Majed Chambah
email: majed.chambah@univ-reims.fr

Keywords:

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Introduction

The motion pictures represent an important cultural heritage. As films age they became prone to all kinds of defects such as dust, scratches, vinegar syndrome and color dye fading. Several film defects such as dye fading and vinegar syndrome are chemically irreversible processes [START_REF] Reilly | Storage Guide for Color Photographic Materials[END_REF][START_REF] Ebu | European Broadcasting Union. Preservation and reuse of motion picture film material for television: Guidance for broadcasters[END_REF] and often a degraded version is the only available record of a film, hence the importance of digital film restoration. In fact, beside the fact that it can tackle artifacts that are out of reach of traditional photochemical restoration techniques, digital film restoration presents the advantage of not affecting the original material, since it works on a digital copy. Several digital restoration techniques (Chambah 2002[START_REF] Chambah | Perceptual Approach for Unsupervised Digital Color Restoration of Cinematographic Archives[END_REF][START_REF] Buisson | Deterioration detection for digital film restoration[END_REF][START_REF] Joyeux | Reconstruction of degraded image sequences. Application to film restoration[END_REF][START_REF] Besserer | Detection and Tracking Scheme for Line Scratch Removal in an Image Sequence[END_REF][START_REF] Kornprobst | Image Sequence Restoration: A PDE based coupled method for image restoration and motion segmentation[END_REF][START_REF] Decencière | Restauration automatique de films de films anciens[END_REF]) have emerged 2 during the last two decades and became more and more automated but restoration evaluation remains still a rarely tackled issue [START_REF] Chambah | Automatic Color Restoration of Faded Pictures and Motion Pictures[END_REF][START_REF] Decencière | Restoration Quality Assessment[END_REF]. Some caution must be exercised when doing quality assessment in the field of restoration. The material to be restored is not coming from a stable technological process that would produce the same basic quality before any degradation. The cinematographic technological methods have not changed notably since the beginning of cinema exploitation, keeping the same basic camera and projector movements, the same format size, the ubiquitous 35mm film, and similar laboratory procedures, but the quality has always evolved. Even if we disregard the relative statistic dispersion of quality due to different working standards and accidental variations, there has been a constant change of quality. Moreover, the historical evolution was not a continued regular improvement towards higher standards. For example, in the early days of color film, economical reasons were a motive to abandon the high quality Technicolor process using three or two strips of film. As a result there was a loss in quality not only in resolution, because of the combination of the three color layers on one strip of film, but also a loss in the gamut of color covered by the system, because of the complicated chemical process used. It is therefore probably not a good choice to use absolute reference metrics when assessing the quality of a specific film. Knowledge of the historical evolutions of the technology suggests that it would be necessary to use historical reference metrics in this case, a puzzling problem. The use of reference free metrics would then be a better choice in the context of automation. Thus, reliable automatic methods for visual quality assessment are needed in the field of digital film restoration. Ideally, a quality assessment system would perceive and measure image or video impairments just like a human being.

Film degradation

The goal here is not to present an exhaustive list of defects. It would be too long, boring and rather confusing. Moreover there is more than one way to present such a list. The point of view of the librarian does not coincide with the point of view of the image processing specialist. In order to avoid a lengthy and possibly controversial enumeration, we will use a structured approach that we claim is better suited for the problem at hand.

Defects affecting the photochemical material

A film is built of a photochemical emulsion deposited on a base. The base must hold firmly the emulsion, stay chemically neutral for this emulsion, stay flexible, avoid deformations and resist to all the environmental and mechanical constraints that the technology imposes on the film: Tight loops during shooting and projection, strong tension during rewinding or laboratory work, immersion in various chemical baths for processing, sudden heating during projection. After all this treatment it must stay in good condition for long period of storage.

Mechanical degradations

The first degradations that the film must endure are caused by improper handling. Even a repeated normal handling of a film is causing a few problems. The main phenomenon is abrasion caused by the dust or by a mechanical protruding piece of fabric in camera, projector or any mechanical device used during the long life of a film.

