Klaasjan Maas

Chloé Wulles

Jose Manuel

Caicedo Roque

Belén Ballesteros

Valentin Lafarge

José Santiso

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

Jose Santiso
email: <jose.santiso@icn2.cat>

Role of pO 2 and film microstructure on the memristive properties of La 2 NiO 4+ /LaNiO 3- bilayers

LaNiO3/La2NiO4 bilayers deposited at varying pO2 conditions resulted in remarkable differences in film microstructure and cell parameters, directly affecting the electrical behaviour of Pt/LaNiO3/La2NiO4/Pt devices. The devices deposited at low pO2 showed the largest memristance. We propose this is due to the formation of a p-type Schottky contact between LaNiO3 and La2NiO4, where the extent of its carrier depletion width can be modulated by the electric-field induced drift of interstitial oxygen ions acting as mobile acceptor dopants in La2NiO4.

Introduction

Mixed ionic electronic conducting materials (MIEC) based on transition metal oxides are key elements in many different applications, such as ceramic gas separation membranes, gas sensing or as cathode materials for Solid Oxide Fuel Cells 1,[START_REF] Kilner | Materials for intermediatetemperature solid-oxide fuel cells[END_REF] . Recently, a new application has come into play for the use as MIECs oxides in resistive switching devices, more particularly, in those based on the valence change mechanism (VCMs). The combined electronic conductivity and oxide ion mobility in these materials are key aspects in the functioning of these memristive devices. The drift of oxygen ions induced by the application of an external electric field triggers internal redox reactions, locally changing the structure and composition of the material and therefore also its electrical properties. Typically, VCMs are based on the formation of conducting filaments across an insulating oxide such as SrTiO3, TiO2, Ta2O5 or HfO2. This insulator is sandwiched between two metal electrodes, the conducting filament is produced by the ion migration of mobile donors such as oxygen vacancies (𝑉 𝑂 •• ) or cation interstitials under high electric fields and can be considered as a soft dielectric breakdown. [START_REF] Lenser | Resistive Switching[END_REF] In addition, interface-type switching has been demonstrated for some MIEC oxides, such as Pr1-xCaxMnO3 (PCMO), where the mobile defects are oxygen vacancies. [START_REF] Baeumer | [END_REF] However, very few studies are devoted to other types of MIEC oxides where the main mobile species are oxygen interstitials (Oi''), as is the case in La2NiO4. 5,6 The additional interstitial oxygen ions (usually expressed as δ) can be stabilized in the form of over-stoichiometric La2NiO4+δ, partially oxidizing Ni 2+ into Ni 3+ and resulting into a larger concentration of electronic (hole) carriers. The p-type semiconducting properties and high oxygen diffusion of La2NiO4+δ make it an interesting material for the fabrication of VCM-based resistive switching devices.

La2NiO4 has a layered Ruddelsden-Popper-type structure consisting of alternate La-Ni-O perovskite and La-O rock salt layers along the c-axis of the structure. 7 The oxygen diffusion has been proven to be highly anisotropic in this layered structure. [8][9][10][11] Indeed, the interstitial oxygen anions located in the rock salt layers diffuse preferentially along the a-b plane of the structure through an interstitialcy mechanism. 8 Similarly, the electronic conductivity takes place preferentially along the perovskite layers, leading to about three orders of magnitude higher electrical conductivity along the a-b plane compared to the c-axis direction at room temperature. 12,[START_REF] Honig | Localization and Metal-Insulator Transitions[END_REF] In this sense the control over the microstructure of the material, and more particularly the preferential orientation of the crystal domains, is essential to understand the memristive properties of this material when used in VCM-type devices. For this study LaNiO3, a perovskite-structure oxide material showing metallic conductivity, [START_REF] Yu | [END_REF] has been selected as template electrode due its chemical affinity and crystallographic match with La2NiO4 and. To the best of our knowledge the electrical and memristive behavior of the LaNiO3/La2NiO4 bilayer interface remains yet to be explored. This work aims at studying the microstructure and electrical properties of LaNiO3/LaNiO4 epitaxial bilayers deposited under different oxygen partial pressure (pO2) during film growth in order to tune the oxygen content and therefore the contact properties and memristive behaviour of these complex-oxide heterojunctions. This study demonstrates that not only the oxygen stoichiometry of the films, but also the overall crystal domain orientations are largely affected by the deposition conditions, leading to substantial differences in the memristive properties of these MIEC-based devices.

Experimental Methodology

LaNiO3 / La2NiO4 epitaxial bilayers were grown by Pulsed laser deposition by using a pulsed UV excimer laser (Lambda Physik Compex Pro 201, =248 nm) in a PLD-Workstation from Surface-Tec, GmbH, and corresponding ceramic targets prepared by solid-state reaction. SrTiO3(001) single crystals (10x10 mm 2 , from Crystec, GmbH) were used as substrates. LaNiO3 was chosen as bottom electrode material, providing a crystal matching for the subsequent epitaxial growth of La2NiO4. The deposition temperatures were 600 C and 700 C for LaNiO3 and La2NiO4, respectively, as previously determined for optimal crystal quality of single layers deposited on SrTiO3 substrates. The values of the oxygen pressure during deposition were 12 mTorr and 70 mTorr. The pO2 range was limited by the PLD system as it can only operate in high vacuum conditions. Other conditions for the sequential deposition of LaNiO3 and La2NiO4

were the same: target-to-substrate distance of 55 mm, and laser fluence of 1.2 J/cm 2 (spot size 7.3 mm 2 ), and pulse repetition rate of 2 Hz. Given the different ablation rates of the two targets and the different pO2 conditions the number of laser pulses was adjusted to obtain an approximate thickness of 30 nm for both LaNiO3 and La2NiO4 layers. Three sets of bilayers were prepared combining the growth of LaNiO3 and subsequent growth of La2NiO4 at either "High" pO2 (70 mTorr) or "Low" pO2 (12 mTorr), resulting in three bilayers with different pressure combinations, namely H/L, L/L and L/H samples. The attempted growth of a H/H bilayer was discarded from the study as it showed insufficient crystal quality. The deposition protocols were designed to avoid as much as possible some parasitic effects. For instance, the difference in the optimal growth temperature between LaNiO3 and La2NiO4 (600 °C and 700 °C, respectively) made it very important to minimize the time interval between subsequent depositions, since LaNiO3 is prone to vary its oxygen stoichiometry and eventually decompose at high temperatures and low pO2, when heating up from 600 °C to 700 °C. Therefore, this was an important parameter to control. For the bilayer grown under constant low pO2 (L/L), La2NiO4 was deposited as soon as the sample reached 700 °C after a heating ramp of 10 C/min (10 min). For the growth of the H/L and L/H bilayers, the pO2 was only changed after having heated up the sample from 600 °C to 700 °C, and then waiting 1 min for the pO2 to stabilize before growing the second layer in different pO2. The samples were all cooled down under the pO2 used for the growth of the top La2NiO4 layer. For a proper comparison single layers of LaNiO3 and La2NiO4 were also grown at the two different pO2 conditions. The XRD measurements (2/ and  scans) were carried out using a Malvern-Panalytical X'Pert-PRO MRD system with 4angle goniometer, and Cu Kα1 radiation (λ = 1.5406 Å, 2xGe(220) monochromator). Microstructural characterization of the film was carried out by high-resolution transmission electron microscopy (HRTEM) using a FEI Tecnai G2 F20 microscope with field emission gun working at 200 kV. Cross-section samples were prepared by FIB on regions with top Pt electrodes (after the electrical measurements).

During film deposition one part of the sample was covered insitu by a metal mask after the growth of the LaNiO3 preventing the growth of the subsequent La2NiO4 layer. In this way, after deposition, there is one part of the sample which has exposed LaNiO3 layer. For the electrical characterisation of the bilayers an array of 50-200 µm diameter Pt electrodes were deposited by e-beam evaporation through a hard mask. Some electrodes were located directly on the exposed part of the LaNiO3 bottom electrode layer, while the rest were on the La2NiO4 top layer. Electrodes of 200 µm in diameter were selected to form the devices, with a distance between bottom and top electrodes varied between 1400 µm and 1800 µm depending on the sample. Electrical measurements were carried out with a Keithley 2604B SourceMeter in a dark box, using two tips to apply a voltage and simultaneously measure the resulting current. The voltage was applied to the Pt/LaNiO3 electrode while the La2NiO4/Pt electrode remained grounded. First, the initial resistance state (IRS) of the device was measured at low voltage. Then, the device was cycled by triangular voltage sweeps (0 V  +Vmax  -Vmax  0 V), while progressively increasing Vmax by 1 V every five cycles, in order to trigger the memristive behavior, until Vmax = 15 V was reached. The current was measured after each cycle (at a voltage of 500 mV) to determine the resistance state. No electroforming step was required to trigger the memristance, and thus these devices can be considered forming-free.

Results and discussion

Sample preparation and characterisation

Structural characterization:

The XRD measurements performed on the bilayers indicate a high degree of orientation of both LaNiO3 and La2NiO4 layers. Only 00L peaks of both compounds were observed in the 2 scans as depicted in Fig. 1(a). The oscillations in the background of the XRD scans confirm the high crystal quality of the layers.

From the position of the peaks it is possible to determine the corresponding out-of-plane parameters, as shown in Table 1. As far as the LaNiO3 layers are concerned, the c-parameter is 3.83 Å for the layer deposited at high pO2 of 70 mTorr (H/L sample), while for the LaNiO3 deposited at low pO2 we obtain c= 3.86 and 3.87 Å, for L/L and L/H samples, respectively. It seems that the c-axis parameter of LaNiO3 layer mainly depends on the pO2 during its growth and less so on the pO2 present during the subsequent growth of La2NiO4 (when grown at a different pO2). The LaNiO3 c-parameter values are consistent with the results obtained for the single layers, as shown also in Table 1, (the corresponding XRD patterns are in the supplementary part: Figs. S2 andS3) in the sense that the lattice parameters measured for films grown in low pO2 are expanded (about 1%) with respect to those obtained at high pO2. This can be explained by a larger oxygen vacancy concentration, as expected for LaNiO3-. The crystal quality of the LaNiO3 layer is preserved at both pO2. However, when comparing the cell parameter values of LaNiO3 in the single layer deposited at low pO2 with those obtained in the bilayers at the same conditions, a slight reduction of the cell parameter between 0.2-0.4% is observed. This reduction has been observed regardless of the pO2 used during the growth of the subsequent La2NiO4 layer. This means that LaNiO3 deposited at low pO2 oxidizes during the deposition of La2NiO4 (either at low or high pO2), while it maintains a constant oxygen content when deposited at high pO2. This is consistent with the observed oxygen transport previously reported in LaNiO3/La2NiO4 bilayers at elevated temperatures. 17 Therefore, the LaNiO3 layers deposited at low pO2 (L/L and L/H bilayers) could potentially develop an oxygen vacancy gradient at the interface with La2NiO4. A different situation occurs for the La2NiO4 layers. At low pO2 a clear c-axis orientation is observed for La2NiO4 (for the H/L and L/L bilayers), with an intense 006 reflection for both films. The c-axis parameter for the L/L samples is about 12.64 Å, similar to that obtained for La2NiO4 single layers directly grown on the SrTiO3 substrate (Fig. S3 and Table 1) for the same pO2 conditions. This could be another indication of a certain oxygen ion transport happening between the LaNiO3 and La2NiO4 layers. However, in this case it could not be ruled out that part of the changes in La2NiO4 cell parameter are due to a different strain when growing on LaNiO3 instead of directly on SrTiO3 substrates.

