B A Domaracka

L Huber

P Avaldi

Bolognesi

S Maclot

R Delaunay

D G Piekarski

A Domaracka

B A Huber

L Adoui

F Martín

M Alcamí

L Avaldi

P Bolognesi

S Díaz-Tendero

P Rousseau

Determination of Energy-Transfer Distributions in Ionizing Ion-Molecule Collisions

HAL is

The understanding of the electronic and nuclear dynamics in molecular systems induced by sudden ionization/excitation, which drives chemical reactions, offers new opportunities for controlled ultrafast chemistry [START_REF]Ultrafast Phenomena in Molecular Sciences[END_REF]. For example, it has been recently observed the charge migration on the fs timescale after hole formation, which triggers atomic motion and molecular fragmentation [2] or the localization of multiple charges in specific molecular groups and the subsequent Coulomb explosion [3,4]. Ultrafast nuclear rearrangement have been also observed in pump-probe experiments [5], such processes compete with the expected charge separation in multiply charged molecules [6][7][8]. Therefore, a detailed knowledge of the response of complex molecular systems to ionization/excitation and its influence on chemical reactivity is still today a relevant topic [9,10]. In this context, recent combined experiment/theory works have been very valuable in providing pictures of the ioninduced ionization/fragmentation of complex molecular systems [7,8,11,12]. However, a meaningful comparison between experimental and theoretical results requires the knowledge of the energy transferred in the collision, which is in fact represented by a wide energy distribution due to interactions at different impact parameters. Pioneering experimental work reported in [13,14] has already been performed in order to determine the actual energy-deposit distributions in ion-molecule collisions, as well as to study its relationship with the observed fragmentation patterns. However, these methods require the knowledge of the initial and final projectile states which is only straightforward in the case of double-electron capture by singly charged ions (e.g. H + → H -), which is more the exception than the rule.

Here we report on the ionization and fragmentation of a DNA building block, the nucleoside thymidine combining i) ion collisions, ii) VUV photoionization along with iii) ab initio calculations. Combining such state-of-the-art techniques, we provide a complete picture of the charge localization and the excitation energy distribution in complex molecular systems after interaction with ionizing radiation. More importantly, it becomes possible to determine the energy deposited in the target as a result of an ionizing collision with ions, which is the primary process associated with radiation damage. With the development of cancer therapies based on ionizing particles, as hadrontherapy [15], a better understanding of the radiation damage via a multi-scale and -disciplinary approach has become unavoidable [16]. At the molecular scale, this relies on the investigation of ionization/fragmentation of molecules of biological interest in the gas phase at different energy ranges [11,[17][18][19].

The experiments have been performed at ARIBE, the low-energy ions facility of GANIL (Caen, France) and at the GASPHASE beamline of the synchrotron radiation facility ELETTRA (Trieste, Italy). Both experiments are based on crossed-beam set-ups using coincidence timeof-flight mass spectrometry. The photoionization experiments are based on state-selected mass spectrometry using photoelectron-photoion coincidence measurements (PEPICO). The effusive beam of neutral thymidine molecule (2'-deoxy-thymidine abbreviated dThy, see structural formula in Fig. 1(a)) was produced by heating a powder in a oven at a temperature low enough to avoid thermal decomposition [20]. Both experimental set-ups have been described in detail elsewhere [21,22] and a brief description is given in the Supplemental Material [23].

In order to have a picture of the stability of the charged thymidine in the gas phase, the mass spectrum of the charged products detected after the production of singly-charged thymidine in the interaction with 48 keV O 6+ ions is shown in Fig. 1(b). The peak located at m/q = 242 amu corresponds to the intact singly charged dThy + and shows that a fraction of the parent population can be stable, at least in the µs timescale, after single ionization. The main peaks among the heaviest fragments are observed at m/q = 117 amu and m/q = 126 amu.

The first one corresponds to the sugar part S + whereas the second one is assigned to the fragment (B+1) + corresponding to an intramolecular rearrangement associated with a hydrogen transfer to the base part B [24]. Both fragments are the result of the glycosidic bond cleavage, an important mechanism in the radiation damage of DNA [24][25][26], and contribute to 8% of the spectrum.

A very small amount of fragments heavier than the base or sugar parts, i.e. loss of neutral fragments keeping intact the glycosidic bond, are also observed. This is due to the large distribution of impact parameters in the case of ion collisions [27] which leads to energy transfers, spanning from few meV to few tens of eV and involves a distribution of vibrational energy transfer and electron captures in various electronic states. Thus, the knowledge of the distribution of the energy transferred to the molecule plays a key role to unravel its fragmentation dynamics.

It is difficult to assess experimentally this energy distribution even if translational spectroscopy can provide it in the case of multiple electron capture [13,14].

A method that can provide direct insight on the frag-mentation following a selected energy deposition is the PEPICO technique, where the kinetic energy of the photoelectrons allows to pin point the energy left in the target [28]. The photoelectron spectrum of thymidine measured at 50 eV is shown in Fig. 1(c). From this spectrum, thirteen photoelectron binding energy values E b , covering the main features, have been chosen to study the evolution of the fragmentation. A simulation of the photoelectron binding energy spectra was carried out by computing the ionization energies for the 31 st highest molecular orbitals using the outer-valence Green's function (OVGF) method [29] in combination with a 6-311G(d,p) basis set of the Gaussian09 package [START_REF] Frisch | Gaussian 09 Revision C.01[END_REF]. The results are plotted in the panel (c). This method incorporates the effects of electron correlation in the computation of molecular ionization potentials as one-particle theory for the description of ultrafast electron charge density dynamics after ionization of an outer-valence electron. The uncertainty of the calculated energies by this method is about ±0.3 eV [START_REF] Yeager | Applied Many-Body Methods in Spectroscopy and Electronic Structure[END_REF]. The Fig. 1(d-g) shows the computed electron densities of four orbitals corresponding to the HOMO, HOMO- selected PEPICO mass spectra shown in the Fig. 1(h-k).

They illustrate that the charge localization after ionization strongly depends on the orbital and may lead to different fragmentation channels. In the first mass spectrum recorded at E b = 7.96 eV (Fig. 1(h)) we observe the peak due to the parent ion, i.e. the singly charged thymidine molecule. Due to the experimental energy resolution in the PEPICO measurements, not only photoelectrons from the ground ionic state but also from deeper orbitals can be detected [23]. This may lead to molecular dissociation and indeed some fragments are also observed PEPICO spectra via a constrained linear least-square regression to the the ion-induced mass spectrum considering eleven most relevant features. The fit parameters represent the contribution of each PEPICO mass spectrum, i.e. the contribution of the fragmentation of a bunch of excited states of the singly charged ion, to the ion spectrum (See method in Supplemental Material [23]). The result is displayed in Fig. 2 as a function of the excitation energy defined as the difference between the energy left in the target and the ionization potential. The energy distribution increases smoothly up to a maximum around 2 -3 eV and then it extends up to 8 eV and likely also above this energy, in a region not investigated in the present PEPICO experiments. Collisions at closer impact parameters can explain the extended tail towards larger deposited energy [27,[START_REF] Hervé Du Penhoat | [END_REF]33]. Penetrating trajectories are associated with large deposit energy of several tens of eV [33]. However, in the present interaction of 48 keV O 6+ with thymidine, peripheral collisions leading to small energy transfer are dominating. This is due to the fact that the electron capture radius is large (∼ 20 a.u. considering the classical-over-barrier model [34]) compared to the molecular size ("diameter" of ∼ 4.5 a.u.). The form of the distribution shown in Fig. 2 is qualitatively similar to those obtained by fitting theoretical fragmentation probabilities to experimental measured branching ratios in small carbon clusters [27] and fullerenes [33,35], in which the energy distribution was the fitting parameter, thus showing that the present results are also compatible with previous empirical estimations. Notice that, although the set of accessible target states can in principle be different in photoionization and collision processes, due to the different conservation rules that can apply in each case, this is not a problem in the present work because the absence of any symmetry in the molecular target does not restrict the number of accessible states in either process. Moreover, the single-electron capture, which is the dominant process at impact energies considered in this work, is not accompanied by excitation of target and projectile electrons [33,35]. Therefore, one can safely assume that the mass spectra resulting from the collision involves the same target states as the PEPICO spectra.

According to the PEPICO results, the maximum in the distribution of excitation energy corresponds to the region of the HOMO-3 state. The charge distribution (Fig. 1e) leads to the cleavage of the glycosidic bond and the production of both (B+1) + and S + fragments as observed in ion-collision. Larger excitation energy will cause further fragmentation. The partial ion yields of the parent ion and the leading fragments in the PEPICO mass spectra are plotted as a function of the binding energy E b in Fig. 3. The parent ion has a maximum yield centered around 8.5 eV and then vanishes above 10 eV, while the partial yields of the main fragments (B+1) + and S + are observed over a wide E b range starting around 8.5 eV and a display maximum around 10 eV (Fig. 3). Secondary dissociation of these fragments is observed for higher E b which corresponds to the tail towards larger excitation energies in Fig. 2. Fragments at m/q = 55 amu have been previously assigned to C 3 H 3 O + arising from the base part [24]. Several pathways leading to this fragment have been calculated, as shown in Fig. 4. The quantum chemistry calculations for the secondary fragmentation rely on an exploration of the potential energy surface in the ground state, i.e. assuming fast redistribution of the excitation energy over the vibrational degrees of freedom. The structure of the neutral molecule in the gas phase, its ionic form and the fragments produced in the relevant exit channels, together with the associated dissociation energy, have been computed with the density functional theory at the B3LYP/6-31G(d,p) level of theory, using Gaussian09 [START_REF] Frisch | Gaussian 09 Revision C.01[END_REF]. The simulations show that fragment C 3 H 3 O + can be produced from the base part B + (panel (b)) and from (B+1) + (panel (c)), but also from the sugar part S + (panel (b)). More surprisingly it is also possible to form this fragment directly from dThy + without glycosidic bond breaking (panel (d)). The second fragment at m/q = 31 is assigned to HOCH + 2 . This fragment arises from the sugar part [25] and certainly from the outside part of the furanose ring [START_REF] Hervé Du Penhoat | [END_REF]. Computed formation mechanisms show that it is possible to obtain this fragment from the sugar part while keeping intact the glycosidic bond (Fig. 4(b) and (d), respectively). Thus, combining the partial ion yields measured in the PEPICO experiments, the calculation of the binding energies of the different ionic states with the OVGF method and the dissociation pathways one can evaluate the contribution of the different fragmentation channels to the distribution of the energy transfer in the ion collision.