Dust, dirt and thin scratches

The dust is causing thin scratches on film and small particles of dust may penetrate the film base or the emulsion. It is a well known fact for film professional that the beginning and end of every spool is more degraded than the middle. This is because of the manual manipulation of the spools during editing, control or projection, causing both tails of the film spool to float for some moment in rooms and to rub all possible surfaces. Elongated scratches Protruding pieces of metal or small defects on metal surfaces, mainly in the camera during shooting, or later in the various equipments is causing the elongated vertical scratches which run along many frames. Jitter (Image vibrations) Repeated loading, unloading, winding and rewinding of the film strips is also damaging the film holes that are supposed to guaranty a stable repeated positioning of the images during projection. When the holes are deteriorated the strong mechanical tensions and alternating movements of the projection are causing an irregular positioning of each frame. But it is also true that early film strip and old camera movement were not as stable as today and were also causes of the same kind of jitter.

Missing parts and missing frames

Of course a severe mistreatment of the film strip or repeated careless manipulations may cause tears and breaks of the strip.

Chemical degradations of the base

Historically three successive materials have been used for the film base: Cellulose nitrate, cellulose acetate and polyester. Unfortunately each of these materials has some drawback. The first base used was highly flammable and could even detonate. In fact cellulose nitrate enters in the composition of the dynamite. The main problem with the cellulose nitrate is a reaction with the metallic boxes used to store the film spools. The parts in close contact with the film become brittle and disappear in dust. Fortunately it does not harm too much the emulsion. In almost all the developed countries nitrate films have been copied to safer film system. Cellulose acetate base films, also called "safety" films, were developed in order to get rid of the flammability problems of the first films. Unfortunately the new base proved not to be totally safe and it is a general sad rule that each new product developed to cure former problems is exhibiting its own set of problems.

Vinegar syndrome

The "vinegar syndrome" is a degradation that affects film material using a cellulose acetate base. It is the hydrolysis of acetate groups which results in the formation of acetic acid (vinegar). This chemical degradation allows also easier access to moisture. The acetic acid increases the rate of hydrolysis and so the hydrolytic degradation assumes an auto-catalytic nature. At the end the base is dissolved and the emulsion remains alone. There is not much to be done at this stage. The main problem at earlier stages is a deformation of the support which causes variable local blurring effect. This problem has been addressed by a digital restoration method explained in (HeIt 2001). The current new base in use is made of polyester. It is much stronger and resistant to mechanical tension and tearing. There are not yet confirmed reports of specific degradations.

Chemical degradations of the emulsion

The emulsion is supposed to be stable once the chemical processing has been completed. For black and white films there is not much risk of degradation if the laboratory work has been done with sufficient care. A good number of very old black and white films have been kept in good conditions for more than a hundred years now.

Contrast saturation

A common degradation seen in old black and white films is the strong saturation of the black and of the white areas with a severe loss of middle tones. Color dye fading But the complex chemical composite of the emulsion in color film is more sensible to the influence of radiation temperature and humidity. Color fading is caused by chemical changes in the image dyes of color films. Many older films have taken on a distinct color cast, caused by the rapid fading of one or two image dyes. Color negative film, color slide film, color print material, interpositives and color motion-picture release print film are all affected in the same way. The fading of one or two chromatic layers of the film results in a drab image with poor saturation and an overall color cast.

Film reproduction degradations

The reproduction mechanism of the cinematographic film is done by optical duplication. Even conducted with special care this operation is provoking degradation in resolution.

Blurring effect

The film projected in cinema theatres is a third generation copy of the original negative used for shooting. From this negative a first interpositive is printed, and then an internegative is obtained which is the base for numerous copies called release prints. A test exposed in the SMPTE Journal of February March 2004, is measuring the relative loss of quality between the original negative and the release prints. A rough measure of degradation may be derived from the MTF curves by stating that the resolution is divided by two.

Halo

During the printing operations a number of other degradations may occur. An improper adjustment of the lighting is causing halos or uneven distribution of the intensity in each frame. Fog An incomplete isolation from interfering lights may cause a fog effect. In general this effect is pretty stable and must not be confused with the moving fog effect caused by an incomplete mixing of the chemicals during processing. Flicker A number of mechanical misalignments during the optical printing may have also caused the phenomenon called flicker. The overall image intensity is varying from frame to frame. It is really difficult to distinguish this from the flicker present in old shooting and caused by the irregular speed in the camera movement.