When growing La2NiO4 at high pO2, in the L/H bilayer, no 006 reflection is observed. Therefore, this La2NiO4 film does not present domains with c-axis orientation, contrary to single La2NiO4 films which showed c-axis orientation when directly grown on SrTiO3 substrates (Fig. S2). This type of scan does not preclude the existence of other orientations of La2NiO4, i.e. aaxis orientation, which would overlap with the intense substrate peaks. In order to clarify whether La2NiO4 grows with a different orientation -scans of asymmetric reflections, such as 101 La2NiO4, were performed at a sample stage tilt angle =16.7, which is the position expected for the a-oriented domains, as depicted in Fig. 1(b). Indeed, the scan over the L/H sample shows the presence of broad 101 peaks with 4-fold symmetry. This indicates the existence of a-axis orientation with perfect in-plane alignment with the substrate [100]/[010] directions. For comparison, the corresponding -scan of the 101 reflection for the c-axis oriented domains in the H/L sample, in this case at =72.1, was also measured indicating again a fourfold symmetry with a large degree of in-plane arrangement. Therefore, the growth of La2NiO4 at high pO2 promotes the growth of a-axis domains, while c-axis orientation is obtained at low pO2.

This situation was different to the direct growth of La2NiO4 on SrTiO3(001) substrates where a high degree of c-axis orientation is achieved at both pO2 values (Fig. S3). Therefore, it is likely that the exposure of underlying LaNiO3 to the high pO2 before the subsequent growth of La2NiO4 affects the LaNiO3 topmost surface to inhibit the perfect cube-on-cube arrangement of perovskite blocks of c-axis oriented La2NiO4 on LaNiO3. In agreement with the La-Ni-O thermodynamic phase diagram [START_REF] Kilner | Materials for intermediatetemperature solid-oxide fuel cells[END_REF] , it is possible that some secondary phases, resulting from LaNiO3 decomposition into La2NiO4 + NiO are also present at the interface, which could alter the subsequent growth of La2NiO4.

Bilayer microstructure:

In order to address the microstructure of the bilayers crosssectional samples were prepared from the same set of bilayers. Figure 2 shows the cross-sections oriented along [100]-zone axis at the interface between bottom LaNiO3 and top La2NiO4 layers for the three different deposition conditions: H/L (top), L/L (middle), and L/H (bottom). The left column shows HRTEM images of the bilayers where clear differences in microstructure reveal top and bottom layers. All the bilayers show highly oriented LaNiO3 with a large density of vertical planar defects, which very likely correspond to Ruddlesden-Popper type defects, a sort of antiphase boundary defect with a displacement of half primitive perovskite cell along the <111> direction, as previously shown in 15,16 .

On the other hand the nano-structural differences of the La2NiO4 films of the bilayers are more pronounced. The H/L sample shows a sharp interface with highly c-axis oriented La2NiO4. The corresponding FFT of the top La2NiO4 layers are shown in the second column in Fig 2 . For the H/L sample the pattern is consistent with a pure c-orientation, and a-or b-axis parallel to the image, aligned with in-plane main [100]/[010] substrate directions. For the L/L sample, the first layers at the interface with LaNiO3 start growing with c-orientation, but after 5-8 nm additional domains with a different orientation start to nucleate, resulting in a FFT which consists of a mixture of c-and a-orientations. The existence of a c-component was already observed in the XRD pattern of this film. However, the existence of a mixed a-orientation was not detected because of the overlap with the substrate peaks. For the L/H bilayer the majority of the La2NiO4 film seems to be formed by narrow columns of a-oriented domains, some of them with the c-axis parallel to the image (c//). This orientation is identified at atomic scale by the arrangement of bright vertical atomic planes corresponding to the a/b plane of the Ruddlesden-Popper structure, while other domains do not show such contrast, presumably because in this case the c-axis lays perpendicular to the image. The corresponding FFT is consistent with pure a-orientation, as already shown by the -scans in Fig. 1(b). The third column of Fig. 2 corresponds to colour-filtered images obtained by selecting the spots in the FFT for either LaNiO3 phase (red), c-La2NiO4 (green) and a (c//)-La2NiO4 (blue). The H/L sample shows full c-La2NiO4 (full green colour), while the L/L sample shows mixed c-and a-La2NiO4 (green with blue patches). The L/H sample shows a columnar arrangement with an intimate mixture of blue and red stripes, with nanometer size domains, corresponding to a-La2NiO4. The red stripes correspond in this case to a-La2NiO4 domains with c-axis pointing perpendicular to the image, which are not distinguished from a standard perovskite structure (red colour), but obviously do not correspond to LaNiO3 although they show the same FFT pattern. The right column in Fig. 2 summarizes these observations in the form of a schematic of the samples' microstructures. The regions where La2NiO4 domains with different orientation meet together form grain boundaries. In the pure c-axis oriented film (H/L) there are not grain boundaries in the La2NiO4 layer, whereas in the mixed a/coriented La2NiO4 layer (in the L/L sample) there are vertical and horizontal (001)/(100) boundaries between a and c domains. In the pure a La2NiO4 layer (in the L/H sample) the in-plane rotation by 90 of their corresponding c-axis defines also vertical (001)/(100) domain boundaries. The vertical and horizontal domains have been marked with green and red colours, respectively, in Fig 2

(right column).

To conclude this section, different growth conditions (pO2) induce remarkable differences not only in the oxygen concentration profiles across the bilayer, with possible gradients at the LaNiO3- /La2NiO4+ interface, but also in the bilayer nano-structure, with the appearance of a particular type of (001)/(100) grain boundaries within La2NiO4. These differences will surely affect the electrical properties of these bilayers and could ultimately dictate the memristive behaviour of Pt/LaNiO3/La2NiO4/Pt devices, as will be discussed in the following section. The I(V) curves measured for the three samples are shown in Fig. 3d to f, corresponding to every fifth cycle at the ±1 V, ±5 V, ±10 V and ±15 V sweeps. Interface-type memristive devices typically show rectifying I-V characteristics in their initial resistance state (IRS). This nonlinear behaviour is usually explained by the presence of a Schottky-like barrier at the metal/oxide interface necessary for the emergence of the memristive behavior . 5 Both the H/L and the L/L samples show I(V) characteristics in their IRS (I-V sweeps between ±1 V). Since Pt forms an ohmic contact with both LaNiO3 and La2NiO4 (as shown in Fig. S4) the nonlinearity can only be explained by the presence of a Schottky-like barrier at the LaNiO3/La2NiO4 complex-oxide interface. The increased nonlinearity measured for the L/L sample could signify the presence of a larger Schottky barrier. On the other hand the L/H sample shows almost linear I(V) characteristics between ±1 V and should therefore present the smallest LaNiO3/La2NiO4 contact resistance. As will be later explained, this difference is likely due to the high electronic conductivity expected for the La2NiO4 top layer of the L/H sample due to its a orientation, high oxygen overstoichiometry (δ) and concomitant high concentration of electronic hole carriers. Other interesting differences can be noticed between the samples when increasing Vmax gradually from ±1 V to ±15 V. Both the HRS and the LRS for all three samples present some form of hysteresis which is most pronounced at Vmax= ±10 V. It should be noticed that the small hysteretic behavior disappears when Vmax is increased to ±15 V for the H/L and L/H samples. The disappearing of the hysteresis seems to be irreversible as the subsequent application of a lower voltage does not restore the original opening in two resistance states (not shown here). On the other hand the L/L sample progressively shows an increasingly rectifying behaviour with two very distinct resistance states. The Low Resistance State (LRS -higher current branch), shows little rectification within the ±2 V sweeping range, ie before the devices switches to the High Resistance State (HRS). Conversely, the HRS shows highly rectifying I(V) characteristics with an increased threshold voltage when compared to the LRS. This hysteretic behavior and the distinct LRS and HRS states are well maintained even for the largest voltage sweep range (±15 V). The evolution of the HRS and LRS values (measured at 500 mV immediately after each sweep) as well as the corresponding HRS/LRS ratios are shown in Fig. S5 for the three bilayer samples. For the H/L and L/H samples both the HRS and LRS show very low values (<10 3 ) below Vmax=7 V. For larger voltage sweep ranges, in the H/L sample both states increase to values up to 10 4 , while for the L/H sample they remain relatively low. However, in both cases the HRS/LRS ratios stay between 1 and 3 for the whole measured range. Interestingly the L/L sample shows a completely different behaviour where both the HRS and LRS values continuously increase all the way from the lowest values of Vmax = 1 V until Vmax=15 V. At Vmax=7 V, the LRS reaches a stable value of around 10 4 , while the corresponding HRS continues increasing above 10 5 . Therefore, the corresponding HRS/LRS ratio shows a pronounced increase from 1-2 below Vmax=7 V up to 20-30 at Vmax=15 V.

Role of the pO2 on the electrical properties of Pt/LaNiO3/La2NiO4/Pt memristive devices

Focusing on the LaNiO3/La2NiO4 interface we interpret the resistive switching behavior based on the mixed ionic-electronic conducting character of La2NiO4+, and its anisotropy, and we present two possible scenarios or mechanisms which can explain the observed memristance. The layered Ruddlesden-Popper structure of La2NiO4+ is expected to induce a high anisotropy both in the electronic transport and oxygen diffusion properties. [10][11][12][START_REF] Honig | Localization and Metal-Insulator Transitions[END_REF] At room temperature the resistivity along the caxis was measured to up to three orders of magnitude larger than that within the basal plane, 12,[START_REF] Honig | Localization and Metal-Insulator Transitions[END_REF] mainly attributed to differences in mobility due to the variation in interplanar separation within the crystallographic structure. In addition, as a p-type conductor, the resistivity highly depends on the oxygen content, and more particularly to the interstitial oxygen content, which is directly related to the concentration of electronic hole carriers. We recently showed that for epitaxial La2NiO4+ thin films prepared by Pulsed Injection Metal Organic Chemical Vapor Deposition, we could easily change the oxygen interstitial concentration by annealing in different gas atmospheres (from H2/Ar to O2), leading to changes in the c-axis out of plane cell parameter from 12.62 to 12.69 Å. 9 The resistivity values of the different films showed three orders of magnitude increase in resistivity (from 5.3 × 10 -3 to 5.7 Ω cm when decreasing the oxygen content (smaller cell parameter). Thus, in the case of the L/H sample studied here a higher electronic conductivity is expected across the top La2NiO4+ film due to: i) an increased oxygen content (compared to the other two samples) leading to a larger hole carrier concentration, and ii) its a-orientation (opposite to the other samples), and thus along the fastest electronic transport path in this anisotropic material. This could significantly reduce the carrier depletion width on the La2NiO4+ side of the interface Schottky contact, therefore reducing the overall contact resistance. Consequently, this prevents the building up of a memristive behavior in this bilayer (small HRS/LRS ratio at Vmax= ±10 V that then disappears at higher voltages). In the following we focus only on the H/L and L/L samples which, do present a Schottky-like characteristics and present the first potential mechanism. Besides the drift of highly mobile electronic charges, typically forming a depletion zone at the interface, in a material such as La2NiO4+ the application of an external electrical field above a certain voltage facilitates the drift of charged oxygen defects (negatively-charged interstitial oxygen, Oi'', in this case). 10 Then, the accumulation of oxygen interstitials at the interface with LaNiO3 may produce a change in the Schottky barrier as well as in the material properties in this location, which result in a change in the contact resistance and ultimately in the overall resistance of the device. Indeed, in the L/L sample, that shows the largest memristive behavior, a strong nonlinearity was observed at low voltage, probably caused by this potential barrier. The resistance-change mechanism in Pt/LaNiO3/La2NiO4/Pt could therefore be explained as follows: Initially, the device is in a HRS. When applying a high enough positive voltage (on the LaNiO3 side) negatively-charged oxygen ions drift towards the LaNiO3/La2NiO4 interface, locally changing the carrier density (the mobile interstitial oxygen ions, Oi'', act as hole dopants in La2NiO4) and reducing the charge depletion width in this location. The device now finds itself in its LRS as the flow of current is now facilitated through a thinner potential barrier (additional conduction mechanisms can be present, i.e. tunneling). The opposite occurs when applying a high enough negative voltage. Oxygen interstitials in La2NiO4 drift away from the LaNiO3/La2NiO4 interface reestablishing the barrier width, and therefore the HRS. Although, in the previous explanation, the oxygen interstitial accumulation region has been limited to the La2NiO4 part at the interface with LaNiO3, a possible exchange of oxygen ions between the LaNiO3- and the La2NiO4+ layers, may result in an extension of the depletion region through an internal redox mechanism, also modifying the contact resistance. This corresponds to the second mechanism proposed. This recombination at the interface of LaNiO3- and the La2NiO4+ layers has been observed at 600 °C by Moreno et al. 17 , and it is likely that the exchange of oxygen ions between the two materials can occur at room temperature, assisted by the electric field and by Joule heating. Finally, it is possible that the two mechanisms evoked previously are cohabitating, both playing a role in the measured change in resistance of the device.

