Kevin Nadaud
email: kevin.nadaud@univ-tours.fr

Caroline Borderon

Raphaël Renoud

Micka Bah

Stephane Ginestar

Hartmut W Gundel

Study of the long time relaxation of the weak ferroelectricity in PbZrO 3 antiferroelectric thin film using Positive Up Negative Down and First Order Reversal Curves measurements

Keywords: Anti-ferroelectric, Thin film, Residual ferroelectricity, Positive up negative 1 down pulse, First order reversal curves technique, Relaxation

The weak ferroelectric contribution to the polarization of antiferroelectric lead zirconate (PZO) thin films has been investigated using PUND (Positive Up Negative Down) pulse measurements and hysterons decomposition by First Order Reversal Curves (FORC) technique. The PUND allows decomposing the measured current, obtained from the polarization-electric field loop measurements, into a switching and a non-switching contribution. We show that the weak ferroelectric phase is enhanced when a large electric field has been previously applied to the material, in order to switch from the antiferroelectric to the ferroelectric phase. Using the PUND measurement at fields below the antiferroelectric to ferroelectric phase transition, the polarization loop corresponding only to the ferroelectric switching contribution has been determined revealing that this contribution to the overall polarization is small. FORC measurements, however, indicate that the ferroelectric phase is present at different fields. At low fields, a quite homogeneous distribution of hysterons exists and at large field, a high concentration of hysterons at a field near to the antiferroelectric to ferroelectric phase transition can be seen. Moreover, when changing the delay between pulses of the PUND and the FORC measurements, we show that this weak ferroelectricity contribution is metastable and decreases with time.

Introduction

Antiferroelectric materials present an important interest for energy storage applications [START_REF] Coulibaly | Enhancement of PbZrO 3 polarization using a ti seed layer for energy storage application[END_REF][START_REF] Sharifzadeh Mirshekarloo | Large strain and high energy storage density in orthorhombic perovskite (pb0.97la0.02)(zr1-x-ySnxTiy)o3 antiferroelectric thin films[END_REF][START_REF] Liu | Energy storage and polarization switching kinetics of (001)-oriented pb0.97la0.02(zr0.95ti0.05)o3 antiferroelectric thick films[END_REF] due to the double hysteresis in the polarization versus electric field loop.

When a high electric field is applied to an antiferroelectric material, a large increase of the polarization is obtained at the antiferroelectric (AFE) to ferroelectric (FE) phase transition, which permits a much higher stored energy compared to the case of linear dielectrics or relaxor ferroelectrics like (1x)Pb(Mg 1/3 Nb 2/3 )O 3x PbTiO 3 [START_REF] Sadl | Energy-storage-efficient 0.9 Pb(Mg 1/3 Nb 2/3 )O 3 -0.1 PbTiO 3 thick films integrated directly onto stainless steel[END_REF]. Compared to hard ferroelectrics (PbZrTiO 3 ) a better efficiency due to a considerably larger released energy can be obtained. In antiferroelectric materials, however, a residual ferroelectric contribution, called weak ferroelectricity, may be present which increases the overall material's losses. This has been largely mentioned in the literature and widely studied using various methods such as low field hysteresis measurement [START_REF] Dai | Weak ferroelectricity in antiferroelectric lead zirconate[END_REF][START_REF] Boldyreva | Microstructure and electrical properties of (120) O -oriented and of (001) O -oriented epitaxial antiferroelectric PbZrO 3 thin films on (100) SrTiO 3 substrates covered with different oxide bottom electrodes[END_REF][START_REF] Pintilie | Coexistence of ferroelectricity and antiferroelectricity in epitaxial PbZrO 3 films with different orientations[END_REF] or impedance spectroscopy [START_REF] Nadaud | Evidence of residual ferroelectric contribution in antiferroelectric lead-zirconate thin films by first-order reversal curves[END_REF][START_REF] Coulibaly | Effect of ferroelectric domain walls on the dielectric properties of PbZrO 3 thin films[END_REF][START_REF] Luo | Rayleighlike nonlinear dielectric response and its evolution during electrical fatigue in antiferroelectric (Pb, La)(Zr, Ti)O 3 thin film[END_REF]. Understanding the influence of the residual ferroelectric phase of antiferroelectrics is mandatory in order to increase the stored energy and the efficiency of the AFE energy storage devices.

Pulse switching of ferroelectric materials has been described by Merz [11] in the fifties of the last century, especially in order to show the stable or not stable partial switching behavior of those materials. Also called PUND (Positive Up Negative Down) method [START_REF] Liu | Energy storage and polarization switching kinetics of (001)-oriented pb0.97la0.02(zr0.95ti0.05)o3 antiferroelectric thick films[END_REF][START_REF] Wang | Paraelectric/antiferroelectric/ferroelectric phase transformation in as-deposited ZrO 2 thin films by the TiN capping engineering[END_REF][START_REF] Mart | Ferroelectric and pyroelectric properties of polycrystalline la-doped HfO2 thin films[END_REF][START_REF] Nadaud | Effect of thermal annealing on dielectric and ferroelectric properties of aerosol-deposited 0.65 Pb(Mg 1/3 Nb 2/3 )O 3 -0.35 PbTiO 3 thick films[END_REF], and when applied to antiferroelectrics, this technique allows detecting a possible residual ferroelectric phase in the AFE material.

An initial pulse (here of negative polarity, rectangular, trapezoidal or triangular) is applied in order to switch all existing ferroelectric domains in a given direction (Fig. 1a).

Then, two pulses of opposite polarity (P and U, here positive) are successively applied and finally, the application of the pulses N and D, again of opposite polarity (here negative) allows obtaining the full hysteresis cycle. The current signal of the respective pulses is recorded.

In the case of the pulses P and N (so-called switching pulses), the measured current is the sum of the contributions (i) leakage current, (ii) capacitive charging,(iii) possible ferroelectric phase switching as the previous pulse was of opposite polarity, and (iv) polarization changes for fields higher than the AFE-FE phase transition field. In the case of a purely capacitive behavior, a triangular applied voltage should result in a constant current: positive if the voltage is increasing and negative if the voltage is decreasing. A deviation of this behavior indicates a polarization change at this particular field, which may include FE switching and AFE-FE phase transition depending on the field amplitude. The example of the current signals obtained for fields below the AFE-FE transition and for a small residual ferroelectric phase in the AFE material is shown in Fig. 1b, which can be used for representing the obtained global I(E) cycle (Fig. 1c). For the pulses U and D (non switching pulses), no FE phase switching should occur as already switched by the previous pulses of the same polarity, and hence the current is the sum of only the three other contributions. By subtracting the non switching currents from the switching currents (i P -i U and i N -i D ), it is possible to extract the pure switching contribution for both polarities to compute the respective I(E) loop (Fig. 1d).

