Fatme Trad

Alaa E Giba

Xavier Devaux

Mathieu Stoffel

Denis Zhigunov

Alexandre Bouché

Sébastien Geiskopf

Rémi Demoulin

Philippe Pareige

Etienne Talbot

M Vergnat

H Rinnert

Influence of phosphorus on the growth and the photoluminescence properties of Si-NCs formed in P-doped SiO/SiO 2 multilayers

Keywords: Si nanocrystals, Phosphorous, Doping, Phase separation, Nanoscale analysis

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

During the last decades, silicon nanocrystals (Si-NCs) have attracted a great deal of attention in various fields like optoelectronics, sensors, plasmonics, and solar cells. 1,2,3,4 Both the quantum confinement effect and the tunable size-dependent bandgap open great opportunities to effectively control the optical absorption and emission of Si-based devices. Doping Si-NCs with either negatively and/or positively charged impurities is a promising route for tailoring their optical and electrical properties. Si-NCs could be efficiently used in all Si-based tandem solar cells, in which n-and p-type layers of Si-NCs can be obtained. Moreover, co-doping is also a promising approach to induce electronic levels in the bandgap, which could provide novel optical emission bands in Si-based materials. Thus, efficient sub-bandgap emission in co-doped Si-NCs dispersed in a colloidal solution 5,6 or prepared by nonthermal plasma synthesis 7 was demonstrated. Although some calculations predicted high formation energies for impurities in semiconductor nanocrystals, 8 experimental works demonstrated that P atoms can be preferentially localized in Si-NCs rather than in the surrounding silicon oxide matrix. 9 These observations are further in good agreement with other calculations. 10,11 Despite many efforts to introduce efficient dopants into Si-NCs and to investigate the doping-induced electronic properties, many questions still exist, in particular related to the effective generation of free carriers. 12,13 The electronic activation of the inserted impurities is often studied through absorption or photoluminescence (PL) measurements. The existence of free carriers in Si-NCs has been successfully deduced from absorption in the infrared range for heavily P-doped Si-NCs. 14,15 Moreover, the activation of dopants in Si-NCs can be argued by the loss of PL emission from Si-NCs, which may be interpreted as a signature of an Auger effect. 15,16 However, as suggested in the literature, the PL quenching could also originate from defects induced by doping. 17 König et al. claimed that P is unlikely to provide electrons to Si-NCs embedded in SiO 2 because P atoms found inside Si-NCs are almost exclusively located in interstitial states. 18 The PL quenching is then due to deep levels induced by P in the surrounding SiO x shell. In addition, a blueshift of the Si-NCs related PL can be interpreted by a preferential quenching of luminescence of the bigger Si-NCs, which are more likely to contain dopants. 15,16 A precise knowledge of the localization of dopants is a key issue to improve the understanding of the doping induced optoelectronic properties. In case of Si-NCs embedded in a SiO 2 matrix, dopants could be localized in the nanocrystal core, at their surface or in the surrounding matrix. Because of the difficulty to experimentally localize the dopant at the nanoscale, the experimental results dedicated to the localization of dopants are still scarce. Among the potential experimental methods, atom probe tomography (APT) is an efficient way to localize dopants. Such a technique has given evidence that P atoms can be inserted into Si-NCs. 19,20 Using APT to investigate Pdoped Si-NCs embedded in a SiO 2 matrix, Gnaser et al. have reported that around 20 % of P atoms are localized inside Si-NCs, whereas 30 % reside at the interface and 50 % remain in the matrix. 21 In addition to the localization of dopants, the potential modification of the growth process and the size of the Si-NCs induced by the impurities are of primary importance to interpret the optical properties of layers containing Si-NCs and dopants. A correlation between the accurate location of dopants and the optoelectronic properties could strongly help to reach a better understanding of the doping at the nanoscale.

In this paper, we report on the influence of P on the optical properties of Si-NCs formed in SiO/SiO 2 multilayers in relation with the growth of the nanocrystals, the defects and the localization of dopants. The optical properties are analyzed by temperature-dependent PL spectroscopy. Structural and chemical analyses at the nanoscale were carried out by scanning transmission electron microscopy (STEM), STEM electron energy loss spectroscopy (STEM-EELS) and APT. We shed light on the strong modification of the phase separation process in Sirich SiO 2 layers, induced by the softening of the oxide and suboxide layers, which is of primary importance to interpret the PL properties of Si-NCs. Our results indicate that depending on the P content, P atoms lead to a significant improvement of the Si-NCs PL properties or to the quenching of the Si-NCs emission.

Experimental

P-doped (SiO/SiO 2 ):P multilayers were obtained by alternating the deposition of SiO and SiO 2 compounds provided by two electron-beam guns on Si(001) substrates held at room temperature. Materials for evaporation are high purity (99.99 %) SiO and SiO 2 grains with a typical size of 2 to 4 mm. The deposition rates of SiO and SiO 2 , equal to 0.1 nm/s were controlled by quartz microbalances. P atoms were supplied by a GaP decomposition source, maintained at a controlled temperature during all the deposition process, hence leading to the incorporation of P atoms in both SiO and SiO 2 layers. The P content was controlled by the temperature of the GaP cell. 22 As the P content cannot be accurately measured by the quartz microbalance during deposition, the P concentration in the layer was measured after deposition by energy dispersive x-ray spectroscopy and atom probe tomography. A calibration curve was then established giving the correspondence between the GaP cell temperature and the global P content in the film. The following exponential dependence was determined: log(C P ) = 16.1-1.78×10 4 /T, where C P is the P concentration given in atomic percentage and T is the cell temperature given in Kelvin. The P atomic concentration was varied from 5×10 -5 to 2.5 at. %, corresponding to cell temperatures ranging from 600 °C to 860 °C. As-deposited samples were amorphous. The samples were then post-grown annealed in a rapid thermal annealing furnace at different temperatures up to 1100 °C for 5 minutes under N 2 flow (200 sccm). Some samples were subjected to a hydrogen passivation process, carried out at 500 °C during 5 min under a flow of forming gas N 2 (90%) / H 2 (10%). STEM and STEM-EELS experiments were carried out using a JEOL ARM 200 F probe corrected with a cold field emission gun. EELS spectrum images (SI) were recorded with a Gatan GIF quantum ER filter. To avoid any phase precipitation due to electron beam damages, experiments were performed at 80 kV with a probe current of about 50 pA for EELS spectrum images. Two EELS-SI were simultaneously recorded for the low-loss part containing the zero-loss and for the core-loss which allows advanced data post processing like energy drift corrections and multiple scattering corrections. The spectra were recorded with an electron beam having a half convergence angle of 24 mrad for a half collection angle of 56 mrad with an energy dispersion of 0.05 eV/channel and a pixel time of 0.005 s. SI were recorded with a spatial sampling of about 0.2×0.2 nm 2 /pixel. A multivariate statistical analysis software was used to improve the quality of the STEM-EELS data by de-noising the core-loss SI. Low-loss spectra were processed by zero-loss extraction and Fourier-log deconvolution for multiple scattering corrections. To draw chemical maps, multiple linear least square fitting was applied on Si L and P L signals. Due to the overlapping of Si L and P L edges, and to separate chemical states of Si (Si 0 and Si IV ), reference spectra were recorded on pure Si, on pure thermal silica grown on a Si wafer and on SiP 2 . The reference spectra were extracted from SI recorded on lamellae with the same thickness, and with the same STEM and spectrometer parameters as that of the samples under study. For TEM and STEM observations, lamellas thinned up to t/ < 0.8 (ratio between the thickness t and the electron mean free path ) were prepared by focused ion beam (FIB) using a FEI Helios Nanolab 600i dual beam microscope.

During FIB operation, the samples were capped by a pure carbon thin layer followed by a Pt-C layer to protect the sample during the first steps of the lamella extraction. After FIB milling, the lamellae were stored in vacuum to prevent oxidation. The 3D high resolution structural and chemical analysis at atomic scale was also conducted by APT. Experiments were carried out using a laser-assisted wide-angle tomographic atom probe (LAWATAP, Cameca) with a femtosecond UV pulsed laser (λ=343nm, 350 fs, 33 nJ). The analysis was performed in a highvacuum chamber under 10 -10 mbar at 80 K. Tip shaped samples were prepared using a ThermoScientific plasma FIB Helios G4 XCe. APT data treatments and 3D reconstruction were obtained using GPM3Dsoft software. The steady state PL was measured using the 325 nm He-Cd laser as an excitation. The PL signal was collected by a monochromator equipped with a 600 grooves/mm grating and measured by a cooled photomultiplier tube operating in the visible-NIR range. The spectra were corrected for the system response.