While the hysteretic behavior is much more pronounced for the L/L bilayer, the comparison of the film's microstructure between H/L and the L/L samples, as observed by HRTEM and depicted in the schematic in Fig. 2, does not reveal any relevant difference at the LaNiO3/La2NiO4 interface itself. Both bilayers show sharp interfaces between c-LaNiO3 and c-La2NiO4 layers across the entire interfacial region. However, the differences in their respective LaNiO3 and La2NiO4 cell parameters seem to indicate a variation in their corresponding oxygen defect concentrations, higher 𝑉 𝑂 within LaNiO3 in the H/L sample, is expected to limit the amount of oxygen ions that can be incorporated into LaNiO3 (from the La2NiO4 film) and their drift within the film towards the Pt electrode. This could be one of the reasons for the smaller HRS/LRS ratios measured for this sample compared to the L/L sample. The second main difference between these two bilayers is found in the nano-structure of the top part of the La2NiO4 film. A mixed orientation of the La2NiO4 domains in the L/L sample generates a high density of (001)/(100) boundaries within the La2NiO4 layer. It is likely that these internal (001)/(100) nano-domain boundaries form additional barriers, effectively blocking both electronic and ionic drift (particularly those boundaries laying perpendicular to the electric field (red colour in the schematic presented in the right column of Fig. 2), and therefore, accumulating traps in the volume of the La2NiO4 layer. If the presence of these domain boundaries helps increasing the rectifying and hysteretic I(V) characteristics, this could bring an additional strategy for engineering film microstructures with enhanced and better controlled memristive properties.

Conclusions

High crystal quality preferentially oriented LaNiO3/La2NiO4 bilayers were grown using PLD under varying pO2 conditions. The results confirmed that the oxygen partial pressure used during the growth of the films clearly impacts their microstructure, severely affecting the domain orientation of the La2NiO4 layer. At the same time electrical measurements carried out on Pt/LaNiO3/La2NiO4/Pt devices fabricated out of these bilayers confirmed important pO2-dependent differences in the resistive switching behavior of these complex-oxide stacks. Two switching scenarios have been proposed to explain the memristive behavior of the bilayers. The proposed mechanisms could be occurring independently or simultaneously, depending on the voltage range. While the first one has been attributed to the electric field-induced drift of oxygen interstitial ions in the La2NiO4 layer, accumulating at the LaNiO3/La2NiO4 interface, and modulating the depletion width of the Schottky barrier present; the second one would be related to the exchange of oxygen ions between the top La2NiO4 and the bottom LaNiO3 films. This study also reveals the presence of a high density of boundaries between domains with c and a-orientation, under particular film growth conditions, which could also play a significant role in the accumulation of oxygen interstitials in the volume of the La2NiO4 layer, possibly improving the memristive properties of these novel bilayer nickelate-based devices.

Figure 1 .

 1 Figure 1. a) High resolution X-ray diffraction 2/ scans in the vicinity of 002 SrTiO3 substrate reflection of the three bilayer samples combination of the growth at low and high pO2. The peaks labelled as 006 La2NiO4 and 002 LaNiO3 correspond to c-axis oriented domains. Note that the weak 006 La2NiO4 peak in the L/H sample is due to the dominant a-axis orientation of this layer. (The peak with * corresponds to a double-crystal diffraction and is an indication of high crystal coherence of the whole bilayer) b) -scans of the 101 La2NiO4 reflections indicating either dominant out-of-plane c-orientation or a-orientation of the La2NiO4 film, for the extreme H/L and L/H samples, respectively.

Figure 3

 3 Figure 3 presents an overview of the electrical response of the three bilayers using circular Pt contacts with a diameter of 200 µm. voltage was applied to the Pt electrodes directly on top of LaNiO3 bottom layer (bright part of the image in Fig. 3a), while the Pt contacts on top of La2NiO4 (dark side of Fig 3a) were grounded, as shown in the schematic of Fig. 3b. The shape of each voltage sweep cycle versus time is depicted in Fig. 3c. The I(V) measurements were reproducible when cycling within a fixed voltage range.The I(V) curves measured for the three samples are shown in Fig.3dto f, corresponding to every fifth cycle at the ±1 V, ±5 V, ±10 V and ±15 V sweeps. Interface-type memristive devices typically show rectifying I-V characteristics in their initial resistance state (IRS). This nonlinear behaviour is usually explained by the presence of a Schottky-like barrier at the metal/oxide interface necessary for the emergence of the memristive behavior .5 Both the H/L and the L/L samples show

Figure 3 .

 3 Figure 3. a) Optical Image of the top surface of the Pt/LaNO3/La2NiO4/Pt sample showing the circular Pt electrodes. b) Schematic of the measuring configuration from grounded top Pt electrode on the La2NiO4 layer to biased Pt electrode on the LaNiO3 bottom layer. c) Applied triangular voltage sweeps. d) to f) Evolution of the I(V) characteristics of the H/L, L/L and L/H bilayers, respectively, when increasing the maximum voltage from left to right (±1 V, ±5 V, ±10 V and ±15 V). The L/L sample shows clear analog-type switching characteristics. (Arrows indicate the direction of the voltage sweep)

Table 1 .

 1 Values of the c parameter of LaNiO3 and La2NiO4 as well as observed orientation of La2NiO4 domains depending on the bilayer deposition conditions (varying the oxygen partial pressure)

	sample	pO2 (mTorr)	LaNiO3 c-parameter (Å)	La2NiO4 c-parameter (Å)	La2NiO4 orientation
		70/10 (H/L)	3.832	12.664	pure c
	Bilayer (LaNiO3/	10/10 (L/L)	3.862	12.637	mixed c and a
	La2NiO4)
		10/70 (L/H)	3.870	-	pure a
	Single layer	10 (L) 70 (H)	3.877 3.830	12.632 12.686	pure c pure c

Acknowledgements

This work has been financed by the ANR funded project "MICROSWITCH" (ANR-14-ACHN-0012) and by the "FUN-TO-BE" CNRS-CSIC supported International Program for Scientific Cooperation (PICS, n°PICS07294). Part of this work was carried out through the access to PLD and HRTEM facilities granted by the NFFA-Europe Infrastructure (proposal ID 886) under Horizon 2020 EU Funding Program. JS acknowledges the grant PID2019-108573GB-C21 funded by MCIN/AEI/ 10.13039/501100011033. This work was also supported in part by the Spanish Ministry of Economy, Industry and Competitiveness (projects MAT2016-77100-C2-1-P), and by the Catalan AGAUR agency (project: 017-SGR-579). ICN2 is funded by the CERCA programme/Generalitat de Catalunya and by the Severo Ochoa programme (SEV-2017-0706).

Authors contributions

Conceptualization: KM, MB,and JS; Thin Film deposition: CW, JCR, VL; TEM analysis: BB, formal electrical characterisation: CW, KM; writing -review & editing: KM, MB, JS.

Conflicts of interest

"There are no conflicts to declare".

Footnotes and References


Benjamin Meunier

Eugénie Martinez

Raquel Rodriguez-Lamas

Dolors Pla

Monica Burriel

Carmen Jimenez

Yoshiyuki Yamashita

Shigenori Ueda

Olivier Renault

Unraveling the resistive switching mechanisms in LaMnO3+-based memristive devices by operando hard X-ray photoemission measurements

Manganite-based devices have shown promising resistive switching properties, which performance strongly depends on the electrodes used to build them. Their nature affects the physical properties of the metal/manganite interfaces, leading to a different resistive switching responses and mechanisms, which are difficult to fully understand using conventional ex situ characterization techniques. Here, the switching mechanisms taking place in LaMnO3+-based memristive devices with a noble metal top electrode have been studied using an operando high energy x-ray photoelectron spectroscopy (HAXPES). The difference between HAXPES spectra obtained for devices in different resistance states, has allowed to elucidate the redox mechanism taking place in the devices. The HAXPES spectra indicate a reduction (oxidation) of the interfacial LMO when switching from high to low (from low to high) resistance states. The presence of an inert electrode instead of an oxidizable one, results in oxygen being released out of the lattice, instead of reacting with the electrode. Complementary, HAXPES energy shifts induced by the electric field has been used to probe the resistance drop inside the device at different position and for different resistance state.

Introduction

Hard x-ray photoelectron spectroscopy (HAXPES) is a powerful tool for operando nondestructive interface characterization of functional devices. The use of hard X-ray synchrotron radiation (2.4-12 keV) allows to gather information from regions up to 5 times deeper in the sample than with laboratory sources (e.g., Al-K), which is of particular interest to study the buried, often critical interfaces of a device covered by a thick electrode. Operando measurements, consisting in biasing a functional device (capacitor, memory cell, etc) while performing HAXPES, are, therefore, powerful in order to understand the changes in elemental composition and charge distribution inside the device during operation. Additionally, it confers to HAXPES analysis the necessary reliability for its application to device technology. This approach was already implemented for studying the interface polarization in a piezoelectric Pt/Ru/PbZr0.52Ti0.48O3 high-density capacitor. [1] Oxide-based memristive devices could also take advantage of the excellent capabilities of operando HAXPES as the resistive switching mechanism occurring in such devices is often triggered by surface/interface processes. HAXPES has been employed to study the buried interfaces of different oxide-based memristive devices such as TiN/Hf0.5Zr0.5O2, [2] Pt/Ti/Pr0.5Ca0.5MnO3, [3] Au/Fe:SrTiO3/Nb:SrTiO3, [4] TiN/Pt/NiO. [5] and Ti/HfOx. [6,7] Only the later work, however, has been subject to measurements in operando mode with in situ electrical characterization, while the others studied these interfaces after cycling operation. [8] Among the plethora of oxides enabling resistive switching (RS), the switching response of the non-stoichiometric transition metal oxide LaMnO3+δ (LMO) has been reported to be governed by the oxygen vacancies concentration in the material. [9,10] The apparent oxygen excess in LMO is ascribed to the formation of La and/or Mn cation vacancy sites (i.e., La1-x□x,Mn1-y□yO3, viz. LaMnO3+δ), where □ represents cation vacancies, and charge neutrality remains fulfilled thanks to the coexistence of Mn 3+ /Mn 4+ oxidation states. [11,12,13,14] Therefore, the valence state of the material is strongly correlated to the oxygen content, and Mn 4+ can be seen as an intrinsic dopant. [15] We recently investigated the microscopic mechanisms of local interfacial RS in LaMnO3+δ by conductive atomic force microscopy (c-AFM) [16] and demonstrated that the change in resistance is associated with a drift of oxygen ions to the surface and the concomitant variation in the Mn valence state, resulting in modifications in the transport mechanism. However, the experimental configuration used in this work presents several specificities (e.g., low current density, bare surface, large active area, etc) that do not match the requirements of a standard functional device. For this reason, we then performed a first set of operando HAXPES measurements on a TiN/LMO/Pt prototypical memory cell. [17] In this work, we monitored the redox reactions occurring at the TiN/LMO interface during RS. In this case, the TiN top electrode acts as a reservoir in which oxygen is stored when LMO is in high resistance state (HRS) and released during the switch back to low resistance state (LRS). From the HAXPES and electrical characterization we deduced that the switching mechanism was localized at the TiN/LMO interface. However other relevant electrode, but noble (e.g. Au, Pt, Pd) or semi-noble (Cu) metals, are also commonly used for building memory, and also display RS behavior. Since such materials have a low oxygen affinity, the question arises on how the resistive switching mechanism is affected by the absence of an oxygen reservoir interlayer.

