A Quattropanni

D Stoeer

T Fix

G Schmerber

M Lenertz

G Versini

J L Rehspringer

A Slaoui

A Dinia

S Colis
email: colis@ipcms.unistra.fr

Bandgap Tuning in Ferroelectric Bi 2 FeCrO 6 Double Perovskite Thin Films

We report in this work on the variation of the optical bandgap and structural properties of epitaxial Bi 2 FeCrO 6 lms grown by pulsed laser deposition on SrTiO 3 (001) substrates. It is shown that the bandgap can be tuned by varying the laser repetition rate during deposition which has a strong impact on the Fe/Cr order inside the Bi 2 FeCrO 6 double perovskite structure. Ab initio band structure calculations unambiguously show that the presence of antisite defects lead to an increase of the gap with about 0.25 eV with respect to the one calculated in the ideal structure. It is also shown that with increasing Fe/Cr disorder the saturation magnetization is strongly reduced along with the dierence between the Fe and Cr valences. These results suggest that the bandgap of Bi 2 FeCrO 6 can eectively be engineered by modulating the deposition conditions, thus paving the way for applications such as photovoltaic conversion, memory writing and direct CMOS integration.

Introduction

Although silicon is by far the most used semiconductor in nowadays electronic devices, oxide materials are becoming more and more appealing because of their wide variety of properties such as transparency, chemical stability, abundance, low price, reduced toxicity or versatility of the fabrication techniques. Most of the limitations originate from their relative large bandgap, and therefore, doping is systematically required to address dierent functions. Oxides can thus be used as transparent conductive materials (Sn-doped In 2 O 3 1,2 ), magnetic semiconductors (transition metal doped ZnO [START_REF] Belghazi | Magnetic properties of Al-doped Zn 0.95 Co 0.05 O lms: Experiment and theory[END_REF][START_REF] Chang | Eect of Co and O defects on the magnetism in Co-doped ZnO: Experiment and theory[END_REF] or TiO 2 [START_REF] Matsumoto | Room-temperature ferromagnetism in transparent transition metal-doped titanium dioxide[END_REF] ), half metallic materials for spintronic applications (La 2/3 Sr 1/3 MnO 3 , 6,7 Sr 2 FeMoO 6 [START_REF] Serrate | Double perovskites with ferromagnetism above room temperature[END_REF][START_REF] Fix | Nanoscale analysis of interfaces in a metal/oxide/oxide trilayer obtained by pulsed laser deposition[END_REF] ), multiferroic materials (BiFeO 3 [START_REF] Wang | Epitaxial BiFeO 3 multiferroic thin lm heterostructures[END_REF] ), photon management materials (rare earth doped ZnO, [START_REF] Balestrieri | Structural and optical properties of Yb-doped ZnO lms deposited by magnetron reactive sputtering for photon conversion[END_REF][START_REF] Balestrieri | Ecient energy transfer from ZnO to Nd 3+ ions in Nd-doped ZnO lms deposited by magnetron reactive sputtering[END_REF] SnO 2 [START_REF] Bouras | Optical and structural properties of Nd doped SnO 2 powder fabricated by the sol-gel method[END_REF][START_REF] Bouras | Structural, optical and electrical properties of Nd-doped SnO 2 thin lms fabricated by reactive magnetron sputtering for solar cell devices[END_REF] or CeO 2 [START_REF] Balestrieri | Photon management properties of rareearth (Nd,Yb,Sm)-doped CeO 2 lms prepared by pulsed laser deposition[END_REF] )... More recently, oxides were found to have a potential interest for photovoltaic solar cells. [START_REF] Ji | Bulk photovoltaic eect at visible wavelength in epitaxial ferroelectric BiFeO 3 thin lms[END_REF][START_REF] Grinberg | Perovskite oxides for visible-light-absorbing ferroelectric and photovoltaic materials[END_REF] The main issue related to these oxide materials is their high bandgap which, in order to t the AM 1.5G solar spectrum, should be lowered to approach that of an ideal absorber in a single junction solar cell (around 1.1-1.3 eV). [START_REF] Shockley | Detailed balance limit of eciency of p-n junction solar cells[END_REF] One of the most popular oxide absorber is Cu 2 O, which has a bandgap of 2.1 eV and a maximum eciency when integrated in solar cells of 8.1%. [START_REF] Minami | Eciency enhancement using a Zn 1-x Ge x -O thin lm as an n-type window layer in Cu 2 O-based heterojunction solar cells[END_REF] More recently, ferroelectric oxides such as Bi 2 FeCrO 6 with a bandgap of about 2 eV, [START_REF] Nechache | Photovoltaic properties of Bi 2 FeCrO 6 epitaxial thin lms[END_REF] BiFe 1-x Mn x O 3 , [START_REF] Xu | Tunable band gap in Bi(Fe 1-x Mn x )O 3 lms[END_REF] Ho-doped Bi 5 Ti 3 FeO 15 , [START_REF] Bai | Photovoltaic behaviors regulated by band-gap and bipolar electrical cycling in Holmium-doped Bi 5 Ti 3 FeO 15 ferroelectric lms[END_REF] BiXO 3 (with X = Fe, Mg, Zn, Cd, Ca, Sr) [START_REF] Yang | Photovoltaic eects in BiFeO 3[END_REF][START_REF] He | Ferroelectric oxides with strong visible-light absorption from charge ordering[END_REF] or KBiFe 2 O 5 (gap of 1.6 eV) [START_REF] Zhang | New high T C multiferroics KBiFe 2 O 5 with narrow band gap and promising photovoltaic eect[END_REF] were also reported to be suited for photovoltaic devices. Although less studied because of stability issues, non-oxide ferroelectrics are also reported to be potentially interesting for photovoltaic applications. [START_REF] Wang | Ruddlesden-Popper perovskite suldes A 3 B 2 S 7 : A new family of ferroelectric photovoltaic materials for the visible spectrum[END_REF][START_REF] Bi | Enhanced photovoltaic properties induced by ferroelectric domain structures in organometallic halide perovskites[END_REF] Due to the ferroelectric character of these materials, no p-n junction is needed to separate the charge carriers [START_REF] Kirchartz | Classication of solar cells according to mechanisms of charge separation and charge collection[END_REF] and photovoltage values much larger than the bandgap are expected. [START_REF] Yuan | Arising applications of ferroelectric materials in photovoltaic devices[END_REF][START_REF] Yang | Above-bandgap voltages from ferroelectric photovoltaic devices[END_REF] Another advantage is that the bandgap (along with other physical properties) can be tuned either by varying the dopant concentration or by changing the concentration of defects. [START_REF] Colis | Structural defects in Sr 2 FeMoO 6 double perovskite: Experimental versus theoretical approach[END_REF][START_REF] Stoeer | Ab initio study of the electronic structure of Sr 2 FeMoO 6 double perovskites presenting oxygen vacancies or/and antisite imperfections[END_REF] As a matter of fact, Nechache et al [START_REF] Nechache | Bandgap tuning of multiferroic oxide solar cells[END_REF] have recently demonstrated the growth of Bi 2 FeCrO 6 with variable bandgaps which exhibited conversion eciencies of 3.3% and 8.1% in single junction and tandem solar cells, respectively. A good understanding of the relationship between the growth conditions, the Fe/Cr ordering in Bi 2 FeCrO 6 and the bandgap is therefore mandatory.

In this paper we show a simple method to obtain thin lms of Bi 2 FeCrO 6 by pulsed laser deposition showing a modulated bandgap between 1.9 and 2.6 eV. This modulation can be easily obtained by varying the laser repetition rate from 1 to 10 Hz during deposition.

In order to better understand the eect of structural defects on the bandgap variation, ab initio calculations were carried out. While the defected structures (including antisite defects) result in larger bandgap values ranging between 2.50 and 2.55 eV, the bandgap of "perfect" Bi 2 FeCrO 6 is found about 2.3 eV. Although this change does not seem important, it is expected to have an essential role on the open circuit voltage of a PV solar cell and therefore on its conversion eciency.

Experimental details

Epitaxial Bi 2 FeCrO 6 lms were grown by pulsed laser deposition on SrTiO 3 (001) substrates starting from in-house sintered targets and using a KrF excimer laser (248 nm). All samples were grown at 750 • C under 10 -2 mbar O 2 atmosphere. The lms were cooled down after deposition at a rate of 5 • C/min in the same atmosphere. The laser power was kept at 26 mJ while its repetition rate was varied from 1 to 10 Hz. This is expected to modulate the concentration of defects, notably antisite defects, in the sample. The lm thicknesses of the samples deposited between 2 and 10 Hz range between 60 and 98 nm, as determined from the X-ray reectivity measurements, although the same number of laser pulses were used in each case. The lm thickness of the sample deposited at 1 Hz is about 34 nm as the number of the laser pulses was four times smaller than for the rest of the series.

