Nelly Hobeika

Helene Chaumeil

Rana Mhanna

Ming Jin

Xingyu Wu

Arnaud Spangenberg

Davy-Louis Versace

Jean-Pierre Malval
email: jean-pierre.malval@uha.fr

Olivier Poizat'

Two-Photon Initiating Efficiency of a Ditopic Alkoxynitrostilbene Reacting through a Self-Regenerative Mechanism

Keywords: Two-Photon Absorption, Nitrostilbenes, Two-Photon Polymerization

The photophysical properties and the photoinitiating reactivity of a ditopic alkoxynitrostilbene were compared to those of its single branch chromophore used as a reference. Whereas a trivial additive effect is observed when considering the one-and two-photon absorption properties, a clear and very significant amplification has been highlighted for the photoreactivity of this free radical photoinitiator which was used as a hydrogen abstractor in presence of an aliphatic amine co-reactant. We indeed demonstrate that the proximity of two nitroaromatics moieties within the same molecular architecture gives rise to an original cycling mechanism based on a stepwise photo triggering of each photoredox center followed by a subsequent regenerative process. The combination of a high two-photon absorption cross-section ( 780nm ≈ 330 GM) with a strong enhancement in photoreactivity makes this nitrostilbene bichromophore a very suitable candidate for two-photon polymerization applications.

Introduction.

Two-photon induced polymerization (2PP) has emerged as a strategic technology for the three-dimension patterning of photoactive materials [1,2] . The 2PP technique consists in tightly focusing of a fs-pulse laser beam into a photoactivable resin so as to promote non linear absorption effects which ensure an intrinsic spatial confinement of the phototriggered reaction.

The 3D trajectory of the focal point which is translated within the resin medium progressively creates a photopatterned object whose geometry and morphology are initially parameterized using a computer-aided-design (CAD) model. Two-photon initiated polymerization has progressed rapidly as an advanced 3D printing technology which offers a high-quality level writing of complex, 3D structures with feature sizes reaching sub-100 nm scale [3,4] . The research in 3D nano/microfabrication is highly active owing to its promising applications such as high-density 3D optical data storage [5] , biomaterials for regenerative medicine [6] , photonic crystals [7] , 3D sensors [8,9] and the like. In the framework of the free radical polymerization, an efficient two-photon initiated reaction implies the use of chromophores which both exhibit a large two-photon absorption cross-section () and an efficient photoreactivity leading to a high yield ( i ) for the generation of primary free radicals. Even though the design and development of direct two-photon cleavable initiators appears as a very appealing approach [10,11] , the bicomponent method is, by far, the dominant strategy due to its very simple processability. This method consists in associating two distinctive components in the same acrylate-based formulation: i) a two photon active dye whose excited triplet states can promote hydrogen transfer reactions. ii) a hydrogen donor co-initiator such as an amine, a thiol or an alcohol which generates the initiating radical. A large set of two-photon absorbing (2PA) systems have been proposed such as fluorone [12] , ketocoumarine [13] , 4,4'dialkylaminostilbene [5] , or 4,7 diaminofluorene [14] . The main guidelines which motivate the choice of the 2PA chromophore rely on the effective conjugated length within the dye, its electronic symmetry (e.g. dipole, quadrupole) and its dimensionality [15] . The nature and the position of electron donor and acceptor substituents are also of primary importance. Of particular interest, the use of ketones, -diketones or nitro functions [16][17][18] as acceptor groups can not only promote a strong charge delocalization within the -conjugated structure of the dye but can also induce an efficient intersystem crossing yielding a large amount of reactive excited triplet species. In line with this approach, we have previously demonstrated that the trans-4-propoxy 4′-nitrostilbene [19] (PNS in Scheme 1) constitutes a highly reactive 2PA initiator when associated with an aliphatic amine. For instance, we showed that the twophoton initiating performance of this nitrostilbene was superior by a factor four to that of a 2,7 diaminofluorene derivative used typically for 2PP applications [14,20,21] . In the present paper, we extend the analysis to a ditopic alkoxynitrostilbene homologue (DPNS in Scheme 1). The linear and nonlinear absorption properties as well as the photoinitiating efficiency of DPNS have been systematically evaluated and compared to those of its single branch chromophore (PNS) used as a reference. Molecular structures of the nitrostilbene derivatives.

Results and Discussion

Ground State Properties. Figure 1 shows the one-(1PA) and two-photon absorption (2PA) spectra of the chromophores in dichloromethane (DCM) along with their respective theoretical absorption spectra calculated at TDDFT-PBE0/6-31G(d,p) level. The corresponding experimental and theoretical data are gathered in Table 1. For both compounds the absorption spectrum is dominated by an unstructured band with  MAX of similar values and with an intensity which is twice higher for DPNS as compared to PNS. The positions of the calculated vertical energy transitions agree very well with the experimental values. character from the donor alkoxy group to the acceptor nitro one. By contrast, the TD-DFT calculations predict that the lowest energy absorption band of DPNS encompasses two strongly allowed electronic transitions from the two quasi-degenerate S 1 and S 2 excited states (E 12 ~ 0.02 eV). Both S 0 → S 1 and S 0 → S 2 transitions correspond to *-type transitions which are centred on each alkoxynitrostilbene fragment and with the same electronic symmetry as that observed for PNS (see Figure 2). According to the fully optimized geometry of DPNS, it should be underline that the flexible covalent linker between the two ditopic dyes maintains an average distance of ca. 10.5 Å between the oxygen atoms of the alkoxy groups and a distance even larger for the nitro centers (i.e. > 25 Å). One-photon absorption properties of the chromophores in DCM and their related TD-DFT vertical electronic transitions. E red = -1.09 V vs. SCE for PNS and E red = -1.11 V vs. SCE for DPNS). These potentials match that of the nitrobenzene [22] which underlines the role of the nitrophenyl subunit as reducible electrophore. Therefore, in the case of PNS, the electrochemical reaction should correspond to a one-electron reduction process generating a radical anion (PNS  -) centred on the nitrophenyl moiety. As shown in Figure 3, the reduction of DPNS clearly leads to more intensive cathodic and anodic peaks than those measured for PNS at the same concentration. This is a clear indication that the two reducible electrophores of DPNS are activated at very close redox potentials. Even though one can not assert the occurrence of a two successive one-electron transfers or a single two-electron one [23] , this behaviour confirms that the two electrophores are equivalent due to the large distance separating the two nitrophenyl moieties.

This large distance should minimise the coulombic repulsion within the electrogenerated dianion (DPNS 2-) and the subsequent structural changes accompanying the injection of two electrons. Therefore these redox centers can operate independently to each other. This very interesting property will be oriented to trigger sequentially each of these photoredox centers upon two-photon excitation as presented hereafter. The 2PA cross-sections of the dyes have been measured using the two-photon excited fluorescence method [24] . The 2PA spectra of the chromophores are superimposed with their respective 1PA ones in Figure 1. According to our spectral resolution, the two derivatives present a single 2PA band in the 690-880 nm range which reasonably matches their lowest energy 1PA band. The maximum 2PA cross-section ( MAX ) for PNS is about 176 ± 13 GM ( MAX ~ 790 nm) in agreement with our previous measurements [19] . Moreover, the superimposition of the 1PA et 2PA spectra confirms that the S 0 →S 1 transition of PNS is two-photon allowed as expected for a * electronic transition with a strong charge transfer character. In the case of DPNS, the additive effect observed for its linear absorption ability as compared to that of PNS can be also extended to its 2PA

properties. Indeed, the  MAX of DPNS is about 332 ± 13 GM ( MAX ~ 780 nm) which is roughly twice higher that of PNS. This effect is obviously ascribed to the increase in the molecular dimensionality on going from PNS to DPNS. However, it should be emphasized that  MAX increases quadratically with the effective number of -electrons [25,26] (N eff ) contributing to the nonlinear response. A counting method for determining this effective number was proposed by Kuzyk et al [25,26] However, the real advantage of this molecular design strategy will appear when considering the comparative photo reactivity of DPNS vs. PNS.

Excited State Properties. The photophysical properties and the relaxation processes associated to the lowest singlet (S 1 ) and triplet (T 1 ) states of PNS and DPNS have been previously characterized [19,27] . As illustrated in Scheme 2, the locally singlet excited ( 1 LE) of the stilbene derivatives undergoes multiple deactivation processes. First it should emphasized that the addition of a nitro group to an aromatic hydrocarbon [28] is known to 'switch-off' the fluorescence emission properties of the chromophore due to highly efficient competing non radiative channels such as internal conversion (IC) and intersystem crossing (ISC). In this latter case, the presence of n* states localized on the -NO 2 group clearly promotes a very fast ISC process whose rate constant is about ~ 10 11 s -1 for PNS in apolar medium [29] .

However, in our case, PNS and DPNS still remain fluorescent. Indeed, upon increasing solvent polarity, the planar 1 LE state undergoes an adiabatic conformational change corresponding to the rotation of the entire nitrophenyl moiety and generating a highly polar twisted internal charge transfer ( 1 TICT) state [19,27] . As a typical example, Figure S1 shows the absorption and fluorescence spectra of both dyes in ethyl ether (low polar solvent) and acetonitrile (highly polar medium). The corresponding spectroscopic data are summarized in Table S1. Both nitrostilbenes display very large Stokes shifts which are even more enhanced when increasing polarity. In parallel, the fluorescence quantum yields (see Table S1) are amplified by two orders of magnitude on going from ethyl ether to acetonitrile. All these effects clearly evidence a strong electronic and geometrical change between ground and emitting states due to the solvent induced stabilization of this emissive 1 TICT state. As shown in Table S1, the fluorescence 'switch on' of PNS and DPNS occurrences to the detriment of the trans-to-cis photoisomerization. This is consistent with the fact that the photoisomerization of the 4,4'alkoxy nitrostilbenes occurs along the T 1 surface [29][30][31] as depicted in Scheme 2. Indeed, this reaction implies a barrierless torsion about the olefinic C=C bond generating an intermediate ( 3 p) with a perpendicular configuration. The rapid equilibrium 3 trans  3 p decays via the ground state of this twisted species into the trans or cis isomer. Photoinitiating Ability of DPNS vs. PNS and Related Mechanism. In order to compare the photoinitiating performance of DPNS with respect to that of PNS, each dye was mixed to a poly(ethylene glycol) diacrylate monomer in presence of N-methyl diethanolamine (MDEA).

