F Chaffar Akkari
email: ferid.chaffarakkari@ipeiem.utm.tn

H Ben Jbara

D Abdelkader

B Gallas

M Kanzari

Effect of angle deposition γ on the structural, optical and electrical properties of copper oxide zigzag (+γ, -γ) nanostructures elaborated by glancing angle deposition

Keywords: Glancing angle deposition, zigzag columns, electrical anisotropy, copper oxide, Cu 2 O, thermal evaporation

In this work, Cu x O thin films were obtained by air annealing of copper thin films deposited on glass substrates using thermal evaporation method by Glancing Angle Deposition "GLAD" technique. The copper was sculptured into a zigzag shape, which presents two columns with inclined angles + and -where is the deposition angle between the incident flux and the substrate normal. Morphological, structural, optical and electrical properties of the obtained thin films were investigated using X-ray diffraction (XRD), UV-Vis-NIR Spectroscopy and electrical resistivity measurements. The XRD patterns revealed that thin films deposited at different incident angles are mainly crystallized in Cu 2 O cubic phase characterized by the preferential orientation along (111) plane. The optical parameters were calculated from the analysis of the transmittance and reflectance spectra in the wavelength range 300-1800 nm. Theabsorptioncoefficient exceeds 10 5 cm -1 in the visible and NIR spectral ranges. Direct band gap energy increases from 2 to 2.54 eV with deposition angle. The in-plane birefringence and the anisotropic resistivity of the Cu 2 O films were also studied. Their maxima were obtained at incident flux angle of = 60°. Therefore, the GLAD technique is a promising way to create zigzag nanostructures with enhanced anisotropic properties.

Introduction

In the recent years, glancing angle deposition (GLAD) technique has attracted interest for designing novel structures of thin films and controlling their morphologies and physicochemical properties [START_REF] Hong | Synthesis of indium nanostructure-laces by multi-step Glancing Angle Deposition[END_REF][START_REF] Siyanaki | The effect of substrate rotation rate on physical properties of cadmium telluride films prepared by a glancing angle deposition method[END_REF]. This technique is a physical vapor deposition (PVD) process that is implemented by oblique angle deposition and/or substrate rotation in front of the deposition source to control the column orientation and porosity of nanostructured films [START_REF] Polat | Structured SiCu thin films in LiB as anodes[END_REF][START_REF] Gallas | Silver square nanospirals mimic optical properties of U-shaped metamaterials[END_REF]. The GLAD technique has been shown to be capable of thin films elaboration with a wide variety of microstructures, such as vertical columns [START_REF] Lee | Ionic conductivity increased by two orders of magnitude in micrometer-thick vertical Yttria-stabilized ZrO 2 nanocomposite films[END_REF], helical [START_REF] Sinaoui | Investigation of growth and characterization of nanostructured CuIn 5 S 8 thin films produced by glancing angle deposition[END_REF], and zigzag shapes [START_REF] Rahchamani | Study of structural and optical properties of ZnS zigzag nanostructured thin films[END_REF], which is hardly obtained from the normal incidence vapor deposition [START_REF] Wang | Structural and optical properties of nanostructured TiO 2 thin films fabricated by glancing angle deposition[END_REF].

For instance, films with a porous structure have been shown to well operate in various applications such as solar cell [START_REF] Zhou | Glancing Angle Deposition of copper iodide nanocrystals for efficient organic photovoltaics[END_REF], photonic crystals [START_REF] Hawkeye | Optimized colorimetric photonic-crystal humidity sensor fabricated using Glancing Angle Deposition[END_REF], optical filters [START_REF] Hawkeye | Coupled defects in one-dimensional photonic crystal films fabricated with glancing angle deposition[END_REF], gas sensors [START_REF] Horprathum | NO 2 -sensing properties of WO 3 nanorods prepared by glancing angle DC magnetron sputtering[END_REF] and photo-catalysis [START_REF] Samadi | Recent progress on doped ZnO nanostructures for visible-light photocatalysis[END_REF]. Thus, many novel physical and chemical properties of thin films can also be easily engineered [START_REF] Barranco | Perspectives on oblique angle deposition of thin films: From fundamentals to devices[END_REF].

Copper oxides (Cu x O) have gained extensive interest because of its important role in many applications, such as catalyst [START_REF] Yang | Microwave hydrothermal disassembly for evolution from CuO dendrites to nanosheets and their applications in catalysis and photo-catalysis[END_REF], gas sensor [START_REF] Umar | CuO nanosheets as potential scaffolds for gas sensing applications[END_REF], optical and optoelectronic devices [START_REF] Wang | Preparation of p-n junction Cu 2 O/BiVO 4 heterogeneous nanostructures with enhanced visible-light photocatalytic activity[END_REF].

Especially, Cu x O thin films are potential for photovoltaic applications, due to their good properties such as high absorption coefficient in the visible region, elemental abundance, non-toxicity and stability with the characteristicofbeing"ultra-low-cost".Their direct optical band gap energy is in the range of (2.1 to 2.6 eV) for Cu 2 O and (1.21 to 2.1 eV) for CuO [START_REF] Gan | Influence of target power on properties of Cu x O thin films prepared by reactive radio frequency magnetron sputtering[END_REF][START_REF] Heinemann | Band structure and phase stability of the copper oxides Cu 2 O, CuO, and Cu 4 O 3[END_REF][START_REF] Mugwang'a | Optical characterization of Copper Oxide thin films prepared by reactive dc magnetron sputtering for solar cell applications[END_REF][START_REF] Halin | Characterization of cuprous oxide thin films prepared by sol-gel spin coating technique with different additives[END_REF][START_REF] De Los Santos | Crystallization and electrical resistivity of Cu 2 O and CuO obtained by thermal oxidation of Cu thin films on SiO 2 /Si substrates[END_REF]. Copper oxide thin films had been fabricated by various techniques such as electrodeposition [START_REF] Mahalingam | Galvanostatic deposition and characterization of cuprous oxide thin films[END_REF], pulsed laser deposition [START_REF] Chen | Controlled growth and characteristics of single-phase Cu 2 O and CuO films by pulsed laser deposition[END_REF], sol-gel technique [START_REF] Halin | Characterization of cuprous oxide thin films prepared by sol-gel spin coating technique with different additives[END_REF], spay pyrolysis [START_REF] Saravanan | Growth and characterization of spray pyrolysis deposited copper oxide thin films: Influence of substrate and annealing temperatures[END_REF] …etc. However, the common technique used is thermal evaporation due to its simplicity, low cost, reproducibility, ease of composition control and large area deposition. In a previous work, copper oxide nanostructures were developed into inclined nanostructures using a deposition angle of +85° and two zigzag shapes using (+85°,-85) (2 directions) and (+85°,-85, +85°) (3 directions) where = 85° is the deposition angle between the incident flux direction and the substrate surface normal [START_REF] Chaffar Akkari | Optical, structural, and electrical properties of Cu 2 O thin films[END_REF]. Another earlier work, where nanorod structures tilted toward the incident deposition direction were made with a stationary substrate at various deposition angles from 0 to 80° [START_REF] Tounsi | Structural and optical characterization of copper oxide composite thin films elaborated by GLAD technique[END_REF]. In the two previous works, the substrate does not rotate around its own axis. In this actual work, copper thin films were prepared by thermal vacuum evaporation method with the GLAD technique, proceeding with double inclined angles + andjust by rotating the substrate holder with azimuthal angle ϕ = 180 °, in order to develop zigzag shapes. Then they were annealed in air, to form copper oxides films. In this paper, we aim therefore to investigate the influence of incident flux angles ± on the structural, optical and electrical properties of these samples. According to these properties, one can optimize the angle of deposition of Cu 2 O nanostructures for a potential application such as gas sensing devices, since the gas atoms can infiltrate easily in nano-structures or photovoltaic solar cells since its band gap energy is suitable for buffer layer in solar cell structure.

Experimental details

The glass substrates were previously cleaned by washing agents (a commercial detergent, acetone, a solution of diluted hydrochloric acid then rinsed with ethanol and deionized water) before being introduced into the vacuum system. Copper thin films were grown on glass substrates of rectangular shape (2.5  1.5 cm 2 ) by vacuum thermal evaporation using the GLAD technique from a tungsten boat. The substrates were placed above the source at a distance of 11 cm. The level of base pressure achieved before the deposition was 5×10 -5 Torr. During our experiments, the deposition angle , which is defined as the angle between the incident flux direction and the substrate normal, had taken the following values 40°, 60°, 75° and 85°. A motorized substrate holder ensured the rotation of the substrate, in order to form + thenangles. The evaporation is stopped during the rotation of the substrate holder with azimuthal angle ϕ = 180°. A schematic drawing of the film deposition system is shown in Fig. 1. The obtained films were inserted into a programmable furnace, and annealed at 250 °C during 2 hours, in order to obtain copper oxide films. The zigzag nanostructures were characterized by scanning electron microscopy SEM (Zeiss ultra 55 microscope) with an operating voltage equal to 5 kV. The crystalline phase and crystal orientation of the GLAD films were examined using a Philips X'Pert X-ray diffractometer with monochromatic CuK α radiation (λ = 0.154056 nm and 40 kV, 30 mA) in the 2θ range 20°-70°.

Optical measurements, reflectance, and transmittance, of the obtained films, were carried out by means of UV-Vis-NIR spectroscopy. The wavelength accuracy is 1 nm in the wavelength region 300-1800 nm by using a spectrophotometer (Shimadzu 3100 S ). The photometric accuracy of these measurements is ±3%. The attained optical constants estimation errors depend on the R measurement accuracy. The experiment was done twice and the nano-zigzag structures were observed in both cases for the same values of deposition angles.

Results and discussion

Morphological properties

Fig. 2.a and b show highly arranged nano-zigzags copper structure deposited at an angle = ± 85 ° and = ± 60 °, respectively. It depicts void separated nano-zigzags; this space becomes greater for the high deposition angles because of the important shadowing effect produced by the GLAD technique, which is the most heuristic phenomenon for the nanostructures elaborated by GLAD.

The shadowing phenomenon begins as soon as the vapor of the material begins to reach the inclined substrate. The first ad-atoms inhibit the incoming atoms that try to reach the substrate and consequently cover a neighboring region of the substrate. The incoming atoms will be forced to stick on the first ad-atomsandthe"nano-rods"begintobecreatedeachtimethe atoms try to reach the substrate and are prohibited by the top of the formed clusters. The films adhere well to the substrates and there is not a detachment phenomenon, in spite of the observed voids by the SEM images. The effect of annealing on the deposited copper samples is clear as shown in Fig. 2.c and d.