In summary, this letter presents the proof-of-concept of a method to accurately determine the excitation energy distribution of complex molecular ions produced in collisions with fast ions without knowledge of the initial and final states of the projectile nor the ionization potential of the target. The method relies on the combination of photon and ion experiments. The additional support of quantum chemistry calculations allows us to rationalize the measured energy distributions in terms of the electronic states of the singly charged ion and fragmentation channels. Thus the combination of ion and electron velocity resolved spectroscopies with in situ photoionization experiments appears as a promising tool to obtain a complete picture of the molecular dynamics that follows a collision with fast ions.

FIG. 1 .

 1 FIG. 1. (a) Structural formula of thymidine. Considering the glycosidic bond cleavage, the fragments produced are noted B and S for the base and sugar parts, respectively. (b) Mass spectrum of thymidine after the ionization by 48 keV O 6+ ions. White peaks around m/q = 180 and 200 amu are due to pollutions. (c) Photoelectron spectrum (PES) of thymidine obtained at 50 eV (black curve). The blue dashed lines show the energy values chosen for PEPICO measurements. Red bars correspond to orbital energy values computed with OVGF method. Panels (d-g) show the electron density of different molecular orbitals. Panels (h-k) show the PEPICO mass spectra recorded for different binding energies of the electron corresponding to closest energies to the orbitals presented.

  . The main fragment corresponds to (B+1) + indicating that the charge is mainly located on the base part as suggested by the electronic density of the HOMO (panel (d)). In the next mass spectrum, measured at E b = 10.34 eV, two mains peaks are observed with similar intensities. They are assigned to the fragment (B+1) + and S + as observed in the case of the fragmentation induced by multiply charged ions. The similar intensities show that the charge has almost the same probability to be located on each one of the two moieties of the molecule as shown by the non preferential charge localization in the orbital HOMO-3, in panel (e), and neighboring orbitals[23]. The panel (j) presents the mass spectrum recorded at E b = 11.24 eV. The same main peaks are present, but the fragment S + is now prominent. This is consistent with the preferential charge localization of the associated orbital HOMO-6 (panel (f)), although the neighboring orbitals can contribute to S + and (B+1) + peaks[23]. At larger excitation energy (Fig.1k), the mass spectrum is characterized by a strong fragmentation showing a redistribution of the transferred energy leading to the cleavage of several bonds in the molecule.Using the energy selected PEPICO mass spectra we can evaluate the excitation energy distribution in an ion collision. This is achieved by fitting the results of the

FIG. 3 .

 3 FIG. 3. Partial ion yields of some products as a function of the binding energy obtained from PEPICO. (a) Ionized thymidine (dThy + ) and base fragments ((B+1) + and 55 + ). (b) Ionized thymidine and sugar fragments (S + and 31 + ). Each point of the curves represents the areas of a gaussian fit to the mass spectra peaks. Error bars are estimated to 20% of the value due to the fitting method.

FIG. 4 .

 4 FIG. 4. (a) First vertical ionization potential and the most stable conformer of the singly charged thymidine (dThy + ). (b), (c) and (d) Energy levels of some fragmentation pathways obtained after the exploration of the potential energy surface of the most stable conformer of the singly charged thymidine. The calculated barriers of all pathways are not shown here for sake of clarity. (e) Determined distribution of excitation energy.

We thank F. Noury, S. Guillous, R. Richter, M. Coreno and K. Prince for support as well as L.J. Avaldi for advices in the statistical analysis of the data. We acknowledge the allocation of computer time at the Centro de Computación Científica at the Universidad Autónoma de Madrid (CCC-UAM). This work was partially supported by the projects FIS2013-42002-R and CTQ2013-43698-P (MINECO), NANOFRONTMAG (CAM), PHC Galilée 28137PE and an Advanced Grant of the European Research Council XCHEM No. 290853. Research was conducted in the framework of the International Associated Laboratory (LIA) "Fragmentation DY-NAmics of complex MOlecular systems -DYNAMO" and in the COST actions XLIC (CM1204) and Nano-IBCT (MP1002). D.G.P. acknowledges the FPI doctorate grant of the Universidad Autónoma de Madrid. S. D.-T. gratefully acknowledges the "Ramón y Cajal" program of the Spanish Ministerio de Educación y Ciencia.


T Aghavnian

J.-B Moussy

D Stanescu

Rachid Belkhou

N Jedrecy

Hélène Magnan

P Ohresser

Ph M.-A Sainctavit

Antoine Barbier
email: abarbier@cea.fr

M.-A Arrio

Ph Sainctavit

Determination of the cation site distribution of the spinel in multiferroic CoFe 2 O 4 /BaTiO 3 layers by X-ray photoelectron spectroscopy

Keywords:

est destinée au dépôt   Determination of the cation site distribution of the spinel in multiferroic CoFe2O4 / BaTiO3 layers by X-ray photoelectron spectroscopy

Introduction

The ferroelectric perovskite BaTiO 3 coupled with the ferrimagnetic spinel CoFe 2 O 4 is a well referenced system for the study of the magnetic states upon multiferroic couplings [1,2]. Such a study requires to fully analyze the magnetic properties of the spinel of such systems, among them the spinel inversion factor.

We already know that bulk CoFe 2 O 4 has several key properties such as strong magneto-crystalline anisotropy, high magnetostriction and saturation magnetization, as well as high coercive fields at room temperature [3,4]. Bulk CoFe 2 O 4 crystallizes in the spinel structure with a lattice parameter of 0.838 nm. The minimal cubic unit cell (i.e. gathering the 4 rhombohedral unit cells imposed by the fcc lattice) contains 8 formula units and the spinel space group is Fd 3m. The Co and Fe cations can occupy either tetrahedral Td or octahedral Oh sites of the oxygen sublattice. Their distribution follows the formula [Fe y Co 1-y ] Td [Fe 2-y Co y ] Oh O 4 , with the spinel inversion parameter y being 0 for a normal spinel and 1 for an inverse one. Lattice energy calculations [5] show that Co 2+ ions favor Oh sites in CoFe 2 O 4 , thus an inverse spinel structure is predicted. In practice y is usually smaller than 1, especially in thin films [START_REF] Matzen | Films ultraminces épitaxiés de MnFe2O4, CoFe2O4 et NiFe2O4 pour le filtrage de spin à température ambiante[END_REF].

To the bests of our knowledge, a precise analysis of the cation distribution inside the spinel of BaTiO 3 -CoFe 2 O 4 epitaxial thin layers multiferroics has not been looked into by most of the main references on the subject of magneto-electric coupling [2,[START_REF] Vaz | [END_REF]8]. However, the distribution of those cations within the spinel structure strongly affects the electric and magnetic properties of the samples. For instance, the resulting magnetic moment will be different depending on the cation distribution (an illustration of the concept is detailed for MnFe 2 O 4 ferrite in Ref. [9]).

It is therefore of high importance to determine this ion site distribution in order to deduce the global magnetic moment, as well as for any magnetic characterization, and in particular in the context of magnetoelectric coupling. X-ray magnetic circular dichroism (XMCD) is considered the most precise technique to determine the inversion parameter; however it requires access to a synchrotron beamline. Alternatively, authors often choose to estimate the cation distribution using X-ray photoelectron spectroscopy (XPS), as it is a common apparatus for spectroscopy in laboratories, and since it is obvious that the spectra have to contain all necessary information like in the X-ray absorption spectroscopy (XAS) case. Nevertheless, from previous reports no clear and reliable strategy for the XPS fitting procedure was established. It is still unclear how to correctly exploit XPS spectra to extract that data, especially considering that a given fitting method tends to have a non-negligible influence on the result, and methods seem to vary from paper to paper [10,11], sometimes leading to controversies [12,13]. The lack of direct comparison with XMCD does not allow to conclude on the pertinence of the proposed approaches.

Our aim is to define a reliable strategy for XPS data processing based on physical considerations and consistent with XMCD measurements, in order to extract the adequate cation distribution in CoFe 2 O 4 thin films. For that purpose, we use well-defined epitaxial thin films of CoFe 2 O 4 (0 0 1) grown on BaTiO 3 underlayers. The choice of analyzing spinel samples with different Co/Fe ratios was adopted to insure the stability of the method for a series of samples for which the cation organization is likely to vary. We expect that the present method can be extended to a large variety of other spinels.

Maghemite is the oxidized form of magnetite where all the divalent iron have been replaced by trivalent ones. Indeed three Fe(II) ions can be replaced by two Fe(III) ions and a vacancy. Magnemitization of spinel is observed when part of Fe(II) ions are replaced by Fe(III). By convention, we discuss the degree of spinel inversion independently of the degree of maghemitization, which becomes an additional parameter when Co/Fe differs from 0.5. In the same manner, we define the cationic stoichiometry as the Co/Fe ratio regardless of the divalent/trivalent distribution.

Experimental

Three samples of CoFe 2 O 4 /BaTiO 3 /SrTiO 3 (0 0 1) were prepared by Atomic-Oxygen assisted Molecular Beam Epitaxy (AO-MBE) [14][15][16].

The growth was performed in a dedicated AO-MBE setup working in the 10 -10 mbar pressure range equipped with a radiofrequency plasma source, an Auger electron spectroscopy (AES) analyzer, a reflection high energy electron diffraction (RHEED) gun and connected in ultra high vacuum conditions to a separate ultrahigh vacuum chamber with an X-ray photoelectron spectroscopy apparatus. During growth, the samples rotate continuously around the surface normal to obtain homogenous films.