Results and discussion

(SiO/SiO 2 ):P multilayers were prepared using SiO and SiO 2 thicknesses equal to 2.5 and 5 nm, respectively. The number of bilayers is 20, hence a total film thickness of 150 nm for all samples is considered in this study. Figure 1 shows cross-sectional STEM-EELS images for (SiO/SiO 2 ):P multilayers annealed at 1100 °C having a phosphorus content of 7.8×10 -3 at. % (Fig. 1(a)) and 4×10 -1 at. % (Figs. 1 (b)-(e)). Si atoms in a Si matrix (Si 0 ), Si atoms in silica (Si 4+ ), as well as phosphorous atoms are shown in red, green and blue colors, respectively. Fig. 1 (a) shows that Si-NCs are formed in the deposited SiO layers and that the size distribution seems to be rather narrow, as expected for such multilayers structures. 23,24 Moreover, this analysis shows that the multilayer structure is preserved for such a low P content. However, for larger P contents, as high as 4×10 -1 at. %, the multilayer structure can hardly be distinguished and some disorder is evident, as shown in Figs. 1(b)-(c) by mapping the same sample area for visualization of Si atoms in Si-NCs and Si in the surrounding SiO 2 matrix, correspondingly. To detect clearly P atoms by STEM EELS analysis, several samples with different P concentrations have been carefully studied. For P concentrations below 6×10 -2 at. %, it is not possible to extract any significant signature of P atoms from the EELS spectra, which corresponds probably to the detection limit of the technique. However, starting from a concentration of 6×10 -2 at. % (images not shown here), P atoms can be detected. They are almost randomly localized in the multilayer structure: in the amorphous SiO 2 matrix, close to the Si-NCs/SiO 2 interface or inside Si-NCs.

The situation changes for higher P contents. Indeed, for a P concentration of 4×10 -1 at. %, P atoms are preferentially localized inside Si-NCs, as evident from the superposition of Si 0 and P related maps shown in Fig. 1(e). However, as this view corresponds to a 2D projection, the localization of P atoms at the interface cannot be fully excluded, even if elemental profiles across Si-NCs do not show the double peak characteristic of core-shell like microstructure with an over concentration of phosphorus at the SiO 2 /Si-NCs interface. To verify these results and to get a better understanding of the localization of P atoms, APT experiments were carried out.

Figure 2 shows the APT results obtained for (SiO/SiO 2 ):P multilayers annealed at 1100 °C in case of an undoped sample as compared with three samples with different P concentrations equal to 7.8×10 -3 , 6.10 -2 and 4×10 -1 at. %. Cross section volumes extracted from the 3D reconstruction for the different samples revealing the position of Si and P atoms in 21×21×3 nm 3 boxes are represented in Figs. 2(a-d). It should be noted that P atoms cannot be detected in the sample containing 7.8×10 -3 at. % of P which is below the detection limit of P by APT in our samples. For all samples, Si-NCs are clearly identified as Si atom dense spots. The average diameter and size distribution of Si-NCs are almost independent of the P concentration up to a value of 6×10 -2 at. %. For a P concentration of 4×10 -1 at. % (see Fig. 2(d)), Si-NCs are significantly larger. The size distributions shown in Figs. 2 (e)-(f) clearly indicate that the average size increases from 2.5 nm to 3.1 nm when the P concentration increases from 7.8×10 -3 at. %. to 4×10 -1 at. %. The expected mean size, controlled by the SiO thickness is 2.5 nm, which is indeed observed for undoped multilayers and for P contents below or equal to 6×10 -2 at. %. In addition, the size distribution for the highest P concentration becomes broader, suggesting that the size of the Si-NCs is not fully controlled by the SiO 2 barrier layers in presence of a high quantity of impurities. APT allows determining the localization of P atoms for concentrations higher or equal to 6×10 -2 at. %. For this concentration, the 2D map (Fig. 2 (c)) clearly indicates that P atoms can be found in the SiO 2 matrix and that some P atoms are localized inside Si-NCs or at the matrix/Si-NCs interface. P atoms seem to be almost randomly localized. On the contrary, for higher P contents, i.e. 0.4 at. % of P in average, the sample is characterized by a preferential localization of P atoms inside Si-NCs, as shown in Fig. 2(d). From the APT analysis, the fraction of P atoms remaining in the SiO 2 matrix can be estimated to about 40 %. To determine the accurate localization and concentration of P atoms inside Si-NCs, an erosion profile was performed on more than one hundred of Si-NCs and the characteristic profile is shown in Figure 3.

The erosion profile highlights the evolution of the composition of each species from the Si-NCs core to the SiO 2 matrix. [START_REF]Atom probe tomography: put theory into practice[END_REF] The position zero on the x-axis corresponds to the interface between Si-NC and SiO 2 matrix. Concerning P atoms, the shape of the erosion profile shows a significant increase of the P composition in the core of Si-NCs, unambiguously demonstrating that most of P atoms are indeed localized inside Si-NCs, reaching a raw concentration as high as 8 at. %. It must be mentioned that a high concentration of O atoms is also measured inside the Si-NCs. This observation is due to the local magnification effect occurring during APT analyses of multi-phased systems. [START_REF] Vurpillot | [END_REF] A correction of this measurement artefact (see ref. 27) allows to estimate a P concentration of 11 to 12 at. % inside the Si-NCs, far beyond the solubility limit of P in bulk Si. Even if the erosion profile does not show any sign of self-purification effect, which is characterized by a P composition peak at the Si-NC/SiO 2 interface, it cannot be excluded that some P atoms are localized at the Si-NCs surface. Moreover the interface between the Si-NC and the surrounding matrix is not sharp. As shown on the erosion profile, the evolution of the Si concentration from the minimum value to the maximum value extends over 2 nm. The spatial evolution of the P content from almost zero in SiO 2 up to the highest value of about 8 at.% in Si-NC also extends over 2 nm which suggests that some P atoms are located at the interface between Si-NCs and the SiO 2 matrix.

The above presented structural and chemical study at the nanoscale shows that the introduction of P impurities in the SiO/SiO 2 multilayer strongly affects the growth of Si-NCs for P concentrations higher than around 0.1 at.%. For low P concentrations, i.e. below 0.1 at. %, the Si-NC mean size is controlled by the SiO 2 barrier layers and the multilayer structure leads to a relatively narrow Si-NC size distribution. P atoms are almost randomly localized in the matrix, inside Si-NCs or at the SiO 2 /Si-NC interface. The distribution of the number of P atoms per Si-NC is broad and approximately one third of nanocrystals contains 1 P atom. Beyond this concentration, the Si-NC mean size is no longer controlled by the multilayer structure, and the size distribution becomes broader. Moreover, P atoms are preferentially localized inside Si-NCs and P concentrations as high as 10 at. % can be found in some Si-NCs. These results indicate that, while thermodynamics is not favorable to the location of P atoms inside Si-NCs, high concentrations of P atoms can be nevertheless inserted inside Si-NCs. The driving force that controls the localization of dopants and the enriched distribution of P inside Si-NCs is probably the solubility limit of phosphorus, which is very different in Si and in SiO 2 . The impurity redistribution at the SiO 2 /Si interface is well known and if boron segregates preferentially in SiO 2 , both arsenic and phosphorus are known to accumulate at the Si surface. In particular it has been shown that the phosphorus concentration can be four orders of magnitude higher in Si than in SiO 2 at the SiO 2 /Si interface after an annealing treatment at 1100 °C. 28 For the weakest P concentration used in our study, the P content in the multilayer is assumed to be lower than the solubility limit of P in SiO 2 which explains an almost random localization of P in the layer. For higher P concentrations, phosphorus moves to Si, where the solubility limit is higher. It is worth noting that alloying at the nanoscale has not been observed in this study, while for higher P contents, the formation of SiP 2 nanocrystals inside a SiO 2 matrix has been reported. 29 For P concentrations larger than 0.1 at.%, the Si-NCs size increase and size distribution can be interpreted by a softening of the SiO 2 matrix induced by P impurities, as already suggested. 30 Our results are in good agreement with the P-induced Si-NCs size increase from 4 to 8 nm obtained from the phase separation in P-doped silicon-rich silicon oxide produced by cosputtering with a P concentration of 0.5 at.%. 31 Moreover, the P-induced softening effect probably leads to an increase of the diffusion length of P atoms in the matrix and contributes to a higher concentration of P atoms inside Si-NCs.

The influence of P atoms on the optical properties of Si-NCs was studied by PL spectroscopy. PL measurements were carried out at room temperature on intrinsic multilayers and those containing P atoms for different annealing temperatures and different P concentrations.