Issues in restoration evaluation

The code and the image In the traditional photochemical world, in order to evaluate the state of a film for restoration or to assess its quality of conservation, it is only necessary to have a look at it. A quick examination of the base gives an indication of its aging conditions. A projection is the only way to judge the quality. The film itself is the recording support (the base), the recording method and medium (the emulsion), the storage medium, the viewing support and the reproduction support. The digital film is dissociated from the recording, viewing and storing methods and supports. The digital code is not the viewed image; it is only the algorithmic potentiality of an image, when arranged and transmitted for display.

The examination of the code itself does not indicate if there are defects, because the code itself is built to stay unaffected through multiple copies by virtue of its algorithmic nature. It is only because the digitally coded image is the representation (translation) of a bi-dimensional arrangement exhibiting such and such regularities that it is possible to speak of defects and to evaluate a quality. These regularities, defined a priori, are those of a photochemical image recorded by a specific camera, reproduced by some laboratory process, kept on a certain base and digitized on a specific modern system. All these equipments have their characteristics and the processes took place at certain dates and all this may be known or not. We have seen all the transformations and degradations which are occurring to the film up to, and including, the digitization. We think that the question of how to evaluate the quality of a digitized film may be addressed by considering the questions of control strategy, microstructures and macrostructures.

Control strategy

The first consequence of the separation of the code from the support is the necessity for a control strategy quite different from the traditional one. It seems evident that the digitization process must be fully analyzed and all its characteristics documented in full details. We have seen earlier how the coding domains are different and may give rise to quite different quantization noises for example. The earlier techniques may not be known and may be only approximately dated, but this stage at least must be imperatively known. The basic characteristics are the sampling method and dimension, the encoding domain, the original quantization. This may include also knowledge of the characteristics of the film being digitized. The nature of the film is important because negative or positive film have really different contrast. The color process is also interesting; it gives an indication of the possible color gamut. The sampling of a piece of film without any image impressed on it gives indication of the density of the base, the absolute minimum density. Lastly a sampling of a moderately dense flat image gives an indication of the grain size. Moreover, if a quick look at the film base is enough to judge of the conservation quality of one spool, there is nothing comparable in the digital world. It seems therefore imperative to do a search through all the images of the film or at least through a subset offering all the conditions for completeness of control. This is the condition for being able to know precisely the full range of the individual characteristics, amount of noise, grain characteristics, maximum contrast range, largest color gamut, etc.

Degradations or artistic distortions ?

A complete and thorough control strategy cannot solve everything. Another difficulty is coming from the artistic nature of the cinematographic work. Contrary to an industrial controlled environment, the artistic nature of the film work is generating considerable variations in the characteristics of the recorded images. This is creating some puzzling problems for the quality assessment. An example will illustrate clearly this point. If one is doing quality control on numerous sample images and finds a certain part of a film to be affected by a specific color problem. This part exhibits the same grain and noise values than the rest of the film. The scene compositions are not differing from any other scenes. But the contrast level is slightly less important and the color saturation is low with a general blue dominant color.

6 Is this a defect? A professional look at the sequence shows that it is a group of "day for night effect" scenes in the film. But surely it could have been a bad copy or a bad transfer. How to solve this puzzle? One possible answer to this question is to look at the structure of a film. If the transitions between normal quality scenes and scenes having this specific distortion are happening at the same location than the film scene transitions, it is certainly an artistic choice.