The main features of the samples at different stages of their growth are detailed in Table 1. They mainly differ by having different Co/Fe ratios. Two of the SrTiO 3 (0 0 1) substrates were 1% Nb doped (for samples number 1 and 2), the last was undoped (sample 3). All three substrates were annealed in air at 900 • C for 3 h a few weeks before deposition in order to restore the nominal stoichiometry and get rid of the polishing induced defects. We first performed the epitaxial growth of BaTiO 3 (0 0 1) on SrTiO 3 (0 0 1) in the conditions detailed in [14]. Ba and Ti were deposited from individual Knudsen cells. The growth process was monitored in real time by RHEED, which is a technique that permits to control the crystal quality. The stoichiometry of the samples was checked by AES and XPS after the growth. After the BaTiO 3 (0 0 1) crystalline growth the samples were extracted from the AO-MBE chamber and mounted on a different sample holder optimized for CoFe 2 O 4 growth, then later reintroduced in the same epitaxy chamber. The carbon observed by AES on the BaTiO 3 , due to air-exposure, was removed by 45 min high brilliance oxygen plasma exposure. Then the ferrite growth was performed as in Ref. [15], and at a temperature of 450 • C. Then again, real time RHEED and post-growth in situ AES and XPS techniques were used.

The XPS apparatus uses twin anodes with a XR3 X-ray source and the hemispherical analyser Alpha110 from society Thermo Electron Corporation. XPS spectra were obtained by using Al-K˛ (1486.6 eV) and Mg-K˛ (1253.6 eV) X-ray sources. The precaution of using two sources is to insure that the Auger or photoemission peaks cannot be confused. A wide scan of the samples was first realized (0-1200 eV) to detect all the species in the sample. Then we focus on Fe2p, Co 2p and O1s. Those specific energy intervals were then scanned multiple times -typically 10 scans for Co and Fe, 5 for O -in order to maximize the signal to noise ratio. The energy resolution of the XPS spectra is typically around 0.1 eV. Analysis was performed using the software "Avantage Data Spectrum Processing" TM .

The XMCD measurements were performed on DEIMOS (Dichroism Experimental Installation for Magneto-Optical Spectroscopy) beamline at synchrotron SOLEIL (Saint-Aubin, France). DEIMOS is a beam-line dedicated to the study of the magnetic and electronic properties using polarized light. The main advantage of XMCD over other techniques to investigate magnetic behavior is its chemical and orbital selectivity. The energy range is 350-2500 eV, which allows the study of L 2,3 edges of Fe and Co and the K edge of O. The energy resolution is E/ E from 6000 to 10,000. The overall properties of the beamline [17] give us very clear spectra despite the small thickness of the thin film of spinel of 6 nm. Measurements were performed at 4 K. The cryo-magnet delivers ±7 T along the beam direction and ±2 T perpendicular to the beam. All our XMCD measurements we carried out at 4 K, with an incidence angle of 30 • with respect to the surface and a magnetic field of 5 T along the beam direction.

Results

The RHEED patterns were recorded during the deposition of both BaTiO 3 and CoFe 2 O 4 . An example of final RHEED patterns is reported in Fig. 1. It indicates that the samples are epitaxial and single crystalline. Also, we can observe from the comparison of sample 3 (non-stoichiometric, Co/Fe = 0.63) with a reference stoichiometric sample (Co/Fe = 0.50) that the RHEED patterns are similar. Moderate stoichiometry changes do not seem to affect the observations made using the RHEED technique. X-ray reflectivity measurements performed at 8 keV gave us an average deposition rate of 0.06 nm/min for BaTiO 3 and of 0.13 nm/min for CoFe 2 O 4 . The deduced thicknesses of deposition are given in Table 1.

Because our aim is to determine the cation distribution between octahedral and tetrahedral sites in the spinel using XPS, it is necessary to have a reliable and well accepted benchmark. Thus this distribution was first determined using XMCD measurements on DEIMOS beamline. The measurements were performed at the L 3 edge of Fe around 716.7 eV. The experimental plots are presented in Fig. 2, left, for the different samples. The similar form of the XMCD plots indicates that the average electronic structure for Fe is similar for the three samples. In addition, one notices that the samples are inverse spinel. Those plots were fitted using a linear combination of the theoretical [16,18] spectra of Fe 3+ in octahedral and tetrahedral sites and of Fe 2+ in octahedral sites (Fig. 2, middle), as these three components are sufficient since Fe 2+ is not found in tetrahedral sites [16]. The calculated Fe L 2,3 theoretical absorption spectra taken from [19,16] originate from a crystal field atomic multiplet approach, taking into account the occupied 3d electron configurations, the crystal field, the spin-orbit coupling and electron-electron interactions within Fe, and the hybridization of 3d electrons to other valence electrons.

The least squares method was applied using two parameters, the Fe 3+ proportion on the total of Fe 2+ and Fe 3+ cations (close to 1 as Fe 2+ appeared neglectful), and the Fe 3+ Oh proportion on the total of sites Fe 3+ Oh and Fe 3+ Td. This method was used to conclude on the best fitting values (Fig. 2, right).

The percentage of Fe 3+ in octahedral sites is respectively of 55%, 60% and 55% for samples number 1, 2 and 3. We thus conclude that the CoFe 2 O 4 thin layers grown on BaTiO 3 crystallize mainly in the inverse spinel structure. The XPS spectra that we measured for Co and Fe are presented in Fig. 3. The main components are the peaks corresponding to electronic levels 2p 3/2 and 2p 1/2 and the associated shake-up satellites for both metals. The shake-up peaks are attributed to the transition of an electron from a 3d orbital to the empty 4s orbital during the ejection of the core 2p photoelectron [20].

If the obtained spectra can be similar for different laboratories, the exploitation method tends to vary from paper to paper in literature. Zhou et al. [10] fit XPS spectra on bulk and powder CoFe 2 O 4 and differentiates Td and Oh sites, but do not use the same background as us for iron nor fit shake-up satellites. Wang et al. [11] fit also Co and Fe with two peaks, and add the fitting of the shake-up satellite, but only consider the 2p 3/2 ion levels and not the 2p 1/2 as well. The following paragraphs detail our approach towards the exploitation of those XPS spectra.

First of all, beyond the question of site occupation, the XPS spectra allow us also to estimate the stoichiometry of the samples. This was done using a Shirley-type background by fitting the Co2p 3/2 and Fe2p 3/2 peaks (octahedral sites and tetrahedral sites and shakeup) and deducing their area. Lets notice that this decomposition has more validity than trying to highlight a separation between different oxidation states of iron as XMCD measurements have shown that Fe 2+ is less than 1% of Fe 3+ . This area is then corrected using the corresponding Scofield factors [21]. We chose to estimate the Co/Fe ratio by calculating the one between the Scofield-corrected intensity area of Co2p 3/2 and Fe2p 3/2 . The choice of 2p 3/2 over 2p 1/2 was because of the higher intensity of signal giving us more precision. This ratio is found to be respectively of 0.28, 0.48 and 0.63 (Table 1) for samples 1, 2 and 3.

Experimentally, it is easier to discriminate the different contributions to the peaks for the Co spectra than for the Fe spectra. This observation was mostly confirmed during the fitting process with the "Avantage Data Spectrum Processing" TM software, which was more reproducible for Co. The main Co2p 3/2 peak was decomposed into two different peaks (Co2p 3/2 Oh and Co2p 3/2 Td (Fig. 4)) however the main Fe2p 3/2 peak was too broad and did not allow an obvious discrimination of individual components (Fe2p 3/2 Oh and Fe2p 3/2 Td) by this fit technique. As a consequence, we chose the XPS spectra for Co as a reference for estimating the spinel inversion parameter, rather than Fe. This decomposition in Oh and Td peaks is common on Fe [22], and we proceeded by analogy to do the same on Co.

We search a methodology allowing to fit the Co2p XPS spectrum in a reproducible way based on physical properties in terms of electronic structure in order to quantitatively assess the ratios of Oh and Td sites. This method is based upon the following constraints. First of all, we use a Shirley background that covers all the energy range from Co2p 3/2 to Co2p 1/2 . As a guard rule, for all fits of all elements, we used a Shirley background. The next step is to impose an identical spinel inversion parameter for the Co2p 3/2 and Co2p 1/2 states, which implies that the ratio between the areas of the XPS peaks of Co in Td and Oh sites, Co-Td/Co-Oh, must be the same for both Co2p 3/2 and Co2p 1/2 . This constraint is relaxed at the end of the fit and the fit is accepted if there is no more than 2% shift from a central value. The third constraint derives from the rigid band approximation [23] linked to the continuity in the perceived environment for the different sites. It implies to keep, for the decomposition of Co2p 3/2 and Co2p 1/2 in peaks associated to Oh and Td sites, the energy gap between both sites E Co-Oh -E Co-Td constant for each electronic configuration. This constraint is relaxed at the end of the fit and the fit is accepted if there is no more than 0.1 eV shift from a central value. The shake-up is a broad peak consisting of a mix of different peaks, and is considered as well represented by a unique peak. We fix no constraints on its energy positioning for the same reasons.

The fit is performed using symmetric peaks in order to limit the number of fitting parameters, even if we cannot exclude some asymmetry. However this method is suitable to give us the relative area of octahedral and tetrahedral peaks which makes a physical sense.

And indeed all these constraints put together give us a unique and perfectly reproducible fit for the Co2p spectra (Table 2). Whatever the initial position before the process, the fit always converges on the same values. This fully justifies to take Co as a reference. However, if we use the same fitting procedure for fitting the Fe2p lines, we observed that the results are strongly dependent of the initial parameter. It is maybe due to larger peaks and presence of different components (Fe 2+ and Fe 3+ ). Moreover, in the same energy range, there is also the KLL Auger of oxygen (Figs. 3 and5) which is mixed with the shake-up of Fe2p 1/2 . Because an inverse spinel has 100% of its Co in Oh sites (y = 1), and a normal spinel 100% in Td sites (y = 0), we deduce the spinel inversion parameter (y) to be equal to the occupation percentage of the cobalt octahedral sites. From the fit of Co 2p spectra, this percentage was found to be, for samples number 1, 2 and 3, of 80.8%, 80.6% and 82.4% respectively. Those results are within a 2% error margin linked to the shift after releasing constraints.

We tried to fit the Fe2p XPS spectra in the same manner, but the reproducibility was a problem. Indeed, we obtained good fits corresponding to significantly different inversion ratios. As a consequence, those spectra could not be used as the reference for our XPS method. However, as we examine the same sample, it is possible to impose the supplementary constraint of having the same spinel inversion parameter as determined from the Co2p spectrum. Using this added constraint, we can find a good fit that corresponds to the expected results, and which is moreover unique (Fig. 5 and Table 2).

