to finish the proof of Theorem 2. Let

T = θ ∈ [0, π 2 ]; S β (Σ - β ) is above Σ for all β ∈ [0, θ] .
Since Σ ⊂ {z > 0}, we may choose θ 0 > 0 small enough such that Σ - θ 0 ⊂ R n+1 \ B ρ θ 0 . By Lemma 7, we have [0, θ 0 ] ⊂ T . The set T is then a closed interval of the form [0, θ 1 ]. Let us notice that when we symmetrize with respect to Π θ , the image of a point on ∂B ρ stays on ∂B ρ , so that points in ∂Σ cannot be sent to interior points of Σ and interior points of Σ cannot be sent to ∂Σ.

Then, if θ 1 < π 2 , by Lemma 7, there is a point P ∈ Σ - θ 1 such that one of the following occurs:

• P / ∈ Π θ 1 , S θ 1 (P ) ∈ Σ and S θ 1 (Σ - θ 1 ) is on one side of Σ. • P ∈ Π θ 1 , Σ is orthogonal to Π θ 1 at P and S θ 1 (Σ - θ 1
) is on one side of Σ.

In the first case, if P / ∈ ∂Σ, then maximum principle gives that Σ is symmetric with respect to Π θ 1 . If P ∈ ∂Σ, then, by free boundary hypothesis, Σ and S θ 1 (Σ - θ 1 ) are normal to ∂B and thus tangent, since S θ 1 (Σ - θ 1 ) is on one side of Σ. As a consequence, the boundary maximum principle implies that Σ is symmetric with respect to Π θ 1 .

In the second case, if P / ∈ ∂Σ, the boundary maximum principle implies that Σ is symmetric with respect to Π θ 1 . If P ∈ ∂Σ, then as S θ 1 (Σ - θ 1 ) is on one side of Σ, we can locally parametrize Σ and S θ 1 (Σ - θ 1 ) over a quarter of the tangent plane T P Σ by two functions u and v such that u ≤ v, u(P ) = v(P ), and ∇u(P ) = ∇v(P ). Moreover, at P , the tangent vector P is an eigenvector of the second fundamental form of Σ (see (2.1)). Thus the Hessian of u and v at P coincide. So, applying Serrin's corner maximum principle [START_REF] Serrin | A symmetry problem in potential theory[END_REF] to v -u, we obtain that u ≡ v and Σ is symmetric with respect to Π θ 1 .

In any case, we obtain that Π θ 1 is a plane of symmetry of Σ, which is not possible by Lemma 7. This gives that θ 1 = π 2 . Then S π/2 (Σ ∩ {x ≥ 0}) is above Σ. The same argument gives that S π/2 (Σ ∩ {x ≤ 0}) is above Σ. As a Σ is symmetric with respect to {x = 0}.

By changing the coordinate system, we obtain that Σ is symmetric with respect to any vertical hyperplane passing through the origin and then invariant by rotation around the vertical z-axis: Σ is a catenoidal surface. where X is smooth vectorfield and q a smooth function. Then, given a compact domain U ⊂ Σ, there exists a constant C > 0 (not depending on u) such that, for any p, q ∈ U , we have u(p) u(q) ≤ C.

No such statement seems to appear in the literature. Similar results appear in [START_REF] Chen | A remark on the Harnack inequality for non-self-adjoint evolution equations[END_REF][START_REF] Wang | Global heat kernel estimates[END_REF], however they are not directly applicable here because of certain hypotheses.

Proof. In order to prove such an estimate, it is enough to prove an upper bound on |∇ ln u|. Let v = ln u. We have where B ∂Σ is the second fundamental form of ∂Σ and H is an upper bound for the principal curvatures of ∂Σ. So, choosing φ such that ∂ ν φ+2H < 0, we can ensure that ∂ ν (φw) is negative. Thus the maximum cannot be on the boundary of Σ. Let us compute ∆(φw) by using Bochner formula: ∆(φw) = w∆φ + 2(∇φ, ∇w) + φ∆(|∇v| 2 ) = w∆φ + 2(∇φ, ∇w) + 2φ (∇v, ∇∆v) + |∇ 2 v| 2 + Ric(∇v, ∇v) ≥ w(∆φ -2Kφ) + 2(∇φ, ∇w) + 2φ|∇ 2 v| 2 + 2φ(∇v, ∇(-q -(X, ∇v) -w)), where -K is a lower bound on the Ricci tensor. We also have (∇v, ∇(X, ∇v)) = (∇ ∇v X, ∇v) + (X, ∇ ∇v ∇v) ≤ Kw + (∇v, ∇ X ∇v)

= Kw + 1 2 (X, ∇w),
where K is also chosen to be an upper bound for the tensor (∇ • X, •).