In order to obtain a sufficient signal to noise ratio, the switching contribution needs to be high as compared to the capacitive and the leakage contributions. Otherwise, some current subtraction prior to the amplification and acquisition can be used [START_REF] Martin | A new technique based on current measurement for nanoscale ferroelectricity assessment: Nano-positive up negative down[END_REF]. This method has been widely used for ferroelectric materials in order to remove excessive capacitive and leakage contributions and to study the switching times of the different contribution to the polarization [START_REF] Liu | Energy storage and polarization switching kinetics of (001)-oriented pb0.97la0.02(zr0.95ti0.05)o3 antiferroelectric thick films[END_REF][START_REF] Wang | Paraelectric/antiferroelectric/ferroelectric phase transformation in as-deposited ZrO 2 thin films by the TiN capping engineering[END_REF][START_REF] Mart | Ferroelectric and pyroelectric properties of polycrystalline la-doped HfO2 thin films[END_REF][START_REF] Nadaud | Effect of thermal annealing on dielectric and ferroelectric properties of aerosol-deposited 0.65 Pb(Mg 1/3 Nb 2/3 )O 3 -0.35 PbTiO 3 thick films[END_REF][START_REF] Martin | A new technique based on current measurement for nanoscale ferroelectricity assessment: Nano-positive up negative down[END_REF][START_REF] Si | Ultrafast measurements of polarization switching dynamics on ferroelectric and anti-ferroelectric hafnium zirconium oxide[END_REF] To the best of our knowledge, however, it has never been used for studying the weak ferroelectricity in antiferroelectrics.

Ferroelectric materials can also be investigated by First Order Reversal Curves (FORC) measurements which permit the decomposition of the P(E) loop into elementary hysteresis contributions called hysterons [START_REF] Mart | Ferroelectric and pyroelectric properties of polycrystalline la-doped HfO2 thin films[END_REF][START_REF] Mitoseriu | Hysteresis and tunability characteristics of Ba(Zr,Ti)O 3 ceramics described by first order reversal curves diagrams[END_REF][START_REF] Fujii | Thickness Dependence of Dielectric Nonlinearity of Lead Zirconate Titanate Films[END_REF][START_REF] Mitoseriu | First order reversal curves diagrams for describing ferroelectric switching characteristics[END_REF][START_REF] Ricinschi | First order reversal curves diagrams applied for the ferroelectric systems[END_REF]. This method also gives information on the degradation of the material and the presence of a possible residual ferroelectric phase in AFE materials which otherwise is difficult to see from the P(E) loops [START_REF] Nadaud | Evidence of residual ferroelectric contribution in antiferroelectric lead-zirconate thin films by first-order reversal curves[END_REF][START_REF] Fujii | Thickness Dependence of Dielectric Nonlinearity of Lead Zirconate Titanate Films[END_REF][START_REF] Zhu | Influence of Mn doping on domain wall motion in Pb(Zr 0.52 Ti 0.48 )O 3 films[END_REF][START_REF] Stoleriu | Analysis of switching properties of porous ferroelectric ceramics by means of first-order reversal curve diagrams[END_REF][START_REF] Hoffmann | Insights into antiferroelectrics from first-order reversal curves[END_REF].

E I i P i U i N i D (d) E I i P -i U i N -i D
The standard implementation of this method is to vary the electric field from one saturation state to another [START_REF] Fujii | Thickness Dependence of Dielectric Nonlinearity of Lead Zirconate Titanate Films[END_REF][START_REF] Mitoseriu | First order reversal curves diagrams for describing ferroelectric switching characteristics[END_REF][START_REF] Ricinschi | First order reversal curves diagrams applied for the ferroelectric systems[END_REF] but it is also possible to use a bipolar waveform with an increasing amplitude [START_REF] Nadaud | Evidence of residual ferroelectric contribution in antiferroelectric lead-zirconate thin films by first-order reversal curves[END_REF][START_REF] Cima | Characterization and model of ferroelectrics based on experimental preisach density[END_REF]. The advantage of the bipolar waveform approach is the existing delay between the different pulses allowing to investigate temporal relaxation of the switching processes which can reach several tens of second in ferroelectric [START_REF] Borderon | Dielectric long time relaxation of domains walls in pbzrtio 3 thin films[END_REF] and in antiferroelectric [START_REF] Chan | Temporal effect of low-temperature ferroelectric behaviors in pb0.97la0.02(zr0.60sn0.30ti0.10)o3 ceramics[END_REF][START_REF] Chan | Temporal effects in dielectric properties of some antiferroelectric complex perovskites[END_REF] materials.

In this paper, the complementary PUND and FORC measurements are used in order to obtain a better understanding of the weak ferroelectricity in antiferroelectric PbZrO 3 , visible at low fields and to study especially its temporal relaxation and the dependence on the applied electric field amplitude. In the first part, conventional P(E)-loops (at low and large fields) are used in order to show the weak ferroelectric phase enhancement in the material after the realization of a P(E) major loop. The ferroelectric switching contribution to the polarization is studied by PUND measurements, for different amplitudes of the electric field and with different delay time between the pulses, in the second and third part, respectively. Using these measurements, the P(E) loop of the switching contribution has been extracted in order to see the effect of amplitude and delay and to study the evolution of the antiferroelectric-ferroelectric transition fields. In the last part, FORC measurements are presented in order to gain better understanding of the phenomena that appear at low and high fields, and to correlate them to the results of the PUND measurements.

Experiments

Antiferroelectric lead zirconate (PbZrO 3 ) thin films are prepared by a sol-gel process using multi-step spin coating [START_REF] Coulibaly | Enhancement of PbZrO 3 polarization using a ti seed layer for energy storage application[END_REF][START_REF] Nadaud | Evidence of residual ferroelectric contribution in antiferroelectric lead-zirconate thin films by first-order reversal curves[END_REF][START_REF] Coulibaly | Effect of ferroelectric domain walls on the dielectric properties of PbZrO 3 thin films[END_REF] is then added in the solution, as well as ethylene glycol [HO -CH 2 -CH 2 -OH] used for reducing the appearance of micro-cracks [START_REF] Yu | Effects of poly(ethylene glycol) additive molecular weight on the microstructure and properties of sol-gel-derived lead zirconate titanate thin films[END_REF] in the films and improves the solution stability [START_REF] Yi | An acetic acid/water based sol-gel PZT process i: Modification of Zr and Ti alkoxides with acetic acid[END_REF]. The solution has been deposited on alumina substrates precoated with a titanium and platinum layers in order to form the bottom electrode. Twelve layers result in an overall film thickness of 800 nm. Square platinum electrodes of 0.1 mm width are deposited by RF magnetron sputtering in order to realize a Metal-Insulator-Metal (MIM) capacitor and to allow the dielectric characterizations of the fabricated thin films.

Structural characterizations reveal the studied thin film presents the (100) and (111) crystallographic orientations [START_REF] Coulibaly | Enhancement of PbZrO 3 polarization using a ti seed layer for energy storage application[END_REF] and presents a columnar structure [START_REF] Nadaud | Evidence of residual ferroelectric contribution in antiferroelectric lead-zirconate thin films by first-order reversal curves[END_REF][START_REF] Nadaud | Dielectric, piezoelectric and electrostrictive properties of antiferroelectric lead-zirconate thin films[END_REF].