Concerning the Fe2p fitting, we found that it is important to determine the background over a large range including the huge peak at ≈748 eV corresponding to the KLL Auger peak of oxygen, which contributes massively to the background. It is here a better solution than switching to an Al K˛ source to eliminate the Auger contribution of O, as the Auger contribution of Co would then overlap in energy with Fe2p.

Discussion

The difference of binding energy between octahedral and tetrahedral sites for samples number 1, 2 and 3 are respectively of 2.75 eV, 2.23 eV and 2.63 eV for Fe2p (average values of Fe2p 3/2 and Fe2p 1/2 ), within a 0.1 eV margin, and of 2.40 eV, 2.35 eV and 2.38 eV for Co 2p. These values are close to the one found for CoFe 2 O 4 nanoparticles [10] (3 eV for Fe2p and 2.10 eV for Co2p), where the amount of normal spinel structure is higher than in our samples. Comparing our results with other experiments on similar samples can give discrepancies [24] since some do not take into account the Co 2p 1/2 peak or have not fitted the shake-up satellite. It so seems that taking into account all parameters -background, electronic levels 2p 3/2 and 2p 1/2 , shake-up satellites -is key in finding a unified method of XPS analysis.

The fact that the difference in binding energy between octahedral site and tetrahedral site in XPS is bigger that the difference of absorption edges of the same sites is not really surprising. Indeed, the screening of the core hole is reflected in the spectra in a completely different manner for the two processes.

The inversion parameter y obtained using the XPS method, of 80.8%, 80.6% and 82.4% for samples number 1, 2 and 3, can be linked to the estimated percentage of Fe 3+ in octahedral and tetrahedral sites. If we consider that in an inverse spinel, Fe 3+ is in similar proportion in both Oh and Td sites, and that in a normal spinel all Fe 3+ are in Oh sites, we find that %Fe(Oh) = (y × 0.5 + (1y) × 1) × 100 which is 59.6%, 59.7% and 58.8%. Each of those results is less than 5% different from results from XMCD -55%, 60% and 55% for samples number 1, 2 and 3 -which is in the error margin of the XPS technique.

We can also observe that the spinel inversion parameter y seems fairly independent of the Co/Fe ratio, so of the stoichiometry of the spinel. This result is quite surprising [25]. Indeed, for stoichiometries with low Co 2+ cations content, we would have expected iron cations to replace some of the Oh sites, modifying thus the inversion parameter. Or a maghemitization of the crystal, which would mean cationic vacancies but would imply also a small change of the inversion parameter.

The results indicate that the spinel of our BaTiO 3 -CoFe 2 O 4 multiferroic thin layers crystallizes in a mostly inverse configuration and thus of high magnetic moment.

If XMCD remains a recognized method when looking for a precise determination of the spinel inversion parameter y, we show that XPS spectra coupled to physical constraints for the analysis, can give a complementary estimation. Both XMCD and XPS methods yield very similar results. This convinces us that XPS can be an excellent synchrotron free approach, when a 5% error on the sought information remains an acceptable precision.

Conclusion

We have grown CoFe 2 O 4 /BaTiO 3 /SrTiO 3 samples by oxygenassisted molecular beam epitaxy, fully controlled in crystalline structure and stoichiometry. The spinel film of CoFe 2 O 4 was studied both in XMCD and XPS. To perform that study, we focused on the Fe spectra in XMCD and the Co spectra in XPS. To fit correctly the Fe spectra in XPS, it appeared necessary to perform the scan on a large band of energy around Fe to obtain the actual background. Using physical considerations to impose constraints to the fit of the XPS spectra, we managed to obtain quantitative values of the distribution of the Co and Fe cations between octahedral and tetrahedral sites of the spinel. Those values were fully consistent with the XMCD results obtained on the same samples, therefore confirming our approach. We also highlighted the inverse nature of the spinel. This method can be of major utility to obtain routinely those values in laboratories equipped with an XPS apparatus, rather than having to perform the determination on a synchrotron beamline. It is also likely applicable to any other spinel ferrites such as

Fig. 1 .

 1 Fig. 1. RHEED patterns after the deposition of CoFe2O4 on BaTiO3 -stoichiometric sample (top), sample 3 (bottom), [1 0 0] direction (left), [1 1 0] direction (right).

Fig. 2 .

 2 Fig. 2. Experimental XMCD spectra at the L2,3 edge of Fe in the samples (a); theoretical XMCD plots of Fe 3+ Oh, Fe 3+ Td and Fe 2+ Oh (b); from Ref. [19], fitted XMCD spectra of the sample 2 recorded at the L2,3 edge of Fe, obtained by the least-squared combinatory method (c).

Fig. 3 .

 3 Fig. 3. XPS spectrum of Co in CoFe2O4 for sample 3 (a). XPS spectrum of Fe in CoFe2O4 for sample 3 (b).

Fig. 4 .

 4 Fig. 4. XPS spectrum of Co in CoFe2O4, with the best fit peaks for sample 3.

Fig. 5 .

 5 Fig. 5. XPS spectrum of Fe in CoFe2O4 -decomposition in individual contributions of the best fit for sample 3.

CrFe 2 O 4 ,

 24 MnFe 2 O 4 , NiFe 2 O 4 , CuFe 2 O 4 .

Table 1

 1 Main characteristics of the CoFe2O4/BaTiO3/SrTiO3 samples.

	Sample	1	2	3
	BaTiO3 thickness (nm)	15	13	11
	CoFe2O4 thickness (nm)	6	6	6
	XPS Co/Fe	0.28	0.48	0.63
	Spinel inversion parameter yXMCD	0.90	0.80	0.90
	Spinel inversion parameter yXPS	0.808	0.806	0.824
	% Fe 3+	Oh by XMCD	55%	60%	55%
	% Fe 3+	Oh by XPS	59.6%	59.7%	58.8%

Table 2

 2 Binding energy position of the fitting peaks in XPS for sample 3.

	Sample 3	Co	Fe
	2p 3/2 Oh	786.95 eV	717.50 eV
	2p 3/2 Td	789.35 eV	720.15 eV
	2p 3/2 shake-up	793.35 eV	725.65 eV
	2p 1/2 Oh	802.35 eV	731.65 eV
	2p 1/2 Td	804.6 eV	733.05 eV
	2p 1/2 shake-up	809.5 eV	740.6 eV

Acknowledgements

We acknowledge SOLEIL for provision of synchrotron radiation facilities and are grateful to the SOLEIL-DEIMOS beamline staff members who helped achieve those experiments. This work was partly supported by the CNano-MAEBA grant, for which we are also grateful.


Sophie Cerf
email: sophie.cerf@gipsa-lab.fr

Mihaly Berekmeri
email: mihaly.berekmeri@gipsa-lab.fr

Bogdan Robu
email: bogdan.robu@gipsa-lab.fr

Nicolas Marchand
email: nicolas.marchand@gipsa-lab.fr

Sara Bouchenak
email: sara.bouchenak@insa-lyon.fr

Cost Function based Event Triggered Model Predictive Controllers Application to Big Data Cloud Services

High rate cluster reconfigurations is a costly issue in Big Data Cloud services. Current control solutions manage to scale the cluster according to the workload, however they do not try to minimize the number of system reconfigurations. Event-based control is known to reduce the number of control updates typically by waiting for the system states to degrade below a given threshold before reacting. However, computer science systems often have exogenous inputs (such as clients connections) with delayed impacts that can enable to anticipate states degradation. In this paper, a novel eventtriggered approach is proposed. This triggering mechanism relies on a Model Predictive Controller and is defined upon the value of the optimal cost function instead of the state or output error. This controller reduces the number of control changes, in the normal operation mode, through constraints in the MPC formulation but also assures a very reactive behavior to changes of exogenous inputs. This novel control approach is evaluated using a model validated on a real Big Data system. The controller efficiently scales the cluster according to specifications, meanwhile reducing its reconfigurations.

I. INTRODUCTION

The dynamic management of the size of Big Data Cloud services clusters is a primary issue. Clusters that are too large have high monetary and energetic costs while too small ones bring poor performance, often resulting in financial and commercial penalties [START_REF] Evolven | Downtime, outages and failures -understanding their true costs[END_REF]. Moreover, a high frequency of cluster reconfigurations requires intensive usage of communication channels, implies additional costs and degrades performances in case of bottlenecks. From the point of view of the low-scale user, the example of Amazon EC2 [START_REF]Amazon EC2[END_REF] public Cloud where one can rent resources by the hour is a relevant example where scaling up and down the cluster too often is not beneficial.

Solutions have been proposed to automatically scale the cluster, most of them are heuristic ones based threshold crossing by a relevant metric (request rate, CPU utilization, response time) [START_REF] Ali-Eldin | Efficient provisioning of bursty scientific workloads on the cloud using adaptive elasticity control[END_REF], [START_REF] Nathuji | Q-clouds: Managing performance interference effects for qos-aware clouds[END_REF]. Amazon Auto Scaler [START_REF]Amazon EC2[END_REF] is perhaps the most popular implementation of such an approach for public Clouds, it provides the basic mechanism for reactive controllers but its configuration is based on best-effort policies. More advanced reactive techniques have also emerged which use queuing theory to decide capacity requirements [START_REF] Gandhi | Autoscale: Dynamic, robust capacity management for multi-tier data centers[END_REF]. Predictive methods, that foretell future disturbance behavior and adapt the scaling of the cluster accordingly, have also been considered. They are based on machine learning [START_REF] Matsunaga | On the use of machine learning to predict the time and resources consumed by applications[END_REF], Markov and Fast Fourier Transforms [START_REF] Gong | Press: Predictive elastic resource scaling for cloud systems[END_REF] or use wavelets to provide predictions [START_REF] Nguyen | Agile: Elastic distributed resource scaling for infrastructure-as-a-service[END_REF]. Solutions which combine reactive and predictive actions using control theory have also been developed, see [START_REF] Ali-Eldin | An adaptive hybrid elasticity controller for cloud infrastructures[END_REF] or [START_REF] Berekmeri | A control approach for performance of big data systems[END_REF] where a PI controller and a feedforward action are used. However, none of the previous solutions consider the cluster reconfiguration rate. This case is an emblematic example of the control problem of systems with costly reconfigurations that one can find for instance in computer science or robotics.