In the present contribution we report the recent work on M/LMO/Pt stack with M = Au or Pd by means of operando HAXPES. We focus here on the behavior of a memory cell with a chemically inert electrode, compared to the previously reported LMO covered by an oxidizable TiN electrode. We propose a model explaining how an oxygen reservoir at the interface with the electrode is created during RS in the presence of a noble metal as top electrode. The results clearly illustrate that the electrode material has an impact on the nature of the oxygen reservoir necessary for multiple RS operation and thus on the associated switching mechanism itself. The LMO thin film (100 nm-thick) used for this study was grown by pulsed injection metalorganic chemical vapor deposition following the strategy III of Ref. [18] The structural characterization of the LMO thin film is presented in Figure S1 of Supporting Information. In order to both match the HAXPES requirements and allow in situ electrical characterization, the device depicted in Figure 1(a) was fabricated. The active region consists of a thin (~7 nm) metal layer (Au or Pd) deposited directly on a 100x100 µm² LMO surface. Outside this region, the top electrode was electrically isolated from LMO by a 50 nm thick, sputtering-deposited Si3N4 insulating layer. 300 m away from this active area, a thick current collector of the same material as the electrode was deposited, wire-bonded to the sample-holder, and eventually connected to a sourcemeter (Keithley 2635B). The Pt bottom electrode was also wire-bonded to the grounded contact pad of the sample-holder. Once in the HAXPES chamber, an additional small series resistance was measured due to the long distance between the chamber and the end station control cabin. By comparing the resistance measured directly at the sample's output and for a sample mounted on the sample holder in the analysis chamber, the total external resistance was measured to be Rext = 2.5 .

Experimental

HAXPES measurements were performed at the BL15XU beamline of the SPring-8 synchrotron facility (Hyogo, Japan). The photon energy was set to 7933.5 eV (as determined from the Fermi level of the Au top electrode of one of the device) The X-rays were monochromatized by a Si(111) double crystal and post monochromator (channel cut crystal). The beam cross-section was estimated to 10030 µm 2 , with take-off angle of 70°, the effective footprint of the beam on the sample surface was 400x30 m². The photoelectrons were detected by a high-resolution hemispherical analyzer (VG Scienta R4000) providing an overall energy resolution of 240 meV as measured by the Fermi cutoff of an Au film which also defined the binding energy (BE) scale. The projected beam could be selectively directed onto either a single memory device or the bare LMO film.

Results and discussion

Resistive switching of the M/LMO/Pt devices (M=Au,Pd)

Once the sample described in Figure 1a was mounted and introduced into the analysis chamber under Ultra High Vacuum (UHV) condition (~10 -7 Pa), electrical measurements were performed. Figure 1b shows the characteristic hysteretic current-voltage (I-V) curve measured in-situ by cyclic voltammetry for Au top electrode-memristive device. The device exhibits an asymmetric bipolar counter-clockwise-type switching without requiring an electroforming step. Noticeable differences are observed between the SET and RESET processes. An abrupt RESET from LRS (RLRS = 7.2 ) to HRS (RHRS = 22 k) occurs for positive polarity at ~2.0 V for a corresponding high to low resistance ratio of four orders of magnitude. Conversely, the SET process requires higher voltage in negative polarity and, instead of a single abrupt switch, it presents several consecutives steps starting at ~ -6 V. The compliance current was set to 10 mA in negative polarity in order to avoid the oxide electrical breakdown at high voltage (1A expected at -10V for R = RLRS). With this limitation, a decrease of the resistance of only two orders of magnitude is observed after the SET process. However, similar measurements performed ex situ (Figure S2) on other devices reveal that the resistance can be decreased to its initial value by increasing the current limit and that, in these conditions, several SET-RESET cycles can be obtained.

The log-log representation of the I-V curve in Figure 1c shows two linear regimes associated with two slopes: m = 1 and m > 1. The m = 1 case corresponds to ohmic transport in the device. It is observed in the initial LRS branch and at low voltage in HRS (|Vappl.| < 1 V). The presence of a non-ohmic behavior after RESET indicates a modification of the transport mechanism due to the switching process, as discussed in our previous works. [16,17] In situ electrical measurements have been performed the same way on the memristive device with a Pd top electrode. The corresponding log-log I-V curve depicted in Figure 1d shares similar features with the Au top electrode one, i.e. an ohmic behavior in the LRS and at low voltages in the HRS. The most noticeable difference between the two metal electrodes is the high to low resistance ratio of "only" two orders of magnitude for Pd with RLRS = 8.6  and RHRS = 225  compared to the 4 orders of magnitude observed for Au, although the associated conduction mechanism is assumed to be the same irrespective of the electrode material. These observations are also supported by the I-V curves obtained on LMO samples from similar ex situ measurements in standard metal/LMO/Pt devices with square top electrodes (see FigureS2 of Supporting Information). The similarity in the I-V characteristics demonstrates that the samples are not affected by the operando conditions (UHV and high energy radiation), at least as far as the first resistive switching operations are concerned. Having an ohmic behavior in low voltage range for both electrode in HRS and LRS indicates that the transport and switching mechanisms are different from the one reported when using TiN electrode in which case RS was driven the change of the electrode/interface state and resistance. [17] Moreover, the devices with inert metal electrodes (Au or Pd) present a larger resistive switching amplitude (of several orders of magnitude) compared to the one observed in our previous study using TiN (RHRS/RLRS=4).

HAXPES study of M/LMO/Pt devices in the virgin state

Prior to the operando measurements, the LMO bare surface was analyzed. Figure 2 presents the O 1s, La 3d5/2 and Mn 3s core levels. A Shirley-type background was subtracted from all the spectra. Peak fitting was performed using pseudo-Voigt functions.

The O 1s spectrum in Figure 2a. consists of two contributions, Olatt. (529.4 eV) and Ointer. (531.8 eV) corresponding to lattice oxygen from the LMO and interfacial oxygen (oxygen outside of the lattice, for instance adsorbed oxygen on a bare surface), respectively. This model for O 1s deconvolution in LMO has been widely used by electrochemists to study the LaMnO3 as oxygen catalyst. [19,20] The remarkable catalytic activity and stability of LMO is directly related to the noticeable and stable presence of adsorbed oxygen on its surface. [21] The La 3d photoelectron peak of lanthanum compounds shows very prominent satellite features due to core-hole screening processes triggered by charge transfer between the unoccupied 4f valence orbitals and 2p orbitals of oxygen. [22,23] Thus, in the resulting La3d5/2 spectrum, a main peak (cf 0 ) and two satellite peaks (cf 1 L) show up, where c and L denote a core-hole in the La 3d5/2 states and a ligand O 2p hole, respectively. [24] In our previous study, we demonstrated the direct relationship between the oxygen content in the La surrounding and the ratio of the satellites/main peak areas. [16] The Mn 3s core levels are split into two contributions due to differences in the 3s-3d electrostatic exchange interaction between spin-up (3s ↑ ) and spin-down (3s ↓ ) states after photoemission. The energy separation between the 3s ↑ and the 3s ↓ component in the 3s core level spectrum may be expressed by the Van Vleck's theorem as: [25] ∆𝐸 𝑀𝑛3𝑠 = (𝑛 + 1)𝐺 2 (3𝑠3𝑑)

Where n is the number of unpaired 3d electrons and G² the Slater exchange integral.

The energy separation increases linearly against n as long as the nature of the ligands is the same. [26] Data reported in the literature give an average multiplet splitting of 5.4 eV for pure Mn 3+ states and 4.4 eV for pure Mn 4+ states. [27,28] Thus, in our case, the estimated valence of manganese is 3.2 ±0.05 for a measured peak separation of EMn3s = 5.17 eV.

According to the device description of Figure 1, after fabrication the LMO surface was locally covered by a thin metal layer, either Au or Pd depending on the device. In this configuration, electrical measurements and HAXPES analyses can be performed simultaneously in so-called operando conditions. Figure 3 presents the Au 4f7/2 and Pd5/2 3d HAXPES spectra of the corresponding memristive devices for different applied voltages (Vappl. = 0, ±1V). The series of spectra provides evidence that the operando measurements are performed in reliable conditions. When a bias is applied, the photoelectrons kinetic energy is modified by the electrostatic potential and the measured energy levels are shifted accordingly. However, the shifts observed on the HAXPES spectra are smaller than the applied biases, due to the external series resistance resulting in a small bias drop across the circuit. The effective potential observed on the samples (V) for Vappl = +/-1V is equal to +/-0.7eV. The energy shift of a specific core level may be interpreted as the electrical potential "seen" at the location of the corresponding element in the device. As the metal electrode, Pd or Au, is located on top of the whole device (see Figure 1a), the observed shift of each spectrum is directly related to the total resistance of the sample. Thus, for an applied voltage Vappl and a HAXPES energy shift V (= electrical potential), the equivalent resistance is calculated with the following potential divider relationship:

𝑅 = 𝑅 𝑒𝑥𝑡 . 𝑉 (𝑉 𝑎𝑝𝑝𝑙 -𝑉) ⁄ (2)
with the external resistance Rext = 2.5  corresponding to the external resistance of the overall setup (wires, contacts, etc).

Considering the Pd 3d spectrum, the amplitude of the energy shift is the same for positive and negative polarities and is determined as 0.72 eV in the virgin state. The corresponding resistance for the whole Pd/LMO stack is RPd+LMO = 6.4 . The energy shift for Vappl = +1 V gives a value of the LMO resistance with Au electrode of RAu+LMO = 6.1  thereby similar to RPd+LMO with Pd electrode. The total resistance RLMO+Rext = 8.6  is also in good agreement with the one electrically measured in LRS.

Operando measurements of the Pd/LMO/Pt stack

From the HAXPES measurements under applied electric field it was observed that not only the spectra of the metal electrode shift due to the applied electric field, but also those characteristics of the LMO film itself. For the LMO elements, unlike for Au or Pd, the peak shifts also include chemical state-and/or composition changes due to resistive switching triggered by the electric field.

For operando HAXPES measurements, we address only the results from the Pd/LMO/Pt device, as in the case of the Au top electrode, both La 3d and Mn 3s spectra overlap Au peaks, thereby hampering data interpretation (the La 3d core level is partially hidden by the asymmetric tail of the Au 4s signal and the Mn 3s peak overlaps the Au 4f line). HAXPES measurements were carried out on a Pd/LMO/Pt device before and after switching. The La3d5/2 and O1s core levels were measured, while the Mn 3s core-level was not recorded due to an overlapping with the Pd 3d one.