The electric measurements have been performed with an AixACCT TF2000 ferroelectric analyzer. The conventional P(E) loops, presented in the first part, have been measured using a sine waveform and the applied voltage has been varied from 2 V to 40 V with a step of 2 V. The AFE-FE transition fields correspond to the maxima of the respective current waveforms. In the case of the PUND pulse measurements, a triangular waveform with a voltage sweep from 10 V to 70 V, corresponding to an electric field from 125 kV cm -1 to 875 kV cm -1 , and a step of 10 V have been applied, with the sequence de-scribed Fig. 1. The delay between the pulses ∆t has been varied from 1 µs to 1000 s and the duration of the pulses 2t r has been adjusted for a constant slew rate of 4 V ms -1 . For the construction of the P(E) loops of the pure switching contribution, the non-switching current is subtracted from the switching current signal and integration as a function of time to obtain the polarization.

The waveform used for the FORC measurements correspond is similar to this PUND method, except that only the switching pulses P and N are applied. The delay between the positive and negative pulses has been varied from 1 µs to 1000 s and the duration of the pulse slew rate has been kept constant. The FORC distribution extraction method is presented elsewhere [START_REF] Nadaud | Evidence of residual ferroelectric contribution in antiferroelectric lead-zirconate thin films by first-order reversal curves[END_REF][START_REF] Cima | Characterization and model of ferroelectrics based on experimental preisach density[END_REF]. The applied voltage has been swept from 2 V to 70 V with a step of 2 V corresponding to an electric field varying from 25 kV cm -1 to 875 kV cm -1 .

Reduction of the step voltage permits increasing of FORC distribution resolution and hence to better discern the residual ferroelectric phase.

Results and discussions

In the first part, conventional P(E) loops are presented in order to show the current peaks associated to the weak ferroelectricity in the studied PZO thin films. In the second and third parts, a precise characterization of this weak ferroelectricity has been done using the PUND method by varying respectively the electric field amplitude and the delay between pulses. Each time, the PUND measurements are presented separately for low and high fields. Finally, the FORC measurements are used in order to show the effect of the delay between pulses on the high field ferroelectric contribution.

Large field P(E) loops

The polarization has been measured as a function of voltage with an amplitude swept from 2 V to 70 V with a step of 2 V, corresponding to an electric field of 25 kV cm -1 and 875 kV cm -1 applied to the 800 nm thick sample.

The polarization versus electric field loops obtained from a virgin sample are shown in Fig. 2a. The typical double hysteresis of an antiferroelectric material is well visible.

The maximum value of polarization is 26 µC cm -2 which is close to the value reported for PZO thin films [START_REF] Ayyub | Ferroelectric behavior in thin films of antiferroelectric materials[END_REF]. The antiferroelectric to ferroelectric and ferroelectric to antiferroelectric transition fields are respectively E + AF = 628 kV cm -1 and E + FA = 244 kV cm -1 for the positive and E - AF = -580 kV cm -1 and E - FA = -229 kV cm -1 for the negative polarity.

In the case of this first measurement, the significant increase of the polarization is visible at field above ±575 kV cm -1 , indicating the onset of a rapid antiferroelectric to ferroelectric phase transition. The current loops corresponding to these first applied electric fields cycles are shown Fig. 2c When repeating the measurement for a second time, similar values of the maximum polarization and transition fields are obtained (Fig. 2b), however, the curves are more regularly spaced and the sharp increase of the polarization at driving fields between 575 kV cm -1 and 650 kV cm -1 is not visible (Fig. 2d). This difference is attributed to the enhanced weak ferroelectricity and will be detailed below, discussing the low field P(E)-loops.

Low fields P(E)-loops

The polarization and current loops of Fig. 2 for electric fields below 500 kV cm -1 are shown in Fig. 3a and3c for the first applied cycles (i.e. new sample) and in Fig. 3b and 3d for the second cycles (i.e. after the exposition of the sample to the maximum electric field of 875 kV cm -1 ). At low electric fields, and especially for the first applied cycles (Fig. 3c), the current describes an ellipse with the principal axis near to the xaxis, which is typical for a capacitive behavior when using a sinewave excitation and indicates a low leakage current. In the case of the cycles after a major loop with a driving field amplitude of 875 kV cm -1 had been applied, a more open P(E) loop, (Fig. 3b) and a current peak at ±100 kV cm -1 (3d) are observed, less visible but already existing for the first cycles. This current peak is attributed to the weak ferroelectric contribution [START_REF] Dai | Weak ferroelectricity in antiferroelectric lead zirconate[END_REF][START_REF] Boldyreva | Microstructure and electrical properties of (120) O -oriented and of (001) O -oriented epitaxial antiferroelectric PbZrO 3 thin films on (100) SrTiO 3 substrates covered with different oxide bottom electrodes[END_REF][START_REF] Pintilie | Coexistence of ferroelectricity and antiferroelectricity in epitaxial PbZrO 3 films with different orientations[END_REF] in the AFE thin film. This indicates that the ferroelectricity is present into the material, even before the application of the large electric field of 875 kV cm -1 and is largely enhanced by the application of a large electric field. The opening of the P(E)-loops in the case of the second series indicates higher losses, which can be attributed to the more important switching contribution.

For certain compounds of PZO-based antiferroelectric materials [START_REF] Zhang | Electric-field-induced antiferroelectric to ferroelectric phase transition in polycrystalline NaNbO3[END_REF][START_REF] Li | Electric field induced metastable ferroelectric phase and its behavior in (Pb,La)(Zr,Sn,Ti)O 3 antiferroelectric single crystal near morphotropic phase boundary[END_REF][START_REF] Yang | Metastable ferroelectric phase in lanthanum-doped lead zirconate titanate stannate antiferroelectric ceramics[END_REF][START_REF] Zhuo | Field induced phase transitions and energy harvesting performance of (Pb, La)(Zr, Sn, Ti)O 3 single crystal[END_REF][START_REF] Gao | Electric field induced phase transition and domain structure evolution in (Pb, La)(Zr, Sn, Ti)O 3 single crystal[END_REF], the whole material stays in the ferroelectric phase when the electric field applied during the initial AFE-FE phase transition is released, and the transition is hence irreversible. In our case, the phase transition is reversible and the major part of material goes back to the antiferroelectric phase, giving rise to the typical double hysteresis loop. One part of the material, however, stays in the ferroelectric phase thus enhancing the weak ferroelectricity.

One can note that the weak ferroelectricity is only visible around E = ±100 kV cm -1 if a larger electric field is applied during the cycle. This suggests that the weak ferroelectricity switching depends on the electric field amplitude previously applied and has a time relaxation.