As previously indicated, such an aliphatic amine which is added in a large excess (5 wt %) is

  abs p p t i I M R k k 1 2 2          
typically used as H-donor reactant. It should be underlined that the concentration of DPNS was adjusted to half that of PNS in order to obtain isoabsorbing formulations when irradiated at 365 nm (see inset Figure 5). The photopolymerization reaction of the laminated acrylate films was monitored by real-time FT-IR and the conversion rates of acrylate double bonds are shown in Figure 5. The conversion time-profile of DPNS formulation clearly presents a much faster dynamics as that observed for PNS. For instance, the formulation with DPNS reaches 90 % of C=C bonds conversion after 60 s irradiation whereas only 55 % conversion is observed for the PNS formulation. The initiation efficiency  i of these photoinitating systems can be evaluated according to the following equation [START_REF] Fouassier | Photoinitiator, Photopolymerization and Photocuring: Fundamentals and Applications[END_REF] : (eq. 1)

In this equation, R p corresponds to the rate of polymerization and I abs is the light absorbed by the sample. k p and k t are the rate constants for the polymerization propagation and termination steps respectively. Such kinetics parameters are strongly dependent on t he structure of the monomer (M). In our precise case, they can be kept constant for both formulations. Based on this formalism, we show that DPNS exhibits an initiation efficiency which is 6 times higher than that of PNS. In order to explain such a significant amplification in the photoreactivity of DPNS, the photolysis of the nitrostilbenes has been performed in N 2 -saturated ACN solutions in presence of MDEA and methyl methacrylate (MMA). Figure 6 shows the evolution of the absorption spectra of the nitrostilbenes in N 2 -saturated ACN solutions when irradiated at 365 nm in presence of the distinctive reactants. First it should be indicated that the photoreaction which induces dyes photobleaching mainly occurs at the triplet state [START_REF] Görner | [END_REF]34] even though the presence amine also leads to a fluorescence quenching of the nitrostilbene chromophores. In this latter case, the Stern-Volmer plots obtained from both steady-state (I 0 /I) and fluorescence lifetime ( 0 /) measurements show linear correlations as typically illustrated in Figure S2 for DPNS in ACN. The Stern-Volmer constants derived from the slopes of these correlations are slightly higher for DPNS as compared to PNS with values of ca. 4.4 x10 9 M -1 s -1 and 2.9 x 10 9 M -1 s -1 respectively. Note Whereas the first kinetics can be confidently assigned to the fast equilibrium reaction accompanying the trans-to-cis photoisomerisation process, the second kinetics correspond more specifically to the photoreduction step and the subsequent reactions following this latter process. According to the half-times of this sequence, it is clear that DPNS undergoes a faster photobleaching than PNS. The photoreduction of the 4-alkoxy 4'-nitrostilbene in presence of aliphatic amine has been thoroughly investigated on the basis of transient absorption and ESR methods [19,[START_REF] Görner | [END_REF]34] . The stepwise photo induced mechanism can be summarized as follows:

3 Ar-NO 2 + MH → Ar-NO 2  -+ MH  + (1) Ar-NO 2  -+ MH  + → Ar-N  O 2 H + M  (2)
Upon excitation of the nitrostilbene, an electron transfer is promoted from the amine reactant to the excited triplet state of the dye (step 1). The generated amine radical cation and nitro radical anion can consequently react together through a proton transfer reaction and then produce a nitro H-adduct radical (Ar-N  O 2 H) and a -aminoalkyl radical (step 2). This latter radical whose presence has been previously demonstrated by ESR spin trapping methods [19] is highly reducible [35] and can further reduce a second nitrostilbene dye at the ground state (step 3) as follows:

Ar-NO 2 + M  → Ar-NO 2  -+ M  + (3) Ar-NO 2  -+ M  + → Ar-N  O 2 H + P (4)
These two steps also lead to the formation of a second nitro H-adduct radical and other byproducts related to the oxidation of M  (step 4). The crucial role of MDEA as a sacrificial reactant should be underlined here since it provides two electrons and two protons as depicted in Scheme S1. Note that a similar mechanism has been proposed for the photocatalytical reduction of a nitrobenzene series using triethanolamine as reducing agent [36] . The steps (3) and ( 4) are very interesting since they are globally equivalent to the two previous steps but with the difference that the steps ( 3) and ( 4) do not require any excited species. Therefore, according to this global mechanism, two nitro H-adduct radicals are ultimately produced. This species is relatively long living. Indeed, the half-time for the nitro H-adduct radical of the 4-alkoxy 4'-nitrostilbene was estimated in the 1-100 ms range in Ar-saturated ACN [34] .

Moreover, it has been demonstrated that these nitro H-adduct radicals undergo an efficient disproportionation [34] yielding a nitroso compound concomitantly with the regeneration of the nitrostilbene dye as follows : clearly opens an alternative reaction route for the consumption of the -aminoalkyl radical which exhibits a very high reactivity upon addition reactions [37] onto the C=C bonds of the acrylate monomers. These addition reactions certainly compete with the step 3 and should reduce the dyes photobleaching process. This assumption is perfectly in line with the increase of the photobleaching half-time of PNS ( 1/2 = 882 s) which is multiplied by a factor 3.6 and that of DPNS ( 1/2 = 603 s) whose amplification is even higher with a factor 7.1. Of particular interest, the reduction of the photobleaching rate for DPNS is more significant than that observed for PNS. This effect suggests that the disproportionation step still remains active for DPNS. In this case, the disproportionation of the two nitro H-adduct radicals should mainly proceeds intramolecularly allowing the regeneration of one nitrostilbene branch. By contrast, the disproportionation step appears unlikely for PNS due to its diffusion-limited character.

Ar-N  O 2 H + Ar-N  O 2 H → Ar-NO + Ar-NO 2 + H 2 O ( 5
For DPNS, the occurrence of this regenerative process clearly constitutes an additional source for the further production of -aminoalkyl radicals and can thereby explain the higher initiation efficiency observed for DPNS as compared to PNS. We have therefore demonstrated that DPNS is a more performant free radical photoinitiator than PNS upon one photon excitation. The initiation efficiency ( i ) of the bichromophore is about 6 times higher than its reference. Such a significant difference can be extended to two-photon initiated polymerization.

Two-Photon Polymerization Properties. The two-photon initiating performances of PNS

and DPNS have been evaluated by two-photon patterning polymer lines using the same diacrylate formulations which were prepared for the one-photon polymerization experiments.

The methodology simply consists in focusing a fs-pulse laser beam ( exc : 780 nm) into the resin using a 40X, 0.65-NA objective. The nonlinear absorption of the nitrostilbenes is then promoted which initiates the polymerization. . .

th i P C R   
Therefore P th and [R 0 ] can be connected by the following equation [1,38] :

(eq. 2)
Where C,  and  i denote respectively the concentration, the 2PA cross-section and the initiation efficiency of the photoinitiator. As shown in Figure 7, the two-photon polymerization thresholds have an estimated power value of 7 mW and 3 mW for PNS and DPNS formulations, respectively. Considering that: i) the 2PA cross-section of DPNS is ~ twice higher that PNS ii) the concentration of the DPNS is half that of PNS in this set of formulations. It can be inferred ). According to this methodology, we estimate that the initiating efficiency of DPNS is ~ 5.4 higher than that measured for PNS which nicely corroborates the comparative photoreactivities of the two nitrostilbenes upon one photon excitation. From a practical point of view, the two-photon initiating performance of a photoinitiator corresponds in fact to the product of its initiation efficiency ( i ) with its two-photon absorption ability (). In this context, it can be inferred that the two-photon initiation property of DPNS clearly outperforms that of PNS by about a factor of one order magnitude. Finally, to qualitatively demonstrate the applied potential of DPNS as suitable two-photon active initiator, we have inscribed into the DPNS diacrylate formulation two arbitrary 2D-microstructures upon excitation at 780 nm (Figure 8A-B

Conclusion.

The one-and two-photon absorption properties of a ditopic alkoxynitrostilbene were compared to those of its single branch chromophore used as a reference. The molecular association of two D--A stilbenes with a long saturated linker maintains the electronic integrity of each subunit. An additive effect for the contributions of each single branch to the one and two-photon absorption is observed for the bichromophore which displays a significant 2PA cross-section of about ~ 330 GM at 780 nm. This amplification is much higher when considering the photoreactivity of the bichromophore which can be used as a H-abstractor in presence an aliphatic amine. The initiation mechanism involves a successive photoactivation of the two nitro centers whose proximity allows a final disproportionation step leading to a regenerative process. By taking advantage of a synergy combining a 2PA additive effect and a strong enhancement in the photoreactivity we show that the two-photon initiation efficiency of DPNS is one order of magnitude higher than its reference system.

4.

Experimental Section.

Materials. The synthesis and the characterization of PNS and DPNS are detailed in reference [19] and [27] ( = 0.52) [39] . The phosphorescence measurements were performed using the same spectrofluorometer which is also equipped with a Xe-pulsed lamp operating at up to 25 Hz.

Luminescence measurements were performed in glassy matrix of 2-methyltetrahydrofuran (2MTHF) at 77 K. The samples were placed in a 5-mm diameter quartz tube inside a Dewar filled with liquid nitrogen.

Fluorescence lifetime measurements. The fluorescence lifetimes were measured using a

Nano LED emitting at 372 nm as an excitation source with a nano led controller module, Fluorohub from IBH, operating at 1MHz. The detection was based on an R928P type photomultiplier from Hamamatsu with high sensitivity photon-counting mode. The decays were fitted with the iterative reconvolution method on the basis of the Marquardt/Levenberg algorithm [START_REF] Connor | Time correlated single photon counting[END_REF] . Such a reconvolution technique allows an overall-time resolution down to 0.2 ns.

The quality of the exponential fits was checked using the reduced  2 ( 1.2).

Cyclic voltammetry.

The cyclic voltammetry experiments [START_REF] Malval | [END_REF] (using a computer-controlled Radiometer Voltalab 6 potentiostat with a three-electrode single compartment cell; the working electrode was a platinum disk; a saturated calomel electrode (SCE) used as a reference was placed in a separate compartment with a salt bridge containing the supporting electrolyte) were performed at 300 K, in N 2 -saturated dichloromethane with a constant concentration (0.1 M) of n-Bu 4 NPF 6 (TBAPF 6 ). Ferrocene (Fc) was used as an internal reference.