The crystalline structure of the samples was analyzed by X-ray diraction using a Rigaku SmartLab diractometer equipped with a monochromatic source delivering a CuKα 1 incident beam (45 kV, 200 mA, 0.154056 nm). X-ray θ-2θ, reectivity, rocking curves, and φ-scans analyzes were performed in order to conrm the epitaxy of the lms, check the presence of spurious phases, determine the lm thickness, calculate the in-and out-of-plane correlation lengths and estimate the atomic ordering degree.

The magnetic properties were analyzed at room temperature using a MPMS SQUID-VSM (Quantum Design) magnetometer allowing a maximum eld of 7 T. The magnetic eld was always applied in the lm plane along the [100]SrTiO 3 (001) direction.

The optical bandgap of the lms was determined from spectroscopic ellipsometry measurements performed on a Horiba Uvisel. The dispersion model used was the Adachi new Fourouhi model. [START_REF] Yoshikawa | Optical constants of ZnO[END_REF] The gure of merit related to χ-squared values was always around 0.05.

Structural and optical properties

Bi 2 FeCrO 6 presents a double perovskite structure that can be described in the R3 symmetry (a rh = 0.547 nm and α rh = 60.09 • ). However, this structure can be be fairly approximated with a pseudo-cubic one. [START_REF] Baettig | First principles study of the multiferroics BiFeO 3 , Bi 2 FeCrO 6 , and BiCrO 3 : Structure, polarization, and magnetic ordering[END_REF] The equivalence of the two symmetries can be evidenced in g. 1a where both the hexagonal and the pseudo-cubic lattices are clearly visible. This pseudo-cubic symmetry explains why the Bi 2 FeCrO 6 growth can be performed on four-fold structures such as the one of SrTiO 3 . Fig. 1b shows the X-ray diraction patterns obtained for Bi 2 FeCrO 6 lms deposited at dierent laser frequencies. Except the (00l) peaks of the STO substrate, only the (00l) peaks of the Bi 2 FeCrO 6 phase are observed. Their close position at the left side of the STO peaks is in agreement with the very close lattice parameters of pseudo-cubic Bi 2 FeCrO 6 (0.3930 nm) and STO (0.3906 nm). These patterns also suggests that the lms are grown epitaxially with out-of-plane (OP) lattice parameter c of the Bi 2 FeCrO 6 pseudo-cubic phase oriented along the normal to the surface of the STO substrate. No secondary phases could be observed in the resolution limit of the X-ray diraction technique. [START_REF] Nechache | Epitaxial thin lms of the multiferroic double perovskite Bi 2 FeCrO 6 grown on (100)-oriented SrTiO 3 substrates: Growth, characterization, and optimization[END_REF] The epitaxial growth of Bi 2 FeCrO 6 on STO is clearly shown by the φ scan measurements recorded on the STO(111) and BFCO(111) peaks (g. 1c) of a lm grown at 2 Hz. The four peaks observed upon rotation of the sample indicates a four-fold symmetry of the Bi 2 FeCrO 6 and a cube-on-cube epitaxial relation of BFCO on STO.

In order to have information on the atomic order between Fe and Cr ions inside the Bi 2 FeCrO 6 structure, θ-2θ measurements were carried out along the pseudo-cubic [111]BFCO direction (the sample was tilted by 54.7 • ). Fig. 1d shows the presence of superstructure peaks (SP) at about 19.6 • and 61.2 • showing a chemical order in BFCO, i.e. an alternate stacking of Fe and Cr planes along the [111] direction. Note however that, although it is widely accepted that ordered double perovskite structures give raise to superstructure peaks, [START_REF] Nechache | Bandgap tuning of multiferroic oxide solar cells[END_REF] it is also suggested that such peaks can have dierent origins such as structural modulations (i.e. periodic cationic displacements) that can be found in ferroelectric oxides. [START_REF] Shabadi | Origin of superstructures in (double) perovskite thin lms[END_REF] The lms thickness and the vertical and lateral coherence lengths were obtained from reectivity (not shown here), θ-2θ (g. 1b) and rocking curve (not shown here) measurements. Their variation as a function of the laser frequency is reported in g. 2. Although the same number of laser shots was used for the samples grown from 2 to 10 Hz while the other deposition conditions were identical for all samples, the thickness of the lms decreases while increasing the laser frequency. It is also interesting to note that the vertical correlation length (usually known as the crystallite size as calculated using the Debye-Scherrer formula) is much smaller than the thickness and is decreasing while the thickness decreases. This suggests that although epitaxied on the substrate, the Bi 2 FeCrO 6 lms do not present a columnar growth mode and that a larger concentration of defects is present in the lms grown at large laser frequency. This can be easily understood if we keep in mind that the deposition rate of these lms is larger. It is also interesting to note that the lateral correlation length is larger on the thinner lms. This seems very probably due to the strains induced by the substrate and the progressive relaxation of the BFCO lattice as the lm thickness increases. The large correlation values are compatible with the low lattice mismatch between Bi 2 FeCrO 6 and the substrate and with the epitaxial character of the lms. signicantly from 1.9 to 2.6 eV when the deposition rate increases from 1 to 10 Hz, while the crystallite size slightly decreases. One way that possibly explains the bandgap variation is to consider a modication of the bond between the transition metal and the oxygen, altering in this way the hybridization energy and the Coulomb repulsion. It was recently shown that a d 5 -d 3 system with both cations in 3+ state (such as Fe and Cr in BFCO [START_REF] Nechache | Epitaxial thin lms of the multiferroic double perovskite Bi 2 FeCrO 6 grown on (100)-oriented SrTiO 3 substrates: Growth, characterization, and optimization[END_REF] ), due to homogeneous distribution in the degenerated d orbitals between the two cations, can make a system more rigid [START_REF] Vailionis | Mist strain accommodation in epitaxial ABO 3 perovskites: Lattice rotations and lattice modulations[END_REF][START_REF] Ohtomo | Spontaneous atomic ordering and magnetism in epitaxially stabilized double perovskites[END_REF] and less sensitive to strains. [START_REF] Nechache | Bandgap tuning of multiferroic oxide solar cells[END_REF] Thus, only limited rotation and tilts are possible, which will lead to little changes in the bandgap. [START_REF] Rondinelli | Substrate coherency driven octahedral rotations in perovskite oxide lms[END_REF] Another way to inuence the bandgap is to consider the defects existing in the material. One type of defects that is common to double perovskite structures and that can be usually quantied from the XRD data is the antisite defects. Their concentration is proportional to the ratio of the intensities of the superstructure peak (I SP ) and of the main peak (I (222) or I (444) ) of the double perovskite.

As in our case the BFCO(222) peak is very close to the SrTiO 3 (111) one, the BFCO(444) peak was used for a more precise cation ordering estimate. Fig. 3 shows the variation of the I SP /I (444) calculated from g. 1e (which should be proportional to the Fe/Cr order) as a function of the laser repetition rate. Surprisingly, no clear evolution of the I SP /I (444) ratio is observed with the laser frequency (since the diractogramms in g. 1e are very similar).

This can be explained on the basis of the work of Shabadi et al [START_REF] Shabadi | Origin of superstructures in (double) perovskite thin lms[END_REF] suggesting that, given the low scattering contrast (close atomic form factors) between the Fe and Cr cations, the presence of the superstructure peak (SP, i.e. BFCO(111)) is not a reliable indicator of the Fe/Cr ordering. This is also supported by the fact that ordering should be very dicult to obtain in Bi 2 FeCrO 6 . According to dierent authors, the similar formal valence and ionic radii of Cr 3+ (0.615 Å) and Fe 3+ (0.645 Å) ions hampers the chemical order in Bi 2 FeCrO 6 [START_REF] Nechache | Bandgap tuning of multiferroic oxide solar cells[END_REF][START_REF] Ohtomo | Spontaneous atomic ordering and magnetism in epitaxially stabilized double perovskites[END_REF] contrary to what is observed in other double perovkistes such as Sr 2 Fe 3+ Mo 5+ O 6 [START_REF] Kobayashi | Room-temperature magnetoresistance in an oxide material with an ordered double-perovskite structure[END_REF][START_REF] Anderson | B-cation arrangements in double perovskites[END_REF] where the valences are dierent and the order can be more easily achieved. It can be therefore reasonable to conclude that chemical order in Bi 2 FeCrO 6 should be also accompanied by a valence change of Fe and Cr towards a +2 and +4 state, respectively. The bandgap values were extracted from ellipsometry measurements. The order parameter is dened as the intensity ratio of the superstructure (SP) and BFCO(444) peaks reported in g. 1e.