Microstructures and macrostructures

We have seen that the structure of the film could guide the quality evaluation process. The Technicolor process example given in the introduction will serve to further illustrate the conclusion of the previous section. We were stating that the abandon of the high quality Technicolor process resulted in a loss of quality in resolution but also in the color gamut of the color films. We see here two different kinds of qualities which must be treated separately. It is certain that digital processing may give useful indication when considering the characteristics of noise grain size maximum gradient contours and measures of the same sort. Those are the measurable microstructures of the images. But when it comes to different qualities like the color gamut, the contrast, the luminance, the color dominant, we are considering qualities that are relating to perception and to artistic expression. We must exercise some caution while measuring these characteristics. Their evaluation requires some knowledge of the technical representation systems and of their possible artistic or semantic usage. Without having to investigate fully in the semantic aspects of film, it is probably enough to subordinate the interpretation of these measures to the visible construction of the film. The editing, the scene content to some extent and the camera movements are often macrostructure which may help prevent misinterpretations of the measures. It is interesting to note that these macrostructures are not too difficult to detect automatically.

Cinema image quality evaluation

In the sphere of cinema, the image quality is judged visually. In fact, experts and technicians judge and determine the quality of the film images during the calibration (post production) process. As a consequence, the quality of a restored movie is also estimated subjectively by experts. On the other hand, objective quality metrics do not necessarily correlate well with perceived quality (Wang, 2002). Plus, some measures assume that there exists a reference in the form of an "original" to compare to, which prevents their usage in digital restoration field, where often there is no reference to compare to. That is why subjective evaluation is the most used and most efficient approach up to now. But subjective assessment is expensive, time consuming and does not respond, hence, to the economic requirements. Thus, reliable automatic methods for visual quality assessment are needed in the field of digital film restoration. Ideally, a quality assessment system would perceive and measure image or video impairments just like a human being. Two approaches can be taken: • The "psychophysical approach" or "human visual system approach", which is based on models of the human visual system [START_REF] Winkler | Vision Models and Quality Metrics for Image Processing Applications[END_REF]. Their general structure is usually determined by the modeling of visual effects, such as color appearance, contrast sensitivity, and visual masking, to name a few. Due to their generality, these metrics can be used in a wide range of applications; the downside to this is the high complexity of the underlying vision models. Besides, the visual effects modeled are best understood at the threshold of visibility, whereas image distortions are often supra-threshold. • The "engineering approach" or "imaging system approach", where metrics make certain assumptions about the types of artifacts that are introduced by a specific compression technology or transmission link. Such metrics look for the strength of these distortions in the video and use their measurements to estimate the overall quality. Based on the latter approach, a few studies of restoration quality (mainly on black and white films) are emerging [START_REF] Decencière | Restoration Quality Assessment[END_REF], in order to characterize the impairments to detect (like dust, flickering and scratches). Only limited success has been achieved. This is due to factors like the absence of reference to compare to, the difficulty to characterize precisely the impairments affecting films, the high definition of images that makes defects very visible, the spatiotemporal dimension of the images to restore, and to the lack of correlation between the metrics and the perceived quality. In fact, a metric may indicate that a scratch is less prominent after its correction, but perceptually an ill corrected scratch offends more that the original scratch, since we have been used during decades to see scratched movies. This example illustrates the complexity of some perception mechanisms and the difficulty to set measures that correlate these mechanisms.

AI based reference free quality assessment

The aim of automatic image quality assessment is to replace human judgment of perceived image quality with a machine evaluation. Avoiding inputs from human subjects can lead to deterministic models, yet objective systems should keep human scores as references to ensure consistency with subjective results. The developed system aims at assessing the perceived quality automatically when an enhancement algorithm is applied on an image. It is based on reference free approach. Figure 1 gives a schematic representation of the overall system. The model operates on numerical features extracted only from the image at test, that are used feed the neural network to obtain the quality rating. The subsystem consists of two main parts: feature extraction and an artificial neural network as a combiner. The set of extracted features used to assess the image at test are different designed metrics : Clearness (number of pixels on the edges of the image, mean lightness value on the edges of the image), contrast (global contrast, local contrast), lightness (mean lightness of the image, entropy of the distribution of the lightness, standard deviation of the lightness) and color (scatter of the color dominant, proportion of the color dominant, spatial scatter of the color dominant). In order to train and then evaluate the performance of the proposed model, we need the set of Mean Opinion Scores (MOS) of subjective image quality tests obtained by a group of human observers. After the system was trained, the system evaluates the quality of the image based on the extracted features (global image quality MOS) and computes the correlation with the MOS of subjective assessment. Our experimental results show that the correlation ratio can be up to 0,95 which is very promising. The aim of this paper is not going into technical detail and focus only on the main idea. Further details can be found in [START_REF] Ouni | Évaluation de la qualité des images couleur application à la recherche & à l'amélioration des images[END_REF].

