D Babonneau

S Camelio

G Abadias

D Christofilos

J Arvanitidis

S Psilodimitrakopoulos

G M Maragkakis

E Stratakis

N Kalfagiannis

P Patsalas

Self-Assembled Dichroic Plasmonic Nitride Nanostructures with Broken Centrosymmetry for Second Harmonic Generation

Keywords: Refractory conductors, Plasmonics, Bottom-Up Fabrication, Defects, Second Harmonic Generation 2

TiN and ZrN are emerging as important alternative plasmonic materials. In addition to their well-known assets, they can incorporate point defects that break the centrosymmetry of their cubic crystal structure, making them promising candidates as non-linear optical materials and especially for second harmonic generation (SHG). Their refractory character and chemical stability were obstacles for the bottom-up fabrication of TiN and ZrN nanostructures so far. In this work, it is shown that highly directional dichroic nanostructures of TiN and ZrN may be indeed grown by self-assembly using glancing angle deposition on periodic rippled dielectric surfaces. The produced nitride nanostructures exhibit point defects and exceptional photothermal durability. These nanostructures exhibit strong SHG response when probed by a near-infrared laser. It is shown that SHG is strongly associated with the near-field enhancement due to the localized surface plasmon resonance of the nanostructures. Given that such nanostructures can endure extremely high electric fields, they are expected to be able to emit massive SHG signals and be applied in laser technology as optical components such as polarizers and SHG emitters.

I. INTRODUCTION

Transition metal nitrides (TMNs), predominantly TiN and ZrN, are well-known optical conductors. Concomitantly, surface plasmon polaritons at TiN surfaces and localized surface plasmon resonance (LSPR) at TiN nanoparticles were observed as early as 2004 by Patsalas et al. [START_REF] Patsalas | Interface Properties and Structural Evolution of Nanoscale TiN/Si and TiN/GaN Heterostructures[END_REF] and Reinholdt et al. [START_REF] Reinholdt | Structural, compositional, optical and colorimetric characterization of TiN-nanoparticles[END_REF], respectively. In 2011, TMNs were proposed for the first time as realistic alternative materials for plasmonic devices by Naik et al. [START_REF] Naik | Oxides and nitrides as alternative plasmonic materials in the optical range[END_REF] due to the extraordinary combination of their properties that include refractory character [START_REF] Guler | Refractory Plasmonics[END_REF], chemical and mechanical stability [START_REF] Patsalas | The effect of substrate temperature and biasing on the mechanical properties and structure of sputtered titanium nitride thin films[END_REF][START_REF] Abadias | Structure, phase stability and elastic properties in the Ti1-xZrxN thin-film system: Experimental and computational studies[END_REF][START_REF] Abadias | Stress, phase stability and oxidation resistance of ternary Ti-Me-N (Me = Zr, Ta) hard coatings[END_REF][START_REF] Abadias | Electronic structure and mechanical properties of ternary ZrTaN alloys studied by ab initio calculations and thin-film growth experiments[END_REF], and durability to high electric fields [START_REF] Bagheri | Nonlinear Refractory Plasmonics with Titanium Nitride Nanoantennas[END_REF]; the latter also opens substantial opportunities for non-linear optics [START_REF] Bagheri | Nonlinear Refractory Plasmonics with Titanium Nitride Nanoantennas[END_REF][START_REF] Sato | Broadband Plasmon Resonance Enhanced Third-Order Optical Nonlinearity in Refractory Titanium Nitride Nanostructures[END_REF][START_REF] Kinsey | Effective third-order nonlinearities in metallic refractory titanium nitride thin films[END_REF].