One of the goals of the operando measurements is to track small energy shifts in the photoemission spectra in complement with the chemical environment analysis. Therefore, it requires a proper and rigorous data treatment process. The following assumptions were made and applied as initial constraints during the O 1s fitting process:

-Peak fitting was performed using a combination of pseudo-Voigt functions with the Lorentzian to Gaussian ratio and the peak width kept constant for all the data sets. -In open circuit conditions (Vappl=0 V), the O 1s spectra are fitted using the binding energies from the reference bare surface (see Figure2). -For operando data (i.e. Vappl. = +/-2 V) only the energy is a free parameter. Indeed, for the Pd electrode, the operating voltage is well under VSET or VRESET and no chemical change is expected.

The resulting fits are presented in Figure 4 and the main parameters summarized in Table 1.

The O 1s spectra are dominated by the silicon oxide signal from the partially oxidized Si3N4 layer. Indeed, at the center of the 100x100 m² Au electrode, the beam footprint is 400x30 µm 2 but the effective footprint is much larger due to the beam tail and X-ray diffusion through the surface material, explaining the presence of a Si-O component in the O 1s spectrum. When the device is biased, the Si-O contribution is split into two separate peaks (see Figure 4c-d).

Oxidation of the Si3N4 surface occurs prior to electrode deposition. After device fabrication part of the native oxide is covered by the metal while the majority remains uncovered. Hence when a bias is applied on the top electrode, the covered part sees a larger electrical potential resulting in the shifted Si-O contribution Si-O/Pd in O 1s spectra, while the uncovered surface's contribution will not be shifted, and thus explaining the presence of these two contributions.

The two additional contributions to the O 1s spectrum (light blue and dark green contributions in Figure 4a-d) are specific to the LMO in the memristive device, and match well the Olatt. and Ointer. contributions from the bare surface (see Figure 2). 1. Relative content of each contribution in the O1s spectra (see Figure 4) and their energy shift when a ±2 V voltage is applied. Pristine refers to the bare surface presented in Figure 2.

O 1s

By comparing the O1s spectra measured in LRS and HRS in Figure 4a and 4b, respectively, noticeable differences are observed, though unfortunately partly screened by the large Si-O signal. For more details, Table 1 shows the relative area of each contribution. From LRS to HRS, the Ointer. contribution slightly increases with respect to the Olatt. one. We previously reported such a modification of the surface composition after RESET (LRS to HRS) operation performed on a bare LMO surface using a conductive AFM tip. [16] We concluded by the application of a positive bias (VRESET > 0) oxygen ions drift from the lattice towards the top electrode and remain adsorbed, triggering the change in resistance. At this point one may wonder what is the physical meaning of Ointer. when the surface is covered by an electrode. Some hints may be found in previous studies of oxide-based resistive switches revealing the formation of oxygen in gas form during switching operation. For instance, the growth of microscopic gas bubbles have been observed in Pt/TiO2/Pt during operation. [19] the formation of an oxygen pocket has even led to the deformation of the top electrode in a Pt/SrTiO3 system. [29] In both cases, the electrode was made of a noble metal. However, to the best of our knowledge, such oxygen pockets/bubbles have never been reported using an inert electrode. Therefore, the Ointer contribution is ascribed to adsorbed at the metal/oxide interface, which could eventually lead to the formation of oxygen pockets. Thus, the increase of the Ointer. contribution indicates that oxygen is released from the LMO lattice during RESET stage (VRESET > 0). Complementing the literature example, observations of the surface of Au/LMO/Pt devices after several ex situ I-V cycles show that a lift-off of the electrode had occurred (see Supporting Information in Figure S3 and S4). This deformation and eventual destruction of the top electrode could originate from the expansion of this gas pocket during the electrical measurements. This gas expansion may also be enhanced by Joule heating, keeping in mind that the LRS in LMO involves a large current density.

Figure 4c and 4d present the O 1s measured while applying a negative and positive voltage, respectively. The behavior of the Si-O signal has been previously explained, the shifts measured for the oxidized Si3N4 in contact with the electrode (Si-O/Pd) follows the one observed for Pd 3d. The energy shift of each contribution reported in Table 1 is in good agreement with our model: the deeper the contribution in the device, the lower the corresponding shift. The energy shifts in LRS show that Ointer. and Olatt. are not at the same electrical potential, indicating a noticeable potential drop between the top electrode and the LMO surface, and thus the presence of a LMO interfacial layer. In HRS the energy shifts of Ointer. and Olatt. are the same, but it cannot be concluded that the interfacial resistance vanishes since it may be screened by the large LMO resistance (as expected from Eq.2).

To complement these observations, Figure 4e and 4f show the La3d spectra recorded in LRS and HRS, respectively. By comparing them with the La 3d spectra from the bare surface it is observed that the main peak at 833.7 eV (orange peak) is less intense compared to its two satellites (in grey). This observation can be explained by an increase of the oxygen content.

At the excitation energy of 7.93 keV, the estimated probing thickness (3xIMFP) by La 3d photoelectrons is 18 nm. Due to the presence of the 7 nm-thick top electrode in the device, the probing area into the LMO (about 10 nm) is less deep than when measuring a bare surface. Thus, the difference in the La 3d spectra indicates that the interface with Pd is more oxidized than the bulk LMO, in agreement with a more oxidized LMO surface commonly observed in this material. [30] This relative intensity between the main photoemission line and its two satellites is also the key parameter regarding the spectra modification between the LRS and HRS. A slight decrease of the main peak intensity is observed in HRS (Figure 4f), indicating an increase of the oxygen content in the La neighborhood. From the O 1s analysis it was previously deduced that oxygen ions drift to the top electrode/LMO interface under the effect of the electric field, and part of the oxygen is released from the LMO lattice as gas traps at the interface. The La spectra indicates that, in addition to the oxygen release, an oxygen-rich region is formed at the top LMO interface. Therefore, a good consistency is found among the whole set of photoemission results as interpreted here. Compiling all these observations, a model of the Pd/LMO interface is depicted in Figure 5a.

Based on the operando analysis of the O 1s and La 3d spectra our system may be divided into three different regions: i) bulk LMO with standard stoichiometry, covered by ii) an oxygen-rich LMO region and, iii) the interfacial oxygen-pocket region. Coming back to the proper stoichiometry of LaMnO3+, and considering a homogenous cation concentration, the different regions may be described as:

i) Bulk LaMnO3+: 𝐿𝑎 1-𝑥 □ 𝑥 𝑀𝑛 1-𝑦 □ 𝑦 𝑂 3-𝛿 𝛿𝑉 𝑂 •• ii) Oxygen-rich region: 𝐿𝑎 1-𝑥 □ 𝑥 𝑀𝑛 1-𝑦 □ 𝑦 𝑂 3
being □ the cation vacancies and 𝑉 𝑂 •• the oxygen vacancies. Excess oxygen occupying interstitial sites is highly unlikely in LMO, as in other perovskites, due to steric effects, which implies the presence of cation vacancies. Having an apparent oxygen-rich region indicates the probable presence of oxygen vacancies in the observed bulk LMO. Thus, the described oxygen drift could be equivalently ascribed to oxygen vacancies motion under the applied electric field.

Based on the electrical characterization results, i.e. ohmic transport in HRS and LRS and abrupt SET and RESET processes, it is most likely that the RS mechanism is related to the formation of conductive filaments. The presence of filaments is also supported by the fact that we only observed small variations of the composition between HRS and LRS, despite a very large change in device resistance.

For the LMO in the filaments to be more conductive than the rest of bulk, these filaments needs to have an oxygen-rich stoichiometry with a larger Mn 3+ /Mn 4+ mixed valence state ratio. When a positive voltage (VRESET) is applied, oxygen from the bulk and from the filament drifts toward the top electrode, and consequently oxygen vacancies drift away (see Figure5b). The reduction of the conductive filament drastically reduces its conductivity. Oxygen cannot react with the noble metal electrode and, thus, accumulates at the interface (as deduced from the La 3d spectrum), or is released in gas form between the LMO and top electrode. The resulting state has a slightly more conductive interface but an electrically insulating bulk region, since the filament is broken. The device is therefore in its HRS. When a negative voltage (VSET) is applied oxygen is reintroduced into the material and conductive paths can be formed again: the device is back in its LRS (Figure 5c). Hence, this resistive switching operation can be repeated several times. However, the drawback of this mechanism with inert electrodes is the degradation of the metal/oxide interface due to the accumulation of released oxygen, and the possible formation of gas pockets. Based on these present observations on Pd(Au)/LMO/Pt devices and on our previous results on TiN/LMO/Pt devices [17] , it can be concluded that memristive devices using TiN top electrodes are more reliable for non-volatile memory applications due to more reliable redox processes occurring at the TiN/LMO interface, hindering the existence of gas pockets in the latter case.

Conclusions

In this work we presented an operando HAXPES study of the effect of a chemically inert electrode in a resistive switching device with LaMnO3+ as active material. The use of HAXPES in the present contribution has successfully demonstrated its ability to fingerprint modifications of the chemical composition of the LMO/metal interface during resistive switching. The fine analysis of the spectra recorded during the process reveals small changes in the O1s and L3d spectra indicative of the switching mechanism. We demonstrated that in this case the switching mechanism is different to the one when using a chemically reactive TiN electrode, though it is also triggered by redox reactions and oxygen drift. The main consequence is that oxygen, instead of partially oxidizing the electrode (in the case of the TiN electrode), is here accumulated at the interface, both in a top LMO oxygen-rich layer, and in the form of gas pockets under the metal electrode. This conclusion provides an explanation of the low endurance measured on devices with noble metal electrodes compared to those with TiN, and the severe degradation or lift-off of the electrodes after several RS cycles. These results bring complementary information on the effect of the nature of the top electrode to our previous studies on LMO-based resistive switching devices, providing a good overall view of this system. They show the importance of the choice of the electrode in an oxide-based memory cell device and how it affects both the electrical response and the redox chemistry of the materials that compose the whole system. Although a large HRS/LRS ratio has been demonstrated, desired for ReRAM applications, long time endurance of the device and stability are also critical properties for these devices. Based on our studies, only noble metals would not be suitable to build LMO-based memristive devices; an interlayer acting as an oxygen reservoir (ie. TiN) during operation would be required for optimal operation.

Figure 1

 1 Figure 1 (a) sketch of the operando memristive device where the Pt bottom electrode is grounded and the metal top electrode is biased. (b) In situ I-V characteristics obtained for an Au/LMO/Pt stack with a current compliance of 10 mA. (c) and (d) log(I)-log(V) plots

Figure 2

 2 Figure 2 HAXPES spectra of O1s (a), La 3d5/2 (b) and Mn 3s (c) core levels of a pristine LMO surface.

Figure 3

 3 Figure 3 HAXPES shift of Pd 3d5/2 and Au 4f7/2 for the M/LaMnO3+stacks (M=Au, Pd) for applied voltages of 0V and +/-1V.