The nanostructures become denser because of the incorporation of oxygen into the inter-motif space. As the shadowing effect is appreciable at high deposition angles, the incorporation of oxygen becomes easier and it reaches more depth. It is for this reason that the oxidation is found to be perfect for the sample deposited at = ± 85 °, as shown in Fig. 2.c. The example of Fig. 2.d shows that there are two parts; the upper part of the nano-zigzags has been perfectly oxidized, the lower part is also sufficiently oxidized but not with the same reach. This is due to the shadowing effect, which is not good enough for ordinary and weak angles of deposition and consequently, the oxidation does not occur with the same magnitude. Obviously, the annealing affects the surface state. The SEM images in Fig. 3.a and b -taken at two different magnifications -show a porous surface of nano-zigzags of copper deposited at = ± 85 °. While Fig. 3.c and d show the changes occurred on the surface by annealing effect, it is clear that it becomes a compact surface by oxidation.

Several works have attempted to synthesize Cu 2 O nanostructures using chemical methods [START_REF] Mingzhen | Preparation of Cu 2 O nanorods by a simple solvothermal method[END_REF][START_REF] Cheng | Repeatable synthesis of Cu 2 O nanorods by a simple and novel reduction route[END_REF][START_REF] Sabbaghan | Preparation of Cu 2 O nanostructures by changing reducing agent and their optical properties[END_REF] like solvothermal method [START_REF] Mingzhen | Preparation of Cu 2 O nanorods by a simple solvothermal method[END_REF]. Z. Cheng et al. [START_REF] Cheng | Repeatable synthesis of Cu 2 O nanorods by a simple and novel reduction route[END_REF] achieved a simple and highly repeatable wet chemical approach for the synthesis of Cu 2 O nanorods. Apart from chemical methods, PVD processes were also used to create Cu 2 O nanorods array by anodic oxidation [START_REF] Lv | Fabrication and characterisation of Cu 2 O nanorods array by anodic oxidation method[END_REF]. Nevertheless, in almost all cases, the nanorods are randomly oriented, and thus, physical properties of the sample seem to be isotropic. The GLAD technique makes it possible to develop highly oriented nanostructures, which is an advantage that characterizes this technique and enhances or creates anisotropic properties.

Structural properties

The XRD measurements were used to study the structural properties of air annealed Cu thin films deposited on glass substrates and prepared by GLAD technique. of the Cu 2 O zigzag thin films has gradually reduced at larger inclination angles. We reported in our previous work [START_REF] Chaffar Akkari | Optical, structural, and electrical properties of Cu 2 O thin films[END_REF] that the zigzag configuration (+ 85°,-85 °, + 85 °) enhances the crystallinity, comparing with (+85°,-85°) and +85° configurations, contrary to the present work. The effect of angle deposition to produce nanorod architectures not zigzag shape was studied and it was shown that the maximum intensity of the main peaks (111) assigned to the two phases Cu 2 O and CuO was achieved for an angle of 60° [START_REF] Tounsi | Structural and optical characterization of copper oxide composite thin films elaborated by GLAD technique[END_REF]. The actual zigzag shape given in the present work proves to be an inhibitor of copper oxidation and the oxygen surface diffusion mechanism seems to be held up.

Since, the temperature is an important parameter in the kinetic of the oxidation stage [START_REF] Tounsi | Structural and optical characterization of copper oxide composite thin films elaborated by GLAD technique[END_REF], we guess therefore that, the experiment conditions (250° + 2 hours) may not be suitable for enhancement of the intensity of the (111) peak of the copper oxide phases for the actual zigzag configuration (+ ,-) for > 60°. Although CuO is a secondary phase and appears only for the sample deposited at 75 °, it would be interesting to try other annealing times (thermal oxidation) less than 2 hours to avoid the formation of the blends of CuO and Cu 2 O. Because during the increase of the temperature there is a formation of the Cu 2 O phase on the surface of the copper films. Because of a difference in lattice parameters and so the cell volume of copper and its oxide, this creates a high compressive stress that leads to a high density of grain boundaries of this phase [START_REF]Thermal oxidation of copper for favorable formation of cupric oxide (CuO) semiconductor[END_REF]. These conditions with long oxidation times, convert the top layer of Cu 2 O phase into CuO phase, particularly in the form of nanostructures. Regarding, the intensity ratio I 111 / I 200 , we found that the same behavior of this ratio was shown in a recent study on electroplated Cu films on a Si-interposer substrate exposed in an air environment [START_REF] Lee | Electrical resistance evolution of Cu redistribution layer electroplated on a Si interposer[END_REF]. The decrease of I 111 / I 200 ratio of the electroplated Cu thin film was observed when the exposure time is between 31 and 58 days, which demonstrates that atoms in these samples were rearranged at room temperature from an as-electroplated-Cu film, which is generally unstable [START_REF] Lee | Electrical resistance evolution of Cu redistribution layer electroplated on a Si interposer[END_REF]. The ratio I 111 / I 200 of the redistribution layer-Cu (RDL) films decreases from relatively strong 3.7 to 2.7 through self-rearrangement [START_REF] Lee | Electrical resistance evolution of Cu redistribution layer electroplated on a Si interposer[END_REF]. In the present study, this ratio reaches a value slightly below 2, but it does not require a period longer than 31 days.

In order to study more the structural properties of the elaborated samples, the crystallite size D of the films was calculated using Scherrer'sformuladefinedas [START_REF] Bartram | Handbook of X-Rays[END_REF]:

0.9 cos D   
where D is the average crystallite size,  the wavelength of CuK  radiation,  is the full width at half maximum and  is the Bragg angle. As seen in Table 1, the crystallite size decreases with the increase of incident angle. In the Scherrer's formula, the crystallite size D has been determined using a Gaussian function to fit the peaks in the XRD patterns. Then, to remove the effect of broadening from XRD machine, the presence of micro-strain and/or extended defects (stacking faults, micro-twinning, dislocations, etc.) [START_REF] Karen | Liquid-mix disorder in crystalline solids: ScMnO 3[END_REF][START_REF] Theivasanthi | Electrolytic synthesis and characterization of silver nanopowder[END_REF], we used the Williamson-Hall method. The total broadening  tot is given by the following equations: . From the slope, the strain is calculated and it varies between 8.63×10 -3 and 11.45×10 -3 . The largest strain value is attained for the deposition angle = 75° which exhibits the lowest ratio I 111 / I 200 . The most probable reason is due to the apparition of the monoclinic CuO phase (in = ±75°) that may exert an additional constraint on the (111) planes of Cu 2 O (cubic). From the intercept, we can calculate the crystallite

(1) (2)
(3) size, which decreases from 27 to 21 nm. The results are resumed in Table 1. The crystallite size obtained by the Scherrer's formula was less than that obtained by the Williamson-Hall method because the strain correction factor has been taken into account in the case of the Williamson-Hall method,whereasnotinthecaseoftheScherrer'smethod.

Optical properties

In order to investigate the effect of the deposition angle ± on the optical properties of the Cu x O zigzag nanostructured thin films, their optical parameters such as transmittance, reflectance, absorption coefficient, refractive index, optical band gap, birefringence and Urbach energy were studied.

Fig. 5 illustrates the transmittance T and reflectance R spectra of Cu x O zigzag thin films deposited at different incident angles in the wavelength range 300-1800 nm. The spectra reveal interference effects in the transparency region 500-1800 nm with a fall of transmittance at the band edge, whereas the interference effect disappears in the region of very strong absorption. As can be seen also, the transmittance of Cu x O films increases gradually as the incident angle increases. The transmittance spectrum of copper oxides zigzag GLAD films deposited at = 85° shows a good interference pattern which indicates that the film exhibits an acceptable homogeneity of thickness and a sharp fall of transmittance at the band edge. The degree of slanting at the absorption edge can relate to possible differences in the distribution of defect tail states below the band gap [START_REF] Muiva | Surfactant-assisted chemical bath deposition based synthesis of 1-D nanostructured CuO thin films from alkaline baths[END_REF].

Based on the reflectance spectra of the Cu x O zigzag thin films, the thickness and the refractive index of the different films deposited respectively at = 40°, 60°, 75° and 85°, were calculated from the positions of the interference maxima and minima of reflectance using a standard method [START_REF] Chopra | Thin film phenomena[END_REF]. The estimated thicknesses were compared to the ones measured directly by SEM technique, and the values are close. The values of the refractive index in the transparency region and the thickness of the Cu 2 O zigzag films are illustrated in Table 2. We show an obvious change in the refractive index, which may be attributed to the quantum size effect due to the nanocrystallites Cu x O. Despite the same deposited quantity of copper, the thickness decreases versus the deposition angle from 740 nm at = 40° to 323 at = 85°. Indeed, the bending of the nanosized zigzag becomes more and more important, when the deposition angle increases. For grazing angles of flux deposition, the high degree of bending of the zigzag shapes is reached. Therefore, the bending effect may be the main reason for the thickness decrease. The idea is illustrated in Fig. 6. This decrease of the thickness produces less and broad oscillations in the transparency region of the transmittance spectra.

The optical absorption coefficient α of Cu x O zigzag thin films was calculated from the transmittance T(λ) and reflectance R(λ) data using the following relation [START_REF] Abdelkader | Investigation on optical properties of Sn x Sb y S z sulfosalts thin films[END_REF]:

  2 1 1 ln R dT       
where d is the film thickness. Cu x O zigzag thin films have a high absorption coefficient of (10 4 -10 5 cm -1 ) in the visible and near-IR spectral range. This result is very important because it is known that the spectral dependence of absorption coefficient affects the solar cells conversion efficiency. To obtain an absorber material with high efficiency, we need a great absorption coefficient within a large spectral range for photovoltaic applications [START_REF] Caglar | The effects of Al doping on the optical constants of ZnO thin films prepared by spray pyrolysis method[END_REF]. On the other hand, we notice a shift of the absorption edge towards higher photon energies (Fig. 7.a) that is to say that, the oxide phase is favorably promoted with the increase of the deposition angle. Because the oxygen gets into the nanopores easily for high angles and the depth of the pores is more and more accessible when the deposition angles are stronger. In fact, it is well known that the shadowing effect is enhanced when the flux angle deposition increases, therefore the space between the nanostructure motifs or the zigzag nanorods will be more important. As a result, it facilitates the oxygen insertion. This is not very remarkable in the XRD results, because in most of the cases when the deposition angles are closer to grazing ones, the crystallinity is deteriorated even when we manipulate crystalline phases at the beginning. The second probable reason is the existence of a conflict between the two phases of oxides Cu 2 O and CuO, whereas, for the angle 40 °, one has only one dominant phase. We believe that the combined effect of deposition angle and the coexistence of copper oxide phases leads to the (4) blue shift of the absorption edge. I. Mukherjee et al. [START_REF] Mukherjee | Band gap tuning and room-temperature photoluminescence of a physically self-assembled Cu 2 O nanocolumn array[END_REF] found a similar result for physically selfassembled Cu 2 O nano-column array grown by oblique angle sputter deposition. They explained that the blue shift in nano-columnar structures with high anisotropic crystal orientation could be discussed in terms of the light of Burstein-Moss effect since the loss of charge carriers due to trapping at grain boundaries is expected to be minimized in such structures. This explanation may comfort the fundamental reasons for the already given combined effect.