DFT calculations. The theoretical absorption spectra have been computed based on Density

Functional Theory (DFT) and Time-Dependent DFT (TDDFT). The overall computation strategy was defined as follows: After initial AM1 optimization calculations (vacuum), subsequent optimization of geometrical structures of the derivatives were carried out using the PBE0/6-31G(d,p) level of calculation. (Note that PBE0 is a good functional for the description of photofunctional molecules [42] ). Finally, the TDDFT vertical transitions have been computed using the same level of calculations. All calculations have been performed

using GAUSSIAN 09 package [START_REF] Frisch | Gaussian 09, Revision B.01[END_REF] . A relative two-photon excited fluorescence method [24] was employed to measure the two-photon absorption cross-sections, . The measurements of 2PA cross-sections were performed relative to reference molecules (r) such as fluorescein [24,[START_REF] Albota | [END_REF] in water at pH = 11.

Photoreactions

The value of δ for a sample (s) is given by: Where S is the detected two-photon excited fluorescence integral area, c the concentration of the chromophores, and  is the fluorescence quantum yield of the chromophores. η is the collection efficiency of the experimental set-up and accounts for the wavelength dependence of the detectors and optics as well as the difference in refractive indices between the solvents in which the reference and sample compounds are dissolved. The measurements were conducted in a regime where the fluorescence signal showed a quadratic dependence on the intensity of the excitation beam, as expected for two-photon induced emission. For the calibration of the two-photon absorption spectra, the two-photon excited fluorescence signal of each compound was recorded at the same excitation wavelength as that used for standards (i.e.  exc = 782 nm for fluorescein). The laser intensity was in the range of 0.2-2 x 10 9 W/cm 2 .

The experimental error on the reported cross section is 15 %. Photopolymerization. The photopolymerization was monitored in situ by real-time Fourier transformed infrared spectroscopy with a Thermo-Nicolet 6700 IR-spectrometer. The laminated formulations are sandwiched between two polypropylene films to minimize the inhibiting effect of oxygen. A round teflon spacer is used to maintain a constant thickness of 30 µm. The sample is then positioned between two BaF 2 pellets and irradiated using a 100 W Mercury-Xenon Lamp (LC 9588/02A from Hamamatsu) equipped with a band pass filter centred at 365 nm (A9616-07 from Hamamatsu). The conversion rates are obtained from the disappearance of the progressive vinyl C=C stretching vibration band at 1630 cm -1 .

Microfabrication. The 3D lithographic microfabrication was carried out using a Zeiss Axio Observer D1 inverted microscope. The two-photon excitation was performed at 780 nm using respectively a mode-locked Ti: Sapphire oscillator (Coherent, Chameleon Ultra II: pulse duration: ~140 fs; repetition rate: 80 MHz). The incident beam was focused through a 0.65 NA objective (40 X) which leads to a radial spot size 590 nm at  exc = 780 nm (1/e 2 Gaussian).

A drop of the resin is deposited on a cover slip which is mounted on a 3D piezoelectric stage allowing the translation relative to the laser focal point. The intensity of the entering laser is controlled with the use of an acousto optic modulator. The displacement of the sample and all photonic parameters (i.e. excitation power and irradiation times) are computer-controlled. The 20 3D microstructure is finally obtained by washing away the unexposed monomer resin using ethanol.

5.

Table of Contents

A two-photon activable ditopic alkoxynitrostilbene was developed to promote free radical photoinitiation reactions. Taking benefit of a synergy between high two-photon absorption cross-sections ( 780 nm ~ 330 GM) and a regenerative photoinitiation process, the bichromophore clearly exhibits an amplified two-photon induced reactivity and constitutes a promising candidate for two-photon polymerization applications.

  Scheme 1.Molecular structures of the nitrostilbene derivatives.

Figure 1 .

 1 Figure 1. Experimental one-(full lines) and two-photon (circles) absorption spectra of PNS and DPNS in DCM along with their respective theoretical absorption spectra at TDDFT-PBE0/6-31G(d,p) level (vertical bars).

Figure 2 .

 2 Figure 2.Frontier molecular orbital diagrams of the chromophores with the main transitions corresponding to the longest wavelength absorption band.

Figure 3 .

 3 Figure 3. Cyclic voltamogramms of PNS and DPNS in DCM + (nBu)4NPF6 (0.1 M) on platinum electrode at 200 mV s -1 (concentration of chromophores: 2 x10 -4 M).

  and consists in calculating N eff by geometrically weighting the number of electrons in each conjugated path of the molecule. The normalization of  MAX by N eff 2 gives access to the intrinsic 2PA properties of the dyes which allow a better comparison of 2PA performances. According to this methodology, we show that the intrinsic 2PA cross-sections of PNS ( MAX = 176 GM, N eff = 16) and DPNS ( MAX = 332 GM, N eff = 22.6) have equivalent values of ca. 0.69 GM and 0.65 GM respectively. Therefore, it is clear that the covalent association of two -conjugated nitroalkoxystilbenes into a single supramolecular architecture does not lead to a cooperative effect in term of 2PA ability.

Scheme 2 .

 2 Scheme 2.Schematic representation of the multiple deactivation pathways for the excited singlet and triplet states of the nitrostilbene derivatives.

Figure 4 .

 4 Figure 4.Luminescence spectra of PNS and DPNS in glassy matrix of 2MTHF (77K). Phosphorescence emission was recorded by time-gated method (delay : 1 ms, and time gate : 10 ms)

Figure 5 .

 5 Figure 5.Real-time FTIR kinetics of C=C bonds conversion curves upon irradiation of diacrylate formulations (I365nm = 5 mW cm -2 ). Photoinitiating systems: PNS / MDEA (0.35 wt % / 5 wt %) and DPNS / MDEA (0.4 wt % / 5 wt %). Inset: Absorption spectra of the corresponding resins (30 µm -thick-films).

Figure 6 .

 6 Figure 6. Evolution of the absorption spectra of the chromophores in N2-saturated ACN as function of the irradiation time (irr. = 365 nm, P = 60 mW) : A / Dye. Nitrostilbene derivatives mixed with MDEA (0.1 M). B / Dye. Nitrostilbene derivatives mixed with MDEA (0.1 M) and with MMA (0.25 M). Vertical lines indicate the spectral position of irr.

Figure S3 .

 S3 Figure S3. According to this absorbance time-profile, the amount of by-product increases until ~ 120 s of irradiation then decreases presumably due to its consecutive consumption. Both for PNS and DPNS, the global photoreaction progress can be monitored based on the time-dependent profiles of the absorbance at  MAX (376 nm) which mainly reflect the kinetics for the dyes consumption. These kinetics (insets of Figure 6A) clearly exhibit two distinctive regimes: i) a fast photobleaching step with a half-time of about  1/2 ~ 3 s for both nitrostilbenes ii) a long-time process with  1/2 of about 245 s for PNS and 85 s for DPNS.

)

  Hence the progressive photobleaching of PNS and DPNS in presence of MDEA should involve successively two nitroaromatics redox centers. The main structural difference between PNS and DPNS relies on the presence of two nitroaromatics redox centers which are covalently connected together for DPNS. Such a structure appears particularly well suited for this multistep mechanism and presents very important advantages: i) the cage escape of the -aminoalkyl radical is not necessary between step (2) and step (3) ii) the encountering condition required for the disproportionation in the step (5) is intrinsically fulfilled for DPNS. Therefore this mechanism clearly accounts for the ~ 3 times faster photobleaching observed DPNS in presence of MDEA as compared to PNS. As shown in Figure 6B, the addition of a methyl acrylate (MMA) as third reactant leads to a drastic drop in the production of nitrostilbene by-products whose absorption bands are hardly detected. The presence of MMA

Figure 7

 7 typically shows a symmetrical set of polymer lines pattern obtained by the gradual decrease of the laser excitation power for PNS and DPNS formulations. The progressive decrease of the incident excitation power allows reaching the two-photon polymerization threshold power (P th ) for each formulation. This fundamental parameter corresponds to the local absorbed-energy density below which the polymerization is not observed (see star pictogram in Figure7). From a mechanistic point of view, this polymerization threshold can be related to the minimum amount of initiating radicals ([R 0 ]) allowing the degree of monomer conversion to past the gelation point.

Figure 7 .

 7 Figure 7. Two-photon polymerized lines fabricated for distinctive laser excitation powers (ex = 780 nm, exp = 2 ms). The star pictogram indicates the corresponding value of Pth at 780 nm. Formulations: Poly(ethylene glycol) diacrylate mixed with MDEA (wt 5 %) and with : PNS (0.35 wt %  13.6 mM) and DPNS (0.4 wt %  6.8 mM).

  ). In addition, a second formulation based on pentaerythritol triacrylate (PETIA) monomer was prepared with DPNS and MDEA similarly to the diacrylate formulation. The two-photon polymerization with PETIA leads to the formation of a rigid three-dimensional polyacrylate network. Such a macromolecular structuration should guarantee the integrity of photopatterned µ-structures considering the risk of collapse and deformation during the rinsing step of the µ-structure with ethanol. As an illustration, Figure8Cshows the scanning electron microscopy (SEM) image of a 2D-grid composed of periodic 20 x 20 µm squares and generated with the PETIA formulation. This µ-structure has perfectly resisted to the rising step and displays a goodfidelity replication of the periodic model without any severe deformations.

Figure 8 .

 8 Figure 8. A-B Transmitted light microscope images of periodic 2D µ-structures. Formulation: diacrylate resin with DPNS / MDEA (0.4 wt % / 5 wt %). (ex : 780 nm, P = 5 mW, exp = 2 ms). C. SEM images of µ-grid. Formulation: PETIA with DPNS / MDEA (0.4 wt % / 5 wt %). (ex : 780 nm, P = 5 mW, exp = 2 ms).

  Photoreactions in liquid medium. The photoreduction of the nitrostilbene derivatives in presence of coreactants (i.e MDEA / MMA) was performed in N 2 -saturated ACN under continuous irradiation at 365 nm with an Hg-Xe lamp (LC 9588/01A from Hamamatsu) equipped with a band pass filter centred at 365 nm (A9616-07 from Hamamatsu). The progress of the reaction was monitored via UV-Vis absorption spectra. The irradiated solutions were continuously stirring and the experiments were performed at 25°C.

Table

Acknowledgments

The authors acknowledge the meso-centre 'EQUIPEX@MESO' of Strasbourg University for access to High Performance Calculation resources.

6.