In order to better study the influence of the weak ferroelectric contribution to the overall properties, the PUND method has been used which allow separating the switching and the non-switching contributions. To begin with (section 3.3), the measurements have been done for different electric field amplitudes and a given delay between pulses to determine the minimum electric field necessary to enhance the weak ferroelectricity.

After that, the delay between the pulses has been varied in order to discern stable and unstable ferroelectric switching (section 3.4).

PUND measurement with fixed delay between pulses

To begin with, the delay ∆t between the negative orientation pulse and the PUND measurements as well as the delay ∆t PU between the PUND pulses has been set to 250 µs.

The measured current is plotted as a function of the electric field for fields below the AFE-FE phase transition field (E max ≤ 500 kV cm -1 ) and for fields above (E max > 500 kV cm -1 ) in Figs 4a and 4c and Figs. 4b and4d, respectively.

Below the AFE-FE phase transition and in the case of the first series of applied fields, no notable difference between the switching and the non switching current is visible (Fig. 4a) suggesting that switching of the residual ferroelectric phase does not occur. At electric fields above the AFE-FE phase transition (Fig. 4b), the switching current exceeds the non switching current, well visible for the highest applied fields. In the case of the second series, even below the phase transition, a small difference between the switching and the non switching current appears, visible around ±100 kV cm -1 (Fig. 4c). The broad current peak indicates that there does not exist a sharp but a rather distributed coercive field.

For fields above the AFE-FE transition, switching of the ferroelectric residual phase is hardly visible due to the much more important amplitude of the phase transition current. There is no significant difference between the first and the second measurement series since, for field above the AFE-FE transition, the sample has already been exposed to a large field and changes the phase structure of the material (a part of the material stays in the ferroelectric phase). For both series, the maximum of the phase transition current appears at higher fields for the switching pulses, particularly well visible in the case of the positive pulses P and U (after initial orientation by application of a negative pulse).

The shift of the current maximum is 100 kV cm -1 (from 700 kV cm -1 to 600 kV cm -1 ) for the E + AF field and 35 kV cm -1 for the E - AF field while small for the ferroelectric to antiferroelectric transitions E + FA (below 10 kV cm -1 ). The antiferroelectric to ferroelectric transition hence seems to depend on the initial polarization orientation of the material, which correspond to a memory effect in antiferroelectric materials [START_REF] Li | Memory effect in antiferroelectrics: A systematic analysis on various electric hysteresis loops[END_REF]. The residual ferroelectric cells that switch below the antiferroelectric to ferroelectric transition [START_REF] Nadaud | Evidence of residual ferroelectric contribution in antiferroelectric lead-zirconate thin films by first-order reversal curves[END_REF] may also influence and thus lower the transition field when initially orientated favorably.

Influence of the delay between the pulses of the PUND measurement

In the previous section, the delay between pulses has been kept constant (250 µs). The delay between the initial orientation pulse and the PUND measurement pulses however influences the switching behavior. We have hence varied this delay from 1 µs to 1000 s using a homogeneous repartition in the logarithmic scale. All measurements presented in this section have been performed on an enhanced ferroelectric behavior sample that has already been exposed to an electric field of 875 kV cm -1 . As before, we study (i) ferroelectric switching at low fields and (ii) the AFE-FE transition at high fields.

Low field measurements

The full switching and non switching current waveforms for different delay are shown in Fig. 5a and5b, respectively for a field of 500 kV cm -1 . The non switching current almost does not depend on the delay ∆t PU between the pulses P and U, although it is possible to subtract the non-switching from the switching current in order to amplify the difference. The differential current waveforms for various delay times ∆t between the initial orientation pulse and the PUND measurement pulses and for different applied field are shown in Fig. 6. As the delay between the orientation pulse and the PUND measurement evolves, the differential current signal changes, showing that the orientation of the residual ferroelectric phase is not stable. Independent of the applied field, a maximum of polarization switching is obtained for delay times ∆t ≤ 100 µs. As the delay is getting longer, the differential switching current signal decreases, indicating a partial relaxation of the initial domain orientation. For delay times longer than 100 s, the differential current signal disappears. This indicates that the influence of the negative orientation pulse prior to the PUND measurements has become negligible and that the ferroelectric domains are completely relaxed. A similar behavior has been observed for all applied electric fields.

Integration with time of the differential current waveform signals allows obtaining the P(E) loop of the residual ferroelectric phase which is shown in Fig. 7a for a driving field amplitude of 500 kV cm -1 and for different delay times ∆t. The straight horizontal lines of the P(E) cycles when the applied field decreases to zero are typical for corrected PUND loops [START_REF] Wang | Paraelectric/antiferroelectric/ferroelectric phase transformation in as-deposited ZrO 2 thin films by the TiN capping engineering[END_REF][START_REF] Nadaud | Effect of thermal annealing on dielectric and ferroelectric properties of aerosol-deposited 0.65 Pb(Mg 1/3 Nb 2/3 )O 3 -0.35 PbTiO 3 thick films[END_REF].

The maximum value of the weak ferroelectric contribution P = 0.7 µC cm -2 for a field E max = 500 kV cm -1 is very small compared to the typical values of polarization for ferroelectric or antiferroelectric materials. In accordance to the broad current peaks of Fig. 6, a rather wide distribution of the coercive field (with a mean value around 150 kV cm -1 ) has been found, confirming the poly-crystalline structure of the thin film.

The difference between the maximum and the minimum polarization, ∆P m has been calculated for each delay time between pulses and is reported on Fig. 7b several seconds, indicating that the orientation of the weak ferroelectricity is not stable.

The existence of small ferroelectric clusters in antiferroelectric materials has been shown

previously [START_REF] Nadaud | Evidence of residual ferroelectric contribution in antiferroelectric lead-zirconate thin films by first-order reversal curves[END_REF][START_REF] Coulibaly | Effect of ferroelectric domain walls on the dielectric properties of PbZrO 3 thin films[END_REF], the dimension of those ferroelectric domains however seems to be too small in order to allow self-maintaining of oriented polarization. The mean coercive field value increases with increasing delay ∆t indicating that the ferroelectric cells which do not go back to the antiferroelectric phase need a higher electric field to switch. The polycrystalline structure of the film and the small grain size create different local clamping conditions for each ferroelectric cluster. Each ferroelectric cluster thus has a different polar orientation with respect to the electric field and surrounding environment implying a distribution of coercive field and delay to go back to the antiferroelectric phase.

High field measurements

The current as a function of an applied electric field above 625 kV cm -1 is shown in Fig. 8 for various delay times between the orientation pulse and the PUND measurement.

While successive application of the two positive P and U pulses should result in an identical phase transition current, it can be more particularly noticed that the phase transition field E + AF is approximately 100 kV cm -1 higher for the P-pulse (switching pulse), especially when the delay time ∆t is short (Fig. 8a). This difference decreases with increasing delay ∆t and disappears at longest delay times (Fig. 8d). Only a much smaller difference (around 30 kV cm -1 ) of the phase transition field E - AF between the respective N and D pulse is observed. The ferroelectric to antiferroelectric relaxation (field E - AF ) seems not to be affected, neither by the type of pulse (switching or non-switching) nor by the delay ∆t.