Figure 1 Architecture of the proposed no reference subsystem based on neural network.

Conclusion

In this paper we have addressed the issue of image quality assessment in the field of digital film restoration field. We detailed the defects that affect films and the issues of quality assessment in the field of film restoration. We also presented briefly an IA based image quality assessment system. Even though the developed reference free metrics have shown their efficiency in helping assess objectively the image quality, in helping automatically fine tune restoration algorithm parameters and speed up restoration quality estimation, human validation is still needed and still a lot of research need to be done.


Brice Kengni-Zanguim

Nabil Najjari

Bernard Doudin

Yves Henry

David Halley

Influence of electric polarisation on Coulomb blockade in a superparaelectric clusters assembly

We study electron transport in an assembly of epitaxial Cr2O3 nanoparticles embedded in a MgO tunnel barrier: a non-usual variation in the Coulomb blockade charging energy is observed as a function of the size of the clusters. In striking contrast to the expected increase of charging energy when decreasing the clusters size, an almost constant behaviour is observed. We argue here that the spontaneous super-paraelectric moment carried by the clusters core is the origin of this unusual behaviour since it drives the dielectric constant in this clusters assembly. This phenomenon could be exploited to fabricate devices with single valued Coulomb blockade energy despite a statistical dispersion in the clusters size.

Text

Scaling electronic devices down to nanometric dimensions often leads to electrostatic repulsive effects inside a single nanoparticle or epitaxial cluster, strongly modulating the electron conductance via the clusters as a function of the applied bias voltage. This Coulomb blockade (CB) effect was observed in many devices including conductive clusters -metallic, organic or semi-conductive -embedded in an insulating matrix [START_REF] Devoret | Single Charge Tunneling -Coulomb Blockade Phenomena in Nanostructures[END_REF][START_REF] Kastner | [END_REF] and is extensively exploited for developing single electron transistors (SET) devices. [START_REF] Devoret | Single Charge Tunneling -Coulomb Blockade Phenomena in Nanostructures[END_REF][START_REF] Meirav | Single-electron phenomena in semiconductors[END_REF]4] CB occurs as the energy E = e 2 /C needed to add an electron on a particle of capacitance C becomes significantly larger than the thermal and quantum fluctuation energies. With C ≈ 4π  rl for particles of length scale l, this energy scales inversely with the size of the particle and becomes dominant when l is reduced down to a few nm.

The dielectric environment (of constant r) of the particles can however also possibly change with the length scale of the system, and can be significantly modified though external parameters, as an applied electric field. This can be the case for ferroelectric systems for which there is a growing interest at the nanoscale, since they could be the key to new nanometric devices, for instance high-density non-volatile memories. [5] Preserving a ferroelectric state when decreasing the material dimensions down to a few nanometers is challenging [6,7] but has been recently reported [8] in several high quality ferro-electric oxides [9,10] . The issue of CB effects combined with a dielectric medium polarisation was theoretically discussed by of Udalov et al. [11,12] with one experimental report we are aware of. [13] Those works considered composite devices with nanometric conducting clusters embedded in a ferro-electric matrix, or above a ferro-electric gate. It turns out that the coupling between CB effects and the surrounding electric polarisation is far from being negligible and induces a large modification of the conductance Coulomb diamonds in the clusters.

We recently reported [14][15][16] a system, made of granular Cr2O3 nanoparticles embedded in a MgO insulating matrix, where, schematically, the in-plane interface of flat chromium oxide nanometric epitaxial islands with insulating MgO are in a conducting state whereas the core of the clusters is insulating and carries a spontaneous electric moment. [14,[17][18][19] It provides a unique system with a metallic fraction playing the role of a Coulomb island, next to a dielectric of high polarizability. We show here that the Coulomb charging energy E is nearly size invariant, in striking contrast from the expectations of the size dependence of the capacitance of an island. As the nanoparticles also influence the dielectric environment through their insulating fraction, we discuss here how it can impact the charging energy, explaining quantitatively how it makes it size independent. This phenomenon is highly appealing for developing new devices exploiting CB effects with a well-defined E values, providing a clear electrical spectroscopy signature that is robust versus the particles size statistics.