Dr Klaasjan Maas

Dr Edouard Villepreux

Dr David Cooper

Dr Eduardo Salas-Colera

Dr Juan Rubio-Zuazo

Dr German R Castro

Olivier Renault

Carmen Jiménez

Herve Roussel

Dr Xavier Mescot

Juan Rubio- Zuazo

Dr Carmen Jimenez

Hervé Roussel

Dr Quentin Rafhay

Dr Michel Boudard

Dr Mónica Burriel
email: monica.burriel@grenoble-inp.fr

Dr H Roussel

Tuning Memristivity by Varying the Oxygen Content in a Mixed Ionic-Electronic Conductor

Keywords: Mixed ionic electronic conductors, memristive devices, valence change memories, La 2 NiO 4, neuromorphic

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Introduction.

Valence-Change Memories (VCMs) are a type of bipolar memristive devices in which internal redox reactions govern the resistance-change mechanism. [1] More particularly, in oxides, the motion of oxygen ions is held responsible for the local valence-change (redox reaction) in the cation sublattice, [1] resulting in a change in the electronic conductivity, which magnitude depends on the structure of the device and its constituent materials. VCMs are one of the most promising technologies for the hardware implementation of neuromorphic computing, as they combine a large number of desired properties including speed, scalability, low power consumption and CMOS compatibility [2] . In addition, under certain operating conditions VCMs can present analog conductance (resistance) modulation, a critical requirement for memristors when used for computing. Significant examples of the use of memristors for neuromorphic computing include the simulation of both neuronal and synaptic dynamics including spike-timing-dependent plasticity (STDP) [2] and the full hardwareimplementation of convolutional neural networks (CNNs) for image recognition using memristor crossbars. [2] The VCM mechanism can exhibit both filamentary and interface-type switching, classified as such depending on whether the region of the device responsible for the change in resistance is localized (filamentary) or widespread across the electrode/oxide interface (i.e. interfacetype). [3] Despite filamentary switching being the most commonly used type of VCM, there is currently little control over the formation and properties of the conducting filament, which creation usually requires an electroforming. This destructive biasing step is necessary to give rise to the memory properties and can be understood as a soft dielectric breakdown of the sandwiched material leading to the stochastic creation of filamentary pathways which can connect and disconnect the two electrodes. Conversely, in interface-type VCMs, the resistance-change is usually attributed to a field-induced change of a Schottky barrier at the electrode/oxide heterojunction. [1] The externally-applied electric field induces a drift of oxygen ions which accumulate at the positive electrode, locally changing the junction properties, such as the depletion width of the majority carriers. Besides changing the contact characteristics, the accumulation of oxygen ions near the electrode can also lead to the generation of oxide interlayers. This process mostly depends on the Gibbs free energy for the formation of the metal-oxide phase (from the electrode material). The additional oxide interlayer then adds a series resistance to the device, increasing its overall resistance and can trigger a memristive behavior. Some materials are more prone than others to result in interface-type VCM. In particular, we have used the following criteria to select La 2 NiO 4+δ as valence change oxide for the design of our interface-type VCM,: 1) Oxygen storage: stable material (it should not decompose) when varying its oxygen stoichiometry; 2) Oxygen transport: the material should be able to supply oxygen towards the metal/oxide interface in its bulk, allowing for an homogeneous change in resistance to occur across the entire metal/oxide interface; and 3) Electronic conduction: the material should not be an insulator to prevent dielectric breakdown (in order to avoid filamentary-type switching). In order to extend the range of materials showing interface-type memristance and to adapt the choice to the targeted application and time-response required, these three criteria can applied in the future as selection rules in the search for new memristive oxides.

1.1. La 2 NiO 4+δ , a layered structure with mixed conduction properties.

La 2 NiO 4+δ (L2NO4) has been selected for the design of interface-type VCMs due to its oxygen storage, oxygen transport and electronic conduction properties. L2NO4 is a Mixed Ionic-Electronic Conductor (MIEC), combining good electronic conductivity, with high oxygen exchange and diffusion coefficients. [4] It has a rich phase diagram which allows it to oscillate between two main crystal structures being: I4/mmm, a high symmetry tetragonal phase and Bmab, a lower symmetry orthorhombic phase. This phase transition is both temperature and pO 2 -dependent. [5] L2NO4 also has the particularity of accommodating interstitial oxygen anions within its lattice. This additional oxygen is key to the bulk oxygen conduction exhibited by the material. [6] A large range of oxygen over-stoichiometry can be stabilized in L2NO4 without it decomposing, [7] making it a potentially good oxygen storage medium. The high symmetry tetragonal structure of L2NO4 as well as the location of the interstitial oxygen and the naming convention used throughout the text for the different atoms and bonds composing the material are shown in Figure 1a andb. Concerning its bulk oxygen-ion diffusion capabilities, both experimental and computational evidence has shown that an insterstitialcy diffusion mechanism is the preferred pathway for a fast diffusion along the ab plane (around three orders of magnitude higher than along the c axis). [4,[8][9][10] There is an agreement between the measured and the calculated activation energy for this diffusion process (E a ≈ 0.88 eV) [8,11,12] , which is significantly lower than for other well-known ionic conductors (e.g yttria-stabilized-zirconia has an activation energy of 1 eV [13] ). This low value together with a very high diffusion coefficient is even more relevant when compared to other oxides used as memristive materials which are not good ionic conductors in their pristine state (eg: SrTiO 3 , Al 2 O 3 , Ta 2 O 5 , HfO 2, see ref. [14] for more details). However, it is possible that ionic insulators might in fact show significant oxygen diffusion capabilities after being electroformed or if they contain a high density of defects introduced (intentionally or not) during processing. The presence of oxygen interstitials ( , using the Kroger-Vink defect notation) has to be electrically compensated by positive holes to keep the overall charge-neutrality within the L2NO4 lattice. This charge compensation takes place according to the following equation:

(1)
where refers to molecular oxygen present in the gas phase and where the other defect elements are described by Kröger-Vink notation: is a neutral vacant interstitial site, denotes a doubly negatively-charged oxygen ion in an interstitial site and represents a positively-charged hole carrier. Equation 1shows that the amount of present in L2NO4 is directly correlated to the hole carrier concentration. It is in this sense that can play the role of an acceptor dopant, giving rise to the p-type conductivity reported for this material. [15][16][17] The large decrease in resistivity with increasing oxygen content is clearly demonstrated by Thanh et al., [18] who had also been the sole to report on the resistive switching characteristics of L2NO4-based devices. In this work they used bulk L2NO4 samples with Ag symmetric electrodes and showed a certain influence of the oxygen content in the resistance switching ratio.

Aiming to the real implementation and functional optimization of this MIEC oxide in memristive devices we recently reported for the first time on the switching characteristics of L2NO4 thin film (67 nm) devices with Pt and Ti as inert and active electrodes, respectively.

The n-type TiO x interface layer that is naturally formed at the Ti/L2NO4 interface gives rise to remarkable analog switching capabilities, presenting very gradual set and reset transitions, optimal for the hardware implementation of synaptic functions. [14] However, the role of the interstitial oxygen ions (and concomitant presence of positively-charged hole carrier) in the memristive behavior of these novel VCM devices had not yet been studied. We use a planar, or "top-top" electrode configuration with macroscopic distances (a few hundreds of microns)

between the electrodes to avoid the formation of conductive filaments and to have a good control over the interface-type resistance changes in the memristive devices. In the conventional top-bottom "sandwich-type" configuration the low series resistance of L2NO4

(tens of nm-thick) is expected to lead to high currents, subsequent Joule heating and can ultimately induce an important degradation of the electrode material and/or the sandwiched material. While Cu or Ag-based Electro-Chemical Metallization (ECM) cells have been largely studied for their filamentary resistive switching mechanism, the formation of Pt filament in a Pt/TiO 2 /Pt memristive device has also been reported, [19] suggesting that the majority of electrode materials can migrate through the cell and short circuit the device if exposed to harsh programming conditions.

In this paper, we studied the effect of varying the oxygen content of the L2NO4 mixed conductor on the electrical and memristive properties of planar Pt/L2NO4/Ti thin film devices.

High oxygen contents in the pristine state enhance the memory properties of these devices by offering increased multilevel capabilities as well as a larger programming window.

Results and Discussion

2.1.Deposition of L2NO4 thin films and fabrication of Pt/L2NO4/Ti devices.

The optimal growth conditions for L2NO4 thin films are summarized in Table 1. Using these conditions, it was possible to grow highly oriented films on [1 0 0]-oriented SrTiO 3 (STO) single crystal substrates. Total pressure inside the reactor 5 Torr

The crystal quality and preferential orientation of the L2NO4 films can be observed in Figure 1.d where the θ-2θ X-ray diffraction (XRD) patterns show only the 0 0 l L2NO4 (l even) reflections of the L2NO4 phase in addition to the h 0 0 STO substrate peaks.

2.2.Tuning the oxygen content in L2NO4 through post annealing treatments.

In order to evaluate how the oxygen stoichiometry of L2NO4 affects its conduction and memristance properties, a batch of four L2NO4/STO samples was deposited simultaneously in the PiMOCVD reactor. One of the samples was kept as deposited (pristine) while the other three were annealed in O 2 , Ar or a highly reducing H 2 (6%)/Ar gas mixture at 500 ºC for 1 h prior to top electrode deposition. The diffractogram in Figure 2.a shows a clear displacement of the 006 L2NO4 peak to higher 2θ angles (lower interplanar spacing along the c crystallographic direction of L2NO4) when the samples are subjected to reducing conditions. This is in agreement with a loss of oxygen, and more particularly of interstitial oxygen. [7] The Ni oxidation state was determined from X-ray Absorption Near-Edge Structure S3, for more details). [20] The oxidation state of Ni then allows the calculation of the oxygen off-stoichiometry by chargebalance. The results are shown in Table 2 where they are compared to the values obtained by Woolley et al. who used the same technique to determine the δ value of L2NO4 ceramic samples under different measurement conditions. [20] The c lattice expansion was calculated form the 006 L2NO4 peak position and is plotted as a function of the oxygen off-stoichiometry δ in = 1373 K and pO 2 < 10 -10 bar), [21] no significant difference was observed when comparing the integrated intensity of the diffraction peaks relative to the L2NO4 phase before and after annealing at this relatively low temperature and time (727 K, 1h). We can thus conclude that the post-annealing treatments did not lead to a degradation of the L2NO4 film, as also confirmed by the similar morphologies of the SEM images taken of the surface of each sample (displayed in Erreur ! Source du renvoi introuvable..a-d, SI) and by the fact that no additional peaks were observed by XRD after the treatments.

2.3.The presence of a TiO x functional interlayer.