Since then, the relevant scientific interest, as attested by the number of published papers, shows a geometrical growth with an annual increase over 60%. However, most of these works do not report the realization of TMN nanostructures in microelectronic architectures, instead they are dealing with continuous TMN layers (e.g. Ref. [START_REF] Naik | Titanium nitride as a plasmonic material for visible and near-infrared wavelengths[END_REF][START_REF] Naik | Epitaxial superlattices with titanium nitride as a plasmonic component for optical hyperbolic metamaterials[END_REF][START_REF] Lu | Dynamically controlled Purcell enhancement of visible spontaneous emission in a gated plasmonic heterostructure[END_REF][START_REF] Saha | On-Chip Hybrid Photonic-Plasmonic Waveguides with Ultrathin Titanium Nitride Films[END_REF][START_REF] Chaudhuri | Photonic Spin Hall Effect in Robust Phase Gradient Metasurfaces Utilizing Transition Metal Nitrides[END_REF], computational works (e.g. Ref. [START_REF] Habib | Hot carrier dynamics in plasmonic transition metal nitrides[END_REF][START_REF] Metaxa | Infrared Plasmonics with Conductive Ternary Nitrides[END_REF][START_REF] Lalisse | Plasmonic efficiencies of nanoparticles made of metal nitrides (TiN, ZrN) compared with gold[END_REF][START_REF] Kassavetis | Plasmonic spectral tunability of conductive ternary nitrides[END_REF][START_REF] Patsalas | Conductive nitrides: Growth principles, optical and electronic properties, and their perspectives in photonics and plasmonics[END_REF], works evaluating the plasmonic potential of certain continuous films via their dielectric permittivity (e.g. Ref. [START_REF] Guler | Refractory Plasmonics[END_REF][START_REF] Kassavetis | Plasmonic spectral tunability of conductive ternary nitrides[END_REF][START_REF] Patsalas | Conductive nitrides: Growth principles, optical and electronic properties, and their perspectives in photonics and plasmonics[END_REF][START_REF] Naik | Alternative plasmonic materials: Beyond gold and silver[END_REF][START_REF] Guler | Refractory plasmonics with titanium nitride: Broadband[END_REF][START_REF] Briggs | Temperature-dependent optical properties of titanium nitride[END_REF] and implementing specific figures of merit [START_REF] West | Searching for better plasmonic materials[END_REF][START_REF] Blaber | A review of the optical properties of alloys and intermetallics for plasmonics[END_REF], or colloidal nanoparticles [START_REF] Guler | Nanoparticle plasmonics: Going practical with transition metal nitrides[END_REF][START_REF] Karaballi | Photothermal Transduction Efficiencies of Plasmonic Group 4 Metal Nitride Nanocrystals[END_REF][START_REF] Karaballi | Synthesis of Plasmonic Group-4 Nitride Nanocrystals by Solid-State Metathesis[END_REF] and powders [START_REF] Rodriguez | Stabilizing the Plasmonic Response of Titanium Nitride Nanocrystals with a Silicon Oxynitride Shell: Implications for Refractory Optical Materials[END_REF][START_REF] Barragan | A Non-Thermal Plasma Route to Plasmonic TiN Nanoparticles[END_REF]. This is mostly due to the difficulty of the bottom-up fabrication of nitride nanostructures due to their refractory character and to the covalent metal-nitrogen bonding [START_REF] Matenoglou | Structure, stability and bonding of ternary transition metal nitrides[END_REF] that prevent the melting and resolidification in realistic temperatures, as well as the differing optical properties of ultrathin TiN films compared to their opaque counterparts [START_REF] Patsalas | Interface Properties and Structural Evolution of Nanoscale TiN/Si and TiN/GaN Heterostructures[END_REF][START_REF] Shah | Optical Properties of Plasmonic Ultrathin TiN Films[END_REF]. As a result, only a few experimental works can be found reporting the top-down fabrication of plasmonic devices [START_REF] Hu | Evolutionary Design and Prototyping of Single Crystalline Titanium Nitride Lattice Optics[END_REF][START_REF] Briggs | Fully CMOS-compatible titanium nitride nanoantennas[END_REF][START_REF] Murai | Plasmonic arrays of titanium nitride nanoparticles fabricated from epitaxial thin films[END_REF][START_REF] Guler | Local heating with lithographically fabricated plasmonic titanium nitride nanoparticles[END_REF][START_REF] Dutta | Plasmonic interconnects using zirconium nitride, Conference on Lasers and Electro-Optics[END_REF][START_REF] Kamakura | Enhanced Photoluminescence from Organic Dyes Coupled to Periodic Array of Zirconium Nitride Nanoparticles[END_REF][START_REF] Howell | Fabrication of plasmonic TiN nanostructures by nitridation of nanoimprinted TiO2 nanoparticles[END_REF][START_REF] Bagheri | Large-area fabrication of TiN nanoantenna arrays for refractory plasmonics in the midinfrared by femtosecond direct laser writing and interference lithography[END_REF]. In particular, the patterning methods employed so far for the fabrication of TiN and ZrN nanostructures include direct focused ion beam patterning [START_REF] Hu | Evolutionary Design and Prototyping of Single Crystalline Titanium Nitride Lattice Optics[END_REF], the combination of electron beam lithography [START_REF] Briggs | Fully CMOS-compatible titanium nitride nanoantennas[END_REF][START_REF] Murai | Plasmonic arrays of titanium nitride nanoparticles fabricated from epitaxial thin films[END_REF][START_REF] Guler | Local heating with lithographically fabricated plasmonic titanium nitride nanoparticles[END_REF][START_REF] Dutta | Plasmonic interconnects using zirconium nitride, Conference on Lasers and Electro-Optics[END_REF] or nanoimprint lithography [START_REF] Kamakura | Enhanced Photoluminescence from Organic Dyes Coupled to Periodic Array of Zirconium Nitride Nanoparticles[END_REF][START_REF] Howell | Fabrication of plasmonic TiN nanostructures by nitridation of nanoimprinted TiO2 nanoparticles[END_REF] with reactive ion etching, and laser interferometry patterning with wet chemical etching [START_REF] Bagheri | Nonlinear Refractory Plasmonics with Titanium Nitride Nanoantennas[END_REF][START_REF] Bagheri | Large-area fabrication of TiN nanoantenna arrays for refractory plasmonics in the midinfrared by femtosecond direct laser writing and interference lithography[END_REF].