The extinction coefficient k can be calculated from the relation [START_REF] Kindyak | Optical properties of laser-evaporated CuGaSe 2 films near and above the fundamental absorption edge[END_REF]: As can be seen, the extinction coefficient decreases with increasing the deposition angle.

The absorption coefficient α can be related to the band gap energy E g according to the Tauc model [START_REF] Tauc | Amorphous and Liquid Semiconductors[END_REF]:

() q g h A h E    
where A is a constant, h is the Planck constant and q is equal to 1/2 for a direct gap and 2 for an indirect gap semiconductor. The band gap value E g was determined by extrapolating the linear part of the (αhν) 2 vs. hν curve to the horizontal photon energy axis (Fig. 8). It is clearly seen that the optical band gap increases progressively from 2 eV to 2.54 eV as the deposition angle increases.

The band gap values are summarized in Table 2. Based on the XRD study, the increase in the optical band gap with the deposition angle can be attributed to the decrease in crystallite size, which leads to quantum confinement, the quantum size effect as predicted from the nanocrystalline nature of the films. The confinement of the charge carriers in the nanocrystallites results in the increase of a band bending effect and intrinsic microstrain. In our case, the values of the crystallite sizes are in the same order of magnitude. Therefore, it may not contribute significantly to the band gap variation. The measured band gap value using transmittance and reflectance spectra is the effective band gap. According to the Burstein-Moss (B-M) effect, the measured band gap is the sum of B-M

(5) [START_REF] Sinaoui | Investigation of growth and characterization of nanostructured CuIn 5 S 8 thin films produced by glancing angle deposition[END_REF] shift and the intrinsic band gap. This shift allows for the Cu x O thin films empowering considerably the light of a shorter wavelength to be transmitted, making it supportive in enhancement solar cell efficiency .

The Urbach energy E u or called also Urbach tail is the broadening near the band gap due to the existence within the band gap limit of supplementary density levels with energies, which can give rise to band tailing as has been offered for poor crystalline, disordered and porous materials. In other words, it acts as localized states corresponding to optical transitions between localized states adjacent to the valence band and extended states in the conduction band above the mobility edge.

The Urbach energy E u has been calculated using a linear part of lnα below the optical band gap using the following expression [START_REF] Zaynobidinov | Urbach energy and the tails of the density of states in amorphous semiconductors[END_REF]:

0 h E u e       
where α 0 is a constant. In fact, Urbach energy E u describes the local defects in the material. The values of E u for different deposition angles are classified in Table 2, which clearly show that the Urbach energy increases with the increase of incident angles. This result is in good agreement with the XRD results.

The highly oriented nanostructure of the GLAD zigzag films tends to be anisotropic, thus the optical birefringence can be calculated. Fig. 9 illustrates the in-plane birefringence of Cu 2 O zigzag thin films deposited at different deposition angles , which is defined as the difference between the two in-plane refractive indices [START_REF] Beydaghyan | Enhanced birefringence in vacuum[END_REF],namelyΔn = n x -n y . The two in-plane refractive indices n x and n y are determined based on the reflectance spectra measured in two orthogonal directions of incident linearly polarized light: R x is in the x-direction and R y is in the y-direction. From Fig. 9, it is clear that the in-plane birefringence is weak at a low incident angle, then it increases with increasing deposition angle and reaches a maximum value of Δn = 0.1063 at an angle = 60. This maximum of birefringence can be linked to the geometry of the columns. Generally, at deposition angles higher than 60°, the samples have highly porous structures and this tends to reduce the value of the birefringence. As a conclusion, the glancing angle deposition is a promising way to obtain [START_REF] Rahchamani | Study of structural and optical properties of ZnS zigzag nanostructured thin films[END_REF] larger optical birefringence, and it seems particularly flexible to generate designed devices such as retardation plate and polarizer.

Electrical properties

Fig. 10 shows the resistance versus annealing time curves. It exhibits three main parts. Two stages, which represent the two phases metallic and semiconductor, and a vertical rise that separates the two stages, which indicates the metal-semiconductor transition phase. In the first stage, the samples still have their metallic phase. However, they have different resistances as a function of the deposition angle. Moreover, it is noted that the sample deposited at = ± 85 ° has the highest resistance, although it is still in the metallic phase, this is due to the favored oxidation process by large pores provided by a high deposition angle. That is why this sample passes rapidly from the metal to semiconductor phase. The phase transition duration decreases as a function of the deposition angle. The oxidation mechanism is easier for high deposition angles since the diffusion of oxygen is controlled by the deposition angle. It is important to note that, the sample deposited at = ± 40 ° has the lowest resistance in both stages even after the metal-semiconductor transition phase. Besides, let us recall that the sample deposited at = ± 75 ° has two polycrystalline copper oxide phases Cu 2 O and CuO, this is the reason on which we are based to explain the existence of a slight second transition located at 32 min within the semiconductor phase.

In order to study the electrical anisotropy of the zigzag thin films, resistivity measurements were taken between different locations on the sample (Fig. 11 This abrupt change of electrical properties for GLAD films deposited at incident angles higher than 60° is mainly linked to the porous structure, which becomes more and more important for this values of incident angles. Voids between columns may reduce the electron mean free path and create barriers potential for the conduction mechanism leading to a drop of the electrical conductivity [46].

Conclusion

Copper oxide zigzag thin films were successfully prepared on glass substrates by air annealing of copper films deposited using thermal evaporation method at oblique angle deposition. The X-ray diffraction results show that the elaborated films, for all incident angle deposition, have a polycrystalline nature and grown in a preferential orientation along the (111) plane. The crystallite size of the prepared samples decreases with the deposition angle. The deposition angle = 40 seems to be the adequate parameter to create copper oxide zigzag thin films with a good crystalline features. We have shown that the elaborated films have a good absorption coefficient, higher than 10 5 cm -1 in the visible range and it depicts a blue shift with the deposition angle. Therefore, the band gap increases from 2 eV and 2.54 eV, when increasing the deposition angle. The Urbach energy was found to increase from 259 to 534 meV with increasing of incident deposition angle, which probably related to the increase of disorder and the density of defects in the nanostructures. The maximum of birefringence was obtained for an angle = 60. The electrical measurement brings out the electrical anisotropy of the elaborated samples.

Figures and tables captions

Fig. 4 .

 4 a shows the XRD patterns of Cu x O thin films deposited respectively at = 40°, 60°, 75° and 85°. It can be seen that the Cu x O thin films are strongly grown along the (111) direction at 2θ = 36.5°, which indicates a preferential orientation of the Cu 2 O cubic phase according to the JCPDS card number (01-078-2076) and the RRUFF ID: R050384.1. Besides, we note the appearance of (111) peak assigned to CuO monoclinic phase (JCPDS 45-0937 and RRUFF ID: R120076.9) at 2θ = 38.7°, in addition to Cu 2 O peaks, at deposition angle = ±75°. The relative intensity of the main peak decreases from 1400 to 525 (cps) with increasing the deposition angle. As shown in Fig.4.b, the intensity ratio I 111 / I 200 decreases as a function of the incident flux angle, which indicates that the degree of crystallinity

   is the strain (micro-deformation),  instr = 0.02° is the instrumental broadening and D WH is the crystalline size of Williamson-Hall method. When we plot {( β tot -2 instr )cos 2 θ} versus sin 2 , we get a linear curve with a slope (4)

Fig. 7 (

 7 Fig.7 (b).As can be seen, the extinction coefficient decreases with increasing the deposition angle.

  .a). In Fig.11.b electrical resistances of samples deposited at different incident angles, are shown. It is clear that the electrical resistance increases with increasing deposition angle and reaches a maximum value at an angle of 60°. After that, it decreases with further increase of the deposition angle.

Fig. 1 :Fig. 2 :

 12 Fig. 1: The schematic diagram of the GLAD technique for the synthesis of zigzag (+ ,-) nanostructures.

Fig. 3 :

 3 Fig. 3: SEM cross-section images taken at a tilt angle of 3.5° (a) Cu zigzag nanostructures deposited at = ±85° with a magnification of 101.91 K× and (b) 3.537.56 K× (c) SEM crosssection images taken at a tilt angle of 14.7° of the Cu 2 O film deposited at = ±85° with a magnification of 103.36 K× and (d) 62.75 K×.

Fig. 4 :

 4 Fig. 4: (a) X-ray diffraction patterns of Copper oxide zigzag thin films deposited at different incident angles and (b) Evolution of I 111 / I 200 ratio, I 111 and strain ε function of deposition angle.

Fig. 5 :

 5 Fig. 5: Optical transmittance and reflectance spectra of the copper oxide zigzag thin films function of deposition angle.

Fig. 6 :

 6 Fig. 6: Effect of the deposition angle on the bending of the zigzag motif.

Fig. 7 :

 7 Fig. 7: (a) Absorption and (b) extinction coefficients of copper oxide zigzag thin films deposited at different deposition angles.

Fig. 8 :

 8 Fig. 8: Direct band gap energy of copper oxide zigzag thin film deposited at = ±85°.

Fig. 9 :

 9 Fig. 9: Variation of birefringence Δn as a function of deposition angle.

Fig. 10 :

 10 Fig. 10: Variation of electrical resistance versus annealing time of copper oxide zigzag thin film deposited at different incident angles.

Fig. 11 :

 11 Fig. 11: (a) Layer configuration for the electrical measurements and (b) Variation of electrical resistance at a different position in the copper oxide zigzag thin film deposited at different incident angles.