At p, we have 0 = w∇φ + φ∇w, that is, ∇w = -w ∇φ φ . Thus ∆(φw) ≥ w(∆φ -4Kφ) + 2φ|∇ for some constants a, b, c and d which depend only on X and q. Thus, combining with (A. for some constants A, B, C and D which depend only on K, X, q and φ. This implies that φ(p)w(p) ≤ M for some constant M = M (A, B, C, D) and thus w(q) ≤ M for q ∈ U , since φ ≥ 1 on U .

As a consequence, we have the following Harnack inequality for the Robin boundary condition.

Proposition 9.

Let Σ be a Riemannian manifold with compact boundary and u be a positive solution to ∆u + qu = 0 on Σ, ∂ ν u + u = 0 on ∂Σ, where q is a smooth function. Then, given a compact domain U ⊂ Σ, there exists a constant C > 0 (not depending on u) such that, for any p, q ∈ U , we have u(p) u(q) ≤ C. So the above proposition applies to v and gives the expected result for u.

Figure 2 .

 2 Figure 2. Catenoidal hypersurface C α , α ≈ 0.92

Figure 3 .

 3 Figure 3. Angle between ----→ pR t (p) and the z-axis

Figure 4 .

 4 Figure 4. Configuration if S θ (P ) is not above Σ As a consequence, by integrating (5.1) along ∂B ρ ∩ {z = 0}, we have

Proposition 8 .

 8 Appendix A. Harnack inequality In this paper, we are considering solutions u to some elliptic equations on Σ under the Robin boundary condition ∂ ν u + u = 0. Elliptic regularity theory for this condition can be found in [11, Theorem 2.4.2.6]. Besides, one can also remark that, if d is a smooth function on Σ with ∂ ν d = 1 (for example -d could be the distance function to ∂Σ) and v = e d u, then ∂ ν v = (∂ ν d)e d u + e d ∂ ν u = 0 and v solves some elliptic equation. So results for Neumann boundary data can be translated to the Robin boundary condition.In the proof of Proposition 3, we use a Harnack inequality up to the boundary that can be derived from the following one. Let Σ be a Riemannian manifold with compact boundary and u be a positive solution to ∆u + (X, ∇u) + qu = 0 on Σ, ∂ ν u = 0 on ∂Σ,

u 2 =

 2 -q -(X, ∇v) -|∇v|2 . Now, let w = |∇v| 2 and consider a nonnegative function φ with compact support such that φ = 1 on ∂Σ and φ ≥ 1 on U . Let consider p a point of maximum of φw. We notice that ∂ ν v = 0, so ∇v is tangent to ∂Σ. Thus∂ ν (φw) = w∂ ν φ + 2φ(∇ ν ∇v, ∇v) = w∂ ν φ + 2φ(∇ ∇v ∇v, ν) = w∂ ν φ + 2φB ∂Σ (∇v, ∇v) ≤ w(∂ ν φ + 2H),

  2) and multiplying by φ(p), we obtain at p,0 ≥ 2 n (φw) 2 -A(φw) 3/2 -B(φw) -C(φw) 1/2 -D

Proof.

  As explained above, let d be a smooth function on Σ such that ∂ ν d = 1 and v = e d u. We then have ∂ ν v = (∂ ν d)e d u + e d ∂ ν u = 0. We also have ∆v = e d (u|∇d| 2 + u∆d + 2(∇u, ∇d) + d∆u) = 2(∇v, ∇d) + (∆d -|∇d| 2 -dq)v.

  2 v| 2 -2φ(∇v, ∇q)

			-2	|∇φ| 2
	|∇ 2 v| 2 ≥	1 n	|∆v| 2 =	1 n	(w

φ w + w(X, ∇φ) + 2w(∇v, ∇φ).

At p, ∆(φw) ≤ 0, and so

(A.2) 0 ≥ 2φ|∇ 2 v| 2 + w(∆φ -4Kφ -2 |∇φ| 2 φ + (X, ∇φ)) -2|∇φ|w 3/2 -2φ|∇q|w 1/2 .

We have

|∇ 2 v| ≥ 1 √ n |∆v|.

Then, by (A.1), we have 2 -aw 3/2 -bw -cw 1/2 -d)

L.M. was partially supported by the ANR-19-CE40-0014 grant. A.M. was partially supported by the National Council for Scientific and Technological Development -CNPq, Brazil (Grant 305710/2020-6). The authors were partially supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior -Brasil (CAPES-COFECUB 88887.143161/2017-0).