Figure 4(a) shows the dependence of the PL signal with the annealing temperature for intrinsic multilayers. For the as-deposited sample and the sample annealed at 700 °C, no significant PL is observed. For samples annealed at temperatures between 900 °C and 1000 °C, a very weak and broad PL is observed, which is a signature of amorphous Si nanoclusters formation. 32 The PL band intensity significantly increases with the annealing temperature, reaching the highest intensity for an annealing temperature of 1100 °C. This broad PL band, extending from 1.2 eV to 1.9 eV is attributed to the recombination of electrons and holes confined in Si-NCs. Fig. 4(b) shows the dependence of the PL signal with the annealing temperature for the (SiO/SiO 2 ):P multilayer containing 4×10 -1 at.% of P. The introduction of P atoms in the multilayers leads to a drastic change of the dependence of the PL emission of Si-NCs with the annealing temperature. Indeed, the PL intensity of Si-NCs starts to be intense already after an annealing temperature as low as 900 °C, and the highest PL intensity is observed for an annealing temperature of 1000 °C, i.e. 100 °C below the optimal annealing temperature for the undoped sample. Furthermore, it must be noted that the PL intensity characteristic of this optimal annealing is slightly higher that the highest PL shown by the intrinsic layer. However, the comparison of the PL intensity of both samples annealed at 1100 °C indicates a strong decrease of the PL intensity for the samples containing P atoms, by a factor close to 10. To get a better understanding of the optimal annealing temperature, i.e. the annealing temperature leading to the highest Si-NCs PL intensity, we have analyzed the annealing temperature dependence of the PL for multilayers with different P contents. Fig. 4(c) shows the optimal annealing temperature as a function of the P concentration. For P concentrations lower than 10 -2 at. %, the optimal annealing temperature is almost unchanged, within the accuracy of the measurements, taking into account that samples were annealed at different temperatures with a step of 50 °C. For higher P contents, a decrease of the optimal annealing temperature is unambiguously induced by P doping. For a P concentration of 3.5 at. %, the optimal annealing temperature reaches a value as low as 800 °C. This can be explained by the P-induced softening of the matrix suggested above, which leads to the reduction of the thermal budget requested for the phase separation and crystallization processes followed by the formation of optically active Si-NCs. Fig. 5(a) shows the normalized room temperature photoluminescence spectra of (SiO/SiO 2 ):P samples annealed at 1100 °C for different P concentrations. A P-induced PL redshift is clearly observed. Fig. 5(b) shows the PL energy which corresponds to the PL peak maximum as a function of the P content. For P contents up to 2×10 -2 at. %, the PL energy slightly decreases. However, for higher P, contents a pronounced redshift is observed from about 1.55 eV for the undoped and weekly doped layers to 1.42 eV for the sample doped with 1.1 at. % of P. A spectral shift could be interpreted by the activation of dopants inside Si-NCs. A P-induced PL blueshift was indeed explained by a preferential doping of the largest Si-NCs, which were assumed to be quenched by the Auger effect. 15,16 The observed PL spectral shift obtained with our samples cannot be interpreted by a preferential activation of dopants inside largest Si-NCs, but rather is due to the P-induced increase of the Si-NCs size and the associated weakening of the quantum confinement effect. Indeed, the PL redshift occurs concomitantly with the increase of the Si-NCs size for P contents higher than 0.1 at. %, as demonstrated by the microstructural study.

Fig. 5(b) also displays the dependence of the Si-NCs PL peak intensity with the P concentration. The PL intensity increases with P concentrations up to about 0.2 at.% and then decreases for higher concentrations until quenching occurs. Understanding the behavior of the PL intensity with the P content is not straightforward and should be considered in light of the Pinduced growth modifications. A similar P dependence of the PL intensity has been reported, attributing the PL intensity increase to a reduction of the density of P b centers at the interface between Si-NCs and SiO 2 . 15 The interfacial defects act as non-radiative pathways and the decrease of their density would then result in a higher PL intensity in comparison with the undoped sample. However, another hypothesis could be raised to account for the increase of the PL intensity. As P promotes the phase separation, the addition of P could lead to an increase of the Si-NCs density, which is however not observed.

To investigate more into details the Si-NCs related PL intensity increase, low temperature PL measurements and passivation annealing treatments were carried out. Figure 6 shows the PL spectra measured at 50 K, normalized to the peak at around 1.6 eV, for different P contents. The PL spectra are characterized by two peaks. The contribution at about 1.6 eV is due to the recombination of charge carriers confined in Si-NCs. This peak is slightly blueshifted as compared to the measurements at room temperature, which is well known and can be explained by the temperature dependence of the Si bandgap. 33,34 The second peak at around 1 eV is observed for our samples only at temperatures below about 150 K, and its intensity is a decreasing function of the temperature. This peak is obtained for both undoped and doped samples, which indicates that this PL band is not due to P-induced radiative defects or electronic levels. Moreover, being normalized to the confined charge carriers peak, this near-infrared contribution is a decreasing function of the P content. It is worth mentioning that Fujii et al. have reported a similar PL band emission at low temperatures from Si-NCs embedded in a SiO 2 matrix for undoped, B-doped or P-doped layers. 35,36,37 The origin of this sub-band is referred to the recombination of charge carriers trapped at the interfacial P b centers, at the interface between Si-NCs and the SiO 2 matrix. A P b center is the dominant electrically-active defect at the Si-SiO 2 interface, which introduces energy levels in the band-gap of SiO 2 and participates in the trapping and de-trapping of charge carriers. [START_REF] Li | CCD Image Sensors in Deep-Ultraviolet: Degradation Behavior and Damage Mechanisms[END_REF] The clear decrease of this defect-related contribution with increasing P contents shown in Fig. 6, strongly suggests that P insertion leads to a decrease of the number of P b centers, which then contributes to the increase of the Si-NC related PL intensity shown in Fig. 5(b). The reduction of the number of interfacial defects could be attributed to the P-induced reduction of strain at the Si-NC/matrix interface during the cooling and/or annealing treatments.

To further study the role of P on P b centers, both undoped and doped multilayers have been subjected to a passivation treatment, consisting of an annealing at 500 °C during 5 min under forming gas. Figure 7 shows the low temperature PL spectra of an undoped multilayer annealed at 1100 °C prior to and after a passivation treatment. It can be observed that the P b center-related defect band at ~1 eV is completely suppressed while the Si-NCs related PL is strongly enhanced after the passivation treatment, which leads to a decreasing number of P b centers. This observation highlights the high efficiency of the passivation treatment. The inset of Fig. 7 shows the PL intensity of passivated Si-NCs as a function of the P content. It clearly appears that the PL intensity is a continuously decreasing function of the P concentration, which partially contradicts the evolution shown in Fig. 5 (b) for unpassivated Si-NCs. The reason for that is concerned with the fact, that Si-NCs prepared without any passivation treatment are characterized by the presence of P b centers which limit their PL intensity. On the contrary, for hydrogen-passivated samples, defects are almost all suppressed and the presence of P atoms do not contribute, in that case, to an increase of the PL intensity. Our results suggest that P atoms contribute to the increase of the PL intensity only for layers containing defect centers. We believe that the mechanisms leading to an increase of the PL intensity, induced by forming gas passivation or P doping, are different. While H 2 molecules contained in a forming gas reduce the number of P b centers after their formation during the annealing treatments at 1100 °C, the introduction of P atoms probably limits their formation because of the P-induced modification and softening of the SiO x matrix. It can then be concluded that the introduction of P in passivated multilayers hence mainly contributes to decrease the PL intensity of Si-NCs. Two hypotheses may be put forward to explain the P induced Si-NC PL quenching. The first one is concerned with a non-radiative Auger recombination process, assuming that P atoms are active, while the second one is concerned with deep defect states providing efficient pathways for non-radiative charge carrier recombination. For the former, P is considered as a dopant, located in substitutional states while for the later, P is considered to be in interstitial states or at the surface of the nanocrystals. Unfortunately, if the structural and chemical studies at the nanoscale are highly efficient to localize P atoms, the distinction between substitutional or interstitial sites is not achievable. If P atoms are in substitution and are electrically active, plasmons are expected to appear. Infrared absorption spectroscopy (results not shown here) measurements have been carried out in the 400-8000 cm -1 range and no absorption band has been observed, hence suggesting that no plasmonic related absorption occurs, which is however expected for the highest P concentration used in this work. In addition, it is demonstrated that P atoms play a significant role on the size distribution, inducing an increase of the mean size and an associated decrease of the quantum confined carriers related PL. Furthermore, as highlighted by APT and STEM-EELS analyses, some P atoms are expected to be localized at the interface between Si-NCs and SiO 2 matrix, where they may act as non-radiative centers. The small size of these nanocrystals embedded in a matrix and their high surface to volume ratio may not be favorable to the collective oscillation of free carriers or to an efficient doping.

Conclusions

(SiO/SiO 2 ):P multilayers with different P contents were prepared by co-evaporation of successive layers of SiO and SiO 2 from e-beam guns and P atoms were supplied by a GaP decomposition source. The optical properties were studied and compared with structural and chemical analysis at the nanoscale. The introduction of P atoms is shown to play a significant role on the phase separation and on the growth of Si-NCs within the SiO layers. P atoms induce a softening of the matrix leading to: i) the formation of highly luminescent Si-NCs at lower annealing temperatures in comparison with undoped samples; ii) a PL redshift induced by a decrease of the SiO 2 barrier effect and a subsequent formation of Si-NCs with larger sizes; iii) an increase of the PL intensity induced by a decrease of non-radiative defects at the Si-NCs/SiO 2 interface for average P concentrations below 0.5 at.%, iv) a PL quenching induced not only by Si-NCs having sizes beyond the size limit for quantum confinement but also by P induced defects which may arise from the insertion of P inside Si-NCs. It is worth mentioning that, from the application point of view, P-doping of SiO/SiO 2 multilayers is of clear interest since the PL intensity can be strongly increased and the annealing temperature needed to reach a high PL intensity can be significantly reduced. Photoluminescence intensity (arb. units)

Fig. 1 .