As X-ray diraction measurements did not allow to extract information on the Fe/Cr order, it is interesting to look into the magnetic properties of this material. Indeed, the lack of cationic order in Bi 2 FeCrO 6 is reported to strongly reduce the saturation magnetization because of antiferromagnetic interactions (Fe-O-Fe and Cr-O-Cr) appearing in the system. [START_REF] Shabadi | Origin of superstructures in (double) perovskite thin lms[END_REF] Figure 4 shows the room-temperature magnetization loops recorded for Bi 2 FeCrO 6 lms deposited at a laser repetition rate of 2, 6 and 10 Hz. As observed, the saturation magnetization is small (below 10 emu/cm 3 ∼ 0.12 µ B ), much smaller than the values reported in more ordered lms (about 150 emu/cm 3 ). [START_REF] Nechache | Multiferroic properties-structure relationships in epitaxial Bi 2 FeCrO 6 thin lms: recent developments[END_REF] This indicates a strong disorder in our samples that increases with the laser repetition rate.

Our results clearly indicate that a correlation exists between bandgap and atomic disorder, and gives thus the opportunity to engineer the bandgap by controlling the deposition conditions. These variations are also probably correlated to the Fe and Cr valences. In order to have further insight on the correlation between the defect concentration, the bandgap, the saturation magnetization, and the valence of the transition metal ions, ab initio calculations have been carried out.

Ab initio calculations

According to previous investigations, [START_REF] Baettig | First principles study of the multiferroics BiFeO 3 , Bi 2 FeCrO 6 , and BiCrO 3 : Structure, polarization, and magnetic ordering[END_REF][START_REF] Zhe-Wen | Electronic structure and magnetic and optical properties of double perovskite Bi 2 FeCrO 6 from rst-principles investigation[END_REF] we adopted the R3 symmetry, corresponding to the space group 146, for the Bi 2 FeCrO 6 unit cell. The primitive cell, corresponding to the trigonal representation, contains one Bi 2 FeCrO 6 formula unit and is used for full relaxation calculations. The hexagonal representation allows to build a cell containing three Bi 2 FeCrO 6 formula units with 3 Fe and 3 Cr sites: this cell is used for simulating chemical congurations with exchanged Fe and Cr atoms and allows to build three new "disordered" congurations.

All calculations were performed using the Vienna Ab initio Simulation Package (VASP5.4) [START_REF] Kresse | Ecient iterative schemes for ab initio total-energy calculations using a plane-wave basis set[END_REF][START_REF] Kresse | From ultrasoft pseudopotentials to the projector augmentedwave method[END_REF] allowing to determine accurately the electronic structure and to investigate the magnetic properties of the considered system. It uses so-called Augmented Plane Waves and is based on the Projector Augmented Wave (PAW) [START_REF] Blöchl | Projector augmented-wave method[END_REF] method using pseudopotentials to determine the wave function outside the augmentation region. The HSE06 functional [START_REF] Heyd | Hybrid functionals based on a screened Coulomb potential[END_REF][START_REF] Krukau | Inuence of the exchange screening parameter on the performance of screened hybrid functionals[END_REF] is used in order to extract the bandgap from the calculation without having to play with empirical parameters like in the GGA+U approach. [START_REF] Perdew | Generalized gradient approximation made simple[END_REF][START_REF] Dudarev | Electron-energy-loss spectra and the structural stability of nickel oxide: An LSDA+U study[END_REF] Moreover, we have checked that the bandgap obtained with the HSE06 functional is recovered with the GGA+U method with so large U ef f values (around 7 eV) that the electronic structure is strongly distorted. All calculations are performed with a cuto energy of 400 eV and a k-points sampling of 4×4×4 for the trigonal cell and of 5×5×2 for the hexagonal cell.

After full relaxation, the lattice parameters and the internal structural position of all atoms were obtained and reported in table 1.

The dierent hexagonal cells are built keeping these atomic positions unchanged regardless if the Fe or Cr site is occupied by an Fe or a Cr atom. The "perfectly ordered" conguration, where Fe atoms are situated on Fe sites and Cr atoms are situated on Cr sites, presents alternating pure Fe and Cr (0001) planes: we used the notation FCFCFC for this case where the F and C letter corresponds to the Fe and Cr (0001) planes into the stacking.

The three dierent other cases, approximating Fe-Cr chemical disordered congurations, that can be built into the hexagonal cell are then CFFCFC, FFCCFC and FFFCCC. All these structures, with their respective Fe-Cr congurations, are presented in g. 5. (x,y,z) (z,x,y) (y,z,x) (0.0353, 0.8938, 0.4419) displayed in gs. 6 and 7. These DOS present several common features: (i) the occupied Fe states range mainly from -8 to -6.5 eV as a consequence of the strong localisation resulting from the use of the HSE06 functional, (ii) the unoccupied Fe states present a sharp structure around 2.5 eV which determines the bottom of the conduction band, (iii) the Cr occupied states range mainly from -1.5 to 0 eV and are weakly hybridized with the Fe states, (iv) the minority spin energy bandgap is included into the majority spin bandgap. For the "perfectly ordered" conguration, which has the lowest total energy between all four cases, all Fe (respectively Cr) sites being occupied by Fe (Cr) atoms, they carry a positive (negative) magnetic moment of 4.21 µ B (-2.76 µ B ) giving a total magnetic moment per cell of 6 µ B (table 2). For the "disordered" congurations, the Fe atom on a Cr site carries a negative magnetic moment and, reciprocally, the Cr atom on a Fe site carries a positive magnetic moment, so that the the total magnetic moment per cell is reduced to 2 µ B (table 2). The sign of the magnetic moment changes from one magnetic atomic layer to the next one whatever the nature of the magnetic atom is, i.e. only anti-parallel aligned magnetic moments are obtained for neighboring Fe/Fe, Cr/Cr and Fe/Cr atoms. These results strongly support the reduced saturation magnetization observed in our disordered samples. Considering the DOS (gs. 6 and 7), the minority spin band gap E ↓ g is entirely included into the majority spin band gap E ↑ g so that the band gap for the total DOS is equal to the minority spin band gap. Consequently, the band gap is found equal to 2.30 eV for the "perfectly ordered" conguration whereas it increases to 2.50-2.55 eV (table 2) for the "disordered" congurations in qualitative agreement with the experimental observation. However, the variation of the bandgap remains weak as compared to experiments as a consequence of the weak Fe-Cr hybridization evidenced by the calculations.

This investigation shows also that the used hexagonal cell, which only allows to alter the ordering along the c direction, does not allow considering really dierent "disordered" congurations and larger cells are required in order to model the "disorder" into planes perpendicular to the c direction. Nevertheless, it is worth noting that these calculations are In order to discuss from a theoretical point of view the relation between the valence of the Fe and Cr atoms and the degree of disorder into the Fe-Cr sequence, we have investigated the charge carried by each atom. First, we have done the so-called Bader charge analysis [START_REF] Bader | Atoms in Molecules -A quantum theory[END_REF] on the charge densities determined previously using the code developed by Henkelman et al. 5355 We have found that the Bader charges for Fe and Cr are nearly equal (equal to +1.9 [START_REF] Wang | Quantifying charge ordering by density functional theory: Fe 3 O 4 and CaFeO 3[END_REF] we conclude that the Bader charge analysis is not very helpful for investigating theoretically the charge ordering in such oxide.

In their work, Wang et al [START_REF] Wang | Quantifying charge ordering by density functional theory: Fe 3 O 4 and CaFeO 3[END_REF] demonstrate that charge ordering can be quantitatively investigated by analyzing the Born eective charge (BEC) carried by each atom. Moreover, the BEC determination results from the calculation of the variation of the polarization, which is a global quantity given by integration over the whole cell, and is consequently not sensitive to a dened volume (like a sphere or the Bader analysis volume) around the considered atom.