Pt and Ti electrodes (200 nm thick) were evaporated on top of the L2NO4 thin film to construct the memristive devices. When two inert Pt electrodes were used to construct Pt/L2NO4/Pt devices no memristivity was obtained as, as shown in Figure S2, regardless of the post annealing treatment (O 2 , Ar or H 2 ) the electrical characteristics remain completely ohmic up to ±10 V. Although Pt forms an ohmic contact with L2NO4, [14] Ti tends to naturally form an oxide (i.e. TiO x ) interlayer when put in contact with an oxide. [14,[22][23][24][25] The presence and composition of this interlayer in the Ti/L2NO4/STO stack was analyzed in detail. which grows spontaneously during the evaporation of Ti through the electrode microfabrication process. [14] The uniformity, thickness and chemical nature of this interlayer was analyzed in detail by Electron Energy Loss Spectrum (EELS) using an advanced spectral unmixing technique (known as the Vertex Component Analysis method, or VCA) and focusing in particular on the convoluted O K-edge Energy Loss Spectrum (ELS) (more details about this technique are available in the SI). The optimized results for the spectral unmixing were obtained using four components, corresponding to the four different phases appearing in the EELS maps: one for the STO substrate, one for the L2NO4 oxide thin film, one for the TiO x interlayer and finally one for the Ti top electrode. The resulting C1, C2, C3 and C4 abundance maps and their corresponding spectral components are displayed in Figure 3.b and c, respectively. First, it can be noticed that the calculated ELS corresponding to C1 is remarkably similar to the experimental spectrum measured in epitaxial thin films of defectfree STO, [26] and as expected, the location where this component is most present is in the STO substrate region (Figure 3.b). The second spectral component (C2) does not contain an oxygen fingerprint, it is a pure background signal which implies that there is no oxygen present in the upper part of the Ti electrode. As the corresponding abundance map suggests, component C3 corresponds to the fingerprint of oxygen in a L2NO4 environment, in agreement with the spectrum reported by Ravkina et al. [27] Finally, the last spectrum (C4) confirms the presence of oxygen chemically bounded to Ti in the interfacial region of the sample, forming an amorphous TiO x layer with a shape very similar to the reference spectrum for amorphous TiO 2 reported in [28] .

2.4.Role of the oxygen content in the resistivity of the L2NO4 thin films.

The resistivity values of the different films plotted in Figure 4 show a steady increase in resistivity from 5.3×10 -3 Ω.cm to 1.3×10 -2 Ω.cm when decreasing the δ oxygen content from 0.09 to 0.04. Interestingly, the film annealed in a highly reducing H 2 /Ar atmosphere corresponds to close to stoichiometric L2NO4 (estimated δ = 0.01) and shows an increase in resistivity of over two orders of magnitude to 5.7 Ω.cm. Similar values of resistivity (~5 Ω.cm at 300 K) have been reported for samples annealed in a highly reducing CO/CO 2 atmosphere. [29] These results confirm that the oxygen content has an important influence on the electrical properties of L2NO4 and that changes in the measured film resistivity can be directly related to the changes in the Ni oxidation state, oxygen content and c lattice parameter in the L2NO4/STO thin films (shown in Figure 2.c). annealed films, respectively). This is consistent with the interstitial oxygen ions acting as acceptor dopants through a self-doping process, as previously described, assuming however that the presence of the interstitial point defect has no significant influence in the band gap of L2NO4.

2.5.Role of the oxygen content on the memristive properties of Pt/L2NO4/Ti devices.

To construct the L2NO4-based memristive devices Pt has been used as chemically inactive ohmic contact, so that the Pt/L2NO4 junction offers little resistance to the passage of current.

Conversely, titanium was selected for its high oxygen affinity, making it a chemically active electrode that can easily form a TiO x interlayer at the Ti/L2NO4 junction as shown in Figure 3. This oxygen deficient interlayer may then play the role of an oxygen storage/release medium. The exchange of oxygen between the two materials by reversible redox reactions provides an effective playground for VCMs. A sketch of the Pt/L2NO4/Ti device is presented in Figure 5.a.

To illustrate the key role of oxygen on the memristive characteristics, three Pt/L2NO4/Ti devices with increasing oxygen content in L2NO4 were programmed from their Initial

Resistance State (IRS) to a High Resistance State (HRS) using the standardized initialization procedure which is described in the Experimental Section (and illustrated at the bottom of Interestingly, the oxygen annealed sample does not show the initial decrease in resistance and is directly programmed to a higher resistance state showing first a slow increase in resistance (during the first 40 pulses, approx. 500s) followed by a steeper increase. The initial resistance drop for the two reduced samples could originate from the field-induced drift of the interstitials ( ) present in the sample towards the Ti/L2NO4 interface. This would reduce the Ti/L2NO4 contact resistance (R C,Ti ) due to a local dopant increase at the Ti/L2NO4 interface, hence decreasing the carrier depletion width at this location (schematic illustrations depicting the proposed mechanisms shown in Figure S4). At a certain threshold concentration of (achieved after the first ten write pulses), the oxygen ions could start drifting into the Ti electrode and increase the resistance of the device by increasing the TiO x interlayer thickness and/or its oxygen stoichiometry (making it more insulating). In the case of the oxygenannealed sample, the oxygen concentration at the Ti/L2NO4 interface should already be high initially, so that oxygen ions can start drifting into the Ti electrode and oxidizing it as soon as the first write pulse is applied.

At the end of the initialization procedure, the resistance of the three samples continues to increase spontaneously (i.e. without the application of an external electric field) during the 100 s long readout step. This effect could be explained by the diffusion of the oxygen ions away from the Ti/L2NO4 interface to restore a homogeneous chemical gradient across the L2NO4 film and/or a stable space-charge zone near the Ti/L2NO4 interface (see Figure S4).

Although a certain TiO x formation cannot be excluded and could still occur during the first instants of time, once the temperature has cooled down (due to the absence of Joule heating) and the closest oxygen ions have drifted away, oxygen diffusion is expected to be the main process governing the resistance change. From a physical perspective it is generally assumed that the slowest process limiting the speed of redox-based resistive switching devices is the ion hopping, as there are many hops involved to travel a certain distance, compared to other processes such as redox reactions which only involve a single transition at an interface. [30] By extrapolating the high temperature diffusion coefficients measured by Burriel et al. in epitaxial thin films [9] , the characteristic diffusion length for L2NO4 at room temperature can be estimated, giving a value of 635.3 nm along the ab plane and of 0.86 nm along the c axis for at time of 100 s. This Fickian diffusion process would lower the doping concentration, increasing the depletion width thus also R C, Ti and the total resistance of the device. This effect is more important for the oxygen-annealed sample as a higher gradient of interstitial oxygen ions has to re-equilibrate. The orientation of the film crystal grains is expected to play an important role in the rate of the oxygen drift and diffusion processes (faster along the ab plane). However, to univocally assess the orientation effect on the memristive response of Ti/L2NO4/Pt devices, these would have to be built using La2NO4 single crystals such as the ones used by Burriel et al. [12] to measure the diffusion coefficients (at high temperatures)

along the two perpendicular crystallographic directions of interest.

The oxygen-annealed sample stands out in the relative resistance change it experienced during and after initialization. This sample shows a more than a 20-fold increase in resistance while the argon and hydrogen-annealed samples only show an 82% and 34% increase, respectively.

The larger resistivity of the reduced L2NO4 samples leads to a large voltage drop across the film. Conversely, this voltage drop will be concentrated mostly at the Ti/L2NO4 junction in the case of the more conducting oxygen-annealed sample, especially in the presence of a continuous TiO x interlayer. The formation of a thicker and/or more oxidized (thus more resistive) TiO x interlayer in this sample due to the field-induced drift of oxygen from L2NO4 into the Ti electrode is expected to result in a large relative resistance increase. On the other hand, the lower amount of oxygen ions available in L2NO4 combined with a smaller electric field at the Ti/L2NO4 junction and a higher IRS value results in a smaller relative resistance change for the reduced samples, possibly related to a smaller oxygen ion drift and more difficult oxidation of the Ti electrode. This then explains the smaller programming window measured for the reduced samples. for the oxidized sample can be observed. The absence of rectification can be an important marker of an inhomogeneous Schottky barrier height. [31] The I-V characteristics of the initialized samples are shown in Figure 5.d shows hysteretic memristive behavior with very gradual transitions both for the SET and RESET operations. The inset (corresponding to a magnification of the electrical characteristics at low voltage) puts in evidence that only the oxygen-annealed sample has a significant difference in the HRS and LRS when cycling the devices between +5 and -5 V. The presence of two well differentiated resistance states at low voltages (close to 0 V) is a signature of better memory retention capabilities for this device.

The bipolar characteristics (using write and erase pulses) as well as the relaxation behavior of the L2NO4-based memory were further analyzed for the optimized oxygen-annealed devices for which multilevel, analog-type memory programing capabilities have been demonstrated

(see details in the SI, Figure S5).

Under the ±5 V biasing conditions used (see It can thus be concluded that for the future scaling-up and integration oxygen-rich L2NO4devices are the most promising ones. The dimensions of the devices could be reduced remaining in a horizontal device structure by advanced lithography techniques or moving to a vertical one using, for example, the epitaxial transfer method on Si at room temperature. [32] When reducing the dimensions the voltage drop occurring in each of the localized regions of the device should be well controlled to avoid filamentary switching. In addition, a precise interplay between the TiO x interlayer thickness, stoichiometry and homogeneity and the oxygen content of L2NO4 are certainly crucial parameters that can be further optimized to tune the resistance-change and the retention characteristics of the memristive device.

Conclusion

L2NO4 was selected as mixed ionic-electronic conducting oxide to build memristive devices with tunable oxygen stoichiometry due to its inherent ability to accommodate oxygen interstitials in its structure, due to its fast oxygen diffusion and its semiconducting electronic

Experimental Section

Thin film preparation. L2NO4 thin films (~67 nm) have been deposited on 10x10 mm² (100)SrTiO 3 single crystal substrates purchased from Crystec. The substrates followed a mild 3-step cleaning process (acetone, isopropanol, deionized water) in an ultrasonicating bath before deposition of the L2NO4 film by PiMOCVD (more details about the deposition conditions are given in Table 1). The lamellas for the STEM experiments have been prepared by Focused Ion Beam (FIB) using a FEI strata 400S equipped with a dual beam electron/Ga + ions. The region of interest (ROI) is thinned to approximately 100 nm using a decreasing acceleration, from 16 kV to 2 kV in order to limit the amorphization of the sample. The high resolution STEM images have been acquired on a FEI TITAN Themis microscope with an electron acceleration of 200 kV.