The tunnel junctions are grown by e-beam gun on an MgO [001] substrate by molecular beam epitaxy in ultra-high vacuum: the bottom Fe electrode is deposited at 100°C and annealed at 500°C for one hour yielding an atomically flat surface. A 3 nm MgO layer is then deposited at 100°C, followed by the Cr layer -nominal thickness tCr -and 3 nm MgO at 200°C. Chromium crystallizes in a centred cubic structure with a Cr[100](100) // MgO[110](100) epitaxial relationship, and follows a Volmer-Weber growth mode [20,21] leading to the appearance of flat islands embedded in the smooth MgO barrier. The stack (Fig. 1(a)) is then annealed at 500°C to achieve a complete oxidation of Cr into Cr2O3. [14] The top electrode is made of 5 nm Fe followed by 20 nm Co capped with Pd. Wedge samples with a varying tCr are obtained by progressively masking the sample during deposition of chromium (Fig. 1(b)), providing a straightforward method to vary continuously the average cluster size value on a single processed batch. A complete structural analysis was previously detailed [14] : Chromium was shown to form Cr2O3 clusters inside the MgO barrier, with a 2 nm height and 4-8 nm width. X-ray Absorption Spectroscopy measurements on such clusters indicated a complete oxidation of chromium, mostly into the corundum Cr2O3 phase (about 90%), whose c-axis was in the plane, and (less than 10%) into an oxygen-rich phase close to metallic CrO2 probably located at the edge of the clusters. The metallic part of the clusters is possibly due to the oxygen rich (VO2-like) oxide but the large strained-induced modification of Cr2O3 relative to the bulk -see for instance the appearance of a magnetization-could also make that Cr2O3 locally turns into a conducting phase. Tunnel junction devices in the microns range of lateral size were obtained through successive U.V. photolithography, Ar + ions milling, and SiOx and Ti/Au deposition steps as in Ref. [14]. Electron transport measurements were performed in a He4 static cryostat, in a four-wire configuration. The conductance G per junction surface unit A (Fig. 1c) increases linearly with the nominal deposited chromium thickness. This is consistent with our previous observations [14] : the density of Cr2O3 clusters -in the order of 10 4 /m 2 -and their thickness d -equal to 2 nm-do not change with the amount of deposited chromium, as was also pointed by Koda and coworkers [20,21] on similar systems. As a consequence, only the area S of individual clusters varies, proportionally to tCr, explaining the linear conductance variation observed in Fig. 1(c): the direct tunnelling of electrons between bottom and top electrodes being ruled out, the tunnelling takes probably place via the clusters interfaces whose surface is proportional to tCr . A sketch of an individual cluster is given in Fig. 1(d), supposing a metallic phase at the edge of the cluster. Notice that in an alternative hypothesis the metallic part could be located closer to the cluster core.

The I(V) curves measured at 6.2K -see Fig. 2 theory [START_REF] Devoret | Single Charge Tunneling -Coulomb Blockade Phenomena in Nanostructures[END_REF]14] evaluates the electronic occupation probability on a cluster and, taking into account the tunnelling probability through both MgO tunnel barriers and yields the conductance at a given voltage. A background charge of Q = 0.28 electron for tCr = 0.71 nm reflects the global environment charge in the vicinity of an individual cluster or an intrinsic electric polarisation.