The calculation of the BEC using the HSE06 functional being too much computer time consuming, we have determined the electronic structure with the GGA+U approximation taking U ef f = 4 eV. We have checked that, by comparing to the results obtained with the HSE06 functional, the GGA+U with U ef f = 4 eV method describes accurately the occupied states, and consequently the charge density, even if the bandgap is found around 0.7 eV smaller (i.e. about 1.6 eV). This is motivated by the fact that the polarization and the Born eective charges are obtained by summing over the occupied states. The BEC are obtained using the linear-response approach as implemented into VASP [START_REF] Gajdo² | Linear optical properties in the projector-augmented wave methodology[END_REF] and the polarization is calculated into the framework of the modern theory of polarization. 58 This shows clearly that the BEC dierence between the Fe and Cr ions is directly related to the level of local Fe-Cr "disorder" varying from 0.57 |e| to 0.15 |e| for a limited "disorder" and being equal to 0.07 |e| for the extreme FFFCCC case. These BEC variations have no impact on the polarization found equal to 57 µC/cm 2 for all considered cases. It can be therefore concluded that, for this system, the higher the Fe-Cr "disorder" is, the smaller the dierence between the Fe and Cr BEC is and the lower the energetical stability of the structure is. This result clearly shows a correlation between the chemical order in Bi 2 FeCrO 6 and the valence dierence between the Fe and Cr ions.

Conclusion

In this work, the structural and optical properties of epitaxial single phase Bi 2 FeCrO 6 lms of dierent thickness grown by pulsed laser deposition on SrTiO 3 (001) substrates are reported.

The main result is that the bandgap and the Fe/Cr order of the lms can be tuned by varying the laser repetition rate. It is shown that chemically ordered Bi 2 FeCrO 6 lms have a low bandgap of about 1.9 eV while in a disordered lm this value can reach 2.6 eV. Ab initio band structure calculations conrmed indeed the increase of the bandgap in the disordered state, accompanied by a reduction of the saturation magnetization. It is also shown that the existence of the ordered state in Bi 2 FeCrO 6 is correlated to the existence of a large valence dierence between Fe and Cr. These experimental and theoretical results nally suggest that the Bi 2 FeCrO 6 double perovskite lms can eectively be engineered to obtain a desired bandgap by controlling the deposition conditions, which can open the path to several applications including photovoltaic conversion, memory writing and direct CMOS integration.

Figure 1 :

 1 Figure 1: (a) Sketch of the crystalline structure of Bi 2 FeCrO 6 (BFCO) showing the equivalency of the pseudo-cubic and hexagonal notations. The hexagonal unit cell and the corresponding axis referential are emphasized with black lines. (b) X-ray diraction patterns of BFCO thin lms deposited on SrTiO 3 (001) at dierent laser repetition rates. (c) φ scan measurements recorded on the STO(111) and superstructure BFCO(111) peaks showing a four-fold symmetry of the BFCO structure. The diractogramm recorded on the BFCO peak was shifted along the vertical axis for visibility reasons. (d) X-ray diraction patterns measured along the pseudo-cubic [111] direction of the Bi 2 FeCrO 6 lms deposited on SrTiO 3 (001). (e) Zoom on the superstructure (SP) and BFCO(444) peaks reported in g. (d). For all gures, the crystalline planes of Bi 2 FeCrO 6 have been indexed in the pseudo-cubic structure notation.

Figure 2 :

 2 Figure 2: Variation of the lms thickness, vertical and lateral correlation lengths as extracted from reectivity, θ-2θ and rocking curve X-ray diraction measurements, respectively.

Figure 3 :

 3 Figure 3: Variation of the lms bandgap and order parameter with the laser repetition rate.The bandgap values were extracted from ellipsometry measurements. The order parameter is dened as the intensity ratio of the superstructure (SP) and BFCO(444) peaks reported in g. 1e.

Figure 4 :

 4 Figure 4: Room-temperature magnetization loops recorded for Bi 2 FeCrO 6 lms deposited at a laser repetition rate of 2, 6 and 10 Hz.

Figure 5 :

 5 Figure 5: Ideal (a) and disordered (b-d) crystalline structures used for the ab initio calculations. The F and C letters accounts for Fe and Cr atomic planes. The black lines represent the hexagonal unit cell.

FCFCFCFigure 6 :Figure 7 :

 67 Figure 6: Densities of states (DOS) -total, Fe and Cr atoms projected -for the "perfectly ordered" conguration (FCFCFC) into the hexagonal cell. Positive and negative DOS correspond respectively to majority and minority spins states.

  |e| and for Cr atoms into a Cr-Cr-Fe or Cr-Cr-Fe stacking they are found equal to +0.43, +0.15, +0.25, +0.26, +0.25 and +0.14 |e|, 3. nally, for the Fe atom into a Fe-Fe-Fe stacking, ∆Q * zz = -0.33 |e| and for the Cr atom into a Cr-Cr-Cr stacking, ∆Q * zz = +0.40 |e|.

Table 1 :

 1 Structural data (reduced coordinates) of the R3 symmetry Bi 2 FeCrO 6 trigonal unit cell with a = 0.55135 nm and α = 59.821 • obtained from the full relaxation calculation.

	Atom	Coordinate	Value
	Bi1	(x=y=z)	0
	Bi2	(x=y=z)	0.4966
	Fe	(x=y=z)	0.2253
	Cr	(x=y=z)	0.7194
	O1	(x,y,z) (z,x,y) (y,z,x) (0.4010, 0.5367, 0.9395)
	O2

Table 2 :

 2 Local magnetic moments on the 3 Fe and the 3 Cr sites and spin dependent bandgaps for the four congurations considered in this work. ) and that they do not change when the Fe-Cr sequence is changed. According to a recent similar investigation on Fe 3 O 4 and CaFeO 3 systems,

	conguration	local magnetic moment (µ B )	E ↑ g /E ↓ g
	FCFCFC	4.212 / -2.760 / 4.212 / -2.760 / 4.212 / -2.760 4.15 / 2.30
	FFCCFC	4.212 / -4.168 / 2.807 / -2.762 / 4.175 / -2.804 3.12 / 2.52
	CFFCFC	2.864 / -4.126 / 4.164 / -2.805 / 4.176 / -2.808 3.13 / 2.55
	FFFCCC	4.212 / -4.167 / 4.163 / -2.807 / 2.823 / -2.762 3.08 / 2.50

|e|

Table 3 :

 3 Born eective charges Q * zz on the Fe and Cr atoms for the four cases considered in this work; the variation ∆Q * zz being calculated relative to the same atom into the "perfectly ordered" FCFCFC conguration. All quantities are given in absolute value of the electron charge |e|.

	FCFCFC	Fe	Cr	Fe	Cr	Fe	Cr
	Q * zz	+3.86 +3.06 +3.86 +3.06 +3.86 +3.06
	FFCCFC	Fe	Fe	Cr	Cr	Fe	Cr
	Q * zz ∆Q * zz	+3.67 +3.66 +3.49 +3.21 +3.81 +3.07 -0.19 -0.20 +0.43 +0.15 -0.05 +0.01
	CFFCFC	Cr	Fe	Fe	Cr	Fe	Cr
	Q * zz ∆Q * zz	+3.32 +3.77 +3.64 +3.10 +3.84 +3.31 +0.26 -0.09 -0.22 +0.04 -0.02 +0.25
	FFFCCC	Fe	Fe	Fe	Cr	Cr	Cr
	Q * zz ∆Q * zz	+3.68 +3.53 +3.69 +3.31 +3.46 +3.20 -0.18 -0.33 -0.17 +0.25 +0.40 +0.14

Table 3

 3 gives the BEC obtained on each Fe and Cr atom for the four cases considered in the present work. For the "perfectly ordered" conguration, the Fe BEC are found 0.80 |e| larger than the Cr ones in clear contrast to the result obtained by the Bader charge analysis.For the three "disordered" congurations, depending on the Fe-Cr sequence, the BEC show signicant variations ∆Q * zz : 1. for Fe atoms into a Cr-Fe-Cr stacking, ∆Q * zz are found equal to -0.05 and -0.02 |e| and for Cr atoms into a Fe-Cr-Fe stacking they are found equal to +0.01 and +0.04 |e|, 2. for Fe atoms into a Fe-Fe-Cr or Cr-Fe-Fe stacking, ∆Q * zz are found equal to -0.19, -0.20, -0.09, -0.22, -0.18 and -0.17

Acknowledgement

This work was carried out under the framework of the FERROPV project supported by the French Agence Nationale de la Recherche (ANR) under the reference ANR-16-CE05-0002-01.