Post annealing treatments

The microscope is probe corrected.

EELS composition maps have been obtained using a conventional analysis with the "Gatan GMS 3.20", integrating the signal under the respective element edges: O-K (~532eV), Ti-L2 (~462eV) and L3 (~456eV), Ni-L2 (~872eV) and L3(~855eV) and La-M4 (~849eV) and M5(~832eV) (taking in account the plural scattering).

VCA analysis: An advanced hyperspectral unmixing method was used to observe the variations in the environment of the O K-edge spectrum acquired by EELS by looking at its fine structure. The unmixing of the main end-member components was carried out using the Vertex Component Analysis (VCA) algorithm developed by J.M.P Nascimento and J.M.

Bioucas-Dias. [33] The use of a VCA algorithm constitutes a highly relevant method for discriminating EELS fine structures in high resolution hyperspectral images. [34] The automated determination of the spectral components limits the operator-induced variability and helps determining more precisely the regions which each of the components is present or where a mixture of components co-exist. Further constrains in the automatically generated EELS maps ensure that the results remain physically meaningful. Four components were required for the spectral unmixing to properly define all the pure end-members components present in a consistent and reproducible manner for all the EELS maps regions measured. A spectral unmixing using five components was also attempted but the fifth component generated undesired artefacts (mainly in the form of peak compensation in the background component). More details on the VCA data treatment are given in SI.

Current-Voltage (I-V) characteristics have been obtained using a source-measurement unit (Agilent B1500) using two tungsten-carbide probes operated in a voltage-control mode in a Karl-Suss probe station (acting as a vibration damper, a Faraday cage and a dark box to limit outside disturbances). The voltage is always applied on the Pt electrode while the Ti counter electrode remains grounded. The bipolar sweeps always start at zero bias and go as follows: 0

V  +V max  0 V  -V min  0 V, with a constant sweeping rate of ~350 mV/s. The resistance values extracted from the I-V curves were always calculated at V = +10 mV.

Resistance-time (R-t) characteristics, when operated in pulsed mode, consist in applying a voltage stress at higher voltages (≥1 V) to program the device to a higher or lower resistance state, and reading out the remnant resistance state of the device after each programing pulse at a lower V = +10 mV bias to ensure that no ionic movement would be triggered by the electrical field. The standardized initialization procedure (illustrated at the bottom of Figure

Supporting Information

Figure 1 .

 1 Figure 1. (a) Schematic representation of the tetragonal structure of stoichiometric La 2 NiO 4.00 showing the alternating Perovskite and rock-salt type layers, which is at the origin of the anisotropic conduction properties of the material. (b) Naming convention used in the text for the different atoms and their bounds composing the structure. O app stands for "apical oxygen atom", O eq for "equatorial oxygen atom", O i for "interstitial oxygen atom", Ni-O app for "nickel-apical oxygen bond" and Ni-O eq for "nickel-equatorial

  (XANES) experiments by measuring the displacement of the Ni-K edge, from which the δ value could be estimated. The inset in Figure 2.b shows the shift of the Ni K-edge towards higher energies when the samples are treated in increasingly oxidizing atmospheres, from which the Ni oxidation state was calculated using as references the Ni-K edge positions measured by Woolley et al. (see supplementary information (SI), Figure

Figure 2 .

 2 c. A linear relationship is observed between the c lattice parameter and the oxygen off-stoichiometry (black line). The out of plane lattice parameter increases from c = 12.62 Å to 12.69 Å when the estimated oxygen over-stoichiometry varies from δ = 0.01 to 0.09. Although L2NO4 is expected decompose into La 2 O 3 + Ni at very high temperatures and under very reducing conditions (for example T

Figure 2 .

 2 Figure 2. (a) Out of plane θ -2θ XRD pattern of L2NO4/STO samples with different gas atmosphere anneals (500 ºC/1 h). A clear displacement of the out of plane 0 0 6 L2NO4 Bragg peak to higher 2θ values (lower interplanar spacing) is observed. The out-of-plane c parameter equals 12.62 Å, 12.64 Å, 12.66 Å and 12.69 Å for the H 2 , Ar, pristine and O 2 anneal, respectively. This behavior is expected when interstitial oxygen ions are removed/introduced from/in the lattice. (b) XANES spectra at the Ni K-edge obtained for L2NO4 with different gas anneals. The inset shows a magnification of the edge area, highlighting the shift of the Ni K-edge towards higher energies when the samples are treated in increasingly oxidizing atmospheres. A solid Ni foil was used as reference for the calibration of the curves (dashed-dotted line). The edge position was obtained using the first/second derivative method. (c)

Figure 3 .

 3 a shows the atomic composition maps (for Ti, O, Ni and La) obtained by Electron-Energy Loss Spectroscopy (EELS) on a TEM cross-section specimen taken beneath a Ti top electrode. The L2NO4 film appears continuous, dense and homogeneous in composition (no visible gradient in the O, Ni or La maps), with a thickness of ~67 nm. It can be noticed that an additional inhomogeneous darker contrast is present between the L2NO4 thin film and the Ti top electrode in the Annular Dark Field (ADF) image shown at the top of Figure 3.a. We have previously shown that this interlayer corresponds to a sub-stoichiometric TiO x interlayer

Figure 3 :

 3 Figure 3: (a) EELS composition maps (in atoms per unit volume) for Ti, O, Ni and La taken on a Ti/L2NO4/STO TEM cross-section specimen. The O-K (~532eV), Ti-L2 (~462 eV) and L3 (~456 eV), Ni-L2 (~872 eV) and L3(~855 eV) La-M4 (~849 eV) and M5 (~832 eV) edges were used for quantification. The top image corresponds to a wide angle ADF image of the entire stack showing the region of the sample used for the mapping. (b) Heat maps showing the fractional abundance of the 4 components displayed in (c) which describe the different oxygen environments present in the stack. (c) The hyperspectral O K-edge measured by EELS in each pixel of the image was unmixed using a VCA data processing technique. The resulting component spectra C1, C2, C3 and C4 are fingerprints of oxygen in a STO environment, in the electrode (no oxygen), in the L2NO4 thin film and in a TiO x environment, respectively.

Figure 4 :

 4 Figure 4: Evolution of the L2NO4 film resistivity with increasing oxygen stoichiometry (δ equals 0.01, 0.04, 0.05 and 0.09, respectively).

Figure 5 .Figure 5 .

 55 Figure 5.c). The results for the three Pt/L2NO4/Ti samples with different oxygen contents, all

Figure 5 :

 5 Figure 5: (a) Sketch of the sample showing the biasing conditions. (b) I-V characteristics of the Pt/L2NO4/Ti devices annealed in O 2 , Ar or H 2 /Ar in their IRS. (c) Initialization procedure of the three devices. Only the read pulses are shown, the vertical black arrows show the start and the end of the write sequence (every fifth read pulse is shown to increase readability). An illustration of the programming sequence is shown at the bottom of the figure. (d) I-V characteristics of the three devices after initialization. The inset is a magnification of the I-V curve between -0.2 V and +0.5 V.

Figure 5 .

 5 d) only the oxygen-annealed sample has memory retention, while the changes induced in the reduced samples are volatile. The argon-annealed sample has a hysteretic behavior for V > 300 mV, however the opening of the IV curve disappears at low voltages, while the hydrogen-annealed sample has a very small opening for V > 1 V. Thus we can conclude that oxygen is clearly playing a major role in the memristive properties of the L2NO4-devices. The oxidation/reduction of Ti (TiO x ) could explain the varying retention characteristics. While redox reactions are likely present in the oxygen-annealed case, in the reduced samples the hysteresis could be due to a temporary variation in the hole-doping concentration at the Ti/L2NO4 interface, changing its electrical characteristics in a reversible and volatile manner.

  transport properties. For this highly oriented L2NO4 thin films were first deposited by PiMOCVD on SrTiO 3 single crystal substrates and then subjected to different anneal treatments under oxidizing or reducing conditions. The combination of structural (XRD), chemical (XANES), microscopy (SEM and PEEM) and electrical measurement techniques proved the direct relationship between the changes in oxygen content, cell parameter, Ni oxidation state, work function and resistivity in the L2NO4 films after the anneals. Next, films with three different oxygen stoichiometries were used to build functional memristive devices: platinum was selected as inert ohmic electrode while titanium was selected for its high oxygen affinity, making it a chemically active electrode. As proven by a combination of STEM, EELS maps and VCA analysis titanium formed an amorphous TiO x interlayer at the Ti/L2NO4 junction, key for the appearance of memristance. The initialization of the device achieved by applying a train of negative write pulses on the Pt side (in Pt/L2NO4/Ti) results in a resistance increase in all cases. Remarkably the film with the highest oxygen content presents the highest (2000 %) relative resistance increase and the largest opening of the IV memristance curve after initialization. This study has clearly shown that the additional oxygen ions present in the form of interstitial point defects in L2NO4 give rise to the memristive properties of the device. These promising results offer additional insights into the use of L2NO4-based heterostructures with tuned oxygen content for the development of memristive systems for analog computing and synaptic applications.

  have been used to tune the bulk oxygen content of L2NO4. For this, three different pristine (as deposited) L2NO4/STO samples have been annealed in oxidizing (O 2 ) or reducing (Ar and H 2 (6%)/Ar) atmospheres at 500 °C during 1 h, with a heating ramp of 10 °C/min and natural convection to cool down the sample back to room temperature.Metallic electrodes have been fabricated in high vacuum conditions by e-beam induced metal evaporation (MEB550 from PLASSYS) using a laser-assisted lithography process (µPG 101 from HEIDELBERG Instruments) in local cleanroom facilities. A mild Ar + etch has been used to prepare the exposed surface of the patterned sample and remove potential left-overs of the photoresist before metal evaporation. 200 nm thick Pt and Ti electrodes have been deposited (with a growth rate of 0.5 nm/s) to provide sufficient mechanical resistance and prevent potential degradation during electrical contacting.Phase identification, structural and crystallinity characterization of the samples have been performed using X-ray diffraction (XRD) in θ-2θ geometry (Bruker D8 Advance) with monochromatic CuK α1 radiation (λ = 1.5406 Å).The resistivity of the films has been measured using a 4-probe setup.XANES experiments have been carried out on the CRG beamline SpLine (BM25A) at theEuropean Synchrotron Radiation Facility (ESRF), more details about the experiment are available in the SI.PEEM measurements were carried out with a NanoESCA MkI spectromicroscope (ScientaOmicron) in ultra-high vacuum (base pressure 2×10 -10 mbar) on samples treated beforehand using argon cluster ions in order to remove the adventitious carbon layer without damaging the oxide material underneath. VUV excitation (He I, h=21.22 eV) and aberrationcompensated imaging spectrometer provided secondary electron spectra from which the work function threshold was determined within ±100 meV. The energy scale of the spectra was referenced to the Fermi level of the sample.