Fig. 1 Fig

 1 Fig. 1

  Fig.5

Table 1 :

 1 Estimated values of structural parameters for the copper oxide zigzag thin film deposited at different incident angles.

Table 2 :

 2 Thickness, refractive index, band gap and Urbach energy values of copper oxide zigzag thin films deposited at different incident angles.

Table 1 Deposition angle γ (°) Crystallite size (nm) Scherrer'smethod Crystallite size D WH (nm)

 1

	Strain  (×10 -3 )

Thin film Copper source Current I (A) Substrate Azimuthal angle ϕ = 180° -γ γ Incident flux Motorized substrate holder

/I 200 Incident angle  (°)

evaporated silicon thin films, Appl. Opt. 43 (2004) 5343-5349.

[46] V. Collado, N. Martin, P. Pedrosa, J.Y. Rauch, M. Horakova, M.A.P. Yazdi, A. Billard, Temperature dependence of electrical resistivity in oxidized vanadium films grown by the GLAD technique, Surf. Coat. Technol. 304 (2016) 476-485.


M Bauer

F Cornu

Thermal contact through a two-temperature kinetic Ising chain

Keywords: exact analytical results, thermal contact, Glauber spin dynamics, current fluctuations PACS numbers : 05.70.Ln, 02.50.Ga, 05.60.Cd

We consider a model for thermal contact through a diathermal interface between two macroscopic bodies at different temperatures: an Ising spin chain with nearest neighbor interactions is endowed with a Glauber dynamics with different temperatures and kinetic parameters on alternating sites. The inhomogeneity of the kinetic parameter is a novelty with respect to the model of Ref.

[1] and we exhibit its influence upon the stationary non equilibrium values of the two-spin correlations at any distance. By mapping to the dynamics of spin domain walls and using free fermion techniques, we determine the scaled generating function for the cumulants of the exchanged heat amounts per unit of time in the long time limit.

Introduction

Thermal contact between two macroscopic bodies initially at different temperatures corresponds to a situation where the heat transfer between the two bodies is ensured by a thin diathermal interface. The latter may be an immaterial interface between two solids or a diathermal wall between two fluids. Over a time window during which the two macroscopic bodies have negligible energy variations, they behave as thermostats with constant thermodynamic temperatures, while the interface is a mesoscopic system with traceable configurations. After a long enough time inside the considered time window the interface tends to a stationary non equilibrium state where the instantaneous heat current which it receives from a thermostat has a non-vanishing mean value. Even if the interface is described by a model where its degrees of freedom obey a (deterministic or stochastic) microscopic dynamics, there is no general framework, such as Gibbs equilibrium ensemble theory, which would allow to determine the probability distribution of the interface configurations and the corresponding mean instantaneous heat current. Therefore it is most valuable to exhibit solvable models which would shed some light into the dependence of the heat instantaneous current upon the model parameters and the temperatures of the two thermostats.

Such a solvable model has been introduced by Racz and Zia in 1994 [START_REF] Racz | Two-temperature kinetic Ising model in one dimension: Steady-state correlations in terms of energy and energy flux[END_REF]. They consider a chain of classical spins with periodic boundary conditions and interacting with a nearest-neighbor ferromagnetic (Ising) interaction. They endow it with a Glauber stochastic dynamics with singlespin flips at a time and such that the spins on odd and even lattice sites are flipped by thermostats at two different temperatures. In our view, the model may be seen as a zig-zag shaped chain inside a very thin strip between two half-planes occupied by two different thermostats, and where the odd (even) sites are located on the left (right) side of the strip. The stationary two-spin correlations at any distance as well as higher order spin correlations have been extensively studied in Refs. [START_REF] Racz | Two-temperature kinetic Ising model in one dimension: Steady-state correlations in terms of energy and energy flux[END_REF][START_REF] Schmittmann | Stationary correlations for a far-from-equilibrium spin chain[END_REF][START_REF] Mobilia | Complete solution of the kinetics in a far-fromequilibrium Ising chain[END_REF][START_REF] Mobilia | Exact dynamics of a reaction-diffusion model with spatially alternating rates[END_REF].

We point out that the latter model indeed satisfies two requirements needed for a correct description of a thermal contact between two macroscopic bodies during a transient time window where their temperature can be considered as constant. First the contact must be mediated by changes in the internal energy of the spin interface. Second, if the energies of the macroscopic bodies were kept tracked of, the transition rates for both the interface configurations and these two energies would obey the detailed balance with the microcanonical equilibrium probability distribution for these variables of the whole system. Then, in the infinite time limit the two bodies and the interface would be at the same temperature; in a time window where the energy variations of the macroscopic bodies are negligible, the transition rates for the interface depend only on the temperatures of the macroscopic bodies and they obey the local detailed balance [START_REF] Derrida | Non-equilibrium steady states: fluctuations and large deviations of the density and of the current[END_REF][START_REF] Bauer | Local detailed balance : a microscopic derivation[END_REF] 1 . In the case of a spin interface the two corresponding requirements become: 1) a transition between two spin configurations involves only one thermostat; 2) the transition rate involving a given thermostat obey the detailed balance at the same temperature. For the present two-temperature Ising chain the first requirement is obviously fulfilled2 , and the simplest transition rates which fulfill the second requirement are those chosen by Glauber in the case of an Ising chain in contact with a unique thermostat [START_REF] Glauber | Time-Dependent Statistics of the Ising Model[END_REF]. (We recall that Glauber looked for single-spin flip dynamics such that in the infinite time limit the spin chain indeed relaxes to the canonical equilibrium state at the thermostat temperature and then he chose the simplest transition rates.) On the other hand, the quantities of interest for exchange processes which have been considered over the last three decades, both theoretically and experimentally, are amounts of microscopically conserved entities (matter, energy, ...), which are exchanged over a very long time (in the case of thermal contact the corresponding quantities are the heat amounts received by the interface from each thermostat during a fixed long time interval). They have been focused on because the scaled generating function for their cumulants per unit of time in the long time limit as well as its Laplace transform, the large deviation function of the corresponding time-integrated current, have been shown to obey generic symmetry relations, the so-called fluctuation relations. The latter relations are derived from properties of the system dynamics and they are a milestone of the stochastic thermodynamics theory (For a review see Ref. [START_REF] Seifert | Stochastic thermodynamics, fluctuation theorems and molecular machines[END_REF].)

In this context it is also interesting to have at hand solvable models where the statistics of the time-integrated currents can be calculated. For instance, such a model has been exhibited in the case where the heat transfer between two thermostats is ensured by a wire. The energy quanta are represented as particles whose stochastic dynamics is a Symmetric Simple Exclusion Process (SSEP) with adequate boundary conditions: particles with hard cores hop on the sites of a one-dimensional lattice with the same hopping rate in both directions, but have different in-coming and out-going rates at the two lattice ends in contact with two particle reservoirs with different chemical potentials. Then all cumulants of the heat coming out of one thermostat per unit of time can be calculated in the long time limit [START_REF] Derrida | Current Fluctuations in the One-Dimensional Symmetric Exclusion Process with Open Boundaries[END_REF][START_REF] Bodineau | Cumulants and large deviations of the current through nonequilibrium steady states[END_REF] .

In the present paper we consider a generalized version of the diathermal interface model of Ref. [START_REF] Racz | Two-temperature kinetic Ising model in one dimension: Steady-state correlations in terms of energy and energy flux[END_REF]: the two Glauber dynamics which flip the spins on sites with odd or even indices respectively have not only different temperatures but also different kinetic parameters (see Sec.2). Our aim is to calculate the scaled generating function of the joint cumulants per unit of time for the heats flowing out of the two thermostats in the long time limit.

The difference between the kinetic constants is relevant for two macroscopic bodies made with different materials. Some of the corresponding kinetic effects have been previously investigated in our study of a very simple model for a one-dimensional interface between two half-planes occupied by bodies at two different temperatures : a model of independent two-spin pairs where the leftside (right-side) spin in each pair is flipped only by the thermostat on the same side according to a Glauber dynamics. The whole statistics for the spin configurations and the heat amounts exchanged by every pair with the thermostats have been calculated explicitly [START_REF] Cornu | Thermal contact through a diathermal wall : a solvable model[END_REF].

The method which we use to obtain the scaled generating function for the heat cumulants in the present model is the following. From its definition this function can be obtained as the largest eigenvalue of the modified Markov matrix which rules the evolution of the joint probability for the system configurations and the heat amounts Q o and Q e received on each (odd or even) sublattice.

In order to study energy exchanges more conveniently, instead of considering the spin configurations, we rather formulate the problem in terms of the position configurations for the domain walls, which sit on the dual lattice. In other words we consider the well-known lattice gas representation on the dual lattice ('antiparallel adjacent spin pair' ↔ 'particle' and 'parallel adjacent spin pair' ↔ 'hole') where a particle is in fact a domain wall. The mapping of Glauber spin dynamics to the particle dynamics then includes the possibility for the creation and annihilation of adjacent particle pairs, which correspond to the injection or the loss of energy in the chain respectively3 . Our model with two temperatures and two kinetic parameters is mapped to a reaction-diffusion system with two different creation (annihilation) rates as well as two different hopping rates on spatially alternating sites 4 .

The crucial point is that, in a suitable basis for the representation of the configurations of domain wall positions, the Markov matrix for the evolution of the configuration probability involves only products of two operators : the Markov matrix is mapped to a free fermion Hamiltonian [START_REF] Schütz | Exactly solvable models for many-body systems far from equilibrium[END_REF]. Moreover the mapping can be readily generalized for the modified Markov matrix which rules the evolution of the joint probability for a configuration of domain wall positions and the amounts of heat Q o and Q e . The latter matrix can be diagonalized by using free fermions techniques: Jordan-Wigner transformation and antiperiodic Fourier transform. Thus we obtain a block diagonal matrix, made of 4 × 4 blocks, which can be straighforwardly diagonalized by introducing four pseudo-fermion operators. (In the case of two pseudo-fermion operators and the associated Bogoliubov-like transformation, see for instance Ref. [START_REF] Grynberg | Dynamic Correlation Functions of Adsorption Stochastic Systems with Diffusional Relaxation[END_REF][START_REF] Grynberg | Dynamics of adsorption-desorption processes as a soluble problem of many fermions[END_REF].) The largest eigenvalue of the modified Markov matrix is obtained by filling up all pseudo-fermion states associated with an eigenvalue with a positive real part.