In this paper we propose event-based control techniques to reduce the number of reconfigurations. All the above mentioned control solutions are time-based, hence based on a periodic computation of the control law regardless if it is useful or not. The use of event mechanisms makes us hope for a reduction in the use of resources [START_REF] Aström | Event based control[END_REF], [START_REF] Durand | Further results on event-based PID controller[END_REF] without degrading performances [START_REF] Lunze | A state-feedback approach to event-based control[END_REF] and with stability and robustness guarantees [START_REF] Marchand | A general formula for event-based stabilization of nonlinear systems[END_REF].

However, all the now numerous event-based control strategies in the literature are focused on stability and performance guaranties. Those researches are motivated by the communication or computation reduction in networked controlled systems, which is a different objective from reducing the number of reconfigurations while guaranteeing acceptable performance levels.

Today's triggering strategies are based on level-crossing by the measurement error (see for instance [START_REF] Årzén | A simple event-based PID controller[END_REF], [START_REF] Durand | Further results on event-based PID controller[END_REF]) or more generally by some Lyapunov function (see for instance [START_REF] Velasco | On Lyapunov sampling for eventdriven controllers[END_REF]) or on the vanishing of an event-function related to a Control Lyapunov Function (see for instance [START_REF] Marchand | A general formula for event-based stabilization of nonlinear systems[END_REF]). In all cases, the decision to update or not the control law usually does not use a prediction of what will happen in the future.

Predictive methods that consider explicitly the future evolution of the current state are very encouraging, but until now few methods of this type have been developed. [START_REF] Eqtami | Eventtriggered control for discrete-time systems[END_REF] is an example of this type where the prediction is based on thresholds crossing around the desired states. However, deciding the triggering based on thresholds might not be the best solution in all cases. For instance, an acceptable state at present time but which leads to an undesirable behavior in the future can be detected and then handled by the predictive controller straight away, thus improving performance. Conversely, an undesired behavior in the present time but which leads to a stable situation in the future does not need to be taken care of, thus leading to less events and reconfigurations.

Furthermore, when dealing with multi-input and multioutput (MIMO) systems, the measurement errors alone are not sufficient to decide whether the system is in a critical state and needs a control actuation. Indeed, the multiple signals impact one another and the same value of the error for an output does not necessary reflect a critical situation for the system unless it is correlated to other signals values. For instance, a CPU usage of 90% reflect a normal behavior if memory usage is high, while it is alarming if memory usage is low.

In this paper we suggest an event mechanism based on the optimal cost function for Model Predictive Controllers (MPC) which tackles the issues presented above: the calculation of the cost function which by definition deals with the different interactions of the signals and also the prevision, using the model, of the future behavior of the system over the time horizon. Furthermore, constraints on the command shape (for instance the command should be constant on a time window of n sampling time) supervised by an event mechanism ensure that changes in the command signal are reduced while other specifications still hold.

The remainder of this paper is organized as follows: after the description of a motivating example in Section II, preliminaries on MPC are settled in Section III and the proposed event switching mechanism is detailed in Section IV. After a brief presentation of the MapReduce framework we used for validation in Section V, the validation of the approach is given in Section VI. Conclusion and future work complete the paper in Section VII.

II. MOTIVATING EXAMPLE: BIG DATA CLOUD SYSTEMS

Big Data Cloud services aim to process requests upon huge unstructured databases and use new programming paradigms that run distributed processes on different cluster resources. Consequently, on a simplified view, the more resources that are available for processing your requests the faster your will handle them. However, automatically assigning the right number of resources is a tricky issue. Moreover, as Cloud resources have high energetic and financial cost [START_REF] Lefurgy | Energy management for commercial servers[END_REF], the resource usage should be reduced in order to minimize the utilization cost.

In contrast to classical systems, controlling Big-Data services that run on the Cloud bring novel specific and contradictory constraints. First, the control value (which is the number of available resources) can only be a positive integer. As we better analyze the behavior of such systems, other specific constraints appear. It is worth noticing that any change in the number of resources implies a novel reconfiguration of the system which takes considerable time and costs money due to system unavailability and/or poor performance [START_REF] Lorido-Botran | A review of auto-scaling techniques for elastic applications in cloud environments[END_REF]. Moreover, quick changes in the control value (thus a high rate reconfiguration of the cluster) can overload the communication channels and lead to an accumulation of delays, which in return increases the time needed for processing the requests [START_REF] Maheshwari | Dynamic energy efficient data placement and cluster reconfiguration algorithm for mapreduce framework[END_REF]. Large delays along with overloaded communication channels, besides being a security threat, can increase the probability of hardware/software faults and lead to performance deterioration and skyrocket the financial costs [START_REF] Evolven | Downtime, outages and failures -understanding their true costs[END_REF]. This explains why the biggest public Clouds such as Google App Engine [START_REF]Google App Engine[END_REF] or Amazon EC2 [START_REF]Amazon EC2[END_REF] have a pay as you use billing policy which comes with an indivisible minimum renting time unit. Therefore changes in the control signal should be monitored. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 2,000-node cluster) [START_REF] Ren | Workload characterization on a production Hadoop cluster: A case study on Taobao[END_REF] Cloud services also face multiple disturbances of unusual nature. The workload, which is the amount of requests that are received by a computing system, is an input signal that, usually, we cannot control. For the considered application, workload variations can be categorized as:

Hours of a day

• Small predictable variations: In this case the disturbance can be statistically modeled using a long tailed distribution, for example using a log-normal distribution [START_REF] Ruiter | A workload model for MapReduce[END_REF]. Fig. 1 is an example of a Big data workload over a day period. We can see for instance from 5am-7am that the distribution has a stable mean and low variance. • Large instantaneous variations: In Cloud services large and unpredictable events also occur quite often. This sudden event changes the mean number of requests and affect the variance, as can been seen at 1am in Fig. 1. As we would like to minimize the total utilization cost, our objective in this paper is twofold: 1) reduce the amount of utilized resources as well as the number of cluster reconfigurations (that is to say the number of changes in the input signals but also their absolute values); 2) react very quickly to large disturbances but also be robust enough not to react to the small normal ones. Predictive control coupled with an event-based triggering strategy offers the needed solution as it will be shown in Sections III and IV.

III. PRELIMINARIES ON MPC

We consider a general nonlinear discrete time system described by the equations

x k+1 = f (x k , u k ) y k = g(x k ) (1)
where x ∈ R n x is the state, y ∈ R n y is the measured outputs and u ∈ R n u the control variables. We assume that f (0, 0) = 0 and g(0) = 0. For sake of simplicity, only stabilization at the origin is considered here. The subscript k of a variable stands for its value at time kT , T being the sampling period. We define a time horizon of N ∈ N instants, N > 0. In this section, we assume the system has no delay neither constraint on the control or state values. Extending the proposed results to delayed or constrained systems can however be done as classically in model predictive control. The example given in Section V typically has delays, saturation constraints on the control and positivity constraints on the states values.

Let U ∈ R n u ×N be a control profile over the horizon N defined by:

U := u 0 u 1 • • • u N-1 (2)
For any initial condition x i , and any control profile U ∈ R n u ×N , we define the corresponding state trajectory X ∈ R n x ×(N+1) by:

X(x i ,U) := x 0 x 1 . . . x N with x 0 := x i x k+1 = f (x k , u k ) ∀k ∈ {0, . . . , N -1}
and we associate to this state trajectory, the output trajectory Y ∈ R n y ×(N+1) defined by Y (X) := y 0 y 1 . . . y N with ∀k ∈ {0, . . . , N}, y k = g(x k ). Let J : R n x × R n u ×N → R + be some cost function depending on an initial condition x i and a control profile U. A very common definition of the cost function for a stabilization objective is the quadratic cost of the form:

J(x i ,U) = N ∑ k=1 X| T k+1 (x i ,U)Q x X| k+1 (x i ,U) +U| T k RU| k (3)
where X| k (x i ,U) (resp. U| k ) denotes the k th column of X(x i ,U) (resp. U), Q x and R are positive, definite matrices of appropriate sizes. Note that

U| k = u k-1 and X| k = x k-1 . J(x i ,U) = N-1 ∑ k=0 Ỹ | T k+1 Q Y Ỹ | k+1 +U| T k RU| k (4)
with Q Y and R positive, definite matrices of appropriate sizes,

Ỹ | k := Y | k (X(x i ,U)) -Y re f | k where Y re f ∈ R n y ×(N+1)
is the reference trajectory on the time horizon N. Finally, let us define the set U of controls as:

U (x i ) := U ∈ R n u ×N s.t. X| N+1 (x i ,U) = 0
U is the set of control profiles that satisfy a terminal constraint at the end of the horizon on the state variable. The final constraint is very common for stability purposes [START_REF] Mayne | Receding horizon control of nonlinear systems[END_REF]. Assumption 1: We assume that for all x ∈ R n x , U (x) is not an empty set. Under Assumption 1, the model predictive optimal control problem is formulated as follows:

Û : R n x → R n u ×N x → Argmin U∈U (x) J(x,U) (5)
and the control law for system (1) expressed in a state feedback form is classically:

u(x k ) := Û| 1 (x k ) (6)
Stability of the control law (5) has been studied in the literature [START_REF] Alamir | On the stability of receding horizon control of nonlinear discrete-time systems[END_REF]. If one defines the following operator Π : R n u ×N → R n u ×N that associates to U given by (2) the control profile Π(U) defined by:

Π(U) := u 1 u 2 . . . u N-1 0 n u ×1
the stability relies on the fact that at time (k + 1)T , the translation of one time step of the solution control profile Û(x k ) at time kT , namely Π( Û(x k )), belongs to the set of admissible control U (x k+1 ). This sort of invariance property guarantees that the cost associated to the control at time (k + 1)T is necessarily strictly smaller than the one at time kT and therefore the stability is guaranteed as soon as the optimal cost function Ĵ(x) := J(x, Û(x)) is a Lyapunov function for system (1).