The Cr2O3 clusters diameters follow a Gaussian distribution law centered at 7.1 nm (with a standard deviation of 1.2 nm), extracted from experimental observations. [14] We do not reproduce the background of the whole curve, but the agreement with the shape of peaks in experimental I"(V) curves derivatives confirms that we deal with a CB effect. [14] The positive voltage at which we observe the experimental first peak in

2 2 V I   curves
varies slowly with a slight, linear decrease as a function of tCr (Fig. 3). Relatively to the first peak, the voltage corresponding to the second peak in 2 2 V I   curves follows nearly the same linear variation as a function of tCr, which implies that the energy difference E between the two states (n or n+1 electrons on the cluster) is nearly independent of the clusters size. This is highly intriguing as this value should decrease with increasing the nominal chromium thickness, i.e. when increasing the capacitance associated to the cluster. Indeed, defining ras the relative dielectric constant of the environment, made essentially of insulating Cr2O3 , and S as the clusters individual area, the energy E should schematically vary as the inverse of C =   rS/d for our disc-shaped particles. [START_REF] Devoret | Single Charge Tunneling -Coulomb Blockade Phenomena in Nanostructures[END_REF]14] This should imply a noticeable decrease of E from tCr = 0.08 nm to tCr = 0.78 nm, which is here clearly not experimentally observed. This expectation is confirmed in Fig. 4(a): a simulation using the orthodox model, for clusters of varying diameters, and an average zero environment charge, shows a large decrease of E with tCr. The counter-intuitive experimental behaviour thus suggests that a more subtle mechanism is at play. We propose that the presence of the individual electric dipole P in the clusters [14,15,17] leads to a variation of the r value with clusters size that cancels out the dependence of the capacitance on the clusters surface area. Since the dielectric constant writes   r=   ), were  is the electric susceptibility, we estimate dthe contribution of the weak spontaneous electric polarisation to  in our particular granular systems. As was shown in our former works [14,15] each cluster carries a -weak-spontaneous electric dipole P -in the order of 5 .10 - 29 C.m-induced by the large epitaxial strain -which probably explains the large, voltagedepend, magnetic anisotropy reported in Ref [16]-. This electric polarization could only be indirectly evaluated [14,15] as direct classical observations -as capacitance measurements-are not sensitive enough for such low polarization values. Our previous findings [14] suggested that the individual electric dipole modulus carried by those clusters followed a complex behaviour as a function of the clusters size, because of two competing effects: increasing the clusters area induces a relaxation of epitaxial strain, and thus a lower electric polarisation, but on the other hand the clusters volume gets larger. Within the tCr range considered in the present work [0.08 nm to 0.78 nm], we therefore observe a relatively low change of the magnitude of ║P║, that we neglect for simplicity in the following discussion. Moreover, the low anisotropy energy enables the electric dipoles to rotate due to thermal fluctuations: the whole clusters assembly is in a super-paraelectric state. [22][23][24] The alignment of the electric dipoles along the applied electric field is thus described by a Langevin law as a function of an applied electric field. [14,15] The estimate of d relates to the average electric dipole along z of an individual cluster, Pz, when applying an electric field E along this direction (perpendicularly to the tunnel barrier), which favors the alignment of the individual permanent electric dipoles along this direction. This polarisation can be written:

E V P d z   0 
where V is the volume of an individual cluster. It relates to the magnitude of the spontaneous dipole ║P║ by

) . ( T k E P L P P B z  
where L is the Langevin function defined as L(x) ≡ 1/x-1/tanh(x). We make an approximation, valid for x < 1, by supposing 3 ) ( x x L  (in fact, for an applied voltage of 0.05 V, we can reach x=3: this means that we will here, for simplicity sake, slightly overestimate the slope of L(x) and the subsequent electric susceptibility).

Within this approximation, T k

E P P B z 3 . 2 
and thus:

0 2 0 3 . .          V T k P E V P B z d (2)
Taking our experimental findings d = 2 nm, and a characteristic ║P║ value of 5. 10 -29 C.m [14] , we obtain, at T= 6.2 K, a value of d close to 50, much larger than unity -and the corresponding r value of bulk Cr2O3 equal to 9 [25] -. The contribution of d therefore dominates the dielectric constant and:

0 2 3 .      V T k P B r (3)
Finally, using our estimate for the capacitance of cylindrical clusters, we

P d T k e P S d V T k e E B B        obtain: 2 2 2 2 2 3 3
This expression can be simplified if we define the equivalent electric charge due to the spontaneous polarisation on a cluster horizontal interface