5. c )

 c consists of the initial readout of the IRS at +10 mV during 100 s to test its stability over time, followed by a train of 100 write/read ([-10 V; 1500 ms]/[+10 mV; 250 ms]) pulses and, finally, another 100 s long readout at +10 mV to probe the new resistance state of and measure its stability over time. All the electrical measurements have been carried out in ambient conditions.

Table 1 : Optimized deposition conditions used for the growth of L2NO4 thin films by PiMOCVD.

 1

	Parameter

Table 2 : Measured Ni K-edge position (XANES) and corresponding values of the Ni oxidation state and oxygen over-stoichiometry δ in L2NO4 .

 2

		Material	Annealing conditions	Measurement conditions	Ni K-edge position (eV)	Ni oxidation state	estimated δ value	Reference
	Calibration data	Ni foil NiO LaNiO 3	---	RT in air RT in air RT in air	8333.0 8345.2 8348.2	0 2 3	---	This work [20] [20]
	Ceramic samples	La 2 NiO 4+δ La 2 NiO 4+δ La 2 NiO 4+δ		RT in air at 650 °C in air 8345.5 8346.1 at 650 °C in N 2 8347.2	2.24 2.15 2.08	0.12 0.08 0.04	[20] [20] [20]
		La 2 NiO 4+δ	O 2 , 500 °C during 1 h	RT in air	8345.76	2.17	0.09	This work
		La 2 NiO 4+δ	pristine	RT in air	8345.54	2.11	0.05	This work
	Thin films	La 2 NiO 4+δ	Ar, 500 °C during 1 h	RT in air	8345.47	2.08	0.04	This work
		La 2 NiO 4+δ	H 2 , 500 °C during 1 h	RT in air	8345.27	2.02	0.01	This work

Acknowledgements

This work has been partially supported by the LabEx Minos ANR-10-LABX-55-01 and by two ANR funded projects MICROSWITCH (ANR-14-ACHN-0012) and Alps Memories (ANR-15-CE24-0018). We acknowledge the European Synchrotron Radiation Facility (ESRF), the Spanish Ministry of Science, Innovation and Universities (MCIU), and The Spanish National Research Council (CSIC) for provision of synchrotron radiation facilities and we would like to thank beamline staff for assistance in using beamline CRG BM25-SpLine. E.S.-C. and G.R.C. thank the MCIU and the CSIC for financially supporting this research project and the beamline operation under Grant No. PIE 2010-6-OE-013.

K. Momma and F. Izumi are kindly acknowledged for creating the VESTA [35] software (3D visualization of crystallographic structure).

The authors declare no competing financial interest.


"NLMS_480" -2018/10/29 -14:34 -page 1 -#1

Translation Surfaces and Moduli Spaces

DIMITRI ZVONKINE

Translation surfaces are surfaces glued from polygons by identifying only parallel sides of equal lengths. Many questions about them can be asked in elementary terms. Sometimes these questions can be answered using elementary methods such as polygonal billiards, monodromies and permutations, cutting and pasting. Other times they require advanced methods of dynamics or intersection theory on moduli spaces.

Translation surfaces

Figure 1 represents two polygons whose sides are divided into pairs of same length and direction. By gluing the polygons along these pairs of sides we obtain a translation surface. This is a closed surface with a at metric (a Riemannian metric with zero curvature, locally isometric to the Euclidean plane) and a nite number of conic singularities whose angles are multiples of 2π. In this example, the surface is of genus 3, and it has 2 conical singularities with angles 4π (white dots) and 8π (black dots). In general, given a translation surface of genus g with n conical singularities that have angles 2π(k i + 1), 1 ≤ i ≤ n, the Gauss-Bonnet formula implies k i = 2g -2.

Figure 2. This is the same translation surface as in Figure 1 Any translation surface can be cut into polygons in many ways. We will consider all such cuttings as equivalent; in other words, we are interested in the surface itself and in its metric, not in a particular cutting. For instance, the polygon in Figure 2 is obtained from those in Figure 1 by cutting the rightmost polygon in two and gluing the pieces along distinguished pairs of sides (the gluings are shown as dotted lines). Both gures represent the same translation surface.

Let C be a translation surface and S 1 the circle of directions on the plane. Then C ×S 1 carries a natural time ow: just imagine a ball rolling on the surface C along the geodesic in the given direction. If the ball meets a side of a polygon it reappears from the other side of the distinguished pair. If it hits a vertex, then the ow is not well-de ned. Thus to each translation surface corresponds a dynamical system, and we can ask whether the time ow is ergodic, whether there are periodic trajectories, and so on. See, for instance, the study of periodic orbits in the regular pentagonal billiard [START_REF] Davis | Periodic trajectories in the regular pentagon[END_REF] and the corresponding jewelry at tinyurl.com/yboob8dw. Here, however, we will be more interested in the space H g (k 1 , . . . , k n ) of all translation surfaces of genus g with conical angles 2π(k i + 1), 1 ≤ i ≤ n.

The complex numbers represented by the sides of the polygons form local coordinates on this space. Moreover, they endow it with an integer a ne structure. This means that choosing a di erent cutting of the surface induces an a ne change of local coordinates with integer coe cients. In particular, this allows us to de ne a canonical volume form on the space H g (k 1 , . . . , k n ). While the total volume of this space is in nite, the volume of H ≤1 g (k 1 , . . . , k n ), the space of translation surfaces with area at most 1, was shown to be nite by Masur [START_REF] Masur | Interval exchange transformations and measured foliations[END_REF] and Veech [START_REF] Veech | Gauss measures for transformations on the space of interval exchange maps[END_REF].

Question 1. What is the volume of H ≤1 g (k 1 , . . . , k n )?
One of the ways to answer this question is to count the translation surfaces whose coordinates are Gaussian integers a +bi , a, b ∈ Z. Eskin and Okounkov [START_REF] Eskin | Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic di erentials[END_REF] showed that the number of such surfaces with area "NLMS_480" -2018/10/29 -14:34 -page 2 -#2 up to N grows as V • N 2g -1+n , where V is the volume of the space H ≤1 g (k 1 , . . . , k n ) and 2g -1 + n its complex dimension. Using representation theory and the theory of quasi-modular forms they were able to produce a method to compute the volumes of all spaces H ≤1 g (k 1 , . . . , k n ). This was later re ned to deal with the volumes of connected components of these spaces, classi ed by Kontsevich and Zorich [START_REF] Kontsevich | Connected components of the moduli spaces of Abelian di erentials with prescribed singularities[END_REF].

Let us illustrate the method with the simplest example of genus 1 translation surfaces. Because such a surface has an in nite automorphism group acting by global translations, it is convenient to mark one point on the surface and consider the space of translation surfaces with one marked point. The marked point could be viewed as a conical point with angle 2π.

Every genus 1 translation surface of area N with one marked point can be cut into a parallelogram, as in Figure 3, in a unique way, d being an integral divisor of N , and 0 ≤ a ≤ d an integer. The marked point is the image of the vertices of the parallelogram under the gluing. Thus the number of such translation surfaces is the sum of divisors of N , and the corresponding generating function is the quasi-modular form

G 2 (q ) = - 1 24 + N d |N dq N .
The well-known transformation law for this quasimodular form reads

G 2 e -2πi /τ = τ 2 G 2 e 2πi τ + i π 4 ,
where the rst term is the standard transformation law for modular forms and the second term is the quasi-modular correction. Plugging h = 2πi /τ and recalling that G 2 (0) = -1/24, we obtain the asymptotic expansion

G 2 (e -h ) ∼ π 2 6h 2 -
1 2h as h → 0 + . Note that this is the whole asymptotic expansion: the di erence between the left-hand side and the right-hand side decreases exponentially.

Returning to our original question, we now have

G 2 (e -h ) ∼ π 2 6
1 (1q ) 2 , as h → 0 + , so the coe cients of G 2 (q ) grow as π 2 N /6, the partial sums up to degree N as π 2 N 2 /12, hence π 2 /12 is the volume of the space H ≤1 1;0 .

The group GL(2, R) acts on plane polygons and thus on the space of translation surfaces in the natural way. This leads to new important questions.

Question 2. Describe the orbit closures of the GL(2, R) action in the space of translation surfaces.

We will come back to this later. If the stabilizer of a translation surface is a lattice, it is called a Veech surface. Veech surfaces have not been classi ed so far, but many unexpected examples have been constructed.

Example. Consider the translation surface obtained by gluing together the opposite sides of a regular 2n-gon. One can show that the matrix

1 cotan π 2n 0 1
lies in its stabilizer and deduce that the stabilizer is a lattice. we obtain a ow on the space H g (k 1 , . . . , k n ), called the Teichmüller ow. Masur and Veech [START_REF] Masur | Interval exchange transformations and measured foliations[END_REF][START_REF] Veech | Gauss measures for transformations on the space of interval exchange maps[END_REF] showed that this ow is ergodic on every connected component of this space.

0 < β g ≤ . . . , ≤ β 1 ≤ α 1 ≤ • • • ≤ α g ∈ R
the eigenvalues of A(t )A + (t ). Because A(t ) preserves the intersection pairing on H 1 (C, R) we actually have

α i (t )β i (t ) = 1.
The Lyapunov exponents of the Teichmüller ow are de ned as

λ i = lim t →∞ 1 2t ln |α i (t )|.
By the Oseledets ergodic theorem, these limits exist and are the same for almost all starting points.

Question 4. What are the Lyapunov exponents λ 1 , . . . , λ g for a given connected component of

H g (k 1 , . . . , k n )?
Main results on this topic include the proof by Avila and Viana [START_REF] Avila | Simplicity of Lyapunov spectra: proof of the Zorich-Kontsevich conjecture[END_REF] that the Lyapunov spectrum is simple, that is, 0 < λ 1 < • • • < λ g , and the computation of the sum λ 1 + • • • + λ g by Eskin, Kontsevich and Zorich [START_REF] Eskin | Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic ow[END_REF]. The complete answer, however, is still unknown.

Moduli spaces and Abelian di erentials

The space H g (k 1 , . . . , k n ) can be viewed in a completely di erent way if we realize that a translation surface is the same thing as a Riemann surface with an Abelian di erential (i.e., a holomorphic di erential 1-form).