The paper is organized as follows. The model is defined in Sec.2 and the relaxation time for the mean global magnetization on each sublattice is calculated. Moreover by solving a hierarchy of equations for the stationary global two-spin correlations at any distance (see Appendix A), we determine the mean instantaneous energy current which flows from each thermostat into the spin lattice in the stationary state. The mapping to the dynamics of domain walls on the dual lattice and the associated modified matrix is derived in Sec.3. Its eigenvalues are determined in Sec.4 by free fermion techniques. The cumulants are obtained in Sec.5 and various physical regimes are discussed. In conclusion we summarize some finite-size effects and their possible cancellation in heat cumulants.

Model

Description of a thermalization process

We consider a one-dimensional lattice with a finite even number of sites L = 2N , where each site j is occupied by a classical spin s j (s j = ±1, j = 1, . . . , 2N ) with periodic boundary condition s j+2N = s j . Spins interact via the Ising ferromagnetic nearest-neighbor interaction with coupling K > 0: the energy of a spin configuration s is

E(s) = -K 2N j=1 s j s j+1 . (2.1)
When the spin at site j is flipped, the energy variation of the Ising chain is equal to

∆E(s j → -s j ) = s j s j-1 + s j+1 2 ∆E, (2.2)
with ∆E = 4K. This variation can take the values +∆E, 0, or -∆E, The model is endowed with a stochastic dynamics where the spin flips at odd (even) sites are due to energy exchanges with a macroscopic body at temperature T o (T e ) in the course of a thermalization process of the two macroscopic bodies. The transition rates must obey local detailed balance [START_REF] Derrida | Non-equilibrium steady states: fluctuations and large deviations of the density and of the current[END_REF][START_REF] Bauer | Local detailed balance : a microscopic derivation[END_REF]: the transition rates w(s j → -s j ) and w(-s j → s j ) for two reversed flips of the spin at site j, while all other spins are kept fixed, must obey the ratio

w(s j → -s j ) w(-s j → s j ) = e -βj ∆E(sj→-sj ) , (2.3)
where β j is the inverse temperature of the thermostat acting on site j :

β j = 1/(k B T j )
where k B is Boltzmann constant and T j is equal either to T o or T e , depending on the parity of j. As shown by Glauber [START_REF] Glauber | Time-Dependent Statistics of the Ising Model[END_REF] in the case of a unique temperature, the simplest transition rates which obey the constraint (2.3) read

w(s j → -s j ) = ν j 2 1 -γ j s j (s j-1 + s j+1 ) 2 , (2.4)
while w(-s j → s j ) is given by the latter expression where s j is replaced by -s j . In (2.4) γ j is the thermodynamic parameter at site j,

γ j = tanh β j ∆E 2 (2.5)
where γ j = γ o or γ e , depending on the parity of the site index, and ν j is the kinetic parameter ν j at site j. The latter is not determined by the local detailed balance; it can be interpreted as the mean frequency at which the macroscopic body tries to flip the spin at site j. Therefore we set ν j = ν o or ν e , depending on the parity of the site index (note that when the two kinetic parameters are equal the present model coincides with that of Ref. [START_REF] Racz | Two-temperature kinetic Ising model in one dimension: Steady-state correlations in terms of energy and energy flux[END_REF]). Since the time scale is arbitrary, it is convenient to introduce the dimensionless kinetic parameters ν a , with a = o or e, defined as The probability P (s; t) for the system to be in spin configuration s at time t evolves according to the master equation

ν a = ν a ν o + ν e . ( 2
dP (s; t) dt = 2N j w(-s j → s j )P (s j ; t) -   2N j w(s j → -s j )   P (s; t) (2.7)
where s j denotes the spin configuration obtained from s by changing s j into -s j . The number of configurations is finite, and the transition rates allow the system to evolve from any configuration to any other one after a suitable succession of transitions. Therefore there is a unique stationary solution of the master equation. In the following we focus on the mean values of global quantities and denote • • • and • • • st the expectation values calculated with the time-dependent probability P (s; t) and the stationary probability P st (s) respectively.

Relaxation of the mean global sublattice magnetizations

The transition rates are invariant under a global flip of the spins, so that a configuration and the corresponding one where all spins are flipped have the same probability in the stationary state.

As a result all stationary correlations for an odd number of spins vanish identically; in particular s j st = 0. As a consequence the mean values of the global magnetizations on the two sublattices,

M o = N n=1 s 2n-1
and M e = N n=1 s 2n respectively, vanish in the stationary state,

M o st = M e st = 0. (2.8)
The relaxation of the mean values of global sublattice magnetizations is readily studied. As in the case of the homogeneous spin chain considered by Glauber [START_REF] Glauber | Time-Dependent Statistics of the Ising Model[END_REF], the evolution equation for the mean value of the spin at site j reads d s j dt = -2 s j w(s j → -s j ) .

(2.9)

According to the expression of the transition rates (2.4)

d s j dt = -ν j s j -γ j s j-1 + s j+1 2 . (2.10)
Then the coupled evolutions of the magnetizations on the two sublattices read

d M o dt = -ν o [ M o -γ o M e ] (2.11) d M e dt = -ν e [ M e -γ e M o ]
.

From these equations we retrieve that both mean magnetizations vanish in the stationary state, as predicted by symmetry arguments. The matrix associated with this system of linear equations has two strictly negative eigenvalues 1 2 (ν o + ν e ) -1 ± (ν oν e ) 2 + 4ν o ν e γ o γ e , each of which is associated with a couple of right and left eigenvectors (the eigenvalues are negative because

(ν o -ν e ) 2 + 4ν o ν e γ o γ e = 1 -4ν o ν e (1 -γ o γ e ) < 1)
. For generic values of the initial magnetizations M o and M e , the inverse relaxation time 1/t rel to their stationary value is given by the opposite of the negative eigenvalue with the smallest modulus, and the relaxation time t rel reads

t rel = 2 ν o + ν e 1 -(ν o -ν e ) 2 + 4ν o ν e γ o γ e -1
.

(2.12)

Mean global heat current in the stationary state

The mean instantaneous heat current j k received by the spin chain at site k from the thermostat at temperature T k is equal to the expectation value of the variation of the chain energy when the spin s k is flipped times the transition rate for the flip. According to the expressions for the energy variation (2.2) and for the transition rates (2.4), the mean instantaneous current reads

j k = Kν k [-γ k -γ k s k-1 s k+1 + s k-1 s k + s k s k+1 ] . (2.13)
Therefore the stationary mean value of the global heat current coming from the thermostat acting on spins at even sites, namely J e = N n=1 j 2n , is determined as

J e st = N Kν e [-γ e -γ e D oo 2 + D oe 1 + D eo 1 ] (2.14)
with the following definitions : D oo 2 is the average over the sublattice of odd sites of the stationary correlation between two spins separated by two sites,

D oo 2 = 1 N N n=1 s 2n-1 s 2n+1 st , (2.15)
D oe 1 = (1/N ) N n=1 s 2n-1 s 2n st and D eo 1 has an analogous definition. Similarly the stationary mean value of the global heat current coming from the thermostat acting on spins at odd sites,

J o = N n=1 j 2n-1 , reads J o st = N Kν o [-γ o -γ o D ee 2 + D oe 1 + D eo 1 ] (2.16) with D ee 2 = (1/N ) N n=1 s 2n s 2n+2 st .
The values of the stationary global two-spin correlations D oo 2 , D ee 2 , D oe 1 and D eo 1 can be determined from a hierarchy of equations for similar quantities with two spins at any distance on the lattice. Details are given in Appendix A with the results (A.11), (A.12), (A. [START_REF] Jordan | Über das Paulische Äquivalenzverbot[END_REF], and (A.20) for any distance between spins. From the latter results we get

D ee 2 = γη - γ o 1 + η N -2 - 1 + η N - (2.17) with γ = ν o γ o + ν e γ e , (2.18)
where the dimensionless kinetic parameters have been defined in (2.6), and η -, with 0 < η -< 1, is defined by

√ η -= 1 - √ 1 -γ o γ e √ γ o γ e . (2.19) Moreover D oo 2 = (γ o /γ e )D ee 2 , while D oe 1 = D eo 1 = γ √ η - √ γ o γ e 1 + η N -1 - 1 + η N - . (2.20)
We notice that the dependence upon the kinetic parameters ν o and ν e occurs only through the parameter γ defined in (2.18).

Eventually the stationary mean value of the global heat current received on even sites can be calculated from expression (2.14) and relation (A.18); we get

J e st = N K ν o ν e ν o + ν e (γ o -γ e ).
(2.21)

Similarly the stationary mean value of the global heat current received on odd sites can be obtained from the expression (2.16) ; it proves to be opposite to that on even sites, J o st = -J e st , as it should in the stationary state where the mean energy of the chain is constant. We point out the following remarkable property : though D ee 2 , D oo 2 , D oe 1 and D eo 1 involve finite size corrections (see (2.17) and (2.20)), these corrections cancel one another in the value of the mean global current J e st : J e st is exactly proportional to the size L = 2N of the ring. Moreover it happens to be equal to L times the mean current received by a spin in the independent pair model of Ref. [START_REF] Cornu | Thermal contact through a diathermal wall : a solvable model[END_REF].

Mapping to a reaction-diffusion system with pair creationannihilation

To prepare the study of the heat amounts exchanged with the thermostats we consider a mapping to another model for which the evolution operator is quadratic.

Domain wall system

When two spins on neighboring sites are antiparallel, one may consider that there is a domain wall between them, whereas there is no domain wall when they are parallel. The domain walls sit on the edges of the initial lattice. Labeling each edge by its mid-point, one gets another lattice which we call the dual lattice in what follows. The edge (j -1, j) and the corresponding site on the dual lattice are labeled by j. If s j-1 and s j are antiparallel, s j-1 s j = -1, then the occupation number by a domain wall at site j on the dual lattice is n j = 1, whereas if s j-1 and s j are parallel n j = 0. Thus the correspondance reads

n j = 1 -s j-1 s j 2 . (3.1)
On a ring the number of domain walls is even and L j=1 n j is even. As a result a spin configuration can be characterized either by the set s = {s 1 , • • • s L } of spin configurations or by the knowledge of the value of s 1 and the set of the positions of the domains walls, namely the set of occupations numbers n = {n 1 , • • • , n L }. The energy of the system can be expressed solely in terms of domain walls as

E(n) = -2N K + 2K 2N j=1 n j .
(3.2)

Quantum mechanics notations

In the following we use the quantum mechanics notations, as commonly done in the literature. Then a column vector is denoted as a "ket", | . . . and a row vector is denoted as a "bra" . . . |.