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Laser repetition rate (Hz)


Michel Fliess
email: michel.fliess@polytechnique.edu

Cédric Join
email: cedric.join@cran.uhp-nancy.fr

A mathematical proof of the existence of trends in financial time series

Keywords: Financial time series, mathematical finance, technical analysis, trends, random walks, efficient markets, forecasting, volatility, heteroscedasticity, quickly fluctuating functions, low-pass filters, nonstandard analysis, operational calculus

We are settling a longstanding quarrel in quantitative finance by proving the existence of trends in financial time series thanks to a theorem due to P. Cartier and Y. Perrin, which is expressed in the language of nonstandard analysis (Integration over finite sets, F. & M. Diener (Eds): Nonstandard Analysis in Practice, Springer, 1995, pp. 195-204). Those trends, which might coexist with some altered random walk paradigm and efficient market hypothesis, seem nevertheless difficult to reconcile with the celebrated Black-Scholes model. They are estimated via recent techniques stemming from control and signal theory. Several quite convincing computer simulations on the forecast of various financial quantities are depicted. We conclude by discussing the rôle of probability theory.

Introduction

Our aim is to settle a severe and longstanding quarrel between 1. the paradigm of random walks 1 and the related efficient market hypothesis [START_REF] Fama | Foundations of Finance: Portfolio Decisions and Securities Prices[END_REF] which are the bread and butter of modern financial mathematics, 2. the existence of trends which is the key assumption in technical analysis. 2 There are many publications questioning the existence either of trends (see, e.g., [START_REF] Fama | Foundations of Finance: Portfolio Decisions and Securities Prices[END_REF][START_REF] Malkiel | A Random Walk Down Wall Street[END_REF][START_REF] Paulos | A Mathematician Plays the Stock Market[END_REF]), of random walks (see, e.g., [START_REF] Lo | A Non-Random Walk Down Wall Street[END_REF][START_REF] Taylor | Modelling Financial Time Series[END_REF]), or of the market efficiency (see, e.g., [START_REF] Grossman | On the impossibility of informationally efficient markets[END_REF][START_REF] Rosenberg | Persuasive evidence of market inefficiency[END_REF][START_REF] Taylor | Modelling Financial Time Series[END_REF]). 3 A theorem due to Cartier and Perrin [START_REF] Cartier | Integration over finite sets[END_REF], which is stated in the language of nonstandard analysis, 4 yields the existence of trends for time series under a very weak integrability assumption. The time series f (t) may then be decomposed as a sum

f (t) = f trend (t) + f fluctuation (t) (1)
where

• f trend (t) is the trend,

• f fluctuation (t) is a "quickly fluctuating" function around 0.

The very "nature" of those quick fluctuations is left unknown and nothing prevents us from assuming that f fluctuation (t) is random and/or fractal. It implies the following conclusion which seems to be rather unexpected in the existing literature:

The two above alternatives are not necessarily contradictory and may coexist for a given time series. 5 We nevertheless show that it might be difficult to reconcile with our setting the celebrated Black-Scholes model [START_REF] Black | The pricing of options and corporate liabilities[END_REF], which is in the heart of the approach to quantitative finance via stochastic differential equations (see, e.g., [START_REF] Shreve | Stochastic Calculus for Finance, I & II[END_REF] and the references therein).

Consider, as usual in signal, control, and in other engineering sciences, f fluctuation (t) in Eq. ( 1) as an additive corrupting noise. We attenuate it, i.e., we obtain an estimation of f trend (t) by an appropriate filtering. 6 These filters 1 Random walks in finance go back to the work of Bachelier [START_REF] Bachelier | Théorie de la spéculation[END_REF]. They became a mainstay in the academic world sixty years ago (see, e.g., [START_REF] Bernstein | Capital Ideas: The Improbable Origins of Modern Wall Street[END_REF][START_REF] Cootner | The Random Characters of Stock Market Prices[END_REF][START_REF] Merton | Continuous-Time Finance[END_REF] and the references therein) and gave rise to a huge literature (see, e.g., [START_REF] Shreve | Stochastic Calculus for Finance, I & II[END_REF] and the references therein). 2 Technical analysis (see, e.g., [START_REF] Béchu | L'analyse technique[END_REF][START_REF] Kaufman | New Trading Systems and Methods[END_REF][START_REF] Kirkpatrick | Technical Analysis: The Complete Resource for Financial Market Technicians[END_REF][START_REF] Müller | Das grosse Buch der Technischen Indikatoren. Alles über Oszillatoren, Trendfolger, Zyklentechnik[END_REF][START_REF] Murphy | Technical Analysis of the Financial Markets[END_REF] and the references therein), or charting, is popular among traders and financial professionals. The notion of trends here and in the usual time series literature (see, e.g., [START_REF] Gouriéroux | English translation: Time Series and Dynamic Models[END_REF][START_REF] Hamilton | Time Series Analysis[END_REF]) do not coincide. 3 An excellent book by Lowenstein [START_REF] Lowenstein | When Genius Failed: The Rise and Fall of Long-Term Capital Management[END_REF] is giving flesh and blood to those hot debates. 4 See Sect. 2.1. 5 One should then define random walks and/or market efficiency "around" trends. 6 Some technical analysts (see, e.g., [START_REF] Béchu | L'analyse technique[END_REF]) are already advocating this standpoint.

• are deduced from our approach to noises via nonstandard analysis [START_REF] Fliess | Analyse non standard du bruit[END_REF], which -is strongly connected to this work, -led recently to many successful results in signal and in control (see the references in [START_REF] Fliess | Critique du rapport signal à bruit en communications numériques[END_REF]),

• yields excellent numerical differentiation [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF], which is here again of utmost importance (see also [START_REF] Fliess | Commande sans modèle et commande à modèle restreint[END_REF][START_REF] Fliess | Non-linear estimation is easy[END_REF] and the references therein for applications in control and signal).

A mathematical definition of trends and effective means for estimating them, which were missing until now, bear important consequences on the study of financial time series, which were sketched in [START_REF] Fliess | Time series technical analysis via new fast estimation methods: a preliminary study in mathematical finance[END_REF]:

• The forecast of the trend is possible on a "short" time interval under the assumption of a lack of abrupt changes, whereas the forecast of the "accurate" numerical value at a given time instant is meaningless and should be abandoned.

• The fluctuations of the numerical values around the trend lead to new ways for computing standard deviation, skewness, and kurtosis, which may be forecasted to some extent.

• The position of the numerical values above or under the trend may be forecasted to some extent.

The quite convincing computer simulations reported in Sect. 4 show that we are

• offering for technical analysis a sound theoretical basis (see also [START_REF] Dacorogna | An Introduction to High Frequency Finance[END_REF][START_REF] Lo | Foundations of technical analysis: computational algorithms, statistical inference, and empirical implementation[END_REF]),

• on the verge of producing on-line indicators for short time trading, which are easily implementable on computers.7

Remark 1. We utilize as in [START_REF] Fliess | Time series technical analysis via new fast estimation methods: a preliminary study in mathematical finance[END_REF] the differences between the actual prices and the trend for computing quantities like standard deviation, skewness, kurtosis. This is a major departure from today's literature where those quantities are obtained via returns and/or logarithmic returns, 8 and where trends do not play any rôle. It might yield a new understanding of "volatility", and therefore a new model-free risk management. 9Our paper is organized as follows. Sect. 2 proves the existence of trends, which seem to contradict the Black-Scholes model. Sect. 3 sketches the trend estimation by mimicking [START_REF] Fliess | Non-linear estimation is easy[END_REF]. Several computer simulations are depicted in Sect. 4. Sect. 5 concludes by examining probability theory in finance.

Existence of trends 2.1 Nonstandard analysis

Nonstandard analysis was discovered in the early 60's by Robinson [START_REF] Robinson | Non-standard Analysis[END_REF]. It vindicates Leibniz's ideas on "infinitely small" and "infinitely large" numbers and is based on deep concepts and results from mathematical logic. There exists another presentation due to Nelson [START_REF] Nelson | Internal set theory[END_REF], where the logical background is less demanding (see, e.g., [START_REF]Nonstandard Analysis in Practice[END_REF][START_REF] Diener | Analyse non standard[END_REF][START_REF] Robert | Analyse non standard[END_REF] for excellent introductions). Nelson's approach [START_REF] Nelson | Radically Elementary Probability Theory[END_REF] of probability along those lines had a lasting influence. 10 As demonstrated by Harthong [START_REF] Harthong | [END_REF], Lobry [START_REF] Lobry | La méthode des élucidations successives[END_REF], and several other authors, nonstandard analysis is also a marvelous tool for clarifying in a most intuitive way questions stemming from some applied sides of science. This work is another step in that direction, like [START_REF] Fliess | Analyse non standard du bruit[END_REF][START_REF] Fliess | Critique du rapport signal à bruit en communications numériques[END_REF].