Indeed, since every polygon lies in the complex plane, it inherits the complex coordinate z , that can be taken as a local coordinate on the translation surface. It also induces the holomorphic 1-form dz on the polygon. These local coordinates and holomorphic 1-forms glue nicely along the edges of the polygons: the change of local coordinate from one chart to the other is just a translation. A special treatment is needed at the conical points. If z i is a conical point with angle 2π(k i + 1), we can use w = (zz i ) Denote by M g,n the moduli space of genus g Riemann surfaces with n marked points. This space is well-de ned as soon as the Euler characteristic 2 -2gn of the surface punctured at the marked points is negative. Its points parametrize all possible complex structures on an oriented genus g surface with n distinct marked numbered points, up to a point-preserving isomorphism. For instance, M 0,3 is a point, because all genus 0 Riemann surfaces are isomorphic to CP 1 and any three points can be brought to 0, 1, ∞ in a unique way by an isomorphism z → (az + b)/(c z + d ) of CP 1 . Similarly, the space M 0,4 is isomorphic to CP 1 \ {0, 1, ∞}. Indeed, by an isomorphism of CP 1 we can bring four points x 1 , x 2 , x 3 , x 4 to 0, 1, ∞, t , where

t = x 4 -x 1 x 2 -x 1 : x 4 -x 3 x 2 -x 3 ∈ CP 1 \ {0, 1, ∞} = M 0,4
is the cross-ratio of the four points. Further, the space M 1,1 is the modular gure H/SL(2, Z), where H is the upper-half plane. Indeed, any elliptic curve can be represented as C/(Z + τZ) for τ ∈ H. The SL(2, Z) action corresponds to replacing the basis (1, τ) of the lattice Z+ τZ with the basis (c τ +d, aτ + b), and then re-scaling the lattice so as to bring the rst vector of the basis to 1, the second vector becoming τ = (aτ + b)/(c τ + d ). The three simplest moduli spaces described above are shown in Figure 5.

M 0,3 M 0,4 M 1,1

Figure 5. Three simplest moduli spaces

The moduli space M g,n admits a natural compactication M g,n called the Deligne-Mumford compacti ca-"NLMS_480" -2018/10/29 -14:34 -page 4 -#4 tion. It is the moduli space of stable curves, that is, nodal curves with n distinct smooth marked points that have a nite number of automorphisms. For instance, M 0,4 is CP 1 , where the three extra points 0, 1, ∞ correspond to the three stable curves in the Figure 6. The moduli space M 1,1 is obtained from M 1,1 by adding one extra point. The corresponding curve is CP 1 with points x = 0 and x = ∞ identi ed into a node and marked point at x = 1, see Figure 7. The moduli space M g,n is a smooth Deligne-Mumford stack, or orbifold, of complex dimension 3g -3 + n.

The points of M g,n \ M g,n , that we add to compactify the space, parametrize nonsmooth stable curves. They form a normal crossings divisor in M g,n called the boundary.

The Hodge bundle E g,n → M g,n is the rank g vector bundle over the moduli space whose ber over a point p ∈ M g,n is the space of Abelian di erentials on the corresponding curve C p . The Hodge bundle extends naturally to M g,n . Without going into details of this, let us mention that an Abelian di erential on a stable curve is a meromorphic 1-form on each irreducible component of the curve with at most simple poles at the nodes of the curve, the residues at the two branches of the node being opposite to each other. With this de nition it is easy to check that the Abelian di erentials form a g -dimensional vector space on any stable curve of genus g . For instance, on the genus 1 nodal curve obtained by identifying x = 0 and x = ∞ on CP 1 , all Abelian di erentials are proportional to dx/x.

Once we have introduced the moduli space and the Hodge bundle, we can remark that every space H g (k

{(C, x) ∈ M 2,1 | x is a Weierstraß point}.
The cohomology class Poincaré dual to this divisor was rst determined by Eisenbud and Harris [START_REF] Eisenbud | The Kodaira dimension of the moduli space of curves of genus ≥ 23[END_REF]. It is equal to

3ψ 1 -6 5 - 1 10
.

So what do all these terms mean? The second and the third terms are pictures representing boundary divisors of M 2,1 . The second term is the boundary divisor parametrizing curves with a separating node; the third term is, similarly, the boundary divisor parametrizing curves with a nonseparating node.

In both cases we take the Poincaré dual cohomology "NLMS_480" -2018/10/29 -14:34 -page 5 -#5

classes of these divisors. The class ψ 1 is the rst Chern class 1 of a line bundle L 1 over M 2,1 . The ber of this bundle over a point (C, x) ∈ M 2,1 is the cotangent line to C at x. In general, there are n line bundles L 1 , . . . , L n like that over M g,n , corresponding to the n marked points.

A general method to compute the cohomology classes Poincaré dual to PH g (k 1 , . . . , k n ) in PE g,n was developed by Sauvaget [START_REF] Sauvaget | Cohomology classes of strata of di erentials[END_REF], though it does not lead to a closed formula.

A conjectural closed formula for the cohomology class of the image of H g (k 1 , . . . , k n ) in M g,n was proposed in the appendix of [START_REF] Pandharipande | Tautological relations via r -spin structures[END_REF]. It involves the ψ-classes introduced above, the classes of boundary strata, while their coe cients are some special values of Bernoulli polynomials.

In conclusion, let us mention two developments that connect the two views on the spaces of translation surfaces.

The rst one is a conjecture by Sauvaget relating the cohomology class Poincaré dual to PH g (k 1 , . . . , k n ) to the volume of H ≤1 g (k 1 , . . . , k n ). For n = 1 the conjecture is proved in [START_REF] Sauvaget | Volumes and Siegel-Veech constants of H(2g -2) and Hodge integrals[END_REF].

The second is a result by Eskin, Mirzakhani and Mohamadi [START_REF] Eskin | Isolation, equidistribution, and orbit closures for the SL(2, R) action on moduli space[END_REF] that Anton Zorich, in his review paper [START_REF] Zorich | The Magic Wand Theorem of A. Eskin and M. Mirzakhani[END_REF] called "the magical wand theorem". It states that any GL(2, Z)-invariant closed subset of H g (k 1 , . . . , k n ) is an algebraic subvariety of E g,n de ned over Q; locally, in period coordinates it is an a ne subspace. This is the single most important step towards the classi cation of all GL(2, Z)invariant closed sets.

Figure 1 .

 1 Figure 1. These two polygons can be glued into a translation surface

Figure 3 .

 3 Figure 3. All translation surfaces of g = 1 with 1 marked point look like this

Question 3 .

 3 Restrict the action to SL(2, R). What are the possible stabilizers of translation surfaces? For what surfaces is the stabilizer a lattice in SL(2, R)?

Figure 4 .

 4 Figure 4. An element of SL(2, R) that stabilizes the regular 2n-gon translation surface

Figure 6 .

 6 Figure 6. The three nonsmooth stable curves of g = 0 with 4 marked points

Figure 7 .

 7 Figure 7. The unique nonsmooth stable curve of g = 1 with one marked point

  More precisely, if we consider a small open set U in H g (k 1 , . . . , k n ), we can naturally identify the groups H 1 (C, R) for all translation surfaces C parametrized by U , since these surfaces are obtained from each other by small deformations. Now, pick a point in U and suppose that at some moment of time t the ow brings us back to U . The translation surface has undergone some transformation that acts by a linear map A(t ) on the homology group H 1 (C, R). Denote by

An important question is: what does this ow do to the homology classes of the translation surface as t → ∞?

  1/(k i +1) as the local coordinate at this point. The Abelian di erential dz = d (w k i +1 ) has a zero of order k i at the conical point. Thus we obtain a Riemann surface C with n marked points x 1 , . . . , x n and an Abelian di erential α that has zeros of orders k 1 , . . . , k n at x 1 , . . . , x n .The sides of the polygons are now the periods of α, either absolute or relative between two zeros. The complete set of periods on a basis of the group H 1 (C, {x 1 , . . . , x n }, Z) forms a set of local coordinates on H g (k 1 , . . . , k n ) that are called period coordinates.

  1 , . . . , k n ) of translation surfaces is naturally embedded into the Hodge bundle E g,0 . Speci cally, it is the locus of Abelian di erentials with n zeros of orders k 1 , . . . , k n . Actually, these spaces form a strati cation of the total space of the Hodge bundle over M g,n . We can also view H g (k 1 , . . . , k n ) as a subspace of E g,n if we decide to mark the zeros of the di erential. Note that the projectivization PE g,n is a compact orbifold, and the projectivizations PH g (k 1 , . . . , k n ) are its closed sub-orbifolds. This new setting leads to a new series of questions. Do the spaces H g (k 1 , . . . , k n ) ⊂ E g,n have smooth closures?The answer to this precise question is negative: the closures are not even normal. It is conjectured, however, that their normalizations are smooth. This would provide a natural smooth compacti cation for every space H g (k 1 , . . . , k n ). What is the cohomology class Poincaré dual to PH g (k 1 , . . . , k n ) in PE g,n ? What is the pushforward of this cohomology class to M g,n ?As the simplest example, consider the space H

	Question 5. Question 6.

[START_REF] Davis | Periodic trajectories in the regular pentagon[END_REF]

(2) of Abelian di erentials with a double zero on a genus 2 curve. Every genus 2 curve C carries two linearly independent Abelian di erentials α and β , each of which has two simple zeros or a double zero. Their ratio α/β is a degree 2 map from C to CP 1 or, in more intrinsic terms, to the projectivization of the dual to the space of Abelian di erentials on C . This shows that every genus 2 curve is hyperelliptic. The degree 2 map described above has six rami cation points, called the Weierstraß points. Every Abelian di erential on C has either two zeros whose images under the map α/β coincide, or a double zero at a Weierstraß point. Thus the image of H 2 (2) in M 2,1 is the divisor

Dimitri Zvonkine

Dimitri Zvonkine is a CNRS researcher at the Versailles University, France. He works on moduli spaces, intersection theory, Gromov-Witten invariants, and integrable systems. He was born in Moscow and likes math from the age of three, as documented in his father's book Math from Three to Seven. He likes songs by Georges Brassens, Tom Lehrer, and Yuli Kim, and those from Kabaret Starszych Panów. 1 One of the ways to de ne the rst Chern class of a line bundle L is to construct a meromorphic section of L and take the divisor of its zeros minus the divisor of its poles. The Poincaré dual cohomology class of the di erence between these two divisors does not depend on the section and represents the rst Chern class of L.