The configuration of occupation numbers by domain walls

n = {n 1 , • • • , n L } is represented as a tensor product |n = ⊗ L j=1 |n j , (3.3)
where |n j is a two-component column vector. The convention used for kets associated to vacant and occupied states is

1 0 j = |n j = 0 and 0 1 j = |n j = 1 . (3.4)
This convention is the standard choice of basis in the condensed matter literature on quantum spin chains. With the representation (3.3)-(3.4) the row-column product n ′ |n takes the form

n ′ |n = L j=1 δ n ′ j ,nj
. Therefore the probability of the domain wall configuration n at time t, P (n; t), can be represented as a row-column (scalar) product P (n; t) = n|P t , where |P t is the column vector defined as

|P t = n P (n; t)|n . (3.5)
With the latter definitions the master equation for the stochastic evolution of the probability P (n; t), which takes the generic form written in (2.7) in the case of P (s; t), can be represented as the evolution of the column vector |P t under the Markov matrix M

d|P t dt = M|P t (3.6)
with

n ′ |M|n = w(n → n ′ ) if n ′ = n (3.7) n|M|n = - n ′ =n w(n → n ′ ),
where w(n → n ′ ) denotes the transition rate from configuration n to configuration n ′ .

Markov matrix for the model

For the present model of domain walls the matrix elements of M can be expressed in terms of Pauli matrices. Indeed the operator for the occupation number at site j reads

n j = 1 2 1 j -σ z j (3.8)
where 1 j denotes the identity 2 × 2 matrix at site j, and σ z j = 1 0 0 -1 j ; the operator which changes the occupation number at site j is σ x j = 0 1 1 0 j . By inspection of the transition rates for the spin configurations w(s → s ′ ) given by (2.4), every transition rate for the occupation numbers by domain walls, w(n → n ′ ), can be written as a matrix element n ′ |W|n , where -for a hop of a domain wall from site j to site j + 1

W = ν j 2 σ x j σ x j+1 n j (1 j -n j+1 ) (3.9)
-for a hop of a domain wall from site j + 1 to site j

W = ν j 2 σ x j σ x j+1 (1 j -n j ) n j+1 (3.10)
-for the annihilation of two domain walls at sites j and j + 1

W = ν j 2 (1 + γ j )σ x j σ x j+1 n j n j+1 (3.11)
-for the creation of two domain walls at sites j and j + 1

W = ν j 2 (1 -γ j )σ x j σ x j+1 (1 j -n j ) (1 j -n j+1 ) . (3.
12)

The latter expressions can be written in a more compact form by using the spin-1 2 ladder operators

σ + = 0 1 0 0 j and σ -= 0 0 1 0 j
. They are such that σ x n = σ + and σ x (1n) = σ -. With the convention (3.4) σ + j annihilate a domain wall at site j, while σ - j creates a domain wall at site j.

Eventually the Markov matrix M derived from the master equation for the evolution of the probability of spin configurations (2.7) reads

M = ν 1 + ν 2 2 (3.13) ×   -(1 -γ)N 1 -γ 2N j=1 n j + 2N j=1 ν j σ + j σ - j+1 + σ - j σ + j+1 + (1 + γ j )σ + j σ + j+1 + (1 -γ j )σ - j σ - j+1   ,
where 1 denotes the identity 2N × 2N matrix and γ has been defined in (2.18). The advantage of the domain wall representation with respect to the spin representation is that the Markov matrix is quadratic in terms of operators acting on different sites instead of involving three operators acting on different sites (for the latter case see for instance Ref. [START_REF] Felderhof | Spin relaxation of the Ising chain[END_REF][START_REF] Felderhof | Note on spin relaxation of the Ising chain[END_REF]).

4 Eigenvalues of the modified Markov matrix

Modified Markov matrix

We are interested in the joint cumulants per unit of time for the heat amounts Q o and Q e which are received by the chain from the thermostat acting on spins at odd and even sites during a time t in the long time limit. The corresponding scaled generating function is

g 2N (λ o , λ e ; t) = lim t→∞ 1 t ln e λoQo+λeQe , (4.1)
where λ o and λ e are real parameters. In fact an evolution equation can be written for the probability P (n, Q o , Q e ; t) for the system to be in configuration n at time t and to have received heat amounts Q o and Q e between times 0 and t. By inspection of the transition rates (2.4) and according to the correspondence (3.1), when the spin at site j is flipped under the action of the thermostat at temperature T j , the variation of Q j is equal to +∆E if a pair of domain walls is created at sites j and j + 1 , -∆E if a pair of domain walls is annihilated at these sites, 0 if a domain wall jumps either from j to j + 1 or from j + 1 to j. As a consequence, with a definition for | P t (λ o , λ e ) analogous to that for |P t given in (3.5), namely n| P t (λ o , λ e ) = P (n, λ o , λ e ; t), we get the evolution equation

d| P t (λ o , λ e ) dt = M(λ o , λ e )| P t (λ o , λ e ) , (4.4)
where the so-called modified Markov matrix M(λ o , λ e ) reads

2 ν 1 + ν 2 M(λ o , λ e ) = -(1 -γ)N 1 -γ 2N j=1 n j + 2N j=1 ν j σ + j σ - j+1 + σ - j σ + j+1 + b j σ + j σ + j+1 + c j σ - j σ - j+1 . ( 4

Jordan-Wigner transformation

In order to find the eigenvalues of the modified Markov matrix M(λ o , λ e ) given by (4.5) we take advantage of its structure analogous to a free fermion Hamiltonian and we introduce the following Jordan-Wigner transformation [START_REF] Jordan | Über das Paulische Äquivalenzverbot[END_REF]

f † j = j-1 k=1 σ z k σ - j and f j = j-1 k=1 σ z k σ + j . (4.8)
The operator f † j is indeed the adjoint of f j , because (σ z ) † = σ z and (σ -) † = σ + . Operators σ acting on different sites commute, whereas σ z j anticommutes with σ + j and σ - j ; moreover (σ + j ) 2 = 0 and (σ - j ) 2 = 0. Therefore the operators f j and f † j obey the fermionic anticommutation relations

{f j , f j ′ } = 0 {f † j , f † j ′ } = 0 {f j , f † j ′ } = δ j,j ′ . (4.9)
The occupation number of site j by a domain wall, given by (3.8), also reads n j = σ - j σ + j = f † j f j . The expression (4.5) of the modified matrix M(λ o , λ e ) is rewritten in terms of fermionic operators as

2 ν 1 + ν 2 M(λ o , λ e ) = -(1 -γ)N 1 -γ 2N j=1 f † j f j + 2N -1 j=1 ν j f † j f j+1 -f j f † j+1 + c j f † j f † j+1 -b j f j f j+1 -ν e (-1) N f f † 2N f 1 -f 2N f † 1 + c e f † 2N f † 1 -b e f 2N f 1 (4.10)
where N f = 2N j=1 f † j f j is the total number of fermions. Since the spin system is on a ring, there can be only an even number of domain walls in the system. As noticed above, the operator for the occupation number by a domain wall n j coincides with the number of fermions at site j, f † j f j . Therefore we have to consider the restriction M + (λ o , λ e ) of M(λ o , λ e ) to the sector with an even number of fermions. According to (4.10) the expression of this rectriction is invariant by translation along the ring if the fermionic operators are chosen to satisfy the antiperiodic boundary conditions

f 2N +1 = -f 1 and f † 2N +1 = -f † 1 . (4.11)
Then the restriction reads

2 ν 1 + ν 2 M + (λ o , λ e ) = -(1 -γ)N 1 -γ 2N j=1 f † j f j + 2N j=1 ν j f † j f j+1 -f j f † j+1 + c j f † j f † j+1 -b j f j f j+1 .
(4.12)

Antiperiodic Fourier transform

The next step to the diagonalization is to rewrite the fermionic operators as antiperiodic Fourier transforms which satisfy the antiperiodic boundary conditions (4.11) The wave numbers are of the form q = (2k + 1)π/(2N ) and we work with a complete family of representatives in the set

B(2N ) = {q = (2k + 1) π 2N , k = -N, -N + 1, • • • , -1, 0, 1, • • • N -1}, (4.13)
namely

B(2N ) = {-π + π 2N , -π + 3π 2N , • • • - π 2N , π 2N , • • • π - π 2N }. (4.14)
The operator f j can be written as the antiperiodic Fourier transform

f j = 1 √ 2N q∈B(2N ) e ıqj η q (4.15)
in terms of the wave fermions

η q = 1 √ 2N 2N j=1 e -ıqj f j . (4.16)
Going from (4.15) to (4.16) relies on the identity 2N j=1 e ı(q-q ′ )j = 2N 1 q-q ′ ≡0(2π) , (4.17)

where 1 q-q ′ ≡0(2π) = 1 if qq ′ is equal to 0 modulo 2π and 1 q-q ′ ≡0(2π) = 0 otherwise. All q ′ s in B( 2N) satisfy e ıq2N = -1, and subsequently f j does obey the antiperiodic boundary conditions (4.11). The adjoint operator f † j reads

f † j = 1 √ 2N q∈B(2N ) e -ıqj η † q . (4.18)
These representations are inserted in the expression (4.12). In the term 2N j=1 f † j f j there occurs a summation over all sites of the ring and one uses the identity (4.17). In the other summations one has to distinguish the two sublattices ; for instance one has to consider the sum N n=1 f † 2n f † 2n+1 and then one uses the identity

N n=1 e ı(q+q ′ )2n = N 1 2(q+q ′ )≡0(2π) = N 1 q+q ′ ≡0(π) . (4.19)
According to definition (4.13), the set B(2N ) does not contain the value 0 and the solution of q + q ′ = 0 corresponds to two distinct values q and -q.