Sketch of the Cartier-Perrin theorem 11

Discrete Lebesgue measure and S-integrability

Let I be an interval of R, with extremities a and b.

A sequence T = {0 = t 0 < t 1 < • • • < t ν = 1} is called an approximation of I, or a near interval, if t i+1 -t i is infinitesimal for 0 ≤ i < ν. The Lebesgue measure on T is the function m defined on T\{b} by m(t i ) = t i+1 -t i . The measure of any interval [c, d[⊂ I, c ≤ d, is its length d -c. The integral over [c, d[ of the function f : I → R is the sum [c,d[ f dm = t∈[c,d[ f (t)m(t)
The function f : T → R is said to be S-integrable if, and only if, for any interval

[c, d[ the integral [c,d[ |f |dm is limited and, if d -c is infinitesimal, also infinitesimal.

Continuity and Lebesgue integrability

The function f is said to be S-continuous at t ι ∈ T if, and only if, f (t ι ) ≃ f (τ ) when t ι ≃ τ . 12 The function f is said to be almost continuous if, and only if, it is Scontinuous on T \ R, where R is a rare subset. 13 We say that f is Lebesgue integrable if, and only if, it is S-integrable and almost continuous. 10 The following quotation of D. Laugwitz, which is extracted from [START_REF] Harthong | Comment j'ai connu et compris Georges Reeb[END_REF], summarizes the power of nonstandard analysis: Mit üblicher Mathematik kann man zwar alles gerade so gut beweisen; mit der nichtstandard Mathematik kann man es aber verstehen. 11 The reference [START_REF] Lobry | Nonstandard analysis and representation of reality[END_REF] contains a well written elementary presentation. Note also that the Cartier-Perrin theorem is extending previous considerations in [START_REF] Harthong | Analyse non standard et représentation du réel[END_REF][START_REF] Reder | Observation macroscopique de phénomènes microscopiques[END_REF]. 12 x ≃ y means that xy is infinitesimal. 13 The set R is said to be rare [START_REF] Benoît | Diffusions discrètes et mécanique stochastique[END_REF] if, for any standard real number α > 0, there exists an internal set B ⊃ A such that m(B) ≤ α.

Quickly fluctuating functions

A function h : T → R is said to be quickly fluctuating, or oscillating, if, and only if, it is S-integrable and A hdm is infinitesimal for any quadrable subset. 14 Theorem 2.1. Let f : T → R be an S-integrable function. Then the decomposition (1) holds where

• f trend (t) is Lebesgue integrable, • f fluctuation (t) is quickly fluctuating.
The decomposition [START_REF] Aït-Sahalia | Nonparametric risk management and implied risk aversion[END_REF] is unique up to an infinitesimal. f trend (t) and f fluctuation (t) are respectively called the trend and the quick fluctuations of f . They are unique up to an infinitesimal.

The Black-Scholes model

The well known Black-Scholes model [START_REF] Black | The pricing of options and corporate liabilities[END_REF], which describes the price evolution of some stock options, is the Itô stochastic differential equation

dS t = µS t + σS t dW t (2)
where

• W t is a standard Wiener process,
• the volatility σ and the drift, or trend, µ are assumed to be constant.

This model and its numerous generalizations are playing a major rôle in financial mathematics since more than thirty years although Eq. ( 2) is often severely criticized (see, e.g., [START_REF] Mandelbrot | Mis)Behavior of Markets[END_REF][START_REF] Taleb | Fooled by Randomness: The Hidden Role of Chance in Life and in the Markets[END_REF] and the references therein). The solution of Eq. ( 2) is the geometric Brownian motion which reads

S t = S 0 exp (µ - σ 2 2 )t + σW t
where S 0 is the initial condition. It seems most natural to consider the mean S 0 e µt of S t as the trend of S t . This choice unfortunately does not agree with the following fact: F t = S t -S 0 e µt is almost surely not a quickly fluctuating function around 0, i.e., the probability that

| T 0 F τ dτ | > ǫ > 0, T > 0, is not "small", when • ǫ is "small",
• T is neither "small" nor "large". Remark 2. A rigorous treatment, which would agree with nonstandard analysis (see, e.g., [START_REF] Albeverio | Nonstandard Methods in Stochastic Analysis and Mathematical Physics[END_REF][START_REF] Benoît | Random walks and stochastic differential equations[END_REF]), may be deduced from some infinitesimal time-sampling of Eq. (2), like the Cox-Ross-Rubinstein one [START_REF] Cox | Option pricing: a simplified approach[END_REF].

Remark 3. Many assumptions concerning Eq. (2) are relaxed in the literature (see, e.g., [START_REF] Shreve | Stochastic Calculus for Finance, I & II[END_REF] and the references therein):

• µ and σ are no more constant and may be time-dependent and/or S t -dependent.

• Eq. ( 2) is no more driven by a Wiener process but by more complex random processes which might exhibit jumps in order to deal with "extreme events".

The conclusion reached before should not be modified, i.e., the price is not oscillating around its trend.

Returns

Assume that the function f : I → R gives the prices of some financial asset. It implies that the values of f are positive. What is usually studied in quantitative finance are the return

r(t i ) = f (t i ) -f (t i-1 ) f (t i-1 ) (3)
and the logarithmic return, or log-return,

r(t i ) = log(f (t i )) -log(f (t i-1 )) = log f (t i ) f (t i-1 ) = log (1 + r(t i )) (4)
which are defined for t i ∈ T\{a}. There is a huge literature investigating the statistical properties of the two above returns, i.e., of the time series (3) and (4).

Remark 4.

Returns and log-returns are less interesting for us since the trends of the original time series are difficult to detect on them. Note moreover that the returns and log-returns which are associated to the Black-Scholes equation (2) via some infinitesimal time-sampling [START_REF] Albeverio | Nonstandard Methods in Stochastic Analysis and Mathematical Physics[END_REF][START_REF] Benoît | Random walks and stochastic differential equations[END_REF] are not S-integrable: Theorem 2.1 does not hold for the corresponding time series (3) and (4).

Assume that the trend f trend : I → R is S-continuous at t = t i . Then Eq. (1) yields

f (t i ) -f (t i-1 ) ≃ f fluctuation (t i ) -f fluctuation (t i-1 ) Thus r(t i ) ≃ f fluctuation (t i ) -f fluctuation (t i-1 ) f (t i-1 )

It yields the following crucial conclusion:

The existence of trends does not preclude, but does not imply either, the possibility of a fractal and/or random behavior for the returns (3) and (4) where the fast oscillating function f fluctuation (t) would be fractal and/or random.

Consider the real-valued polynomial function

x N (t) = N ν=0 x (ν) (0) t ν ν! ∈ R[t],
t ≥ 0, of degree N . Rewrite it in the well known notations of operational calculus (see, e.g., [START_REF] Yosida | Operational Calculus: A Theory of Hyperfunctions[END_REF]):

X N (s) = N ν=0 x (ν) (0) s ν+1
Introduce d ds , which is sometimes called the algebraic derivative [START_REF] Mikusinski | Operational Calculus[END_REF][START_REF] Mikusinski | Operational Calculus[END_REF], and which corresponds in the time domain to the multiplication by -t. Multiply both sides by d α ds α s N +1 , α = 0, 1, . . . , N . The quantities x (ν) (0), ν = 0, 1, . . . , N , which are given by the triangular system of linear equations, are said to be linearly identifiable (see, e.g., [START_REF] Fliess | Critique du rapport signal à bruit en communications numériques[END_REF]):

d α s N +1 X N ds α = d α ds α N ν=0 x (ν) (0)s N -ν (5)
The time derivatives, i.e., s µ d ι XN ds ι , µ = 1, . . . , N , 0 ≤ ι ≤ N , are removed by multiplying both sides of Eq. ( 5) by s -N , N > N , which are expressed in the time domain by iterated time integrals. Consider now a real-valued analytic time function defined by the convergent power series

x(t) = ∞ ν=0 x (ν) (0) t ν ν! , 0 ≤ t < ρ. Approximating x(t) by its truncated Taylor expansion x N (t) = N ν=0 x (ν) (0) t ν
ν! yields as above derivatives estimates. Remark 5. The iterated time integrals are low-pass filters which attenuate the noises when viewed as in [START_REF] Fliess | Analyse non standard du bruit[END_REF] as quickly fluctuating phenomena. 15 See [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF] for fundamental computational developments, which give as a byproduct most efficient estimations. Remark 6. See [START_REF] Genc | An Introduction to Wavelets and Other Filtering Methods in Finance and Economics[END_REF] for other studies on filters and estimation in economics and finance.