Figure 4 O

 4 Figure 4 O 1s spectra for the Pd/LMO/Pt stack in (a) LRS, and (b) HRS for different applied voltage (c) Vappl. = -2 V and (d) Vappl. = +2 V. La3d5/2 spectra for the Pd/LMO/Pt stack in (e) LRS, and (f) HRS

Figure 5

 5 Figure 5 (a) Sketch of the proposed physical model used to describe the M/LMO/Pt devices (M = Au, Pd) studied by operando HAXPES. Oxygen drift (green arrows) and equivalent circuit of the structure (b) during RESET for Vappl. > 0 and (c) during SET for Vappl. < 0. For each stage, the dotted red lines represent the initial state and the continuous lines the final one (i.e. LRS and HRS for SET and RESET, respectively)

Acknowledgements

This work has been supported by the ANR funded project "Alps Memories" under grant nr ANR-15-CE24-0018. This project was performed within the framework of a collaboration between CEA-LETI and NIMS institutes. The HAXPES measurements were performed with the approval of NIMS Synchrotron X-ray Station at SPring-8 (Proposal No.2018A4915). A part of this work was supported by NIMS microstructural characterization platform as a program of Nanotechnology Platform" (Project No. 12024046) of MEXT, Japan. Part of this work was performed on the Platform For Nanocharacterization of CEA with the support of the "Recherche technologique de Base" program of the french ministery of Research. Sample fabrication has been performed with the help of the "Plateforme Technologique Amont" of Grenoble with the financial support of the CNRS Renatech network.


Andreas Herzig

Emiliano Lorini
email: emiliano.lorini@irit.fr

Elise Perrotin
email: elise.perrotin@cril.fr

A Computationally Grounded Logic of 'Seeing-to-it-that'

We introduce a simple model of agency that is based on the concepts of control and attempt. Both relate agents and propositional variables. Moreover, they can be nested: an agent i may control whether another agent j controls a propositional variable p; i may control whether j attempts to change p; i may attempt to change whether j controls p; i may attempt to change whether j attempts to change p; and so on. In this framework we define several modal operators of time and agency: the LTL operators on the one hand, and the Chellas and the deliberative stit operator on the other. While in the standard stit framework the model checking problem is unfeasible because its models are infinite, in our framework models are represented in a finite and compact way: they are grounded on the primitive concepts of control and attempt. This makes model checking practically feasible. We prove its PSPACE-completeness and we show how the concept of social influence can be captured.

Introduction

One of the most prominent accounts of agency is Belnap and Horty's logical analysis of 'agent i sees to it that proposition ϕ is true in terms of 'branching time + agent choice' models (BT+AC models) [START_REF] Horty | [END_REF]Horty, 2001]. At the core of this so-called 'stit logic' are the Chellas stit operator, expressing the ability of a group of agents to ensure that a given formula is true regardless of the choices of other agents, and the deliberative stit operator, adding that truth of this formula is not necessary. This framework is very expressive and was put to work to account for other action-based concepts such as obligation [Horty, 2001;Broersen, 2011], influence [START_REF] Lorini | Emiliano Lorini and Giovanni Sartor. A STIT logic for reasoning about social influence[END_REF], social commitment [START_REF] Lorini | Temporal STIT logic and its application to normative reasoning[END_REF] and responsibility [Lorini et al., 2014]. Unfortunately reasoning turned out to be hard: deciding satisfiability of formulas involving the Chellas or the deliberative stit operator is NEXPTIME-complete even without temporal modalities as soon as there is more than one agent [Balbiani et al., 2008]; it becomes 2EXPTIME-complete with temporal modalities 'next' and 'until' [START_REF] Boudou | [END_REF]; and it is undecidable as soon as agency of sets of agents (groups) comes into play, and so again already without temporal modalities [START_REF] Herzig | Properties of logics of individual and group agency[END_REF]. On the other hand, model checking-which is often considered to be an interesting alternative to satisfiability checking [START_REF] Halpern | [END_REF]-is basically unfeasible because BT+AC models are typically infinite.

We here propose to encode BT+AC models in a finite and compact way, so that model checking becomes practically feasible. We do so by grounding the basic concepts of stit logics, namely histories and choices, on the concepts of control and attempt. The former is borrowed from [van der Hoek et al., 2011;[START_REF] Herzig | [END_REF] while the latter is borrowed from [START_REF] Lorini | Emiliano Lorini and Andreas Herzig. A logic of intention and attempt[END_REF]. Both relate agents and propositional variables: when an agent i controls a propositional variable p then she is able to determine the truth value of p at the next state; if this is the case and i attempts to change p then the truth value of p gets flipped at the next state; and if nobody is able and attempting to change the truth value of p then it remains unchanged.

We follow van der Hoek [2011] whose states are not just valuations of classical propositional logic (sets of propositional variables), but come equipped with a function associating to each propositional variable the agent controlling it. We basically augment these models by a function associating to each propositional variable the set of agents attempting to change it. States are therefore triples, and each triple determines a unique next state: all those variables whose change is attempted by some agent controlling it get their truth value flipped, and the other variables keep their truth value. Attempts are naturally viewed as persistent goals: agents abandon the attempt to change p once p has been successfully changed (possibly by somebody else).

Representing control and attempts in this way only allows for reasoning about the next state: as control does not change, attempts that fail at the first step will always fail, and new attempts cannot appear from one state to the next. We go beyond this simple model and introduce higher-order control and attempt: for two (possibly identical) agents i and j: i may control whether j controls a propositional variable p; i may control whether j attempts to change p; i may attempt to change whether j controls p; and i may attempt to change whether j attempts to change p. This allows us to reason about future states beyond the next state: we can define meaningful temporal operators of Linear-time Temporal Logic (LTL). In these models we interpret modal operators of agency of the kind "group J achieves ϕ": we define a Chellas stit operator and a deliberative stit operator. For the resulting logic we establish PSPACE completeness of model checking.

The paper is organised as follows. In Section 2 we introduce the models and in Section 3 language and semantics of our logic of agency based on control and attempt (LACA). In Section 4 we show that they correspond to particular BT+AC models. Section 5 establishes complexity of model checking. Section 6 illustrates LACA by recasting the definition of social influence of [START_REF] Lorini | Emiliano Lorini and Giovanni Sartor. A STIT logic for reasoning about social influence[END_REF]. Section 7 concludes.

Models of Control and Attempt

We define the models of the logic of agency based on control and attempt LACA. Let Agt = {1, . . . , card(Agt)} be a finite set of agent names, typically noted i, j,. . . Groups are subsets of Agt, typically noted J, J ,. . . The complement of J is J = Agt \ J. We write i instead of {i}.

Atoms and Valuations

Let Prp be a set of propositional variables, or facts, typically noted p, q,. . . Atomic formulas, or atoms for short, follow the grammar

α ::= p | c i α | t i α,
where p ranges over Prp and i over Agt. c i α reads "agent i controls α" and t i α reads "agent i attempts to act on α", or "agent i attempts to change the truth value of α". Atoms are therefore sequences of c i and t i that are followed by a propositional variable: the variable the atom is about. Atoms of the form c i α are control atoms and atoms of the form t i α are attempt atoms. The set of all atoms is noted Atm.

States are simply valuations over atoms: subsets of the set Atm of atoms, noted V , V , etc. If α ∈ V then α is true at V , and if α / ∈ V then α is false at V . When t i α is false then i is indifferent about the value of p (as opposed to i wanting to maintain the truth value of p). We say that V and V agree on α if α has the same status in V and V , that is, either α ∈ V and α ∈ V , or α / ∈ V and α / ∈ V . Otherwise V and V disagree on α. Given a valuation V and a group of agents J ⊆ Agt, Att J (V ) = {t i α : i ∈ J} is the set of J's attempt atoms in V and Ctrl J (V ) = {c i α : i ∈ J} is the set of J's control atoms of V , alias J's abilities at V . Example 1. Two agents 1 and 2 are in a room with a window and a radiator. The state of the room is described by the propositional variables w ('the window is closed') and h ('the heating is on'). 1 controls h and 2 controls w. Moreover, 1 has authority over 2 and can order 2 to open or close the window: 1 controls t 2 w. Initially w and h are both false, and 1 is going to try to switch the heating on and to make 2 flip his goal about the window. This is modelled by the valuation

V hw = {c 1 h, t 1 h, c 2 w, c 1 t 2 w, t 1 t 2 w}.
In our example control is exclusive: no variable is controlled by more than one agent. For the sake of generality we do not impose this as a general constraint.

Successor Valuations

A given valuation determines its successor valuation; let us discuss in detail how this should work.

We have said that t i α means that i is going to try to act on α. If i does not control α then this is going to fail; and if i controls α then the effect of i's trying is that the truth value of α gets flipped. For example, the successor of V h 1 = {c 1 h, t 1 h} should contain h and the respective successors of V h 2 = {c 1 h} and V h 3 = {t 1 h} should not. We understand an attempt atom t i α as a persistent goal in the sense of Cohen and Levesque's logic of intentions [1990] that is only abandoned once it has been satisfied: i will basically keep on trying until either the truth value of α has changed, or until an agent j (possibly i herself) terminates i's commitment to achieve α. When i has the goal to change the truth value of p then i typically also has the goal to abandon the goal t i p, i.e., to change the truth value of t i p. Therefore the valuations V h 1 ∪ {c 1 t 1 h} and V h 1 ∪ {c 1 t 1 h, t 1 t 1 h} should have the same successor, namely {h, c 1 h, c 1 t 1 h}.

We consider that abilities (i.e., control) are a priori preserved, unless they get flipped by a successful higher-order attempt. Therefore the successor of the above V h 1 should intuitively be {c 1 h, h} and the successor of V h 2 should be V h 2 . We also consider that attempts should be preserved if they failed. Hence the successor of V h 3 should be V h 3 . In our example it is possible that 1 achieves h (and therefore abandons her goal to flip h) and simultaneously makes 2 adopt the goal to flip h. Consider V h 1 ∪ {c 1 t 2 h, t 1 t 2 h, c 2 h}. On the one hand, 1's successful attempt to act on h will make 1 abandon her goal to act on h, i.e., t 1 h will become false; on the other hand, her successful second-order attempt to act on t 2 h will make the latter true. Hence the successor should be {c 1 h, c 1 t 2 h, c 2 h, t 2 h, h}, and the successor of the latter should be {c 1 h, c 1 t 2 h, c 2 h}. Overall, h was initially false, then became true, and ends up false again.

The changeset of a valuation V is

V ± = {α : c i α ∈ V and t i α ∈ V for some i ∈ Agt}.
The elements of V ± are the goals that should a priori be abandoned: the atoms of V whose change of truth value is directly caused by the agents' successful attempts in V . The definition of the successor of a valuation then is:

succ(V ) = (V ± \ V ) ∪ V \ V ± ∪ {t i α ∈ V : α ∈ V ± , i ∈ Agt} .
Hence (1) everything in the changeset that is false becomes true;

(2) everything that is true and is neither in the changeset nor a goal to be abandoned remains true. Put differently: V and succ(V ) agree on α if and only if α / ∈ V ± and there is no β ∈ V ± such that α = t i β. Some examples are:

succ({c 1 h, c 2 h, t 1 h}) = {c 1 h, c 2 h, h}, succ({c 1 h, c 2 h, t 1 h, t 2 h}) = {c 1 h, c 2 h, h}, succ({c 1 h, c 1 t 1 h, h}) = {c 1 h, c 1 t 1 h, h}, succ({c 1 h, c 1 t 1 h, t 1 t 1 h}) = {c 1 h, c 1 t 1 h, t 1 h}, succ({c 1 h, c 1 t 1 h, t 1 t 1 h, t 1 h}) = {c 1 h, c 1 t 1 h, h}.
The function succ(.) is total: every valuation has exactly one successor. A valuation may have more than one predecessor, as illustrated by the first two lines of our examples. (Another example is succ(∅) = succ(Atm) = ∅.)