 1 Fig. 1. STEM-EELS mapping for (SiO/SiO 2 ):P multilayers annealed at 1100 °C with a

Fig. 2 .

 2 Fig. 2. 2D slice view (21 × 21 × 3 nm 3 ) taken from the 3D reconstructed volume showing both Si

Fig. 3 .

 3 Fig.3. Erosion profile close to a Si-NC obtained by APT analysis for a multilayer annealed at

Fig. 4 .Fig. 5 .Fig. 6 .Fig. 7 .

 4567 Fig.4. Room temperature photoluminescence spectra of samples annealed at different

  Photoluminescence energy (eV)

Acknowledgements:

The authors acknowledge the ANR project DONNA (ANR-18-CE09-0034) and the French PIA project "Lorraine Université d'Excellence" reference no. ANR-15IDEX-04-LUE support. D.Z. gratefully acknowledges the support from the "Make Our Planet Great Again" short-stay program.

Author contributions

A.E.G.: investigation, optical properties, RTA process; S.G.: investigation, optical properties; F.

Conflicts of interest

There are no conflicts of interest to declare.


Mateusz Skomra

Stéphan Thomassé

CONVEXLY INDEPENDENT SUBSETS OF MINKOWSKI SUMS OF CONVEX POLYGONS

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Let X be a finite set of points in the plane. A subset C of X is convexly independent if C forms a convex polygon. We denote by ci(X) the largest size of a convexly independent subset of X. The celebrated Happy Ending Theorem, from Erdős and Szekeres [START_REF] Erdős | A combinatorial problem in geometry[END_REF], asserts that ci(X) goes to infinity when |X| goes to infinity and X does not have three points on the same line. More precisely, the minimum value one can achieve for ci(X) is logarithmic in |X|. To the opposite, one can try to maximize ci(X) when the set X satisfies some geometrical constraint. A well-studied case is when X is the Minkowski sum P + Q := {p + q : p ∈ P, q ∈ Q} of two sets of points P and Q.

Eisenbrand, Pach, Rothvoß, and Sopher [START_REF] Eisenbrand | Convexly independent subsets of the Minkowski sum of planar point sets[END_REF] proved that ci(P + Q) = O(n 4/3 ) when |P | = |Q| = n. This result was complemented by a construction of Bílka, Buchin, Fulek, Kiyomi, Okamoto, Tanigawa, and Tóth [START_REF] Bílka | A tight lower bound for convexly independent subsets of the Minkowski sums of planar point sets[END_REF] showing the existence of such P and Q satisfying ci(P + Q) = Θ(n 4/3 ). Surprisingly, the set Q they use in the extremal constructions can be chosen convex. A natural question is then to ask for the maximum possible value of ci(P + Q) when both P and Q are convex polygons. In 2014, Tiwary [START_REF] Tiwary | On the largest convex subsets in Minkowski sums[END_REF] proposed an upper bound by showing that ci(P + Q) = O (n + m) log(n + m) when P and Q are respectively a convex n-gon and a convex m-gon. He concluded his paper by mentioning that his upper bound seemed very generous, and left as an open problem the existence of a matching lower bound. Our main result in this paper is that Tiwary's proof indeed provides a sharp bound by exhibiting a matching construction.

Theorem 1.1. There exist two families

(P k ) k 1 , (Q k ) k 1 ⊂ R 2 of convexly independent sets such that |P k | = |Q k | = 2 k and ci(P k + Q k ) (k + 2)2 k-1 for all k 1.
An equivalent point of view of Minkowski sums is to consider (P + Q)/2 instead of P + Q, and therefore ci(P + Q) represents the maximum size of a set M of midpoints of P, Q-segments in convex position. If we actually draw all segments corresponding to these midpoints, we get a bipartite graph with vertex set P Q, whose edges are all pairs (p, q) ∈ P ×Q such that (p+q)/2 ∈ M . In our construction, the number of edges of this graph is cn log n, all vertices of P and Q are in convex position, and all midpoints of edges are also in convex position. This kind of drawing was introduced by Halman, Onn, and Rothblum [START_REF] Halman | The convex dimension of a graph[END_REF] where they define a strong convex embedding of a graph G = (V, E) as a function f : V → R 2 such that f (V ) is convex and {(f (x) + f (y))/2 : (x, y) ∈ E} is also convex. They showed that if G admits a strong convex embedding, then |E| 5n -8 when n 3 is the number of vertices. Recently, García-Marco and Knauer [START_REF] García-Marco | Drawing graphs with vertices and edges in convex position[END_REF] reduced this bound to 2n -3. Equivalently their result shows that ci(P + P ) 2n -3 when P is a convex n-gon. Perhaps surprisingly, our construction shows that when slightly relaxing strong convex embedding to only ask convex positions for the two partite sets of a bipartite graph, the bound goes from linear to n log n.

Another very attractive reason motivating the study of ci(P + Q) for convex n-gons is the famous unit distance problem of Erdős and Moser [START_REF] Erdős | Problem 11[END_REF] asking for the maximum number of pairs of points in a convex n-gon P with distance exactly one. The trick is to observe that if two points p, p of P have distance 1, then p -p lies on the unit circle. A careful counting shows, in particular, that ci P + (-P ) is an upper bound on the number of unit distance pairs. Unfortunately, our construction shows that ci(P + Q) cannot be directly used to improve the already known O(n log n) upper bound on the unit distance problem (see Füredi [START_REF] Füredi | The maximum number of unit distances in a convex n-gon[END_REF] and Aggarwal [START_REF] Aggarwal | On unit distances in a convex polygon[END_REF] for a sharper estimate). We do not know however if the graphs G k arising from our construction can be realized as convex unit distance graphs. If not, it would be interesting to find a forbidden pattern in G k . This would extend the list of forbidden matrices provided in [START_REF] Aggarwal | On unit distances in a convex polygon[END_REF], as none of them appears in G k .

Proof of the main theorem

The proof of Tiwary [START_REF] Tiwary | On the largest convex subsets in Minkowski sums[END_REF] is based on the fact that any collection of convexly independent points in R 2 can be split into at most four monotone chains as in Figure 1. We use the name "south-east chain" for the chain that is situated in the bottom-right part of this figure. Our construction is based on this type of chains. More formally, we make the following definition and point out the subsequent lemma. Definition 2.1. A south-east chain is a sequence (a (1) , . . . , a (n) ) ⊂ R 2 of n 2 points in the plane that satisfies the following two conditions. First, the sequence is strictly increasing on both coordinates, i.e., we have a

(1) 1 < a (2) 1 < • • • < a (n) 1 and a (1) 2 < a (2) 2 < • • • < a (n) 2
(where a (k) i denotes the ith coordinate of the point a (k) ). Second, the corresponding sequence of consecutive slopes is strictly increasing, i.e., we have

a (2) 2 -a (1) 2 a (2) 1 -a (1) 1 < • • • < a (k+1) 2 -a (k) 2 a (k+1) 1 -a (k) 1 < • • • < a (n) 2 -a (n-1) 2 a (n) 1 -a (n-1) 1 .
We say that n is the length of a south-east chain (a (1) , . . . , a (n) ). Lemma 2.2. If the points (a (1) , . . . , a (n) ) form a south-east chain, then they are convexly independent, i.e., the convex hull of {a (1) , . . . , a (n) } has n vertices.

Before giving a formal proof of Theorem 1.1, let us explain it intuitively. Our proof is based on some elementary properties of rotations. Let R : R 2 → R 2 denote the counterclockwise rotation by 60 degrees centered at zero. Let a, b, c ∈ R 2 be three points on a plane such that (a, b, c) is a south-east chain. Moreover, suppose that this chain is sufficiently flat (more precisely, that the line defined by extending the segment [b, c] forms an angle smaller than 30 degrees with the horizontal axis). Then, the images (R(a), R(b), R(c)) also form a south-east chain. Moreover, if all the slopes between consecutive points of (a, b, c) are close to 0, then the corresponding slopes of (R(a), R(b), R(c)) are close to √ 3 (in other words, the corresponding segments form angles close to 60 degrees with the horizontal axis). Even more, under these conditions, the triple

1 2 (a + R(a), b + R(b), c + R(c)
) also forms a south-east chain, and its slopes are close to 1 / √ 3 (which corresponds to the angle of 30 degrees). We refer to Figure 2 for an illustration. The same applies to south-east chains formed by more than three points. In Consider now the linear map L ε : R 2 → R 2 , L ε (x, y) := (εx, ε 2 y), where ε > 0. It is immediate to see that this map preserves south-east chains (i.e., an image of a south-east chain under L ε is again a south-east chain). Moreover, this map flattens the chains. In other words, if A is a south-east chain and ε > 0 is small enough, then the image of A under L ε is a south-east chain that is sufficiently flat to apply the previous observations. Moreover, for small ε, the image of A under L ε is contained in a small neighborhood of 0.