Remark 7. See [START_REF] Taylor | Modelling Financial Time Series[END_REF] for another viewpoint on a model-based trend estimation.

Some illustrative computer simulations

Consider the Arcelor-Mittal daily stock prices from 7 July 1997 until 27 October 2008. 16 • the estimation of the trend thanks to the methods of Sect. 3, with N = 2;

1 day forecast

• a 1 day forecast of the trend by employing a 2 nd -order Taylor expansion. It necessitates the estimation of the first two trend derivatives which is also achieved via the methods of Sect. 3. 17We now look at some properties of the quick fluctuations f fluctuation (t) around the trend f trend (t) of the price f (t) (see Eq. ( 1)) by computing moving averages which correspond to various moments

M A k,M (t) = M τ =0 (f fluctuation (τ -M ) -ffluctuation ) k M + 1
where

• k ≥ 2,
• ffluctuation is the mean of f fluctuation over the M + 1 samples, 18• M = 100 samples.

The standard deviation and its 1 day forecast are displayed in Figure 3. Its heteroscedasticity is obvious. The kurtosis

MA4,100 (t) MA2,100(t) 2 , the skewness MA3,100(t)
MA2,100(t) 3/2 , and their 1 day forecasts are respectively depicted in Figures 4 and5. They show quite clearly that the prices do not exhibit Gaussian properties 19 especially when they are close to some abrupt change.

5 days forecast

A slight degradation with a 5 days forecast is visible on the Figures 6 to 10.

Above or under the trend?

Estimating the first two derivatives yields a forecast of the price position above or under the trend. The results reported in Figures 1112show for 1 day (resp. 5 days) ahead 75.69% (resp. 68.55%) for an exact prediction, 3.54% (resp. 3.69%) without any decision, 20.77% (resp. 27.76%) for a wrong prediction.

Conclusion: probability in quantitative finance

The following question may arise at the end of this preliminary study on trends in financial time series:

Is it possible to improve the forecasts given here and in [START_REF] Fliess | Time series technical analysis via new fast estimation methods: a preliminary study in mathematical finance[END_REF] by taking advantage of a precise probability law for the fluctuations around the trend?

Although Mandelbrot [START_REF] Mandelbrot | The variation of certain speculative prices[END_REF] has shown in a most convincing way more than forty years ago that the Gaussian character of the price variations should be at least questioned, it does not seem that the numerous investigations which have been carried on since then for finding other probability laws with jumps and/or with "fat tails" have been able to produce clear-cut results, i.e., results which are exploitable in practice (see, e.g., the enlightening discussions in [START_REF] Jondeau | Financial Modeling Under Non-Gaussian Distributions[END_REF][START_REF] Mandelbrot | Mis)Behavior of Markets[END_REF][START_REF] Sornette | Why Stock Markets Crash: Critical Events in Complex Financial Systems[END_REF] and the references therein). This shortcoming may be due to an "ontological mistake" on uncertainty:

Let us base our argument on new advances in model-free control [START_REF] Fliess | Commande sans modèle et commande à modèle restreint[END_REF]. Engineers know that obtaining the differential equations governing a concrete plant is always a most challenging task: it is quite difficult to incorporate in those equations frictions, 20 heating effects, ageing, etc, which might have a huge influence on the plant's behavior. The tools proposed in [START_REF] Fliess | Commande sans modèle et commande à modèle restreint[END_REF] for bypassing those equations 21 got already in spite of their youth a few impressive industrial applications. This is an important gap between engineering's practice and theoretical physics where the basic principles lead to equations describing "stylized" facts. The probability laws stemming from statistical and quantum physics can only be written down for "idealized" situations. Is it not therefore quite naïve to wish to exhibit well defined probability laws in quantitative finance, in economics and management, and in other social and psychological sciences, where the environmental world is much more involved than in any physical system? In other words a mathematical theory of uncertain sequences of events should not necessarily be confused with probability theory. 22 To ask if the uncertainty of a "complex" system is of probabilistic nature 23 is an undecidable metaphysical question which cannot be properly answered via experimental means. It should therefore be ignored. The Cartier-Perrin theorem [START_REF] Cartier | Integration over finite sets[END_REF] which is decomposing a time series as a sum of a trend and a quickly fluctuating function might be • a possible alternative to the probabilistic viewpoint,

• a useful tool for analyzing -different time scales, -complex behaviors, including abrupt changes, i.e., "rare" extreme events like financial crashes or booms, without having recourse to a model via differential or difference equations.

We hope to be able to show in a near future what are the benefits not only in quantitative finance but also for a new approach to time series in general (see [START_REF] Fliess | Time series technical analysis via new fast estimation methods: a preliminary study in mathematical finance[END_REF] for a first draft).

FiguresFigure 1 :Figure 2 :

 12 Figures 1 and 2 present

Figure 3 :Figure 4 :SamplesFigure 5 :

 345 Figure 3: 1 day forecast -Standard deviation w.r.t. trend (blue -), predicted standard deviation (black --)

20Figure 6 :Figure 7 :Figure 8 :Figure 9 :Figure 10 :

 678910 Figure 6: 5 days forecast -Prices (red -), filtered signal (blue -), forecasted signal (black --)

Figure 11 :Remark 8 .

 118 Figure11:1 day forecast -Prices (red -), predicted trend (blue --), predicted confidence interval (95%) (black -), price's forecast higher than the predicted trend (green △) , price's forecast lower than the predicted trend (blue ▽)

Figure 12 :Figure 13 :Figure 14 :

 121314 Figure 12: 1 day forecast -Zoom of figure 11

The very same mathematical tools already provided successful computer programs in control and signal.

[START_REF] Black | The pricing of options and corporate liabilities[END_REF] See Sect. 2.

[START_REF] Béchu | L'analyse technique[END_REF].[START_REF] Cartier | Integration over finite sets[END_REF] The existing literature contains of course other attempts for introducing nonparametric risk management (see, e.g.,[START_REF] Aït-Sahalia | Nonparametric risk management and implied risk aversion[END_REF]).

See[START_REF] Lobry | Nonstandard analysis and representation of reality[END_REF] for an introductory presentation.

Those data are borrowed from http://finance.yahoo.com/.

Here, as in[START_REF] Fliess | Time series technical analysis via new fast estimation methods: a preliminary study in mathematical finance[END_REF], forecasting is achieved without specifying a model (see also[START_REF] Fliess | Commande sans modèle et commande à modèle restreint[END_REF]).

According to Sect. 2.2 ffluctuation is "small".

Lack of spaces prevents us to look at returns and log-returns.

The effects of the unknown part of the plant are estimated in the model-free approach and not neglected as in the traditional setting of robust control (see, e.g.,[START_REF] Zhou | Robust and Optimal Control[END_REF] and the references therein).

It does not imply of course that statistical tools should be abandoned (remember that we computed here standard deviations, skewness, kurtosis).

We understand by "probabilistic nature" a precise probabilistic description which satisfies some set of axioms like Kolmogorov's ones.


A Khelil

M Braikia

H Naji

L Loukarfi

IMPROVED MIXING USING MULTIPLE SWIRLING JETS

Keywords: Multiple free jets, swirling flow, homogenisation, vanes swirler

The influence of several parameters such as the inclined vanes of diffuser and the sense of rotation of the single or multiple swirling jets, their number and their arrangement on the flow resulting thermally was experimentally investigated. Experiences concerning the fusion of several jets show that gas mixtures are clearly more homogenized under swirling influence. The results of this study show that the gap between the jets and their sense of rotation relative to the central jet, affects the quality of homogenization jets blown. Compared with the single jet, under uniform velocity and temperature, the multi-swirling jet with an angle of greater openness ensures a stable temperature distribution.