In our particular case, changing the control value too often may be financially expensive and needs to be avoided. For this, we propose to remove some degrees of freedom by adding linear constraints on the control variable. Therefore, we extend the set U as follows:

V (x i ) := U ∈ R n u ×N s.t. X| N+1 (x i ,U) = 0 and AU = B (7)
with A and B matrices of appropriate forms and sizes. For instance, in the application of section V such constraints limit the changes in the control signal. We assume that it is not empty:

Assumption 2: We assume that for all x ∈ R n x , V (x) is not an empty set. In order to keep the invariance property on which the stability relies, we define for the first iteration

Λ0 : R n x → R n u ×N x → Argmin V ∈V (x) J(x,V )
and for the following ones

Λ : R n x × R n u ×N → R n u ×N (x,U) → Argmin V ∈V (x)∪{Π(U)} J(x,V )
The control profile for system (1) is then dynamically defined for any x k at time kT by:

Vk := Λ(x k , Vk-1 )
where Vk-1 is the control profile at time (k -1)T . To initialize the process, we take at k = 0, V0 := Λ0 (x 0 ). The control law for system (1) expressed in a state feedback form is then:

u k := Vk | 1 (8)
The corresponding optimal cost is then Ĵk := J(x k , Vk ). Theorem 1: Under Assumption 2, and if J is a Lyapunov function, the control law defined by (8) asymptotically stabilizes system (1). The proof is trivial since by construction the cost function is strictly decreasing. In particular, if one takes a cost function of the form (3), one can prove that:

Ĵk+1 -Ĵk ≤ J(x k+1 , Π( Vk )) -Ĵk = -x T k+1 Q x x k+1 -u T k Ru k

IV. TRIGGERING ON THE COST FUNCTION

The aim of this section is to propose a triggering algorithm for MPC control laws ( 6) and ( 8) presented in the previous section. This strategy will decide when the control profile needs to be recalculated. For sake of simplicity, we assume in the following that the cost function J is of the form (3).

A. Triggering strategy

To any state x k of the system at time kT and any control profile W k-1 at time (k -1)T , we associate two corresponding costs, namely Ĵk defined as previously by Ĵk := J(x k , Λ0 (x k )) and Jk := J(x k , Π(W k-1 )). The first cost Ĵk corresponds to the cost if one updates the control with its optimal value whereas, Jk is the cost obtained keeping the previous control profile. Note that by construction, one has the following inequality: Jk ≥ Ĵk . We can now define the event function e : R n x × R + → {0, 1} by:

e k = 1 if Jk -Ĵk ≥ ε J Ĵk or if k = 0 0 otherwise (9)
The proposed MPC event-triggered control profile W k at time kT is then:

W k = Λ0 (x k ) if e k = 1 Π(W k-1
) otherwise The control law to apply is as previously the first element of the control profile, that is:

w k = W k | 1 ( 10
)
Note that if e = 0 the control profile of the last time period is applied and then

w k = W k | 1 = W k-1 | 2 .
Hence, contrary to usual event-based control, the input control value is not kept constant in the proposed scheme. The control profile is updated when the cost can be reduced by a factor of 1 + ε J with respect to the cost when keeping the same control profile.

B. Stability of the proposed scheme

The proposed scheme asymptotically stabilizes the system at the origin since it ensures the strict decrease of the cost function until the origin is reached:

Theorem 2: Under Assumption 2, and if J is a Lyapunov function, the control law defined by ( 10) asymptotically stabilizes system (1).

Proof: Assume at time kT , the control profile is W k with a cost J(x k ,W k ) defined by [START_REF] Ali-Eldin | An adaptive hybrid elasticity controller for cloud infrastructures[END_REF]. At time (k + 1)T , the state value is given by x k+1 = f (x k ,W k | 1 ). Assume first that e k+1 = 0, then W k+1 = Π(W k ) and

J(x k+1 ,W k+1 ) = J(x k+1 , Π(W k )) = J(x k ,W k ) -x T k+1 Q x x k+1 -W k | T 1 RW k | 1 J is
therefore strictly decreasing in this case. Assume now that e k+1 = 1, the control profile needs therefore to be updated. In that case, W k+1 = Λ0 (x k+1 ) and therefore one has:

(1 + ε J )J(x k+1 ,W k+1 ) ≤ J(x k+1 , Π(W k )) ≤ J(x k ,W k ) -x T k+1 Q x x k+1 -W k | T 1 RW k | 1
In that case again, J is strictly decreasing as long as x k+1 = 0 which ends the proof of stability of the scheme.

Choice of ε J : The threshold ε J from equation ( 9) should be chosen carefully. If it is too small, the controller will unnecessarily react to noise, model uncertainties or observer error (if one), while if ε J is too large we may not react fast enough to disturbances. The tuning of ε J can be done in a training phase using data from an open-loop experiment with disturbances.

V. APPLICATION TO MAPREDUCE FRAMEWORK

As a motivating example for our work we choose to use the MapReduce framework. MapReduce is a parallel and distributed paradigm running on the Cloud, it aims at treating requests of many types upon large databases. It has been developed in 2008 by Google to automatically handle most of the complexity of parallel computing and nowdays it is backed by Big Data industry leaders such as Google, Facebook, Yahoo, LinkedIn (see [START_REF] Dean | MapReduce: simplified data processing on large clusters[END_REF], [START_REF] Shen | Enabling the New Era of Cloud Computing: Data Security, Transfer, and Management[END_REF] among others).

A. MapReduce Modeling

Modeling MapReduce systems is a very tedious task as we need to decide what are the pertinent input signals and then make a detailed analysis concerning their impact on the outputs. Mathematical models resulting from the above mentioned analysis have been developed and experimentally validated in [START_REF] Berekmeri | A control approach for performance of big data systems[END_REF] with real data form a Big Data MapReduce framework executed on Grid5000, a large-scale versatile grid [START_REF] Cappello | Grid'5000: A large scale and highly reconfigurable grid experimental testbed[END_REF]. These models, largely detailed in [START_REF] Berekmeri | Feedback autonomic provisioning for guaranteeing performance in mapreduce systems[END_REF], are SISO first order transfer functions with input/output delays and saturation. [START_REF] Berekmeri | Modeling and control of cloud services : application to MapReduce performance and dependability[END_REF] further extended them to a MIMO model (see Fig. 2) with integral action and saturation. Let us briefly comment on the model behavior without insisting on technical details. On one hand, when the number of resources executing the job (noted R) is low, the service time (the time to process a request) will be long whereas if there are many resources allocated to the job the execution time will be short. On the other hand, if many requests are sent in parallel to a fixed-size cluster, due to concurrency issues, the service time will be longer than if there were just a few. In terms of control theory, we identified the cluster size R as an tunable input, and the workload (the number of concurrent clients C that sent requests to our system) is considered as an uncontrollable but measurable disturbance. The output of our model is the service time which is a measure of the performance of MapReduce systems. Considering that we can also reject some client requests, when resources are not available for example, we have another input variable for our control which is the maximum number of accepted clients MC and another output metric: the availability rate (which is the number of accepted jobs from the overall number of jobs).

MapReduce Model

CONTROL

B. Model predictive controller

The MPC we use has the following constraints:

• availability is a more critical output than the service time, both should be at their reference values at the end of the horizon N (N is 100 times the sampling time here); • the cluster size should be minimized;

• for the constrained scenario defined by equation [START_REF] Berekmeri | Modeling and control of cloud services : application to MapReduce performance and dependability[END_REF], cluster size can only change at regular time intervals. This value is tuned to correspond to the indivisible minimum renting time of resources in a Cloud. • we consider a limited resource configuration where 0 < R ≤ R MAX = 60 nodes. The controlled system is simulated using Matlab Simulink.

VI. VALIDATION USING MAPREDUCE

The validation scenario we consider is the following: the system is launched with 20 nodes (R = 20), 9 clients including one rejected (C = 9, MC = 8). Once the system reaches a stable operating point (at t = 0 min) we start the control with the strong constraint that the outputs of the system should remain at theses steady state values. The disturbance is initially considered as a step to facilitate the analysis of the results, then we validate the method using a 1-day MapReduce workload from an on-line shopping website (Section VI-B).

A. Step response analysis

In the first battery of tests we compare our method referred to as cost based event solution with an error based one and with an MPC time based controller. All these solutions are calculated without specific constraints on the input signals, as described by equation ( 6); we also compare the cost based event triggered controller to its constrained version, where the input signal R is forced to have constant values on large time internals. By error based triggering mechanism we mean that events are triggered when the control law has been applied over the whole prediction horizon or when one of the outputs cross a certain threshold. For the cost based approach we use the cost-function defined by equation ( 3), and we choose the threshold form equation [START_REF] Berekmeri | Feedback autonomic provisioning for guaranteeing performance in mapreduce systems[END_REF] based on training data (not shown here because of space restriction) in order to ensure no false alarms, thus ε J = 100%. We chose the thresholds for the error based method so that the error based and the cost based solutions ensure the same performances.

Results are shown in Fig. 3. For comparable performances in term of reference tracking and inputs variations, we observe that the cost based controller generates only 1 event over a 60-minute period, which represents 99.2% less events than the time triggered solution (120 events in total), and 91% less than the error based controller (11 events). Moreover, due to the dynamics of the system, once the error thresholds are crossed it takes some time to stabilize the output signals inside the threshold-defined bounds which leads to several consecutive events generated by the error based method after each disturbance occurs. However, the cost based method takes into account this dynamic through state prediction and removes the redundant events. Hence, with significantly less events our proposed event mechanism can guarantee comparable performances.

If we take a look to the impact of the constraints, we clearly see that with comparable performances in reference tracking for both outputs and the exact same number of events, the constrained controller imposes 93.4% less changes in the cluster size than its unconstrained version.

Threshold choice: When looking at the cost functions (Fig. 4), we see that the condition for triggering an event gives no false alarm and reacts each time workload conditions change. Moreover, every event is caused by a clear threshold crossing and it is an indicator of the robustness of our method regardless the value of ε J . Furthermore, we see in Fig. 4 the importance of the threshold expressed as a percentage. Shortly after an event, the cost J is way higher than before as it takes time and resources for the system to reach again the specifications, however this high value of J does not necessary reflect the arrival of another disturbance.

B. Evaluation using a real MapReduce workload

In a second time, we validate our proposed solution using a real 1-day MapReduce workload trace that was recorded on a 2,000-node cluster of Taobao (see Fig. 1), an e-commerce website. Table I compares prices of running the workload using the different controllers based on Amazon EC2 pricing (0.1$ per instance-hour).

The constrained cost based solution enables to realize major savings (from 5 to 63%) as it prevents the brief cluster oversizing just after a disturbance occurred, as can be noticed in Fig. 3. Other drawbacks of the time based and other unconstrained methods should also be considered; mainly overloading communication channel, large delays in starting nodes and numerous system reconfigurations.