Suppose now that we are given three south-east chains A, B, C such that C is included in the set of points (A + B)/2. By applying L ε to all three chains, we can suppose that they are arbitrarily flat, and contained in a small neighborhood of 0. Then, we can apply the rotation R to A, B, C. In this way, we obtain three chains A , B , C that are again contained in a neighborhood of 0, but whose slopes are close to √ 3. We now translate the chains as follows. We do not apply any translation to A, we translate B by the vector (0, 2), and C by the vector (0, 1). Then, we translate A by the vector (1, 5 /2), B by the vector (1, 1), and C by the vector (1, 7 /4). This gives the situation depicted in Figure 3. In this picture, the chain A is contained in the small neighborhood of the point marked by A, and the slopes of this chain are close to the slope of the solid line passing through A. The same is true for the chains B, C, A , B , C . In particular, for sufficiently small ε, the concatenation A := (A, B ) of chains A and B forms a southeast chain (because the dashed line from A to B has slope greater than the slope of the solid line passing through A but smaller than the slope of the solid line passing through B ). By the same reasoning, the concatenation B := (B, A ) forms a south-east chain. Denote A = (a (1) , . . . , a (n) ) and A = (a (1) , . . . , a (n) ). By the observation about rotation made above, the sequence

D := ( a (1) +a (1) 2 , . . . , a (n) +a (n)

2

) is a south-east chain, and all the slopes of this chain are close to 1 / √ 3. Moreover, this chain is contained in a small neighborhood of the point ( 1 /2, 5 /4). We marked this chain in Figure 3 using the same conventions as for the remaining chains. By applying the same reasoning as above, the concatenation C := (C, D, C ) is a southeast chain. To summarize, our construction shows the following statement. Before presenting a formal proof, let us discuss the types of graph drawings that we obtain in this way. Here, we are interested in a drawing f : (U V ) → R 2 of a bipartite graph G = (U V, E) such that f (U ) is a south-east chain, f (V ) is a south-east chain, and the midpoints {(f (u) + f (v))/2 : (u, v) ∈ E} also form a south-east chain. The construction described above implies that if G is drawable in this way and G := (U V , E ) is a copy of G, then the graph G := (U V , E) defined as U := U V , V := V U , and

E := E E {(u, u ) : u ∈ U }
is also drawable in this fashion. By starting from a graph

G 1 = {u 1 , u 2 } {v 1 , v 2 }, {(u 1 , v 1 ), (u 2 , v 1 ), (u 2 , v 2 )}
and iterating the procedure, we obtain a family (G k ) k 1 of drawable graphs, each having 2 k+1 vertices and (k +2)2 k-1 edges. Figure 4 depicts this family of graphs for k 3 and Figure 5 depicts a drawing of G 3 using south-east chains.

In the remaining part of this section, we give a formal proof of the argument described above. Let R : R 2 → R 2 denote the counterclockwise rotation by 60 degrees centered at zero, i.e., the linear transformation given Proof. We have ã =

1 2 (a 1 - √ 3a 2 , √ 3a 1 +a 2 ) and b = 1 2 (b 1 - √ 3b 2 , √ 3b 1 +b 2 ). The inequality ã2 < b2 is trivial. Moreover, b1 > ã1 ⇐⇒ b 1 -a 1 > √ 3(b 2 -a 2 )
, which is true by our assumptions. Furthermore, we have

tan( π 3 + θ) = tan( π 3 ) + tan(θ) 1 -tan( π 3 ) tan(θ) = √ 3(b 1 -a 1 ) + b 2 -a 2 b 1 -a 1 - √ 3(b 2 -a 2 ) = b2 -ã2 b1 -ã1 = sl(ã, b) . Similarly, tan( π 6 + θ) = tan( π 6 ) + tan(θ) 1 -tan( π 6 ) tan(θ) = b 1 -a 1 + √ 3(b 2 -a 2 ) √ 3(b 1 -a 1 ) -(b 2 -a 2 ) = b 2 + b2 -a 2 -ã2 b 1 + b1 -a 1 -ã1 = sl a + ã 2 , b + b 2 .
For any ε > 0 we denote by L ε : R 2 → R 2 the linear transformation L ε (x, y) := (εx, ε 2 y). As a corollary of Lemma 2.3 we may now prove the properties of the three transformations of south-east chains discussed above.

To improve readability, we use the following notation: if A is a sequence, then we denote by A(k) its kth element, so that A = A(1), . . . , A(n) . If A is a south-east chain of length n and ε > 0, then we consider the following three sequences:

(1)

A ε := L ε A(1) , . . . , L ε A(n) , A ε := R A ε (1) , . . . , R A ε (n) , A ε := A ε (1) + A ε (1) 2 , . . . , A ε (n) + A ε (n) 2 .
Using this notation, A ε is a chain obtained by flattening A, A ε is the rotated version of this flattened chain, and A ε is the chain formed by taking the midpoints of the two previous chains. The next result follows from Lemma 2.3.

Lemma 2.4. Suppose that A := A(1), . . . , A(n) is a south-east chain.

Then, for sufficiently small ε > 0, the sequences A ε , A ε , A ε are south-east chains. Moreover, for every k ∈ [n -1] we have the equalities

lim ε→0 + sl A ε (k), A ε (k + 1) = 0 , lim ε→0 + sl A ε (k), A ε (k + 1) = tan( π 3 ) = √ 3 , lim ε→0 + sl A ε (k), A ε (k + 1) = tan( π 6 ) = 1 √ 3 .
Proof. It is obvious that the sequence A ε is strictly increasing on both coordinates. Moreover, for every k ∈

[n -1] we have sl A ε (k), A ε (k + 1) = εsl A(k), A(k + 1)
. Hence, A ε is a south-east chain and we have lim ε→0 + sl A ε (k), A ε (k + 1) = 0. To prove the claim for the remaining two sequences, note that for sufficiently small ε > 0, the inequality sl

A ε (k), A ε (k+1) = εsl A(k), A(k+1) < 1 / √ 3 is satisfied for all k ∈ [n-1]. Hence, by Lemma 2.3, the sequence A ε = R A ε (1) , . . . , R A ε (n) is
strictly increasing on both coordinates and the same is true for the sequence

A ε = Aε(1)+A ε (1) 2 , . . . , Aε(n)+A ε (n) 2 . Moreover, if we denote θ (k,ε) := arctan sl A ε (k), A ε (k + 1) < π
6 , then the sequence θ (1,ε) , . . . , θ (n-1,ε) is strictly increasing and Lemma 2.3 shows that

sl A ε (k), A ε (k + 1) = tan( π 3 + θ (k,ε) ) and sl A ε (k), A ε (k + 1) = tan( π 6 + θ (k,ε) ) .
In particular, the sequences A ε , A ε are south-east chains. Moreover, we have lim ε→0 + θ (k,ε) = 0 for all k, which gives the equalities lim

ε→0 + sl A ε (k), A ε (k + 1) = tan( π 3 ) = √ 3 , lim ε→0 + sl A ε (k), A ε (k + 1) = tan( π 6 ) = 1 √ 3 .
We now show how the transformations given in Lemma 2.4 can be used to prove Theorem 1.1. To do so, we take three south-east chains A := A(1), . . . , A(n) , B := B(1), . . . , B(n) , and C := C(1), . . . , C(m) such that C ⊂ (A + B)/2. As discussed before, we let u := (1, 5 /2), v := (0, 2), w := (1, 1) and we consider the chains A ε , A ε , B ε , B ε defined as in (1).

Lemma 2.5. If ε > 0 is sufficiently small, then the sequences

A ε := (A ε , w+ B ε ), B ε := (v + B ε , u + A ε ) are south-east chains. Moreover, the set (A ε + B ε )/2 contains a south-east chain of length at least 2m + n.
In the statement above, w + B ε denotes the sequence obtained by translating every element of B ε by the vector w and (A ε , w + B ε ) denotes the concatenation of two sequences. The same applies to (v + B ε , u + A ε ).