As shown in Fig. 7, switching of the residual ferroelectric phase at low fields is visible.

The electric field amplitude does not increase the low field switching current, indicating that the ferroelectric phase can be fully orientated already by the fields employed for the low field measurements. As already seen before, increasing of the delay time ∆t leads to less switching of the residual ferroelectric phase.

For a better visibility, the phase transition fields E AF and E FA have been extracted for each delay ∆t and are reported Fig. 9. For a delay times between pulses between up to several milliseconds, the transition fields are rather constant and the difference between switching and non-switching is well visible, with an antiferroelectric to ferroelectric transition field which is 90 kV cm -1 higher for the positive pulse and 35 kV cm -1 higher for the negative pulse. The difference of the ferroelectric to antiferroelectric transition fields is much smaller (about 10 kV cm -1 ) and does not depend on the polarity of the pulses.

An asymmetry of the switching and non-switching transition fields has already been reported for Pb 0.99 Nb 0.02 [(Zr 0.6 Sn 0.4 ) 0.95 Ti 0.05 ] 0.98 O 3 ceramics [START_REF] Li | Memory effect in antiferroelectrics: A systematic analysis on various electric hysteresis loops[END_REF]. In the present case, the observed asymmetry also may come from the asymmetry of the sample, i.e. the number of annealing steps for each layer. The last layer has been annealed only one time wheres the first one has been annealed twelve times and the following layers have a successively decreasing number of heat treatments. Annealing in air atmosphere may create oxygen vacancies in the structure which are known to be responsible of imprint and internal bias for ferroelectric materials [START_REF] Le Rhun | Polarization loop deformations of an oxygen deficient Pb(Zr 0.25 Ti 0.75 )O 3 ferroelectric thin film[END_REF][START_REF] Wu | Interface-oxygen-loss-controlled voltage offsets in epitaxial Pb(Zr 0.52 Ti 0.48 )O 3 thin-film capacitors with La 0.7 Sr 0.3 MnO 3 electrodes[END_REF].

For a delay between pulses above several milliseconds, the difference between the switching and the non-switching transition field progressively weakens due to an increase of the non-switching field, while the switching field rather remains constant. Similar to what has been observed for the switching of the weak ferroelectric phase, the transition between the antiferroelectric and the ferroelectric phase is influenced by the delay time ∆t, too. Switching of the residual ferroelectric phase seems to lower the AFE-FE transition field for the successively applied pulse of the same polarity as long as the delay ∆t is sufficiently short to maintain the orientation of the weak ferroelectricity. The AFE-FE transition is hence favored by the previous orientation of the residual ferroelectric phase. The influence of the delay ∆t on the ferroelectric to antiferroelectric relaxation electric field is similar but less pronounced since switching of the ferroelectric phase already has be established by the first pulse (switching-pulse). The internal fields hence are less modified, also limiting the change of the AFE-FE relaxation electric field.

FORC measurement

In order to further study the ferroelectric contribution into the PZO thin film, FORC measurements with increasing delay time between the positive and the negative pulse have been done. The FORC distributions for various delay between the pulses are shown Fig. 10, the P(E) loops which were used for the FORC extraction are given in the supplementary material. The two peaks visible at (E; E r ) = (-350 kV cm -1 ; -650 kV cm -1 ) and

(700 kV cm -1 ; 350 kV cm -1 ) correspond to the antiferroelectric-ferroelectric transitions.

When the delay between pulses changes, the peaks keep its position and the respective intensity is only slightly affected. The width of the peaks is due to the distribution of the transitions fields.

Further to the phase transition peaks, an additional ferroelectric contribution is visible in the bottom-right corner of the FORC distribution at the position (E; E r ) = (700 kV cm -1 ;

-650 kV cm -1 ), highlighted by the red arrow. This ferroelectric contribution can not be discerned by the high field PUND measurements because the switching current peaks occur for the same field than the antiferroelectric to ferroelectric transitions: for positive fields, the ferroelectric phase switches at E + = 700 kV cm -1 , same field than E + AF and for negative field, it switches at E -= -650 kV cm -1 same field than E - AF . Switching of this ferroelectric phase is correlated to the antiferroelectric phase transition [START_REF] Nadaud | Evidence of residual ferroelectric contribution in antiferroelectric lead-zirconate thin films by first-order reversal curves[END_REF] since the field values correspond to the sharp increases of the polarization which occurs for the first measurement series (See section 3.1 Large field P(E) loops) and which is related to the irreversible antiferroelectric to ferroelectric transition. As the delay time between the pulses increases, the position of this ferroelectric contribution remains unchanged, the magnitude of the peaks, however, decreases and finally disappears, indicating relaxation with time of this contribution. This is consistent to what has been found from the high field PUND measurements and indicates that the difference between switching and non-switching transition fields is influenced by this high field ferroelectric contribution.

The very sensitive FORC measurement technique allows evidencing this contribution although not visible from the P(E) loop.

A more subtle evolution in the FORC distribution in area around the E r = -E axis

20

(dotted line in Fig. 10), delimited by the red square can be seen. When the delay between pulses increases, the hysteron density in this area progressively decreases, corresponding to a decrease of the switching polarization contribution studied in the low field section 3.4.1 (decrease of ∆P m with time). In this area, a homogeneous low hysteron density is visible indicating that switching of the weak ferroelectric phase does not appear at a precise coercive field but with rather distributed values, as already seen from the broad current peak of Fig. 6.

Conclusion

In this paper, using the complementary PUND and FORC methods, we study the time and field dependency of the ferroelectricity contribution to the polarization in antiferroelectric PZO thin film. The PUND measurement permits us to show that some residual ferroelectric phase is present in the antiferroelectric thin film and can be switched at low fields. By varying the delay time between the pulses, we show that this weak ferroelectric contribution is not stable and disappears with time. This might be due to orientation relaxation of the ferroelectric domains in the antiferroelectric matrix. Using the FORC measurements, we show that this ferroelectric contribution, visible at low fields, has a quite homogeneous hysteron distribution, corresponding to the observed broad distribution of the coercive field.

The high field PUND measurements reveal that the delay between pulses also affects the antiferroelectric to the ferroelectric phase transition fields which is higher for the first pulse of identical polarity, especially when the delay time is short. This indicates that the previously oppositely oriented ferroelectric phase hinders the antiferroelectricferroelectric transition. As the delay between pulses is getting longer, the difference between the switching and non-switching phase transition field vanishes.

The high field FORC measurements show that there exist a further ferroelectric contribution, which can be switched at fields of the order of the phase transition electric field. When the delay between the pulses increases, the magnitude of this ferroelectric peak decreases, again indicating that switching of this ferroelectric contribution is not stable. This ferroelectric switching contribution, not visible on the P(E) loops, shows the sensibility of the FORC method and its complementarity with the PUND measurements.