VII. CONCLUSIONS AND FUTURE WORKS

In this paper we propose a new event mechanism for model predictive controllers which has two aims:

• reduce the number of events without degrading performances • reduce the changes in the control inputs thus reacting to large disturbances and being robust to smaller ones This latter constraint is particularly relevant in the context of big-data Cloud services where high frequency of cluster reconfigurations has major financial and energetic costs and leads to performances degradation. The proposed event mechanism is a function of the cost function defined in the optimal problem, and is not based on measurement error as has been often done until now. This enables to take into account the prediction of the system trajectory over a time horizon. For handling the reduction of input changes we add constraints to the MPC formulation, allowing changes only at regular time intervals.

Evaluation is done on a model previously validated on a real MapReduce system. It shows that the constrained cost-based event triggered MPC significantly reduced the number of events as well as the changes in the control law. We generate 86% less events compared to an error based method and we obtain at least 8% cost saving when simulating with a real 1-day workload (based on Amazon EC2 pricing).

Work is in progress to validate the constrained cost-based event triggered MPC approach through experiments using the latest MapReduce release on a public Cloud such as Amazon EC2. We expect to have better results due to the reduction of cluster reconfigurations which are even more costly on the real system due to congestion of communication channels leading to performance degradation. Further researches will also lead to an extension of the cost-based event functions to handle other type of disturbances while improving performances.

24 NumberFig. 1 .

 241 Fig.1. Shopping website MapReduce workload (2,000-node cluster)[START_REF] Ren | Workload characterization on a production Hadoop cluster: A case study on Taobao[END_REF]

Fig. 2 .

 2 Fig.2. MapReduce model schema from[START_REF] Berekmeri | Feedback autonomic provisioning for guaranteeing performance in mapreduce systems[END_REF]

Fig. 3 .

 3 Fig. 3. Comparison of time based, error based and cost based controllers performances without constraints on the inputs (see equation (6))

Fig. 4 .

 4 Fig. 4. Cost-based event function

This work has been partially supported by the LabEx PERSYVAL-Lab (ANR-11-LABX-0025-01) funded by the French program Investissement d'Avenir. This work has been supported, in part, by the European FP7 research project AMADEOS Grant Agreement 610535 on Systems of Systems.


L Fruchter

G Collin

D Colson

V Brouet

Sr 2 IrO 4 magnetic phase diagram, from resistivity

Keywords: numbers: 75.30.Kz, 75.47.Lx, 75.70.Tj

We show that the transition to the antiferromagnetic state in zero magnetic field does show up in the transverse resistivity, for which we point out the possibility for a direct spin orientation effect. In an applied field, we propose that the transition is split into two lines, corresponding to in-plane and out-of-plane magnetic ordering. This picture is corroborated by transverse magnetization measurements. The magnetic phase diagram for Sr2IrO4 was investigated, using the angular dependence of the resistivity transverse to the IrO2 planes.

I. INTRODUCTION

In the recent years, iridium oxides have become a new playground for the study of electron correlation effects. Indeed, while extended 5d orbitals reduce the electron-electron interaction, as compared to the 3d transition metal compounds as cuprates, strong spin orbit coupling (SOC) associated to the heavy Ir competes, together with the on-site Coulomb interaction, with electronic bandwidth to restore such correlations [1]. Amongst these compounds, the Ruddlesden-Popper series, R n+1 Ir n O 3n+1 where R= Sr, Ba and n = 1,2,∞, has attracted much of the attention, in particular due to the structural similarities of these perovskites with the cuprates compounds. Sr 2 IrO 4 , where one IrO 2 layer alternates with an SrO layer, is structurally similar to the first discovered cuprate superconductor, (La,Ba) 2 CuO 4 . The physics of the latter is the one of an antiferromagnetic Mott insulator, with a magnetic interaction described within the framework of a spin-1/2 Heisenberg model. It was early proposed that the strong SOC in the iridate perovskite actually allows for an effective localized state different from this spin-1/2 state, entangling spin and orbital degrees of freedom, with total angular momentum J ef f = 1/2. This spin-orbital insulating state was proposed to be the analog of a Mott insulator [1].

The antiferromagnetic order in Sr 2 IrO 4 is now well documented [2,3]. The moments (0.2 µ B /Ir) lay in-plane and order at T N ≃ 240 K. As for La 2 CuO 4 , the loss of the inversion symmetry in the non cubic structure, due to a rotation of the oxygen octahedra, allows for a Dzyaloshinsky-Morya interaction, which in turn induces a canting of the spins and a ferromagnetic component in the IrO 2 planes [4][5][6]. The net moment (0.14 µ B /Ir), which is coupled antiferromagnetically from plane to plane in zero field, align ferromagnetically with an inplane field H ≈ 0.2 T [7]. Recent ab initio computations conclude that the dominant magnetic interaction is of Heisenberg type, with little effect of the geometrical factors on the exchange coupling [8]. As shown in ref. 19, the absence of a critical behavior in the in-plane magnetic correlation length at T N is also in favor of a two-dimensional Heisenberg behavior with large quan-tum fluctuations. On the basis of such a description, it has been proposed that Sr 2 IrO 4 may exhibit electronic properties similar to the ones of the cuprates, including superconductivity [9]. The nature of the insulating state is however the subject of debate. First, the realization of the J ef f = 1/2 state itself may be questioned, as it requires a perfect orbital degeneracy, which is not obtained in Sr 2 IrO 4 where the octahedra are strongly elongated [10]. The location of a metal-insulator transition, either in the paramagnetic state as for a Mott-Hubbard transition [11], or coincident with the magnetic transition as for a Slater-type transition [12] is controversial.

As first noticed by Kini et al [13], no anomalies can be detected in resistivity at T N . The authors proposed that this could result from the fact that localized states shifts the Fermi level away from the band edges affected by spin polarization. A time-resolved optical study found that the metal-insulator transition takes place over a wide temperature range 0.7 T /T N 1.4, thus accounting for the absence of any sharp anomaly in transport and thermodynamic quantities [14]. Well below T N , for T 100 K, large anisotropic magnetoresistance as well as magnetodielectric effects were observed (Refs. 15,16). It was proposed that the magnetoelectric effects result from the competition of antiferromagnetic and ferromagnetic coupling, at low and high temperature respectively [16]. Here, we show that there is actually a small but clear signature of the magnetic transition in zero field in the transverse resistivity. This allows to use magnetotransport as a probe to establish a magnetic phase diagram up to T N for this compound. We suggest that the magnetic transition under magnetic field is split into two lines, corresponding to in-plane and out-of-plane magnetic ordering. In the ordered region, we propose that spin orientation has a direct impact on resistivity.

II. RESISTIVITY AND TRANSVERSE MAGNETOMETRY

The results below where obtained with a Sr 2 IrO 4 single crystal, with dimensions 300 x 200 x 30 µm 3 . It was grown using a self-flux technique in platinum crucibles, similar to the one in Ref. 2. We denote a and b the crystal lattice vectors of the superstructure in the IrO 2 planes [4]. Low-resistance contacts were achieved using silver epoxy annealed at 500 C in oxygen atmosphere. Although we present only the data obtained with one crystal, we have checked that the reported behavior is typical of what can be observed in several other similar undoped samples.

Careful investigation of the c-axis resistivity reveals the existence of an anomaly at a temperature close to the reported T N for the undoped material. The anomaly is actually very small (Fig. 1, inset), but the sharp jump in the resistivity temperature derivative in zero magnetic field unambiguously points towards a well defined phase transition at T N = 217± 0.5 K (Fig. 1). Quantitatively, the evaluation of the resistance change below T N requires subtracting some arbitrary background, as obtained from the high temperature resistivity, and we have used a linear fit of the resistance logarithm to reveal an increase of the resistivity in zero field below the transition (Fig. 1, lower inset). At least close to the transition temperature, little error is likely made due to the sharp transition, and a scaling of the resistivity change may be attempted. This yields, within a ≈ 20 K interval, a scaling exponent 0.55 ± 0.05. We did not observe such an anomaly for the in-plane resistivity. Previous studies did not uncover such a feature, although a tiny specific heat jump could be evidenced at T N (Ref. 16).

Applying a magnetic field along the c-axis and using the same procedure to uncover the resistivity anomaly shows that there is a "splitting" of the transition with the field (Fig. 1). The positive magnetoresistance anomaly shifts to lower temperature (as an example, data in Fig. 1 allows to assign a transition temperature, noted T 3D , 215.5 ± 0.5 K with H = 0.2 T), while a negative contribution to the magnetoresistance shows up above T N (as an example, data in Fig. 1 allows to assign a transition temperature, noted T 2D , 236 ± 2 K with H = 9 T). With the applied field the transition at T 3D retains its sharp character at small field, while the transition at T 2D appears to be smoother. The existence of the upper branch implies that the transition temperature obtained from high field studies (as in magnetometry) must be overestimated. Data in Fig. 1 suggests, in the present case, that a maximum shift ∆T 2D ≈ 20 K is obtained for H ≈ 7 T. With a field applied parallel to the conducting planes, the behavior is found similar, with a transition temperature increase somewhat larger (Fig. 2, open symbols). At T 3D , both the sign of the resistance anomaly and the sharpness of the transition point towards similarity with the zero field low temperature phase, but the anomaly is quickly washed out in a small magnetic field (Fig. 1). However, we find that a kink in the magnetoresistance substitutes to this anomaly (Fig. 3), allowing to extend the definition of T 3D to low temperature in Fig. 3 (circles). This defines two branches emerging from the (T N , H = 0) point (which we denote T 3D (H) and T 2D (H), respectively in Fig. 2).

Finally, investigating lower temperatures in the same way shows that a third characteristic temperature may be evidenced, which coexists with the former in the temperature range 80 K T 175 K (Fig. 3, squares). We denote this temperature T spinf lip (H) and also report it in Fig. 2. For all three branches of the diagram, the characteristic magnetic field allowing to cross a branch is found smaller when applied parallel to the conducting plane (Fig. 3). Then, the crossovers observed in the parallel configuration are not a consequence of the sample misalignment that unavoidably occurs (which we estimate a few degrees), but are genuinely induced by the parallel field component. The large negative magnetoresistance that we observe at low temperature is similar to the one reported earlier by Ge et al below T = 100 K (Ref. 15), although T 3D (H) appears to occur at somewhat lower field in our case (for instance, H ≈ 1 T at T = 50 K in the present case, and H ≈ 3 T in ref. 15). Our data allow to draw the T 3D,spinf lip (H) lines up to temperatures close to T N and indicate that the resistance change at small field at the T 3D (H) line is similar to the one observed at T = T N and H = 0.