We inductively define the n-th successor of V :

succ 0 (V ) = V , succ n+1 (V ) = succ(succ n (V )).
Example 2 (Example 1, ctd.). In the valuation

V hw = {c 1 h, t 1 h, c 2 w, c 1 t 2 w, t 1 t 2 w},
agent 1 is going to try to switch the heating on and to make 2 flip his goal about the window. Both attempts are going to be successful because 1 controls h and 1 controls 2's attempts on w. Hence at succ(V ) the heating is on and the window is open, and at succ 2 (V ) and beyond the heating is on and the window is closed:

succ 1 (V ) = {c 1 h, h, c 2 w, c 1 t 2 w, t 2 w}, succ n (V ) = {c 1 h, h, c 2 w, c 1 t 2 w, w}, for n 2.

Agent Choices

We define the relation ∼ J between valuations as follows:

V ∼ J V if V and V agree on every atom in Prp ∪ Ctrl ∪ {t i α : i ∈ J, α ∈ Prp ∪ Ctrl and c i α ∈ V }.
That is, V ∼ J V iff V and V agree on facts, abilities, and on successful attempts by agents in J to flip facts and abilities at the next step. We call the latter the immediate choices of J at V . Put differently, V ∼ J V when V and V agree on facts and abilities as well as in the way that agents of J collectively influence the evolution of these facts and abilities in the immediate future. For example, for the valuation of Example 2 we have V hw ∼ {1} {c 1 h, t 1 h, c 2 w, c 1 t 2 w, t 2 w}. When V ∼ ∅ V then V and V agree on facts and abilities, abbreviated V ∼ V . The relations ∼ J are reflexive, transitive, and symmetric. Moreover,

V ∼ Agt V implies succ(V ) ∼ succ(V ).
We generalize this and define the relation ∼ n J between valuations as follows, for n ∈ N 0 :

V ∼ n J V if succ n (V ) ∼ J succ n (V ) and for every m < n, succ m (V ) ∼ Agt succ m (V ).
That is, V ∼ n J V if V and V agree on facts and abilities and on immediate choices of all agents up to step n -1; and moreover agree on immediate choices of J at step n.

Hence V ∼ 0 J V iff V ∼ J V . We define ∼ n = ∼ n ∅ . That is, V ∼ n V if succ m (V ) ∼ Agt succ m (V ) for all m < n.

The Logic LACA

We now define language and semantics of LACA.

Language of LACA

The language of LACA, L LACA (Atm), is defined by the following grammar:

ϕ ::= α | ¬ϕ | ϕ ∧ ϕ | Xϕ | Gϕ | [J cstit]ϕ | [J dstit]ϕ,
where α ranges over Atm and J ranges over the powerset of Agt. The formula Xϕ reads "ϕ holds next"; Gϕ reads "ϕ holds henceforth"; [J cstit]ϕ reads "J sees to it that ϕ" (the so-called Chellas stit operator); and [J dstit]ϕ reads "J deliberately sees to it that ϕ" (the deliberative stit operator). The language L LACA (Prp) is the fragment of L LACA (Atm) without control and attempt atoms and is the language of the original group stit logic (in its discrete time version).

When convenient we drop set parentheses of coalitions and write e.g. [1, 2 cstit]p instead of [{1, 2} cstit]p. Historic necessity is defined as ϕ = [∅ cstit]ϕ. The dual of historic necessity is ♦ϕ = ¬ ¬ϕ and that of the cstit operator is

J cstit ϕ = ¬[J cstit]¬ϕ.
We define the set of atoms of a formula ϕ as follows:

Atm(α) = {α}, Atm(ϕ ∧ ψ) = Atm(ϕ) ∪ Atm(ψ), Atm( * ϕ) = Atm(ϕ) for * ∈ {¬, X, G, [J cstit], [J dstit]}.
For example, Atm(c

1 t 2 p ∧ [1, 2 cstit]c 3 p) = {c 1 t 2 p, c 3 p}.

Truth Conditions

Here are the truth conditions for L LACA (Atm) formulas:

V , n |= α if α ∈ succ n (V ), V , n |= Xϕ if V , n + 1 |= ϕ, V , n |= Gϕ if V , m |= ϕ for every m n, V , n |= [J cstit]ϕ if V , n |= ϕ for every V ∼ n J V , V , n |= [J dstit]ϕ if V , n |= [J cstit]ϕ and V , n |= ϕ,
and as usual for the Boolean operators. The idea underlying the Chellas stit modality [J cstit]ϕ is that the current choices of the agents in J guarantee that ϕ is true no matter what the opponents in J attempt. This is the same as: it is not the case that the agents in J are able to achieve ¬ϕ while the agents in J stick to what they currently attempt. We can therefore view [J cstit] as an operator of inability of J's opponents to avoid ϕ. The deliberative stit modality adds to the Chellas stit a negative condition: for J to deliberately see to it that ϕ, ϕ should not be necessarily true.

Example 3 (Example 2, ctd.). We have

V hw , 0 |= ¬h ∧ ¬w ∧ X(h ∧ ¬w) ∧ XX(h ∧ w) ∧ [1 cstit]Xh ∧ X[2 cstit]Xw.
The formula remains true when the cstit operator is replaced by the dstit operator.

A L LACA (Atm) formula ϕ is satisfiable if V , 0 |= ϕ for some V ; and it is valid if V , 0 |= ϕ for every V .
Example 4. In Homer's Odyssey, Ulysses escapes the sirens by removing his own control to follow their calls. This example is considered by the philosopher Jon Elster to be paradigmatic of the idea that self-control can be used for coping with weakness of will [Elster, 1979]. We model two versions of the situation: what might have happened if Ulysses had not been so clever to get tied to the mast and what actually happens. Initially Ulysses (u) is not on the sirens' island and controls whether he is on the island or not (c u OnIs). He can abandon his control (c u c u OnIs). The sirens, for simplicity viewed as a single agent s, can make Ulysses change his mind about swimming to the island (c s t u OnIs) and indeed attempt to do so (t s t u OnIs). This corresponds to the following valuation:

V u 1 = {c u OnIs, c u c u OnIs, c s t u OnIs, t s t u OnIs}. But Ulysses actually decides to abandon his control by getting tied to the mast: the valuation corresponding to Homer's story is

V u 2 = V u 1 ∪ {t u c u OnIs}. Then V u 1 ∼ s V u 2 and: succ(V u 1 ) = {c u OnIs, c u c u OnIs, c s t u OnIs, t u OnIs}, succ(V u
2 ) = {c u c u OnIs, c s t u OnIs, t u OnIs}. Therefore:

V u 1 , 0 |= X[s cstit]XOnIs, V u 2 , 0 |= [u dstit]XX¬OnIs ∧ [u dstit]X[u cstit]X¬OnIs. In V u
1 , the sirens get their wishes and see to it that Ulysses follow their calls. In V u 2 , Ulysses manages to avoid this outcome and (deliberately) sees to it that he stays on his ship by choosing to remove his own control.

Reconstruction of BT+AC Structures

We now assess our control-and attempt-based logic w.r.t. Belnap and Horty's stit logic by establishing the relation with their semantics in terms of BT+AC models. Our construction leads to a discrete time structure and inverses the standard way the definition of BT+AC models proceeds. The latter takes moments that are ordered in time to be primitives, then defines histories, and finally adds agent choices. In contrast, our construction starts from histories (which are induced by valuations via the successor function) and then defines moments, their temporal ordering, and agent choices. We start by recalling BT+AC models.

Discrete BT+AC Models

We recall the definitions from [Horty, 2001], which we adapt to the case of discrete tree structures as done e.g. in [START_REF] Boudou | [END_REF][START_REF] Broersen | [END_REF][START_REF] Schwarzentruber | Franc ¸ois Schwarzentruber. Complexity results of STIT fragments[END_REF]. A discrete BT structure is a pair (W, <) where W is a non-empty set of moments and < is a strict partial order on W that is tree-like 1 and discrete. Histories are maximal sets of linearly ordered moments. The set of all histories of (W, <) is Hist. The set of histories sharing the moment w is H w ⊆ Hist. The (unique) successor of w on history h is succ(w/h). A BT+AC model is a quadruple M = (W, <, Choice, V) where (W, <) is a BT structure, Choice maps every moment-agent pair (w, i) to a partition of H w , and V associates to each p ∈ Prp a set of moment-history pairs: V(p) ⊆ W ×Hist. Essentially, H w represents all possible timelines branching from moment w, each corresponding to different choices of agents at w, and Choice(w, i) represents the possible choices of a single agent i at w, each leading to a different subset of H w . It is assumed that Choice is such that: 2 1. For every Q ∈ Choice(w, i) we have Q = ∅.

1 If w1<w3 and w2<w3 then w1=w2 or w1<w2 or w2<w1. 2 It is often assumed that Choice(w, i) = ∅, but this is redundant.

2. For every w and mapping s w : Agt -→ 2 Hw such that s w (i) ∈ Choice(w, i) (that is, a selection of one choice for each agent) we have i∈Agt s w (i) = ∅.

3. If h and h are undivided at w then h ∈ Choice(w, i)(h). 3The second constraint says that the agents' choices are independent. The third constraint is called 'no choice between undivided histories'. Letting the set of all selection functions at w be Select w = {s w : s w (i) ∈ Choice(w, i), for every i ∈ Agt}, choices of groups of agents J are defined by:

Choice(w, J) = i∈J s w (i) : s w ∈ Select w .
In particular, Choice(w, ∅) = {H w } for the empty group. Moreover, given a history

h ∈ H w , Choice(w, J)(h) = {h ∈ H w : h, h ∈ Q for some Q ∈Choice(w, J)}.
That is, Choice(w, J)(h) is the particular set of histories containing h among the choices of group J, or in other words, the choice of group J at moment w in the history h.

Then the formulas of the language given in Section 3.1 are interpreted at moment-history pairs as follows:

M, w/h |= p if w/h ∈ V(p), M, w/h |= Xϕ if M, succ(w/h)/h |= ϕ, M, w/h |= [J cstit]ϕ if M, w/h |= ϕ for every h ∈ Choice(w, J)(h), M, w/h |= [J dstit]ϕ if M, w/h |= [J cstit]ϕ and
M, w/h |= ϕ, and as expected for the Boolean operators.

In the rest of the section we show how to construct BT+AC structures from the valuations of Section 2. We start by defining histories as mappings from natural numbers to valuations and then show how every valuation generates a history.

From Valuations to BT Structures

In the context of our models of control and attempt, moments can be identified with ∼ n equivalence classes, choices correspond to agents' attempts, and branching histories stem from different choices for agents.

We start with BT structures. Let V 0 be a valuation.

Let V 0 ∼ be the ∼-equivalence class of V 0 : V 0 ∼ = {V : V ∼ V 0 }. The set of moments is defined as W V0 = n∈N0 W n V0 ,
where for all n ∈ N 0 , W n V0 = V 0 ∼ / ∼ n is the quotient operator. We use disjoint union ( ) in order to ensure that two equivalent moments, whereby agents control the same variables and make the same attempts, can occur at different depths of the BT structure. Given a moment w ∈ W n V0 , the set of successors of w for the order < V0 is:

succ(w) = {w ∈ W n+1 V0 : w ∩ w = ∅}.
Fact 1. succ(w) is a partition of w. Proposition 1. < V0 is a tree-like order.

Recall that histories in BT structures are maximal sets of linearly ordered moments. If h is a history of (W V0 , < V0 ), then for all n ∈ N 0 we denote by h(n) the unique member of h ∩ W n V0 . Every valuation V clearly induces a single history h V such that V ∈ h V (n) for all n ∈ N 0 . The other way round, each history is induced by at least one valuation. Proposition 2. For every history h of (W V0 , < V0 ), there exists V ∼ V 0 such that h = h V . Corollary 1. For every w ∈ W V0 , the set H w of histories containing w is exactly the set of histories induced by valuations in w, i.e., we have H w = {h V : V ∈ w}.