Proof of Lemma 2.5. We start by proving that A ε is a south-east chain. By Lemma 2.4, the sequences A ε and B ε are south-east chains for sufficiently small ε. Hence, the sequence w + B ε is also a south-east chain. Furthermore, we have lim ε→0 + A ε (n) = (0, 0) and lim ε→0 + w + B ε (1) = w = (1, 1). In particular, for sufficiently small ε, the sequence A ε is strictly increasing on both coordinates. Moreover, Lemma 2.4 shows the equalities lim ε→0 + sl A ε (n-1), A ε (n) = 0 and lim ε→0 + sl w+B ε (1), w+B ε (2) = √ 3. We also have lim

ε→0 + sl A ε (n), w + B ε (1) = sl(0, w) = 1 .
Since 0 < 1 < √ 3, the sequence A ε is a south-east chain for sufficiently small ε. The proof for B ε is analogous-it is enough to observe that lim ε→0 + v + B ε (n) = (0, 2) and lim ε→0 + u + A ε (1) = (1, 5 /2) to show that B ε is strictly increasing on both coordinates. Then, B ε is a south-east chain by the equalities lim ε→0

+ sl v + B ε (n -1), v + B ε (n) = 0, lim ε→0 + sl u + A ε (1), u + A ε (2) = √ 3, and
lim ε→0 + sl v + B ε (n), u + A ε (1) = sl(v, u) = 1 2 .
It remains to show that (A ε + B ε )/2 contains a south-east chain of length at least 2m + n. To do so, consider the chain C ⊂ (A + B)/2, |C| = m, and let C ε , C ε be defined as in [START_REF] Aggarwal | On unit distances in a convex polygon[END_REF]. Furthermore, let t := (0, 1) = v/2 and z := (1, 7 /4) = (u + w)/2. Since the transformations L ε and R are linear, we have t

+ C ε ⊂ (A ε + v + B ε )/2 ⊂ (A ε + B ε )/2 and z + C ε ⊂ (u + A ε + w + B ε )/2 ⊂ (A ε + B ε )/2
. Moreover, we define the chain A ε as in (1), we let s := ( 1 /2, 5 /4) = u/2, and we note that s

+ A ε ⊂ (A ε + u + A ε )/2 ⊂ (A ε + B ε )/2. Hence, the sequence D ε := (t + C ε , s + A ε , z + C ε ) is contained in (A ε + B ε )/2
and its length is equal to 2m + n. Therefore, it is enough to prove that D ε is a south-east chain. The proof is similar to the proofs before. By Lemma 2.4, the sequences t + C ε , s + A ε , and z + C ε are south-east chains for sufficiently small ε. Moreover, we have the equalities

lim ε→0 + t + C ε (m) = (0, 1), lim ε→0 + s + A ε (1) = lim ε→0 + s + A ε (n) = ( 1 /2, 5 /
4), and lim ε→0 + z + C ε (1) = (1, 7 /4), which show that D ε is strictly increasing on both coordinates for sufficiently small ε. Furthermore, we use Lemma 2.4 to observe that lim ε→0

+ sl t + C ε (m -1), t + C ε (m) = 0, lim ε→0 + sl s+A ε (1), s+A ε (2) = lim ε→0 + sl s+A ε (n-1), s+A ε (n) = 1 √ 3 , and lim ε→0 + sl z + C ε (1), z + C ε (2) = √ 3.
To finish, we have lim

ε→0 + sl t + C ε (m), s + A ε (1) = sl(t, s) = 1 2 , lim ε→0 + sl s + A ε (n), z + C ε (1) = sl(s, z) = 1 , and 0 < 1 2 < 1 √ 3 < 1 < √ 3.
Hence, for sufficiently small ε, the sequence D ε is a south-east chain.

As a corollary, we may prove our main theorem.

Proof of Theorem 1.1. As noted in the introduction, it is enough to prove the claim for (P k + Q k )/2 instead of P k + Q k . We show, by induction over k, that we can find two south-east chains P k , Q k , each of length 2 k , such that (P k + Q k )/2 contains a south-east chain of length (k + 2)2 k-1 . Let P 1 := (0, 0), (2, 1) and Q 1 := ((0, 2), [START_REF] Bílka | A tight lower bound for convexly independent subsets of the Minkowski sums of planar point sets[END_REF][START_REF] Erdős | Problem 11[END_REF]). The sequences P 1 and Q 1 are south-east chains and the set (P 1 + Q 1 )/2 contains a south-east chain of length three. Moreover, given P k , Q k we may apply Lemma 2.5 to obtain two south-east chains P k+1 , Q k+1 , each of length 2 k+1 , such that (P k+1 +Q k+1 )/2 contains a south-east chain of length at least (k + 2)2 k + 2 k = (k + 3)2 k . Therefore, the claim follows from Lemma 2.2.

Final remarks

Let us discuss some natural questions for further research. Firstly, we do not know how far from optimal is our construction. More precisely, consider the function f : N * → N * defined as

f (n) := max{ci(P + Q) : P, Q ⊂ R 2 are convexly independent and |P | = |Q| = n} .
By joining our analysis with the result of Tiwary [START_REF] Tiwary | On the largest convex subsets in Minkowski sums[END_REF], we obtain the asymptotic bound f (n) = Θ(n log n). One can ask for an optimal constant C such that f (n) Cn log n. Secondly, we wonder if our family of graphs G k = (U k V k , E k ) can be still embedded in the plane if we impose a stronger condition on the midpoints of the edges. For instance, we may ask for an embedding such that U k and V k are south-east chains and the midpoints of E k are contained in some convex curve of small degree, such as a circle or a parabola. Our interest in this question is motivated by the following observation: if G k are realizable in the south-east quadrant of the plane in such a way that the midpoints of E k are on the unit circle, then we obtain a Θ(n log n) bound for the convex version of the unit distance problem, proposed by Erdős and Moser [START_REF] Erdős | Problem 11[END_REF] (see [START_REF] Füredi | The maximum number of unit distances in a convex n-gon[END_REF][START_REF] Aggarwal | On unit distances in a convex polygon[END_REF] for more information on lower and upper bounds for this variant of the unit distance problem). Indeed, suppose that h k : (U k V k ) → R 2 is an embedding of G k such that h k (U k ) and h k (V k ) are south-east chains, h k (U k ), h k (V k ) ⊂ {x ∈ R 2 : x 1 0, x 2 0}, and h k (u)+h k (v) 2 2 = 1 for all (u, v) ∈ E k . Then, the points -h k (U k ) form a monotone concave chain in the north-west quadrant of the plane and h k (V k ) form a monotone convex chain in the south-east quadrant of the plane, which implies that the collection {-h k (U k ), h k (V k )} is convexly independent. Therefore, the polygon P k := conv{-h k (U k )/2, h k (V k )/2} has 2 k+1 vertices and Θ(k2 k ) diagonals of length one. As noted in the introduction, the existence of such an embedding cannot be excluded using the current list of forbidden matrices [START_REF] Aggarwal | On unit distances in a convex polygon[END_REF]. We were neither able to construct these embeddings nor find a new forbidden pattern that occurs in G k .

Figure 1 .Figure 2 .

 12 Figure 1. Splitting convexly independent points into monotone chains.

Figure 3 .

 3 Figure 3. The construction of south-east chains.

Figure 4 .

 4 Figure 4. Graphs that can be drawn using south-east chains.

  Given three south-east chains A, B, C such that C is included in the set of points (A + B)/2, we can construct three south-east chains A, B, C such that C is included in (A + B)/2 and |A| = |B| = |A| + |B|, |C| = 2|C| + |A|. Thus, if we suppose that |A| = |B| = n, then we have |A| = |B| = 2n and |C| = 2|C| + n. By iterating this reasoning, we obtain the claimed bound Θ(n log n).

Figure 5 .Lemma 2 . 3 . 1 √ 3 =

 52313 Figure 5. Drawing of G 3 using south-east chains.

LAAS-CNRS, Université de Toulouse, CNRS, Toulouse, France

Univ Lyon, EnsL, UCBL, CNRS, LIP, F-69342, LYON Cedex 07, France

Institut Universitaire de France E-mail addresses: mateusz.skomra@laas.fr, stephan.thomasse@ens-lyon.fr.


Cyril Eraud
email: cyril.eraud@oncfi.gouv.fr

Adrien Dorie

Anne Jacquet

Bruno Faivre

The crop milk: a potential new route for carotenoid-mediated parental effects

Allocation of carotenoid pigments, either through nutritional provisioning or from endogenous reserves, makes up a form of non-genetic parental investment to progeny that may contribute to fitness. To date, carotenoids derived from endogenous reserves have been acknowledged as important vectors in translating only avian female phenotype and environmental conditions experienced prior to laying. Here, we show that in columbidae, crop milk delivered by both parents may provide chicks with a large amount of endogenous carotenoids at the postzygotic stage. Major carotenoids were xanthophylls and beta-carotene, but their concentrations showed large variation among individuals. Interestingly, a large amount of this variation was explained by brood identity, suggesting either environmental influences and/or phenotypic influences on a parent's ability to transfer these biomolecules. Our study therefore illuminates a potential new route for endogenous carotenoid-mediated parental effects.

Non-enzymatic antioxidants such as carotenoids are key resources that can shape the development of a growing animals. They serve for several crucial functions such as the maintenance of redox homeostasis, the regulation of immune defences, the colouration of future ornaments, gene expression or the synthesis of growth factors [START_REF] Hofmann | Retinoids in development[END_REF]Eichele 1994, M0ller et al. 2000). There fore, early access to these biomolecules may strongly affect offspring viability and later performances.