The quality of TMN films can vary significantly depending on the particular choice of one of the many available growth techniques. There is, unfortunately, a feature that is present in all manufacturing schemes: TMNs typically contain a considerable amount of point defects that are known to affect the properties of the host crystal [START_REF] Balasubramanian | Energetics of point defects in rocksalt structure transition metal nitrides: Thermodynamic reasons for deviations from stoichiometry[END_REF][START_REF] Tsetseris | Trapping and release of impurities in TiN: A first-principles study[END_REF].

Such difficulty in producing defect-free, ultra-thin TMNs, due to their refractory character, may turn from a curse into a blessing in the field of non-linear optics. The included point defects break locally the centrosymmety of the cubic structure, as attested by the Raman spectra of TMNs [START_REF] Spengler | Raman scattering, superconductivity, and phonon density of states of stoichiometric and nonstoichiometric TiN[END_REF][START_REF] Patsalas | Zirconium nitride: A viable candidate for photonics and plasmonics?[END_REF]; the symmetry breaking is an essential prerequisite for the manifestation of second harmonic generation (SHG) [START_REF] Obermeier | Modal Symmetry Controlled Second-Harmonic Generation by Propagating Plasmons[END_REF][START_REF] Psilodimitrakopoulos | Ultrahigh-resolution nonlinear optical imaging of the armchair orientation in 2D transition metaldichalcogenides[END_REF].

In this work, a large-scale, bottom-up, maskless, and etchless fabrication process of strongly dichroic and plasmonic TiN and ZrN nanostructures is presented. It is based on glancing angle deposition (GLAD) [START_REF] Camelio | Anisotropic optical properties of silver nanoparticle arrays on rippled dielectric surfaces produced by low-energy ion erosion[END_REF][START_REF] Babonneau | Tunable plasmonic dichroism of Au nanoparticles self-aligned on rippled Al2O3 thin films[END_REF] and reactive magnetron sputtering on prepatterned substrates [START_REF] Babonneau | Formation of nanoripples on amorphous alumina thin films during low-energy ion-beam sputtering: experiments and simulations[END_REF]. The produced TiN and ZrN nanostructures have the form of aligned nanowires (NWs) with a more-or-less granular morphology, and they exhibit strong dichroism in the visible and near-infrared (NIR) ranges, as well as exceptional photothermal durability. The coherency of the nanostructures may be compromised in the case of multi-stack deposition that results in strong scattering of light. Thus, strong SHG may be present in such systems. Its occurrence is closely coupled with the plasmon resonance while its directionality is associated with the coherence of the nanostructures.