In order to simplify the following discussion, we assume from now on that N is even. Then B(2N ) does not contain π/2 and all values q and πq, are also distinct. After a symmetrization over the values q and πq the matrix M + (λ o , λ e ) appears as a sum of contributions each of which involves only the operators associated with the wave numbers q, πq, -q and -(qπ). Let us introduce the first quadrant in the set B defined as

QB(2N ) = {q = (2k + 1) π 2N , k = 0, 1, • • • (N/2) -1} = { π 2N , 3π 2N • • • π 2 - π 2N }. (4.20)
Then the expression (4.12) for M + (λ o , λ e ) can be rewritten as

2 ν 1 + ν 2 M + (λ o , λ e ) = -N 1 + q∈QB V † q T A q (λ o , λ e ) V q , (4.21)
where V q is the column vector

V q =     η q η q-π η † -q η † π-q     , (4.22)
V † q T denotes the transposed row vector corresponding to the column vector V † q built with the adjoints of the components of V q , and

A q (λ o , λ e ) =    
-γ + cos q ıa q sin q ıc ′ q sin q c q cos q -ıa q sin q -γcos q -c q cos q -ıc ′ q sin q -ıb ′ q sin q -b q cos q γcos q -ıa q sin q b q cos q ıb ′ q sin q ıa q sin q γ + cos q

    , (4.23)
with

a q = ν o -ν e (4.24) b q = ν o b o -ν e b e b ′ q = ν o b o + ν e b e c q = ν o c o -ν e c e c ′ q = ν o c o + ν e c e ,
where the b a 's and the c a 's are defined in (4.6) and (4.7).

Diagonalization of

A q (λ o , λ e )
The characteristic polynomial of A q , det (A q -α1 q ), proves to be a second order polynomial in α 2 , with a constant which is a squared quantity, They read

det (A q -α1 q ) = α 4 -2Dα 2 + F 2 . ( 4
D = 1 + (ν o -ν e ) 2 + ν o ν e 4γ o γ e cos 2 q + (1 -2 cos 2 q)θ(λ) (4.27)
and

F = ν o ν e 4(1 -γ o γ e cos 2 q) + θ(λ) , (4.28)
where the function θ(λ) vanishes when λ is set to zero :

θ(λ) = 2 (1 -γ o γ e )(cosh λ -1) + (γ o -γ e ) sinh λ . (4.29)
The coefficient F can be rewritten as

F = ν o ν e 2 + 2γ o γ e (1 -2 cos 2 q) + (1 -γ o γ e ) cosh λ + (γ o -γ e ) sinh λ , (4.30)
and the property (1γ o γ e ) > |γ oγ e | for γ o < 1 and γ e < 1 ensures that F > 0. The squared roots of the characteristic polynomial are

α 2 = D ± D 2 -F 2 = D + F 2 ± D -F 2 2 . (4.31)
where √ • • • denotes a possibly complex square root. Let us introduce the notations R ± (q, λ)

= 1 2 (D ± F ), namely R + (q, λ) = 1 + ν o ν e θ(λ) sin 2 q (4.32) R -(q, λ) = (ν o -ν e ) 2 + ν o ν e 4γ o γ e -θ(λ) cos 2 q.
(4.33)

We notice that, since F is positive, R + > R -.

Note that 1 + ν o ν e θ(λ) > 0 because of the definition (4.29) of θ(λ) and the identities 1 -2ν o ν e (1γ o γ e ) > 0 and (1γ o γ e ) > |γ oγ e | for γ o < 1 and γ e < 1. According to (4.32), R + can be rewritten as R + = 1sin 2 q + 1 + ν o ν e θ(λ) sin 2 q and we conclude that R + > 0.

On the other hand, by virtue of definitions (2.5) and (4.29), Eventually the four eigenvalues of A q (λ o , λ e ) are

4γ o γ e -θ(λ) = 2 cosh ((β e + β o )∆E/2) -cosh λ -(β e -β
α 1 = R + + R - α 2 = R + -R - α 3 = -α 2 α 4 = -α 1 , (4.35)
In these expressions R + denotes the usual positive square root of the positive number R + , whereas R -is either real positive or purely imaginary depending on the sign of R -, i.e. on the values of λ and q. As noticed previously R + > R -, and in the case where R -is real, all α k 's, with k = 1, . . . , 4, are real and α 1 > α 2 > 0 > α 3 > α 4 .

Largest eigenvalue of the modified matrix M

+ (λ o , λ e )
For the sake of conciseness we omit all dependences upon λ o and λ e in the present section. We denote by D q the diagonal matrix built with the eigenvalues α 1 (q), . . . , α 4 (q) of the matrix A q (λ o , λ e ).

The matrix A q reads A q = P q D q P -1 q ,

where the k th column of P q is made with the components of a (column) right eigenvector of A q associated with the eigenvalue α k (q), and P -1 q is the inverse matrix of P q . Let ξ k (q) denote the k th component of the column vector P -1 q V q , while ξ ⋆ k (q) denotes the k th component of the row vector V † q T P q , where V q and V † q T are defined in (4.22). With these definitions, the relation (4.36) implies that

V † q T A q V q = 4 k=1 α k (q) ξ ⋆ k (q) ξ k (q). (4.37)
The operators ξ k and ξ ⋆ k obey the anticommutation rules {ξ k , ξ k ′ } = 0, {ξ ⋆ k , ξ ⋆ k ′ } = 0 and {ξ k , ξ ⋆ k ′ } = δ k,k ′ . However, since A q is not hermitian (for the usual scalar product), P q is not unitary and the operator ξ ⋆ k is not the adjoint of ξ k . Nevertheless the anticommutation rules are enough to ensure that the spectrum of the operator ξ ⋆ k ξ k is the set {0, 1}. Then, according to the expressions (4.35) of the α k 's, the value of the righthand side of (4.37) with the largest real part is equal to the sum of two eigenvalues and proves to be real positive,

α 1 + α 2 = 2 R + (q) > 0. ( 4
ν o + ν e 2   -N + 2 q∈QB R + (q)   , (4.39)
where R + (q) is given by (4.32). We notice that if λ o = λ e = 0, the modified Markov matrix M(λ o , λ e ) becomes the usual Markov matrix for the evolution of P (n; t); since θ(0) = 0, we retrieve that the largest eigenvalue of the Markov matrix is 0. Moreover the eigenvalue closest to 0 is obtained by setting ξ ⋆ 3 (q)ξ 3 (q) equal to 1 for q = 0, and we retrieve the value (2.12) for the inverse relaxation time of the magnetizations on the two sublattices.

5 Cumulants of heat amounts per unit of time in the long time limit

Scaled generating function for joint cumulants

According to the remark at the end of section 4.1, the scaled generating function for the joint cumulants of Q o and Q e coincides with the largest eigenvalue of M + (λ o , λ e ). By virtue of (4.39) it reads

g 2N (λ o , λ e ) = ν o + ν e 2   -N + 2 q∈QB 1 + ν o ν e θ(λ) sin 2 q   , (5.1)
where QB(2N ) is defined in (4.20) and θ(λ) is given in (4.29). The joint cumulants per unit of time in the long time limit are determined from the relation

lim t→∞ 1 t Q p e Q p ′ o c = ∂ p+p ′ g 2N (λ e , λ o ; t) ∂λ p e ∂λ p ′ o λe=λo=0 , (5.2)
where the index c refers to the truncation of the mean value Q p e Q p ′ o involved in the definition of the cumulant. The fact that g 2N depends only on the difference λ eλ o entails the properties

lim t→∞ 1 t Q p e Q p ′ o c = (-1) p ′ lim t→∞ 1 t Q p+p ′ e c (5.3)
and, in particular,

lim t→∞ 1 t Q p o c = (-1) p lim t→∞ 1 t Q p e c . (5.4)
These properties are linked to the fact that the interface energy can take only a finite number of values whereas the cumulants have no upper bounds in the infinite time limit.

For the sake of completeness we point out that, according to (4.34), θ(λ) depends on λ through the function cosh λ -(β eβ o )∆E/2 ; therefore the scaled generating function satisfies the symmetry g 2N (λ o , λ e ) = g 2N (β oλ o , β eλ e ), which is in fact a consequence of the local detailed balance (2.3). Since g 2N (λ o , λ e ) depends only on the difference λ eλ o , this entails a symmetry for the scaled generating function for the cumulants of Q e , g e 2N (λ e ) = g 2N (0, λ e ), namely the symmetry g e 2N (λ e ) = g e 2N (β eβ oλ e ). Then the corresponding large deviation function for the time-integrated current J e = Q e /t, which can be obtained as the Legendre-Fenchel transform of g e 2N (λ e ), obeys the fluctuation relation f (J e )f (-J e ) = (β oβ e )J e .

Cumulants for heat amount Q e

We give the explicit expressions for the first four cumulants of Q e per unit of time. They are determined as

∂ n g 2N (0, λ e )/∂λ n e | λe=0 = ∆E n × ∂ n g 2N (0, λ e )/∂λ n | λ=0 , where λ = (λ e -λ o ) ∆E
and ∆E = 4K is the energy gap in the chain energy. The function θ(λ) defined in (4.29) can be rewritten as

ν o ν e θ(λ) = 2A[cosh λ -1] + 2B sinh λ (5.5)
where the parameters A and B are those introduced in Ref. [START_REF] Cornu | Thermal contact through a diathermal wall : a solvable model[END_REF] for a model with only two spins, namely

A = ν o ν e (1 -γ o γ e ) (5.6) B = ν o ν e (γ o -γ e ). (5.7)
Then, according to (5.1), the first four cumulants of Q e read

lim t→∞ Q e (ν e + ν o )t = N 2 BS 2 × ∆E (5.8) lim t→∞ Q 2 e c (ν e + ν o )t = N 2 AS 2 -B 2 S 4 × ∆E 2 lim t→∞ Q 3 e c (ν e + ν o )t = N 2 B S 2 -3AS 4 + 3B 2 S 6 × ∆E 3 lim t→∞ Q 4 e c (ν e + ν o )t = N 2 AS 2 -(3A 2 + 4B 2 )S 4 + 18AB 2 S 6 -15B 4 S 8 × ∆E 4 ,
where

S 2n (N ) = 2 N (N/2)-1 k=0 sin 2n (2k + 1)π 2N .
(5.9)

The structure of the cumulants in terms of the coefficients A and B is similar to the structure found for the two-spin model of Ref. [START_REF] Cornu | Thermal contact through a diathermal wall : a solvable model[END_REF] as well as for the ring of Ref. [START_REF] Cornu | Glaubers Ising chain between two thermostats[END_REF]; indeed, in the three cases the scaled cumulant generating function depends on λ e only through the same function θ(λ e ∆E).