As noticed above, the procedure to evaluate the resistance change at the transition is quite arbitrary, which may be a problem in the case of a smooth variation. Angular dependent magnetoresistance, however, confirm the general trend for T 2D (H). Indeed, rotating a large magnetic field around the c-axis reveals the existence of a four-fold contribution to the (negative) magnetoresistance (Figs. 5). The angular-dependent contribution is typically only a few percent of the total magnetoresistance in Fig. 3, and is found maximal along the a and b axis of the crystal. The temperature dependence of the four-fold component extracted in Fig. 5 confirms, with no need for a high-temperature background fit, the existence of an onset at T 2D . A two-fold angular component is also present, which vanishes with increasing temperature simultaneously with the four-fold component. However, while the four-fold component temperature dependence appears similar to the one of some order parameter (as for the zero field anomaly), the two-fold component saturates with decreasing temperature.

Finally, we have performed torque measurements on this crystal, Γ = µ 0 m × H, which provides the magnetization component transverse to the applied magnetic field. Typical results are shown in Fig. 4 for three points in the H -T plane, representative of three distinct regimes in Fig. 2. In the paramagnetic domain, torque is a pure four-fold sine, suggesting a simple anisotropic magnetic susceptibility contribution, as for a two-dimensional system. In the high field regime, a distorted sine indicates that the magnetic moment may no longer be decomposed into two independent linear contributions, which could sign a smooth rotation of the magnetization. Finally, in the intermediate field regime, the torque signal develops a discontinuity (for the data shown, at ≈ 12 deg. from the c-axis), which possibly originates from a spin flip.

III. DISCUSSION

We first comment on the lower line, T spinf lip (H). The problem of the observation of a very large magnetoresistance associated to the presence of a weak ferromagnetic state induced by the magnetic field was already encountered in the case of cuprates [6,17], and received a quantitative interpretation in the case of La 2 CuO 4 [18]. Two contributions should actually be distinguished: one due to the change in the transfer integrals and linked to the orientation of the IrO 6 octahedra, as proposed in Ref. 15, the other one as a pure spin configuration effect. Could the latter contribute in the present case ? In La 2 CuO 4 , in the orthorhombic phase, the CuO 6 octahedra are tilted from the CuO planes by α ≈ 3 deg. (inducing a ferromagnetic component perpendicular to the planes). This allows an antisymmetric superexchange term in the spin Hamiltonian, which would otherwise be zero due to symmetry in the tetragonal phase [6]. In Sr 2 IrO 4 , IrO 6 octaheadra are tilted in the IrO planes by α ≈ 11 deg. (inducing a ferromagnetic component in the planes) and the compound is tetragonal. This also destroys the inversion center which exists midway between the Ir atoms, and allows for a non zero antisymmetric superexchange term [4] (Fig. 6). Equivalently, due to the tilt of the IrO 6 octahedra, the transfer integral between one Ir atom and its four nearest neighbors in the next adjacent plane are unequal. As a consequence, interchanging magnetic sublattices, as could be induced by a spin flip, may strongly influence the transverse conductivity in this case also.

In Ref. 18, it was proposed that the transverse resistivity is controlled by the localization length in the 3D variable-range hopping regime (VRH). In the present case, resistivity can be fitted with the conventional expression for three-dimensional VRH, ρ ∝ exp (T 0 /T ) 1/4 , using T 0 = 6 10 5 K (where T 0 ∝ λ -3 , and λ is the transverse localization length -see Ref. 18 and Refs therein). This yields for the ratio of the hopping length to the localization radius l/λ ≈ (T 0 /T ) 1/4 ≈ 7, indicating that charge hopping from impurity centers is controlled by the pure material transfer integrals between sites [18]. Using λ ∝ t 1/2 , where t is some effective transfer integral between planes, we expect in this case a relative change δρ/ρ ≈ -3 8 (T 0 /T ) 1/4 δt/t. According to Ref. 18, a flip of the spins required to align ferromagnetic moments in the plane is associated to a change δt/t

≈ 1 2 (J ⊥ /J )(κ /κ ⊥ ) 4 (m ⊥ /m ) 2
, where m (⊥) is the in-plane (transverse) effective mass, J (⊥) is the in-plane (transverse) exchange coupling, and κ (⊥) is the corresponding reciprocal lattice constant. We have κ /κ ⊥ = 1.8, J ⊥ /J ≈ 10 -5 (a value comparable to that for La 2 CuO 4 , Ref. 19) and m ⊥ /m > 20 (this is evaluated by the ratio of the bandwidth for J=1/2 along ΓX and NC [12]). This yields δρ/ρ = 1 2 δt/t > 6 10 -2 . Though this is only a rough estimate and magnetic configurations for both cases are different, this illustrates that an effect comparable to the one that we observe may be expected from the spin contribution alone. We expect this contribution to be significant at the T spinf lip (H) line, where there is a field-induced ferromagnetic moment [15]. Then, within this hypothesis that the resistivity change results from the larger transverse transfer integral associated to magnetic ordering, we may tentatively relate the observed scaling of the resistivity to a critical exponent. We expect the resistance change to be proportional to the phase transition order parameter associated to interplane spin ordering, M (this may be assumed in the framework of a two-fluid model, for which there is an amount n ∝ M of ordered moments associated to a larger transfer integral). As a result, the scaling exponent for the resistivity is identical to the conventional exponent β for the order parameter. The value obtained, β ≃ 0.55, is close to the one for a mean field type transition (β = 0.5). In Ref. 19, the transverse fluctuation correlation length above T N yielded a critical exponent ν = 0.75 ± 0.05. This value is far off the mean field value (ν = 0.5). This discrepancy could sign the limit of the present analysis for the resistivity scaling, made within a simple static picture. Also, the scaling in Ref. 19 relates to long range correlations (≃ 3-20 c), while we expect resistivity to be essentially driven by magnetic correlations at the scale of the interplane distance.

We now comment on the upper line, T 3D (H). In Ref. 15, it was proposed that the occurrence of a large magnetoresistance with no relevance to the magnetization could be interpreted as a the result of the concomitant rotation of the spins in a spiral configuration under applied field, and of the Ir-O bond (due to spin-orbit coupling [5]), charge hopping being in turn controlled by the orbitals configuration. While our data do not allow to conclude on the validity of such a scenario, our data brings some new insight on the nature of the T 3D (H) line. Results in Fig. 1 suggest that T 3D (H) retain the second order phase transition character of the zero field transition. In particular, crossing this branch, the resistance change is observed positive (although the overall magnetoresistance is negative, see Fig. 3). Measurements of the angular dependence of the magnetoresistance along constant T and H intervals crossing T 3D (H) bring additional information (Fig. 7). The striking features of such results are: i) a large angular susceptibility of the transverse magnetoresistance develops at the crossing of the line along the a and b directions; ii) just below this line (at 200 K in Fig. 7), there is a two-fold periodicity (we have checked, deliberately tilting the crystal and observing no qualitative change in the R c (θ) behavior, that this cannot be due to the sample misalignment), and the peaks in angular susceptibility are hysteretic.

We propose that a magnetic ordering transition along c-axis may account for these observations. Indeed, such a transition would have a large effect on the transverse resistivity by the virtue of the mechanism described above. Then, ferromagnetic domains being linked to the possibility to order magnetism from plane to plane, it is natural to expect their signature to show up below this transition, as well as some potential a/b unbalance due to inequiva-lent domains. Finally, the susceptibility to the magnetic field at the transition should be the larger when the field is applied along one of the two possible directions for the ferromagnetic moment. The scenario in Ref. 15 proposed that a c-axis field induces an in-plane rotation of the spins, while at a constant angle from the c-axis. If so, we expect a reduction of the ferromagnetic component with the applied field. It is then possible that this reduction favors magnetic decoupling along the c-axis. To the credit of our picture, preliminary measurements on Sr 2-x La x IrO 4 indicate -beside the known T N decreasethat the T 3D (H) line is suppressed in zero field (i.e. the positive transverse resistivity anomaly is replaced by the negative one, as in an applied field). This suggests that disorder may also contribute to weaken the c-axis magnetic ordering, just as for Sr 2 Ir 1-x Rh x>0.03 O 4 exhibits a modified c-axis magnetic ordering to the so-called 'AF-II' phase with ferromagnetic inter-plane order [20]. The observation of a smooth crossover distinct from the sharp transition to the 3D ordered state seems also to corrobo-rate the one that the in-plane magnetic fluctuation correlation length decreases above T N much less rapidly than the out-of-plane one, a feature interpreted as the consequence of a two-dimensional S= 1/2 quantum Heisenberg model [19].

The overall picture would then be the following: in zero field, we observe a single transition temperature at T N were 3D magnetic order establishes. Applying a magnetic field splits the transition into two branches: a near vertical branch at T > T N at which AF order is established in the plane, and a low temperature one, where a long range c-axis magnetic order is established. A second line in the ordered phase delimits field-induced ferromagnetic interplane ordering. We propose that magnetotransport is influenced strongly by direct spin reorientation effects, in addition to bond reorientation ones. Our data clearly contradicts the general belief that there would be no influence of the magnetic state on the transport properties. We have proposed that this influence should be evaluated here considering hopping of the localized charge.

FIG. 1 :FIG. 2 :FIG. 3 :FIG. 4 :

 1234 FIG.1: c-axis resistance anomaly, as obtained subtracting a high temperature linear logarithmic resistance (dashed line in lower inset). Upper inset: scaling of the resistive anomaly. The magnetic field is applied along the c-axis.

FIG. 5 : 1 FIG. 6 :FIG. 7 :

 5167 FIG. 5: Upper panel: angular dependence of the magnetoresistance (in-plane field H = 0.2 T); θ is the angle between a(b) and the magnetic field. Lower panel: Symbols: four-fold and two-fold (dotted line) components of the angular c-axis magnetoresistance (left scale). Full line: zero field c-axis resistance anomaly, as in Fig. 1