Carotenoids are fat-soluble pigments which cannot be synthesised de novo by vertebrates but must be acquired from the diet [START_REF] Olson | Costly sexual signals: are carotenoids rare, risky or required?[END_REF]. In early life, access to this resource then relies exclusively on parental provi sionning. Consequently, allocation of carotenoids makes up a form of non-genetic parental investment to their progeny that may contribute to fitness [START_REF] Blount | Why egg yolk is yellow[END_REF]. In birds, parental allocation of carotenoids may occur at two distinct stages. Obviously, parents may allocate carotenoids at the postzygotic stage, through nutritional provisioning, but they may also allocate this resource earlier (at prezygotic stage) through the deposition of large amount of carote noids in egg yolk [START_REF] Blount | Why egg yolk is yellow[END_REF][START_REF] Royle | Maternally derived androgens and antioxidants in bird eggs: complemen tary but opposing effects?[END_REF].

If the amount of carotenoids either allocated at pre-or postzygotic stages appears to affect a large array of fitness related traits (see [START_REF] Blount | Carotenoids and egg qualiry in the lesser black-backed gull Larus fuscus: a supplemental feeding study of maternal effects[END_REF], 2003a[START_REF] Saino | Experimental manipulation of egg carotenoids affects immunity in barn swallow nestlings[END_REF][START_REF] Mcgraw | Maternaly derived carotenoid pigments affect offspring survi val, sex ratio, and sexual attractiveness in a colorful songbird[END_REF][START_REF] Biard | Carotenoid availabiliry in diet and phenotype of blue and great tit nestlings[END_REF], carotenoid pigments allocated at the prezygotic stage differ sharply from those provided through nutritional provisonning as they are directly derived from parental endogenous (maternal) resources [START_REF] Blount | Why egg yolk is yellow[END_REF]. Proximate and ultimate implications of the allocation of endogenous carotenoids have therefore attracted considerable attention among evolutionary biologists, especially ornithologists, in recent years. In particular, significant advances have been made in understanding the importance of endogenous carotenoids for offspring development, as well as the proximate factors that shape the pattern of investment of this resource. For instance, studies have shown that carotenoid supply is limited for laying females and importantly, that the amount of these biomolecules invested in embryos is traded off against physiological functions related to self-maintenance [START_REF] Blount | Carotenoids and egg qualiry in the lesser black-backed gull Larus fuscus: a supplemental feeding study of maternal effects[END_REF][START_REF] Saino | Carotenoid concentration in barn swallow eggs is influenced by laying order, maternal infection and paternal ornamentation[END_REF]. This has lead to the emergence of a widely acknowledged idea that endogenous carotenoids act as important vectors in translating female's phenotype and environmental conditions experienced prior to laying [START_REF] Blount | Why egg yolk is yellow[END_REF][START_REF] Blount | Carotenoids and egg qualiry in the lesser black-backed gull Larus fuscus: a supplemental feeding study of maternal effects[END_REF]. However, the eventuality that endogenous carotenoid-mediated effects may arise at the postzygotic stage, and the way that male phenotype and environmental experience are implicated, have been largely neglected. The main reason for this is that in birds, egg yolk is the only known pathway for transferring endogenous carotenoids to progeny.

In some bird species including the greater flamingo Phoenicopterus ruber roseus, the emperor penguin Apteno dytes farsteri and all columbidae, chicks are fed on a secretion which is produced at the expense of parents' body fluids [START_REF] Denbow | Gastrointestinal anatomy and physiology[END_REF]. In columbidae, this secretion is a semi-solid substance derived from the sloughed off cells of crop squamous epithelium which are engorged with nutrients transported to them by blood vessels (Bharati et al. 1997). Named "crop milk", this substance is provided by both parents and contains nutrients such as fat, proteins, minerals, enzymes and free amino acids. However, the crop milk is devoided of carbohydrates and calcium {Davies 1939), suggesting the existence of selective pathways for some nutrients that support chicks' development in early life. Hence, we may hypothesize that crop milk may also be a mean for parents to allocate endogenous carotenoids to their progeny, therefore allowing the translation of both their' phenotype and environmental conditions experienced after laying. The presence of carotenoids in crop milk is currently unknown. The aim of the present study was therefore to investigate for the presence of such pigments in the crop milk. To achieve this, we performed biochemical analyses of crop milk dispensed to Eurasian collared dove nestlings Streptopelia decaocto. In addition, we explored potential sources of variation in carotenoid content of crop milk dispensed to chicks. More specifically, we focused on sources usually retained in studies investigating variation in egg carotenoids, such as differences among nests, seasonal variation and within brood variation.

Material and methods

General field procedures for data collection

We conducted the study on a free-living population of Collared dove in Western France (46°N, 0.5°W), from Aug. to Sept. 2005. The species typically lays 2 eggs and during the first 4-5 d of life, chicks are exclusively fed on crop milk dispensed by both parents {Cramp 1985).

The study area was surveyed daily to record nest building and laying. After clutch completion, 24 nests were subse quently checked every 2-d using a mirror fixed on a telescopic cane, to determine hatching time and order. Chicks were individually marked on their right tarsus using a non-toxic permanent marker. Collecting crop milk from breeding adults is not easy since they do not store large amounts of crop milk (Bharati et al. 1997). Therefore, crop milk was obtained by performing a biopsy on 2-d-old chicks, using 5 ml sterile syringes with 18-gauge needles. All nests were sampled at the same moment during the day {12:00-14:00) to avoid within day variation {Horak et al. 2004). Before performing a biopsy, the skin was carefully disinfected using 90% vol. alcohol. A small amount of crop milk (300-500 µl) was then gently sucked into the syringe and transferred into a vial which was immediately kept on ice until storage at -80°C. We found no evidence that biopsies caused extra-mortality since all chicks fledged successfully.

Preparation of crop milk and carotenoid analysis

Crop milk samples were homogenized by vortexing with three glass beads. A small fraction (186 ± 20 mg) of the homogenate was then weighted {±0.01 mg) and used for carotenoid extraction. Before extraction, water was removed from the crop milk by freeze-drying samples at -50°C for 12 h (Telstar, Cryodos-50). Residual material was gently pulverized, and carotenoids were extracted twice. In practice, 500 µl of hexane +0.01 % of 2,6-di-ter-butyl p-cresol {to prevent degradation of carotenoids) were added to dry residuals. The mix was vortexed for 2 min, sonicated for 5 s {Sun et al. 2006), and then centrifugated for 15 min at 1,500 g and 4° C. The supernatant was removed and stored separately. This extraction procedure was repeated on the remaining residue. Supernatants from the two extrac tions were then pooled, and the liquid evaporated under nitrogen gas and no light. For analysis, the pigments were resuspended in 50 µl of a solution of methanol and ter butyl-methyl-ether (50:50). HPLC pigment separations were performed by injecting these 50 µl of pigments suspension into a 717 Waters Plus Autosampler fitted with a stainless steel ProntoSIL C18 reversed-phase column (Bischoff, Leonberg, Germany). The mobile phase was a mixture of methanol, ter-butyl-methyl-ether and deionized water delivered by a 600 E Multisolvent pump at a constant flow rate of 1 ml/min {see [START_REF] Gaillard | Carotenoids of two freshwater amphipod species ( Gammarus pulex and. roeselt) and their common acanthoce phalan parasite Polymorphus minutes[END_REF]) for details about the mobile phase gradient). Absorption spectra were collected using a Waters 996 photodiode array detector, and chromatograms were plotted at 450 nm. Peak identification was performed by comparing retention times and absorp tion properties with pure standards purchased from Extra synthese (Genay, France). Major carotenoid pigments were quantified using an internal standard of �-cryptoxanthin and separate calibration curves established from injection of pure standards [START_REF] Gaillard | Carotenoids of two freshwater amphipod species ( Gammarus pulex and. roeselt) and their common acanthoce phalan parasite Polymorphus minutes[END_REF].

Statistical analyses

Carotenoid concentration was expressed in ng per mg of fresh and dry mass of crop milk. Total carotenoid content was assessed by summing concentrations of major carote noids. General Linear Mixed Models were used to examine the effects of nest identity, hatching order and hatching date on carotenoids concentration in crop milk samples. GLMMs were fitted with total {or specific) carotenoids concentration as the dependent variable, hatching order as a fixed factor, hatching date as a covariate, plus their interaction. To account for the lack of independ ence between chicks raised by the same parents, models were fitted with nest identity as a random factor. GLMMs were performed on carotenoid concentrations expressed per unit of fresh and dry milk. However, since analyses shared similar conclusions whatever the unit used, we present only results of GLMMs fitted with carotenoid concentrations in dry milk (log10+ 1) as the dependent variable.