INTRODUCTION

Swirling jets are usually used for practical heat transfer devices. These, they present an interesting flow pattern for practical and theoretical standpoints. Practically they appear in aircraft combustors and a residential/commercial burner where swirl helps to improve mixing and stabilizes the flame. They are also widely used in industrial burners of power station furnaces or gas turbine combustors to provide stable, high intensity, short flames with wide radial development resulting from good fluid mixing [1,2]. There are still many questions concerning the understanding of the effect of swirl on heat transfer and behaviour of the induced flow [3]. The induced air entrainment by the rapid decrease of the average velocity implies a significant transfer of radial momentum. In general, experimental results show that the swirl increases the spread of the jet. This finding is further accentuated when the swirl number increases [START_REF] Florent | Utilisation de mélange de jets hélicoïdaux pour le chauffage de grands bâtiments, SFT[END_REF][START_REF] Loukarfi | Quelques configurations de jets triples tourbillonnaires en vue de l'homogénéisation thermique d'une ambiance[END_REF][START_REF] Loukarfi | [END_REF][7].

Understanding the effects of swirl, particularly on the entrainment rate of air and on rate of stratification of multiple jets which affects the efficiency of the air conditioning process is of great importance. [8][9]. Multiple swirling jets have been used for enhancing heat transfer in rapid heating, ventilation systems and cooling systems. When high heat transfer coefficients are needed over a wide area, Consequently the fusion of many swirling jets becomes interesting to study. The multiple swirling free jet studies show that the swirling jets will develop more rapidly than the jets without swirl. Note that the number of jets causes a decrease in velocities, and the distance between blowing orifices also diminishes velocities while delaying jets fusion. Far from the orifice and for large numbers of swirl, velocity profiles of multiple jets tend increasingly to those of the single jet. The interaction between the jets allows distribution of velocities in the mixing zone. This region is the seat of normal stresses and maximum shear. The profiles obtained near the swirler have irregularities due to the geometry and conditions of blowing. The decrease in axial temperature for the multijet seems to be exponential. It is less marked when the resulting jet encounters an obstacle. [10][11][12][13][START_REF] Durao | Part I[END_REF][15]. Optimization of these parameters would then influence the quality of the stratification made in the atmosphere treated suggesting the applicability of the interaction of swirl jets in the heating and space cooling of high-rise. According to the literature available there is no advanced research being done, on the multiple swirling jets applied to climatic comfort. The majority of the existing literature deals with multiple swirling jets in various geometric or dynamic conditions, applied to the improvement of combustion.

The aim of this study is to examine the influence of various parameters such as the sense of rotation of the single or multiple swirling jets, their number and arrangements on the flow resulting thermally. Optimization of these parameters would then influence the quality of the stratification made in the atmosphere treated allowing choosing the optimal configuration of interest to industrial.

EXPERIMENTAL SETUP

The experimental facility consists of a chassis on which is fixed a square Plexiglas plate. On the latter, 7 devices blowing hot air (hair dryer type TEFAL-1500) are fixed and directed downwards, and the lower part of these devices is used to fix different types of diffusers with inclined vanes, depending on the studied configuration. Temperatures and velocity of the flow is measured by a thermoanemometer (type velocicalc+ model [START_REF]Velocicalc Plus Air velocity Meter, Operation and Service Manual 1980321[END_REF]) which is a high-precision multifunctional instrument. The data can be viewed on screen, printed or downloaded to a spreadsheet program allowing us to easily transfer data to a computer for statistical treatment. The accuracy is of order ( ) 0.015 m s ± for velocity and ( )

0.3 C ± °
for temperature from thermal sensor [START_REF]Velocicalc Plus Air velocity Meter, Operation and Service Manual 1980321[END_REF]. Note that the thermal sensor is supported by rods that are easily guided vertically and horizontally to sweep the maximum space in the axial and radial directions. The swirling free jet used here is different from the conventional jet because of the existence a tangential component velocity. To obtain this kind of flow, one can use rotary mechanical systems or a static system; it is the latter that we have chosen here. It is composed of inclined vanes (see Fig. 1) which is put in the generating tube jet. The application of a tangential velocity component to the flow (W) provides a rotation to flow fluid, which can be by a non-dimensional swirl number (S). This number is defined as the ratio of the axial flux of tangential momentum to the product of the axial momentum flux and a characteristic radius [START_REF] Gupta | Swirl flows[END_REF].

2 2 = ∫ ∫ h h n n R R n R R S UWr dr R U rdr 1
where R n and R h are the radius of the centre body and the inlet duct, respectively and if the axial and azimuthally velocities are assumed to be uniform and the vane are very thin. In this study, the flow rate was adjusted at Reynolds numbers (Re 0 ) ranging from 10000 to 30000. Study has been done under uniform heat flux condition for each diffuser (S <1.3, Qm ≤ 54 g/s, x/D and r/D= 1 to 10) and air was used as working fluid.

Where: S is the swirl number .

Qm is masse flow rates (g/s).

D is inner diameter of one diffuser (m).

X is axial coordinate of the air flow (m).

R is radial coordinate of air flow (m).

We define a dimensionless temperature quantity Tr ( ) ( )

a max a T T T T Tr - - = (2)
Where:

T is jet temperature (°C).

T a is ambient temperature (°C).

T max is maximum temperature (°C).

Figure 1. Schematic of generator device swirling.

RESULTS AND DISCUSSION

Linear configuration of three swirling jets

Fig. 2 shows the device of triple swirling jets with two different temperature blowing (A2 and B2) and the space between jets axes is equal to 2D. The radial temperatures profiles associated with configurations A2 and B2 at location x/D = 8 is presented in Fig. 3. One can stress that the temperature varies slowly in cases A2 and B2. Also, the triple swirling jets with configuration A2 allows a more thermal homogenization with a more important spreading of the resulting jet (see Fig. 3, legend A2).

Plane configuration with four jets arranged in triangle

In the Fig. 4, we give the equilateral triangular arrangement of the configuration with four jets including a central jet. The space between jets axes is equal to 2D (see Fig. 4). Fig. 5 shows the axial evolution of temperature for cases of the single central jet, the three jets and the four jets, respectively. For the case of the central jet, we note that the temperature decreases rapidly. In the case where only the peripheral jets are active, the temperature begins to increase because of the mixture and then decreases. However, where all four jets are active, the temperature decreases less rapidly compared to the first case. Indeed, the central jet modifies the adjacent behaviour of peripheral jets and improves the homogenization of flow and the temperature decreases monotonically. This may explain the monotonic decrease of the temperature The development of the radial temperature at position x/D=8 is shown in Fig. 6 for the same cases as above. Again, we note that the temperature decreases in the radial direction. This trend is more pronounced in the case of the single central jet. As can be seen, evolution of the radial temperature remains almost constant because of the effect of central jet.

Plane configuration with five jets

Fig. 7 shows schematically the studied configuration, which consists of 4 jets surrounding the central jet and the space between jets axes is equal to 2D. The peripheral jets are inclined by φ=0°, 30°, 45° and 60°, respectively. The effect of the inclination of the peripheral jet is investigated at the location x/ D=8 (see Fig. 8). Analysing this figure, one can note that the decrease of the temperature is pronounced when the inclination is large, while allowing a stable distribution. Moreover, the gap between blowing orifices do not exceed a limited value. Otherwise the interaction between the jets will not have the desired effect for a good homogenization.

Plane configuration with 7 jets

The configuration consists of 7 swirling jets including a central jet and the space between jets axes is equal to 2D (see Fig; 9). Fig. 10 shows the radial evolution of the dimensionless temperature for different positions x/D. We note that all curves decrease from the origin (x/D=0.0) except for the curve at x/D=1. Away from the orifice, the radial profiles decrease slowly becoming almost constant at x/D=10. At this station, the thermal homogenization is almost complete (see Fig. 10).

Conclusion

The multiple swirling jets associated with the choice of the position of blowing air can significantly improve the quality of thermal homogenization of the treated space. The interaction between swirling jets induces the redistribution of temperature in the mixing zone as for velocities [3], while allowing the spreading of the resulting jet. Also, in all studied configurations, the central jet has a significant role in the enhancement of the homogenization. The optimization of parameters such as the diffuser's geometry, the inclination of the initial velocity, the spacing between jets, the number of blowing jets, the inclination of lateral jets from the central axis of the resulting jet and the relative rate between the central jet and peripheral jets contributes to improve the quality of thermal air homogenization in the conditioned space.

Figure 2 .Figure 3 .

 23 Figure 2. Configurations with swirling triple jet.

Figure 4 .Figure 5 .Figure 6 .

 456 Figure 4.Quadruple swirling jet configuration.

Figure 7 .Figure 8 .

 78 Figure 7. Schematic view of the configuration with five swirling jets.

Figure 8 .Figure 10 .

 810 Figure 8. Seven swirling jets arrangement.