Q by d Q P  
, which leads to:

2

) ( 3 Q e T k E B    (4)
A value of ║P║ = 5. 10 -29 C.m corresponds to Q = 0.16 e and (4) yields a E value close to 60 meV at 6K, in reasonable agreement with the experimental value of 90 meV. As can be inferred from (4), if Q is independent of tCr, E does not vary with clusters size. This model, which schematically reflects the main mechanism explaining the experimental low variation of E with tCr in Fig. 3, was checked by means of numerical simulations of E using the same orthodox model as described above (Fig. 4). We supposed, due to polarisation, a constant surface electric charge Q on the clusters. This simulation reproduces the trend experimentally observed as a function of tCr since it yields an almost constant E value as well as E1 and E2

values close to the experimental ones -even though the linear variation with tCr is not fully reproduced-.

As a conclusion, we showed that the presence of a weak ferroelectric polarisation in strained Cr2O3 clusters has a dramatic influence on the Coulomb blockade energy of electrons tunnelling via those clusters: the increase of the capacitance of the clusters when increasing their size is compensated by a decrease of the dielectric constant r due to the spontaneous electric polarisation of the clusters dielectric fraction. This leads to an almost constant energy difference between the first and second CB peaks as a function of the clusters area over a whole decade. This effect originates from the super-paraelectric property of the clusters assembly and the subsequent original dielectric constant behaviour that increases on decreasing the cluster size. This might also explain how measurements performed on average assemblies of clusters reveal well-defined Coulomb blockade: whatever the size involved in charge transfer, the CB energies involved vary little with size. We believe that this effect could be exploited to fabricate devices with a given CB energy, circumventing the unavoidable clusters size dispersion in large assemblies. Furthermore, a less naive model should take into account the electric -or magnetic-polarisation switch induced by a supplementary electron on a cluster. A model, similar to the ones performed by Udalov et al. [11,12,26] would more accurately describe the involved physical mechanism and could perhaps explain the hysteretic conductance that we already reported at low temperature in those systems. [14] For such purposes, the fabrication of 3-terminal devices will clearly provide better experimental insight, and possibly open the door to robust and multi-functional devices taking advantage of CB effects in clusters carrying a spontaneous electric polarisation.

Obtaining such devices will require the increase of the intensity of CB peaks relative to the background conductance in I(V) curves: high quality and thicker MgO barriers could for instance favour the decrease of this background signal. Supposing a constant electric polarisation in the clusters, i.e. that the surface charge Q is centred on a constant value equal to 0.28 e, E is nearly constant.

  (a)-are nearly symmetric in voltage and do not exhibit striking features, contrary to the second derivative curves (Fig. 2(b)) which present resonance peaks almost symmetric in positive and negative voltages. The two well-defined peaks in the positive and negative voltage ranges are located at voltages in the order of ± 40 meV and ±130 meV. A simulation of the expected peaks in a CB effect in such clusters is given in the inset of Fig. 2(b): this simulation, based on an orthodox

Figures:Figure 1 .

 1 Figures:

Figure 2 . 2 K

 22 Figure 2. (a) I(V) curve and (b)

Figure 3 .

 3 Figure 3. Energy of the measured first and second CB peaks in

Figure 4 :

 4 Figure 4: Calculation of the Coulomb blockade peak positions and of their difference E, with a Gaussian distribution of the clusters diameter and a standard deviation of 1.2 nm. The clusters density was taken equal to 1.3 10 16 m -2 , the MgO and Cr2O3 dielectric constants equal to respectively 9.8 and 5.5. (a) Supposing a Q0 environment charge on the clusters following a Gaussian distribution centred on zero with Q0 =0.2 e, E noticeably decreases with tCr.(b)

Acknowledgements

We thank the STnano clean room facility for technical support and partial support from the French National Research Agency (ANR) through the Programme d'Investissement d'Avenir under contract ANR-11-LABX-0058_NIE and ANR-17-EURE-0024 (B.K.) within the Investissement d'Avenir program ANR-10-IDEX-0002-02. We also are grateful to G. Weick for stimulating discussions.