Figure 1 :

 1 Figure 1: Applied PUND signal (a) and measured current (b), as function of time for fields below the AFE-FE transition. Global I(E) cycle (c) and I(E) cycle of only the ferroelectric switching contribution (d). t r is the raise time of the pulse and ∆t the delay between the orientation pulse and the PUND measurement, and ∆t PU the delay between the pulses P and U as well as N and D. Arrows indicates the sense of the cycle.

Figure 2 :

 2 Figure 2: P(E) loops of the first (a) and second (b) series of cycles and corresponding current of the first (c) and second (d) series.

Figure 3 :

 3 Figure 3: Polarization (a)(b) and current (c)(d) for the first series of measurement (a)(c) and for the second series of measurement, after a major loop with a driving field amplitude of 875 kV cm -1 (b)(d).

Emax = 125 kV cm 1 1 SwitchingFigure 4 :

 114 Figure 4: PUND current measurements as a function of the applied electric field of the first series (a),(b) and of the second series (c),(d). For fields below (a),(c) and above (b),(d) the AFE-FE phase transition. The switching pulses are shown in full line and the non-switching pulses in dashed lines.

Figure 5 :

 5 Figure 5: Measured switching current (a) and non switching current (b), at 500 kV cm -1 , for various delay ∆t PU between the pulses P and U as well as N and D of the PUND measurement.

Figure 6 :

 6 Figure 6: Differential current waveforms I SW -I NS for different applied fields and various delay times ∆t between the negative orientation pulse and the PUND measurements.

Pm (µC cm 2 )Emax = 125 kV cm 1

 21 Emax = 250 kV cm 1 Emax = 375 kV cm 1 Emax = 500 kV cm1

Figure 7 :

 7 Figure7: Reconstructed P(E) loops concerning only the switching contribution for difference value of the delay between pulses and for a maximum electric field of 500 kV cm -1 (a). Difference between maximum and minimum polarization as a function of the delay between pulses for the reconstructed P(E) loops (b).

1 t

 1 E max = 625 kV cm 1 E max = 750 kV cm 1 E max = 875 kV cm E max = 625 kV cm 1 E max = 750 kV cm 1 E max = 875 kV cm 1 E max = 625 kV cm 1 E max = 750 kV cm 1 E max = 875 kV cm 1

Figure 8 :Figure 9 :

 89 Figure 8: Measured current as a function of the applied electric field for the PUND measurement for various delay times between pulses, for electric field above 625 kV cm -1 . Switching pulses are in full line and nonswitching pulses are in dashed lines.

E r (kV cm 1 )Figure 10 :

 110 Figure 10: FORC distributions for different delay between the positive and the negative pulse. The dashed gray line shows the E r = -E axis. The red arrows point toward the position of the high field ferroelectric contribution peak and the red square indicates the low field hysterons. Density values below 1 × 10 -6 C V -2 are displayed in white.

  . Firstly, lead acetate trihydrate [Pb(CH 3 CO 2 ) 2 • 3 H 2 O] is dissolved in de-ionized water and acetic acid. Zirconium isopropoxide [Zr(O(CH 2 ) 2 CH 3 ) 4 ]

Acknowledgments

This work has been performed with the means of the technological platform SMART SENSORS of the French region Pays de la Loire and with the means of the CERTeM (microelectronics technological research and development center) of French region Centre Val de Loire.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


Anthony Gauvan

Sharp weak-type estimate for maximal operators associated to Cartesian families under an arithmetic condition

Given a set of integers A ⊂ Z, we consider the smallest family B A n-1 invariant by translation which contains the rectangles

for any a = (a1, . . . , an-1) ∈ A n-1 and where I k = [0, 2 k ] for k integer. We prove that if the set A contains arbitrary large arithmetic progression then the maximal operator MB

A n-1 associated to the family

We work in the Euclidean space R n and for U ⊂ R n we denote by |U | its n-dimensional Lebesgue measure. We let I be the family containing every bounded intervals of the real line so that the family I n defined as

stands for the family containing every axis parallel rectangles in R n . Given an integer k ∈ Z, we let I k be the dyadic interval [0, 2 k ] and we use the following notations:

• for â in Z n , we consider the dyadic rectangle R â defined as

• for a in Z n-1 , we consider the dyadic rectangle R a defined as R a := I a1 × • • • × I an-1 × I -(a1+•••+an-1) .

Introduction

The strong maximal operator M n associated to the family I n is a fundamental operator in harmonic analysis: it is defined for f ∈ L ∞ and x ∈ R n as

M n f (x) := sup x∈R∈I n 1 |R| R |f |. 1
In their seminal work [START_REF] Jessen | Note on the differentiability of multiple integrals[END_REF], Jessen, Marcinkiewicz, and Zygmund proved the socalled strong maximal Theorem which specifies the sharp boundedness property of M n near L 1 .

Theorem 1 (Strong maximal Theorem). For any function f and any t > 0, the following estimate holds

|{M n f > t}| n R n |f | t 1 + log + |f | t n-1
.

It is not difficult to see that Theorem 1 is optimal in the following sense: given any convex increasing function φ : [0, ∞) → [0, ∞) such that lim ∞ φ(x) x(1 + log + x) n-1 = 0, the operator M n cannot be bounded from the Orlicz space L φ to L 1,∞ that is to say we cannot have for any f and t

|{M n f > t}| n R n φ |f | t .
Indeed, it is simple to check that given any integer m ≥ 1, the following estimate holds M n 1 Q ≥ 2 -m n m n-1 2 m |Q| where Q is any axis parallel cube in R n . Obviously more general maximal operators associated to geometric sets can be considered: given a family B included in I n and invariant by translations, one can form in the same fashion the maximal operator M B defined as

M B f (x) := sup x∈R∈B 1 |R| R |f |.
For example in R 3 , motivated by problematics arising in the theory of singular integrals, Zygmund considered the family B Z composed of axis every axis parallel rectangles whose side lengths are of the form

t × s × √ ts
for any t, s > 0. Zygmund asked if one could prove that the maximal operator M BZ associated to the family B Z is bounded from L 1 1 + log + L 1 to L 1,∞ ? Loosely speaking, Zygmund expected such result because a rectangle in B Z , as a plane rectangle, is only defined by two parameters (up to translations). Cordoba answered positively to the question and the following conjecture was formulated.