Results

Carotenoid pigments were detected in all crop milk samples. Mean concentrations {±SE) were 3.81 (±0.37) and 20.50 (±2.48) ng.mg-1 for fresh and dry milk, respectively. HPLC analysis revealed that xanthophylls were the major carotenoid pigments, accounting on average for 94. 86% ( ± 1.54) of the total {Table 1). Xanthophylls were predominantly zeaxanthin, lutein and dehydrolutein

Discussion

Crop milk collected from 2 d old collared dove chicks contained carotenoid pigments. As crop milk is dispensed and produced by both parents at the expense of their body fluids [START_REF] Denbow | Gastrointestinal anatomy and physiology[END_REF], these results suggest that crop milk is a potential post-hatching route for carotenoid mediated parental effects. While the physiological basis of crop milk production differs from that of mammalian milk, these results parallel those reporting the presence of carotenoids in colostrum and breast-milk [START_REF] Patton | Carotenoids of human colostrum[END_REF][START_REF] Schweigert | Effect of parturition on levels of vitamins A and E and of Beta-carotene in plasma and milk of mares[END_REF][START_REF] Macias | Changes in the concentra tion of carotenoids, vitamin A, alpha-tocopherol and total lipids in human milk throughout early lactation[END_REF]). Several xanthophylls found in crop milk were previously shown to improve important health and fitness-related functions when transferred in large amounts into eggs, or when provided through an enriched diet. For instance in birds, exposure to large amounts of lutein and zeaxanthin were shown to improve immunological defences and sexual attractiveness (Blount et al. 2003b[START_REF] Saino | Experimental manipulation of egg carotenoids affects immunity in barn swallow nestlings[END_REF], antioxidant functions [START_REF] Blount | Carotenoids and egg qualiry in the lesser black-backed gull Larus fuscus: a supplemental feeding study of maternal effects[END_REF], post-fledging survival prospects [START_REF] Mcgraw | Maternaly derived carotenoid pigments affect offspring survi val, sex ratio, and sexual attractiveness in a colorful songbird[END_REF] or colouration of growing feathers [START_REF] Biard | Carotenoid availabiliry in diet and phenotype of blue and great tit nestlings[END_REF]. As precursor of vitamin A, P-carotene is also known to enhance health related functions such as resistance to infections, growth and antioxidant defences [START_REF] Bendich | Biological actions of carotenoids[END_REF]). However, carotenoid levels in fresh crop milk were sharply below those reported in eggs of bird species [START_REF] Royle | Maternally derived androgens and antioxidants in bird eggs: complemen tary but opposing effects?[END_REF][START_REF] Blount | Carotenoids and egg qualiry in the lesser black-backed gull Larus fuscus: a supplemental feeding study of maternal effects[END_REF] including the collared dove (119 ± 5 .24 ng x mg -1 of fresh yolk; Eraud et al. unpubl. data). It is therefore unclear how variation in crop milk carotenoids may affect offspring phenotype. Nevertheless, the daily crop milk intake by 1-5 d old squabs can be up to 72% of their body weight [START_REF] De Cock | Morphology of the crop and crop milk in the pigeon ( Columba livia domestica)[END_REF], so the total amount of carotenoid intake may therefore reach important levels when summed over the first days of life. Indeed, on the basis of a daily crop milk intake of 8 g (Cramp 1985), we calculated that the total amount of carotenoids ingested by 1-5 d old collared dove chicks would reach up to 152.4 µg. This corresponds to about half of the total amount of carotenoids transferred into egg yolk (�300 µg for a yolk weighting 2.5 g; Eraud et al. unpubl. data). Accordingly, carotenoids provided by crop milk may represent a non-negligible source of antioxidants during the crucial period of early life, and may also potentially affect the carotenoid status of chicks. In support of this, human studies have shown that carotenoids transferred during the first few days of lactation have the potential to correct abnormally low plasma carotenoid concentrations in the neonate [START_REF] Ostera | Influence of breast feeding on the restoration of the low serum concentration of vitamin E and beta carotene in the newborn infant[END_REF]). In addition, crop milk quality may have also the potential to affect carotenoid status later in life, because in birds, it has been shown that early exposure to carotenoids have long lasting effects on carotenoid accumulation in internal tissues and fluids (Blount et al. 2003a).

We found no evidence that carotenoid content of crop milk differed between related siblings, or was affected by hatching date. However, brood identity explained a significant amount of the variation among samples. Potential explanations for this finding may include differences among pairs to assimilate and incorporate carotenoids into their crop milk, or environmental hetero geneity in carotenoid resources that supply parents. Since parents were not bled, direct evidence that their carotenoid status affects crop milk content has not been presented. Nonetheless, the effect of brood identity indirectly sup ports the idea that the carotenoid content of crop milk may reflect the parent's environmental experience and/or phenotype. In addition, human studies have shown that the carotenoid content of breast milk reflects the carote noid status of mothers. Indeed, beta-carotene supplemen tation in lactating women has been shown to increase milk beta-carotene concentration [START_REF] Canfield | Beta carotene in breast milk and serum is increased after a single beta-carotene dose[END_REF][START_REF] Rice | Maternal vitamin A or beta-carotene supplementation in lactating Bangladeshi wo men benefits mothers and infants but does not prevent subclinical deficiency[END_REF].

To date, carotenoids derived from endogenous reserves have been acknowledged as important vectors in translating only avian female phenotype and environmental conditions experienced prior to laying. Interestingly, our results challenge this view, suggesting that in some species, endogenous carotenoids may also mediate both paternal effects and environmental conditions experienced after laying. However, it remains to be elucidated whether parental fitness may benefit from endogenous carotenoid transfer to their progeny by an additional route other than through egg yolk; and whether changes in parental carotenoid status after laying may also have the potential to improve or impair offspring performance. The extent to which crop milk carotenoid content is traded off against parental functions related to self-maintenance; and the the extent to which crop milk carotenoid content modulates the expression of fitness-related traits in offspring such as growth, antioxidant or immunological defences, then remain worthy of future investigation.

Fig. 1 .

 1 Fig. 1. HPLC chromatogram of the most abundant carotenoid pigments present in a sample of crop milk obtained from a 2 d old chicle a) dehydrolutein, b) lutein, c) zeaxanthin, d) anhydrolutein, e) �-crypytoxanthin (internal standard), and f) �-carotene.

Table 1 .

 1 Level of major carotenoids (dehydrolutein, lutein, zeaxanthin, anhydrolutein and �-carotene) detected in crop milk collected on 2 d old chicks of the Eurasian collared dove (n =48). Means are given±SE.

		Mean concentration (ng x mg-1)		Frequency (%)
	Carotenoids	Fresh crop milk	Dry crop milk	Of occurence	Of total carotenoids
	Dehydrol utei n Lutein Zeaxanthin Anhydrolutein �-carotene	0.53±0.08 1.02±0.12 2.03±0.23 0.05±0.01 0.18±0.07	3.01 ±0.68 4.83±0.53 11.11 ± 1.51 0.27±0.05 1.28±0.72
	Total Xanthopylls Total Carotenoids	3.63±0.37 3.81 ±0.37	19.21 ±2.25 20.50±2.48
	(Table 1). Mean P-carotene concentrations across all samples were 0.18 (±0.07) and 1.28 (±0.72) ng.mg-1
	for fresh and dry milk, respectively. When P-carotene was
	detected, it accounted on average for 16.42% (±3.49) of
	total carotenoids, with an upper value reaching 23.92%. Crop milk concentrations of total carotenoids showed
	large variation among individual samples. Estimates ranged from 0.61 to 12.08 ng x mg -I for fresh milk, and from
	2.73 to 75.61 ng x mg -I after freeze-drying. Interestingly,
	a significant amount of this variation was explained by
	brood identity (GLMM; Percent of total variance: 66.33%;
	F23,47 =4.77, P <0.001). We found a similar effect of
	brood identity when performing separated analyses for
	each carotenoid species (GLMMs; all F23,47 values >4.02,

(Fig.

l

, Table

1

). The rarest carotenoid pigment was P carotene (Table

1

) and was detected in only 31 % of samples all P values �0.001), excepted for P-carotene (GLMM; F23,47=1.34, P =0.254). Furthermore, concentrations of total carotenoids in crop milk collected on related siblings were significantly correlated (Spearman rank coefficient;

  GLMMs; hatching date. all F1,23 values < 1.86, all P values >0.186; hatching rank x hatching date: all F1,23 values <2.59, all P values �0.122).

	100.00 93.75 97.92 62.50 31.25	13.17±1.11 28.35±2.11 51.21 ±2.42 2.13±0.46 5.13±1.54
	total or specific carotenoids (

Acknowledgements -Experiments were carried out in compliance with European legal requirements and national permission (European convention, ETS no. 123, B. F. permit no. 21-CAE-085, C. E permit no. 16-0NCFS). This study was funded by Office National de la Chasse et de la Faune Sauvage, Region Bourgogne, the Agence Nationale de la Recherche and European Social Funding (Research grant to C. E. and Research contract to B. F.). We are also grateful to Kevin J. McGraw for his valuable help in carotenoid identification. Earlier versions of the manu script were improved by Michael Cherry.