The coefficients S 2n (N ) (with N even) can be calculated explicitly. In the special case N = 2, a direct calculation leads to S 2n (2) = (1/2) n . For any N ≥ 2, by extending the summation up to N , rewriting sin u = [e ıue -ıu ]/(2ı), using the binomial formula and an identity similar to (4.17), we get that

if n < N S 2n (N ) = 1 2 2n [2n]! (n!) 2 = W 2n , (5.10)
where W 2n denotes the normalized Wallis integral, W 2n = (2/π) π/2 0 (sin q) 2n dq. The first four values of the latter integrals are For n larger than N , a finite-size correction arises. For instance,

W 2 = 1 2 , W 4 = 3
if N ≤ n < 2N S 2n (N ) = W 2n 1 -2 [n!] 2 (n -N )!(n + N )! . ( 5.12)
From the study of the S 2n (N ) we get that all cumulants of order n smaller than N , the number of sites connected to a given thermostat, are strictly proportional to the size L = 2N of the chain. Finite-size corrections appear only in cumulants of order n ≥ N .

In particular, S 2 (N ) = 1/2 for all N , by virtue of (5.10)-(5.11) valid for all N ≥ 2 with N even. Therefore the first cumulant lim t→+∞ Q e /t given in (5.8) does coincide with the expression (2.21) of the mean global instantaneous current J e st in the stationary state. We have already pointed out that the latter mean global current contains no finite-size corrections. If N > 4 the first four cumulants are given by (5.8) where the S 2n (N ) are to be replaced by the corresponding W 2n given in (5.11).

Eventually the dependences upon the thermodynamic parameters, γ o and γ e , and the kinetic parameters, ν o and ν e , arise only through the coefficients A and B. This is in contrast with what happens for another Ising chain model, where both thermostats act on every spin [START_REF] Cornu | Glaubers Ising chain between two thermostats[END_REF]: for this model the dependence upon the combination γ of the thermodynamic and kinetic parameters defined in (2.18) also arises in the coefficients Σ n (N, γ) which replace the coefficients S n (N ) in the expressions (5.8) for the cumulants.

Various physical regimes

According to the last remark of the previous section, the discussion of the various physical regimes is the same as that performed in the case of the two-spin model of Ref. [START_REF] Cornu | Thermal contact through a diathermal wall : a solvable model[END_REF].

At equilibrium γ o = γ e and, according to (4.29), θ(λ) = 2(1γ 2 e )(cosh λ -1). Therefore only cumulants of even order do not vanish. According to (5.8) the first two cumulants of even order read The probability distribution of Q e is not a Gaussian, since all cumulants of even order have nonvanishing values.

When a thermostat has a kinetic parameter far larger than the other one, the scaled generating function becomes proportional to θ(λ),

Conclusion

In the present paper we have investigated the heat currents in an Ising spin ring where alternating spins are coupled to two macroscopic bodies at different temperatures and with different kinetic parameters. The stationary mean values of the global two-spin correlations at any distance have been calculated. The dependence upon the kinetic parameters arises only through the linear combination γ defined in (2.18)5 . The finite-size corrections in the global two-spin correlations disappear in the mean instantaneous global heat current coming out of one thermostat. The scaled generating function of the joint cumulants per unit of time for the heat amounts exchanged with the two thermostats over a long time have been calculated exactly. At leading

. 6 )

 6 They satisfy the relation ν o + ν e = 1 and, apart from the arbitrary time scale, the model has only three independent parameters, γ o , γ e and ν o .

. 5 )

 5 The coefficient b j (c j ) is equal either to b o or b e (c o or c e ) according to the parity of j; with the notation a = o or e b a = (1 + γ a ) e -λa∆E (4.6) and c a = (1γ a ) e λa∆E . (4.7) According to the evolution equation (4.4) the Laplace transform P (n, λ o , λ e ; t) is equal to n| exp[ M(λ o , λ e )t]| P t=0 (λ o , λ e ) . Thus the scaled generating function g 2N (λ o , λ e ; t) given by (4.2) is equal to the largest eigenvalue of the matrix M(λ o , λ e ) which rules the evolution of | P t (λ o , λ e ) .

  .25) Moreover both coefficients D and F 2 depend on the parameters λ o and λ e only through the difference λ = (λ eλ o ) ∆E. (4.26)

ν o ν e 1 -γ 2 e×

 2 N ∆E 4 .

g( 1 -ν s ( 1 -

 11 2N (λ) = 1 8 N ν s θ(λ),(5.14)where ν s is the kinetic parameter of the slower thermostat. As a consequence g 2N (λ) coincides with the scaled generating function of a continuous-time random walk, because θ(λ) can be rewritten asθ(λ) = 2 p + e λ + p -e -λ -(p + + p -)(5.15)with the probabilitiesp + = (1 + γ o )(1γ e )/2 and p -= (1γ o )(1 + γ e )/2.As a consequence all cumulants of even (odd) order are equal to the same value when they are measured in unit of ∆E: for all p ≥ γ o γ e ) N.In the same kinetic regime, if one thermostat is at zero temperature, for instance γ o = 1, then the scaled generating function coincides with that of a continuous-time Poisson process, because θ(λ) = 2(1γ e ) e λ -1 .(5.17)As a consequence all cumulants of Q e in unit of ∆E are equal to the same value: for all p ≥ γ e ) N.(5.18)

  o γ eθ(λ) > 0 only if -β o ∆E < λ < β e ∆E, and we infer from (4.33) that R -can take both signs.

	cosh (β e ∆E/2) cosh (β o ∆E/2)	o )∆E/2	,	(4.34)
	so that 4γ

  .38) Eventually, according to (4.21), the largest eigenvalue of M + (λ o , λ e ) is

In Ref.[START_REF] Derrida | Non-equilibrium steady states: fluctuations and large deviations of the density and of the current[END_REF] the terminology "generalized" detailed balance is used.

We notice that the first requirement is not satisfied by the Ising chain model of Ref.[START_REF] Cornu | Glaubers Ising chain between two thermostats[END_REF], where two thermostats at different temperatures act on every spin : the latter model does not describe a situation of thermal contact.

The latter correspondance has been used for instance in Ref.[START_REF] Farago | Injected Power Fluctuations in 1D Dissipative Systems[END_REF] for the calculation of the scaled generating function of the energy injected in an Ising spin ring through the random flips of one spin while all other spins evolve according to a Glauber dynamics at zero temperature which dissipates energy along the ring.

In Ref.[START_REF] Mobilia | Exact dynamics of a reaction-diffusion model with spatially alternating rates[END_REF] the mapping has been used in the reverse sense in order to study the relaxation towards the stationary state for the reaction-diffusion system from results obtained for the Ising spin chain dynamics with two temperatures but a unique kinetic parameter.

When the kinetic parameters are set equal, our result are compatible with those of Ref.[START_REF] Schmittmann | Stationary correlations for a far-from-equilibrium spin chain[END_REF].

order in the ring size they prove to be proportional to the ring size, as it is the case for the model where two thermostats act on the same site [START_REF] Cornu | Glaubers Ising chain between two thermostats[END_REF]. Moreover, if the order of the cumulant is lower than the number of spins connected to one thermostat, the finite-size corrections again disappear exactly, and the cumulant is strictly proportional to the ring size.

We notice that the proportionality to the ring size at leading order in the size has already been observed for the cumulants of two other kinds of cumulative quantities when the system is homogeneous (only one temperature and one kinetic parameter) and seen as a Simple Exclusion Process with pair creation and annihilation [START_REF] Popkov | Large deviation functions in a system of diffusing particles with creation and annihilation[END_REF]: the two cumulative quantities are the difference between the numbers of domain wall jumps in the clockwise and anticlockwise directions respectively, and the number of pair annihilations. This is in contrast with the case of the purely diffusive Simple Exclusion Process on a ring, where the cumulants for the difference between the numbers of jumps in the two directions and the cumulants for the total number of jumps are proportional to powers of the ring size which increase with the order of the cumulants [START_REF] Appert-Rolland | Universal cumulants of the current in diffusive systems on a ring[END_REF].

An interesting open problem is the calculation of the heat cumulants in another solvable model for thermal contact : two joined semi-infinite Ising chains coupled to thermostats at two different temperatures [START_REF] Lavrentovich | Energy flux near the junction of two Ising chains at different temperatures[END_REF][START_REF] Lavrentovich | Steady-state properties of coupled hot and cold Ising chains[END_REF]. Then the mean global current which flows from one thermostat to the other through the junction between the two half-chains is obtained by summing the mean currents received by all spins in a semi-infinite Ising chain. The intrinsic inhomogeneity of these currents would have to be dealt with by specific methods.

A Two-spin correlations

By analogy with the homogeneous case [START_REF] Glauber | Time-Dependent Statistics of the Ising Model[END_REF] the evolution equation for the two-spin correlations reads d

Inserting the expression (2.4) for the transition rates we get

The latter equations imply that D ee 2 , D oo 2 , D oe 1 and D eo 1 involved in the expressions (2.14) and (2.16) for the mean global currents J e st and J o st are to be determined from a hierarchy of equations for the two-spin quantities

with an analogous definition for D eo 2p+1 , and

We notice that if the initial probability distribution for the spin configurations is translationally invariant, this property is conserved by the evolution under the transition rates and the stationary two-spin correlation s k s k+p st depends only on the difference p between the site indices; then it is equal to one of the D's defined in (A.3), (A.4) and (A.

while comparison of equations (A.9) and (A.10) leads to the relation valid for

By taking into account these relations in (A.7) and (A.9) we have to solve the coupled equations for the D ee 's and the D oe 's

where the second equation is to be supplemented by the extra boundary conditions for p = 0 and p = N -1 respectively. The latter conditions are derived from (A.9) and (A.6), and D ee 2 . The recursive equations (A.13) are linear and their generic solution, which depends on the two parameters D oe 1 and D ee 2 , can be looked for as linear combinations of two linearly independent solutions. Because of the invariance of these equations under the translation over two sites, one can look for independent solutions which are also eigenfunctions of the translation operator on each sublattice, namely solutions of the form f ee 2p+2 = ηf ee 2p and f oe 2p+3 = ηf oe 2p+1 . These solutions can be written as

By inserting the latter expressions into the recursive equations (A.13) one gets two coupled linear equations for a and b. The latter do not vanish if η is equal to one of the two values

The model is defined for 0 < γ o < 1 and 0 < γ e < 1, so that η + = η -. Then in the two solutions of (A.15) with η + and η

Then the generic solution of (A.13) reads

In fact the solution can be written only in terms of η -(with 0 < η < 1) by using the relation η + η -= 1. Then the boundary conditions (A.14) determine the values of a + and a -. After straightforward calculations and use of the relation (A.20)

The latter formulae are compatible with the expressions of the stationary two-spin correlations determined in Ref. [START_REF] Schmittmann | Stationary correlations for a far-from-equilibrium spin chain[END_REF] in the case where ν o = ν e .