Conjecture 1 (Zygmund's conjecture I). Let {φ i : i ≤ n} be n positive real functions depending on k variables, increasing in each variable separately and assuming arbitrarily small values. Consider then basis B φ of all axis parallel rectangles in R n whose side lengths are of the form

φ 1 (t) × • • • × φ n (t)
for any t ∈ R k . In this situation, the maximal operator M B φ is bounded from

L 1 1 + log + L 1 k-1 to L 1,∞
It appears that stated in this form, Zygmund's conjecture is false and this was proved by Soria in the simplest case n = 3 and k = 2. More recently, Rey exhibited new class of counter examples to this conjecture in [START_REF] Rey | Another counterexample to Zygmund's conjecture[END_REF]. However, those negative results do not necessarily indicate that the idea behind Zygmund's conjecture is false but rather that it is not correctly formulated. Indeed, in his beautiful article [START_REF] Stokolos | On the differentiation of integrals of functions from Lϕ(L)[END_REF], Stokolos proved the following Theorem (for the sake of clarity, we omit the geometric description of this Theorem) thanks to the key idea of crystallisation that he developed.

Theorem 2 (Stokolos). Given any family B of axis parallel rectangles in R 2 which is invariant by translations, there exists an integer k ∈ {1, 2} such that the maximal operator M B is sharply bounded from

L 1 1 + log + L 1 k-1 to L 1,∞ .
Here naturally, given an arbitrary positive number k ≥ 0, we say that the maximal operator M B is sharply bounded from L 1 1 log + L 1 k to L 1,∞ when:

• For any function f and any t > 0, the following estimate holds

|{M B f > t}| n,B R n |f | t 1 + log + |f | t k .
• given any convex increasing function φ

: [0, ∞) → [0, ∞) such that lim ∞ φ(x) x(1 + log + x) k = 0, the operator M B is not bounded from the Orlicz space L φ to L 1,∞ .
In regards of Theorem 2, Stokolos proposed the following reformulation of Zygmund's conjecture in [START_REF] Stokolos | Zygmund's program: some partial solutions[END_REF].

Conjecture 2 (Zygmund's conjecture II). Given any family B invariant by translations included in I n , there exists an integer

1 ≤ k ≤ n such that the maximal operator M B is sharply bounded from L 1 1 + log + L 1 k-1 to L 1,∞ .
So far, this conjecture has not been refuted and let us give non trivial examples:

• In R n , given an integer 1 ≤ k ≤ n, consider the family I n (k) defined as

I n (k) := {I 1 × • • • × I n ∈ I n : # {|I j | : 1 ≤ j ≤ n} ≤ k} .
In [START_REF] Zygmund | A note on the differentiability of integrals[END_REF], Zygmund proved that the maximal operator M n,k associated to the family

I n (k) is bounded from L 1 1 + log + L 1 k-1 to L 1,∞
and easy computations show that those bounds are actually sharp.

• In R n , fix n arbitrary infinite sets of integers {A i : 1 ≤ i ≤ n} and denote by B the smallest family invariant by translations which contains the dyadic rectangles

R a = I a1 × • • • × I an for any a = (a 1 , . . . , a n ) ∈ A 1 × • • • × A n .
In [START_REF] Stokolos | On weak type inequalities for rare maximal functions in R n[END_REF], Stokolos proved the maximal operator M B associated to the family B is sharply bounded from

L 1 1 + log + L 1 n-1 to L 1,∞ .
As explained by Stokolos, this classical result illustrates the fact that a simple rarefaction of the family I n into a family of the form B does not improve the boundedness property of the maximal operator M B associated.

• In R 3 , given an infinite set of integers S ⊂ N, consider the family B composed of every axis parallel rectangles whose side lengths are of the form t × s × 2 j s for any t, s > 0 and j ∈ S. In [START_REF] Hagelstein | Sharp weak type estimates for a family of Zygmund bases[END_REF], Hagelstein and Stokolos proved that the maximal operator M B is sharply bounded sharply bounded from

L 1 1 + log + L 1 2 to L 1,∞ .
• In R 3 , given an infinite set of integers S ⊂ N, consider the family B composed of every axis parallel rectangles whose side lengths are of the form

s × 2 j s × t
for any t, s > 0 and j ∈ S. In [START_REF] Dmitrishin | Sharp weak type estimates for a family of Soria bases[END_REF], Dmitrishin, Hagelstein and Stokolos proved that the maximal operator M B is sharply bounded sharply bounded from

L 1 1 + log + L 1 2 to L 1,∞ .

Result

When studying maximal operators of the above type, a classic argument shows that given an arbitrary family B included in I n and invariant by translations, one can suppose that any rectangle R ∈ B is -up to translations -of the form

R â = I a1 × • • • × I an ,
for some â ∈ Z n . Hence, given an arbitrary infinite set of integer A ⊂ Z, let us denote by B A n-1 the smallest family invariant by translations which contains the dyadic rectangle

R a = I a1 × • • • × I an-1 × I -(a1+•••+an-1)
for any a = (a 1 , . . . , a n-1 ) ∈ A n-1 . We will detail why it seems interesting to study families of the form B A n-1 for infinite A ⊂ Z but observe for the moment that for any a ∈ A n-1 , we have

|R a | = 1.
We prove the following Theorem.

Theorem 3. If the set A ⊂ Z contains arbitrarily large arithmetic progressions then for any integer m ≥ 1, there exists a set E such that

M B A n-1 1 E ≥ 2 -m n m n-1 2 m |E| .
In particular, the maximal operator M B A n-1 associated to the family

B A n-1 is sharply bounded from L 1 1 + log + L 1 n-1 to L 1,∞ .
Thanks to the inclusion B A n-1 ⊂ I n and the strong maximal Theorem 1, it suffices to focus on the sharpness of this bound. Let us discuss about the arithmetic condition: given any integer m, we suppose that A contains a set

{u 0 < • • • < u m-1 } such that for any 0 ≤ k ≤ m -1 u k = u 0 + k(u 1 -u 0 ).
Let us insist that without this arithmetic condition on the set A, it is for the moment difficult to study the sharp boundedness property of the operator M B A n-1 . We will exploit the additive structure of A in order to create a resonance on n-th axis: given an arbitrary m ≥ 1, the set E will be constructed as the Cartesian product of one dimensional crystals i.e. we will set

E = C 1 × • • • × C n .
A one dimensional crystal is a Cantor-like set whose structure is adapted to the specific scales ; we precise this in the following. If it is clear that the first crystals {C i : i ≤ n -1} should be defined according to the set A, the last crystal C n must be defined with great care since we do not have freedom on this axis. Finally, the fact that A contains arbitrarily large arithmetic condition is assured if, for example, it has a strictly positive asymptotic upper density thanks to Szemerédi's Theorem.

Cartesian families

The family B A n-1 defined above is a specific type of Cartesian families of rectangles and it would be desirable to obtain more informations on maximal operators associated to such families in order to progress on Zygmund's problem. More precisely, one could try to tackle Zygmund's problem with the additional hypothesis that the family B is invariant by central dilations i.e. if for any R ∈ B and any λ > 0, we suppose that we have λR ∈ B.