From Valuations to Agent Choices

We now show how to construct a full BT+AC model M V0 = (W V0 , < V0 , Choice V0 , V V0 ) from a valuation V 0 . We have already defined W V0 and < V0 . As all valuations in an equivalence class for ∼ agree on propositional variables, V V0 is naturally defined by

V V0 (p) = {w/h : w ∈ W n V0 for some n ∈ N 0 , h ∈ H w , and p ∈ succ n (V ) for V ∈ w}. If i is an agent and w ∈ W n V0 , we define Ctrl i (w) = {α ∈ Prp ∪ Ctrl : there is a V ∈ w such that c i α ∈ succ n (V ) and t i α ∈ succ n (V )}.
Intuitively, the set Ctrl i (w) is the set of possible immediate choices of agent i at w. We now define Choice V0 (w, i) for w ∈ W n V0 as:

Choice V0 (w, i) = {{h V : V ∈ w and Att i (succ n (V )) ∩ (Prp ∪ Ctrl) = Γ} : Γ ⊆ Ctrl i (w)}.
Proposition 3. Consider w ∈ W V0 . Then: 1. Choice V0 (w, i) is a partition of H w ; 2. Any H ∈ Choice V0 (w, i) is nonempty; 3. Choice V0 (w, i) satisfies the "no choice between undivided histories" constraint; 4. For every selection function s w : Agt -→ 2 Hw such that

s w (i) ∈ Choice V0 (w, i) we have i∈Agt s w (i) = ∅.
The choices of group J at w are:

Choice V0 (w, J) = {{h V : V ∈ w and ∀i ∈ J, Att i (succ n (V )) ∩ (Prp ∪ Ctrl) = Γ i } : ∀i ∈ J, Γ i ⊆ Ctrl i (w)}. Given a history h V with w = h V (n), Choice V0 (w, J)(h V ) corresponds to fixing the immedi- ate attempts of agents of J at h V (n) to those of succ n (V ): Choice V0 (w, J)(h V ) = {h V : V ∈ w and succ n (V ) ∼ J succ n (V )} = {h V : V ∼ n J V }. Proposition 4. M V0 = (W V0 , < V0 , Choice V0 , V V0
) is a BT+AC structure satisfying all constraints of Section 4.1.

Equivalence

We are ready to show that all standard stit validities are also valid in models of control and attempt.

Proposition 5. Let ϕ ∈ L LACA (Prp) be a formula and V 0 ∈ 2 Atm be a valuation. For all n ∈ N 0 and

V ∼ V 0 , M V0 , h V (n)/h V |= ϕ iff V , n |= ϕ.

Beyond Standard Validities

The above inclusion is strict: there are LACA-valid L LACA (Prp) formulas that not valid in BT+AC models. First of all, standard validities in BT+AC models with discrete time such as [J cstit]Xϕ → X ϕ. are also LACA valid. Beyond that, facts and abilities are moment determinate in LACA, as expressed by the implications α → α and ¬α → ¬α, for α ∈ Prp ∪ Ctrl. This property is discussed as optional in the stit literature [Horty, 2001].

Here are three formulas that are not standardly valid but are valid in LACA. The implication Xp → p, for p ∈ Prp expresses that facts don't change by themselves, and

(¬p ∧ Xp) → i∈Agt [i cstit]
Xp expresses that every fact change is caused by some agent. Finally, the equivalences

[i cstit](p ∨ q) ↔ ([i cstit]p ∨ [i cstit]q) and [J cstit]p ↔ i∈J [i cstit]
p are valid because in LACA, agency is based on a relation between individuals and atoms.

Complexity of Model Checking

The model checking problem we are interested in is the following: given a valuation V ⊆ Atm, a time point n ∈ N 0 and formula ϕ ∈ L LACA (Prp), do we have V , n |= ϕ? In this section we prove that it is PSPACE-complete.

Relevant Valuations

A key factor in the complexity of evaluating formulas is the infinity of valuations that need to be checked when evaluating G and [J cstit] operators. Let us investigate whether we can restrict ourselves to considering only a finite number of valuations. First, we define the length of an atom of Atm to be the length of the prefix of the propositional variable:

p = 0, c i α = 1 + α , t i α = 1 + α .
For example, c 1 c 2 t 3 p = 3. We extend the definition of length to finite valuations: V = α∈V α .

We can now establish that finite valuations only have a finite number of successors: Proposition 6. Let V ⊆ Atm be a finite valuation.

1. card(V ) card(succ(V )); 2. Either V > succ(V ) or succ(V ) = V .
Given a valuation V and a group of agents J, considering all valuations such that V ∼ n J V amounts to considering variations on J's attempts as well as on attempts of agents of J outside of their immediate choices. There is actually no need to consider variations on attempt atoms that are not a suffix of any control atom in V .

Proposition 7. Let V be a valuation, J be a group of agents and ϕ be a formula of L LACA (Prp). Then V , n |= [J cstit]ϕ iff V , n |= ϕ for all V such that V ∼ n J V and V and V differ only by attempt atoms t i α such that c i α is a suffix of a control atom in Ctrl Agt (V ).

PSPACE-Completeness

Using the results of the previous section, we now prove that model checking formulas of L LACA (Prp) in our framework is PSPACE-complete. Proposition 8. The L LACA (Prp) model checking problem is in PSPACE-complete.

Proof sketch. The membership proof is based on an algorithm computing whether V, n |= ϕ. The number and size of alternative valuations that need to be checked in the cases of Gϕ, [J cstit]ϕ and [J dstit]ϕ are limited by Propositions 6 and 7.

Hardness is shown through a polynomial reduction of the problem of model checking Quantified Boolean Formulas (QBF), which is known to be PSPACE-complete.

Application: Modelling Influence

In this section we we put our logic LACA into practice and show how it allows us to formalise [START_REF] Lorini | Emiliano Lorini and Giovanni Sartor. A STIT logic for reasoning about social influence[END_REF] definition of social influence. Their concept is based on the idea that for the influencer i to induce an influencee j to make a certain choice, i has to change j's choice context, so that a choice differing from j's preferred option without this modification becomes preferable to j. Agent j's choice context includes both j's available choices determined by what agent j controls (i's choice set) and j's preferences. "i influences j so that ϕ" is defined as:

[i infl j]ϕ = [i cstit]X(Rat j → [j dstit]ϕ),
where Rat j expresses that j is rational. For agent i (the influencer) to induce agent j (the influencee) to see to it that ϕ, i's actual choice must guarantee that every rational choice of j will make ϕ true. The reason why agent j's action is modeled by the deliberative stit operator is that for agent j to be influenced by agent i, j should have a choice available in his choice set possibly leading to ¬ϕ; i.o.w., influence requires that the consequence of the influencee's action could have been avoided, had the influencee made a different choice.

Let us see how we can express that an agent j is rational. Simply supposing that Rat j is yet another special propositional variable (as done in [START_REF] Lorini | Emiliano Lorini and Giovanni Sartor. A STIT logic for reasoning about social influence[END_REF]) fails to do the job because that variable would not be related to the agent's choices. Instead, we suppose that for every agent i there is a formula Rat i characterising the states where an outcome with maximal utility obtains next: an agent is rational in a valuation if the successor valuation is ideal for the agent. One possibility is that Rat i is of the form Xϕ: i prefers ϕ to be true next; in particular, when i prefers ϕ to be true from the next state on then Rat i contains XGϕ. Another possibility is that Rat i characterises, for each action in i's choice set, under which conditions it is rational for i to try to perform it. Then Rat i is typically a conjunction of equivalences of the form t i p ↔ ϕ i p where ϕ i p does not involve attempt atoms. For example, if i prefers to make p true whatever the values of the other variables are then Rat i contains t i p ↔ ¬p. Example 5 (Example 1, ctd.). Suppose agent 2 wants to to save energy: she prefers that the window is closed (w) when the heating is on (h); otherwise she does not care. We therefore have Rat 2 = X(h → w). Consider the valuation

V hw 1 = {c 1 h, t 1 h, c 2 w, c 2 t 2 w}.
Hence 1 is going to try to switch the heating on, which, intuitively, should influence 2 so that the window is going to be closed. In terms of actions: 1's switching the heating on is going to make 2 close the window. We therefore expect V hw 1 , 0 |= [1 infl 2]Xw, i.e., V hw 1 , 0 |= [1 cstit]X(Rat 2 → [2 dstit]Xw). Let us check in detail that this is indeed the case. First, when interpreting the modality [1 cstit] at V hw 1 we check whether V hw 1 ∪ S, 0 |= X(Rat 2 → [2 dstit]Xw), for every S ⊆ {t 2 α : α ∈ Atm} ∪ {t 1 α : c 1 α / ∈ V or α = t i β}. Thanks to Proposition 7, we know that it is enough to consider variations on t 2 w and t 2 t 2 w. Let us consider the three sets V hw 1 , V 2 = V hw 1 ∪ {t 2 w}, and V 3 = V hw 1 ∪ {t 2 t 2 w}. Their respective successors are succ(V hw 1 ) = {h, c 1 h, c 2 w, c 2 t 2 w}, succ(V 2 ) = {h, c 1 h, c 2 w, c 2 t 2 w, w}, succ(V 3 ) = {h, c 1 h, c 2 w, c 2 t 2 w, t 2 w}.

Observe that succ(V 2 ) is obtained from V hw 1 by 1 switching the heating on and 2 simultaneously closing the window, so to speak anticipating 1's influence. Both (V 2 , 1) and (V 3 , 1) satisfy Rat 2 , and it therefore remains to check whether V 2 , 1 |= [2 dstit]Xw and V 3 , 1 |= [2 dstit]Xw, which are indeed the case. Therefore V hw 1 , 0 |= [1 infl 2]Xw.

Conclusion

We have presented a 'stit' framework in which the notion of action is conceptually and computationally grounded on the notions of control and attempt. We have proved that, unlike the standard 'stit' framework in which model checking is unfeasible because histories are infinite, model checking in our framework is PSPACE-complete. In future work, we expect to broaden its scope of application by formalizing further action-related concepts that have been investigated in the literature on 'stit' logic. The latter include the concept of controllability, as an agent i's capacity to influence another agent j to perform a certain action [START_REF] Lorini | Emiliano Lorini and Giovanni Sartor. The ethics of controllability as influenceability[END_REF], as well as the concept of responsibility in its active form (i.e., responsibility for action) and passive form (i.e., responsibility for omission) [Lorini et al., 2014;[START_REF] Lorini | Emiliano Lorini and Francois Schwarzentruber. A logic for reasoning about counterfactual emotions[END_REF]]. As we have shown in Section 6, our logic LACA offers a minimalistic and computionally grounded framework for modeling these concepts. Moreover, we plan to come up with an axiomatisation of the validities of our discrete group stit logic. Last but not least, we plan to enrich our temporal stit framework with past tense modalities for 'yesterday' (Y) and 'always in the past' (H) and to formalize the concept of 'achievement stit' studied in [Belnap et al., 2001] that better approximates agent causation.

Proceedings of the Thirty-First International Joint Conference on Artificial Intelligence

Two histories h and h are undivided at a moment w ∈ W if either w and succ(w/h) are both on h , or w has no successor. That is, if w has a successor then w has the same successor in h and h .Proceedings of the Thirty-First International Joint Conference on Artificial Intelligence

Acknowledgments

We thank the reviewers for their comments. Our research was partly supported by the ANR-3IA Artificial and Natural Intelligence Toulouse Institute (ANITI), the EU ICT-48 2020 project TAILOR (No. 952215) and the ANR chair BE4musIA.