In the same spirit than the dyadic reduction made above, one can show that such a family B is always generated by a family of dyadic rectangles of unit volume {R â : â ∈ F } and so where the set F ⊂ Z n is included in

F ⊂ {x 1 + • • • + x n = 0} .
In the following, let us denote by D n,1 (0) the family {x 1 + • • • + x n = 0} that we identify with dyadic rectangles of unit volume anchored at 0. One can easily inject Z n-1 into D n,1 (0) as follow: given a ∈ Z n-1 , it suffices to define the rectangle R a ∈ D n,1 (0) defined as

R a := I a1 × • • • × I an-1 × I -(a1+•••+an-1)
.

The fact that we have normalized along the n-th axis is not canonical but this is irrelevant for our problem. It would be interesting to give the sharp boundedness properties of maximal operators associated to Cartesian families.

Definition 4 (Cartesian family). Given (n -1) arbitrary sets of integers

{A i ⊂ Z : 1 ≤ i ≤ n -1} ,
we say that the smallest family invariant by translations and central dilations that contains the family of dyadic rectangles of unit volume

{R a : ∀a ∈ A 1 × • • • × A n-1 } ⊂ D n,1 (0)
is a Cartesian family. We denote this family B A1ו••×An-1 and the maximal operator associated by

M BA 1 ו••×A n-1 or M A1ו••×An-1 .
It turns out that even in dimension n = 3, it is difficult to handle Cartesian families. Shall we understand if Zygmund's problem is purely geometric or requires arithmetic tools, the following problems should be addressed:

• Can one specify the sharp boundedness property of the operator M L×Z where the set L is defined as

L = 2 k : k ≥ 1 .
The set L is a typical example of set of integers which do not have additive structure.

• Can one specify the sharp boundedness property of the operator M L×L associated to the family L × L ?

• Generally, can one specify the sharp boundedness property of the operator M A×B where A and B are arbitrary (infinite) sets of integers?

n and we detail then different geometric properties of those sets that will be useful in the following. Given an integer a ∈ Z, we denote by I a the dyadic interval [0, 2 a ] and by O a the oscillation at scale a which is a subset of R defined as

O a := k∈Z (k2 a+1 + I a ).
A one dimensional crystal is defined as follow.

Definition 5 (Crystal). Fix a finite set A in Z that we denote

A = {a 1 < • • • < a m }.
We define the crystal C(A) as

C(A) := I am ∩ 1≤i<m O ai .
An n-dimensional crystal is simply by definition the Cartesian product of n one dimensional crystals i.e. a set E of the form

E := C 1 × • • • × C n .
We will usually denote a n-dimensional crystal by the letter E or Y and one dimensional crystal by C. We define the notion of primitive rectangle associated to a n-dimensional crystal.

Definition 6 (Primitive rectangle). Given an n-dimensional crystal E, there exists a biggest dyadic rectangle R anchored at 0 included in E: we say that it is the primitive rectangle of associated to the crystal E.

We have the following disjointness property and we omit its proof since it is well known in the literature, see [START_REF] Stokolos | On weak type inequalities for rare maximal functions in R n[END_REF] for example.

Proposition 1 (Disjointness property). Fix a finite number of n-dimensional crystals {E i : i ≤ N } and suppose that the primitive rectangles associated

{R i : i ≤ N } are independent i.e. for any i ≤ N we have R i - j =i R i ≥ c n |R i | .
In this situation, we have the following estimate

i≤N E i ≃ n i≤N |E i | .
In order to exploit n-dimensional crystals with a given maximal operator, we need to detail a specific property of those sets which indicate that, in some sense, they are well distributed at specific scales. The following notation will be useful: given a set of integer

A = {a 1 < • • • < a m } and 1 ≤ i ≤ m, we denote by A[i] the set A[i] := {a i < • • • < a m } .
Consider an n-dimensional crystal E defined by n sets of integers {A k : k ≤ n} i.e. the crystal

E = C(A 1 ) × • • • × C(A n ).
Fix then a multi-indice î ∈ N n and consider the crystal

Y ( î) := C(A 1 [i 1 ]) × • • • × C(A n [i n ]).
Of coursed we have supposed that

i k ≤ #A k . It is clear that we have E ⊂ Y ( î)
and denote then by m the integer satisfying

Y ( î) = 2 m Y ( î) ∩ E .
We claim the following.

Proposition 2 (Homogeneity property). If we denote by R the primitive rectangle associated to Y ( î), we have

Y ( î) ⊂ M R 1 E ≥ 2 -m .
Here, M R stands for the maximal operator associated to the smallest family invariant by translations and which contains the dyadic rectangle R.

Proof of Theorem 3

With Propositions 1 and 2 at hands, we are ready to prove Theorem 3. Given an arbitrary large integer m ≫ 1, we construct an n

-dimensional crystal E ⊂ R n such that M B A n-1 1 E > 2 -m n m n-1 2 m |E| . By hypothesis, A contains an arithmetic progression of length m that we denote {u 0 < • • • < u m-1 } . For s ∈ {0, 1, . . . , m -1}, define h s := (n -1)u 0 + (u 1 -u 0 )s and consider then the one dimensional crystals X := C(u 0 < u 1 < • • • < u m-1 ) and Z = C(-h m-1 < • • • < -h 0 )
. Thanks to the crystal X and Z, we define the n-dimensional crystal E as

E := X n-1 × Z ⊂ R n .
Let us prove as claimed that we have

M B A n-1 1 E > 2 -m m n-1 2 m |E| .
Given a positive multi-indice i = (i 1 , . . . , i n-1 ) such that

i 1 + • • • + i n-1 = s ≤ m -1
consider the crystal Y (i) defined as

Y (i) = C(u i1 < • • • < u m-1 )ו • •×C(u in-1 < • • • < u m-1 )×C(-h s < • • • < -h 0 ).
First observe that we have E ⊂ Y (i) and also

|Y (i)| ≃ 2 -i1 . . . 2 -in-1 2 -(m-1-s) |E| ≃ 2 -m |E| .
Observe now that the primitive rectangle associated to Y (i) is the rectangle

R(i) = I ui 1 × • • • × I ui n-1 × I -hs .
The crux of the argument lies in the fact that we have

h s = u i1 + • • • + u in-1
since i 1 + • • • + i n-1 = s and that {u k : 1 ≤ k ≤ m} is an arithmetic progression. Hence the primitive rectangle R(i) satisfies

R(i) ∈ B A n-1
We apply now Proposition 2 which yields

Y (i) ⊂ M R(i) 1 E > 2 -m ⊂ M B A n-1 1 E > 2 -m .
To conclude, it is not difficult to see that the family of rectangles

{R(i) : i ≥ 0, i 1 + • • • + i n-1 ≤ m -1}
is independent and so applying Proposition 1 we obtain i

Y (i) ≃ i |Y (i)| ≃ m n-1 2 m |E| .
This concludes the proof of Theorem 3.

The concept of crystallisation was developed by Stokolos in[START_REF] Stokolos | On the differentiation of integrals of functions from Lϕ(L)[END_REF] and is fundamental for our purpose: we introduce the notion of crystal in dimension 1 and