II. RESULTS AND DISCUSSION

a. Fabrication Concept and Morphology of Nanostructures

The steps of the reported self-assembly process are summarized in Fig. 1a. The selection of substrate is not critical as long as it is atomically flat. In this work, two substrates were implemented: fused silica for optical transmission measurements and SHG experiments, and conductive n-type Si(100) wafers (covered with a 2-nm thick native oxide layer) for Raman spectroscopy experiments, as well as for morphology characterizations via Atomic Force Microscopy (AFM) measurements. Amorphous alumina layers, 170 nm thick, were deposited on top of the various substrates by dual ion beam sputtering (DIBS) as described in more detail in Ref. [START_REF] Babonneau | Formation of nanoripples on amorphous alumina thin films during low-energy ion-beam sputtering: experiments and simulations[END_REF]. The second step of the process is the patterning of the amorphous alumina thin film by 1 keV-Xe + ion bombardment at oblique incidence (55 o ) to form periodic nanoripples [START_REF] Babonneau | Formation of nanoripples on amorphous alumina thin films during low-energy ion-beam sputtering: experiments and simulations[END_REF]. The third step is the GLAD of TiN and ZrN on the rippled alumina surface via reactive magnetron sputtering. This step is particularly challenging for TiN and ZrN due to the combination of the reactive process associated with GLAD geometry, the high melting points of TiN and ZrN [START_REF] Patsalas | Conductive nitrides: Growth principles, optical and electronic properties, and their perspectives in photonics and plasmonics[END_REF], the cohesive energies of Ti and Zr [START_REF] Kaxiras | Atomic and Electronic Structure of Solids[END_REF], the low work function [START_REF] Koutsokeras | Structure, electronic properties and electron energy loss spectra of transition metal nitride films[END_REF], and the ionization potential [START_REF] Matenoglou | Plasma energy and work function of conducting transition metal nitrides for electronic applications[END_REF] of TiN and ZrN compared to Ag and Au.

As a result, the TiN and ZrN reactive sputtering growth lies into the Zone 1 of limited diffusion and of high density of structural defects, according to the updated structural zone model of Anders [START_REF] Anders | A structure zone diagram including plasma-based deposition and ion etching[END_REF], in contrast to Au and Ag that lie in Zone 3 of high crystallinity due to diffusion of adatoms [START_REF] Anders | A structure zone diagram including plasma-based deposition and ion etching[END_REF]. In order to overcome these obstacles, the GLAD deposition (5 o angle of incidence) of TiN and ZrN was performed by reactive magnetron sputtering in the metallic cathode mode (i.e. strictly preventing poisoning of the target). The reactive gas (N2) flow was delivered exclusively close to the substrate by a dedicated gas line monitored by a real-time mass spectrometer, and the ion bombardment conditions were kept mild and below the subplantation threshold [START_REF] Patsalas | Surface kinetics and subplantation phenomena affecting the texture, morphology, stress, and growth evolution of titanium nitride films[END_REF] by applying a bias of -60 V to the substrate during growth. The substrate temperature was tuned to 500 o C, in order to maximize the surface diffusion of the deposited adatoms [START_REF] Anders | A structure zone diagram including plasma-based deposition and ion etching[END_REF][START_REF] Patsalas | Surface kinetics and subplantation phenomena affecting the texture, morphology, stress, and growth evolution of titanium nitride films[END_REF], while preventing the oxidation of the deposited nanostructures via diffusion of oxygen from the underlying alumina into the nitride (a process that might occur at 600 o C [START_REF] Abadias | Stress, phase stability and oxidation resistance of ternary Ti-Me-N (Me = Zr, Ta) hard coatings[END_REF]). The deposition conditions used for this step are summarized in Table 1. The geometry of this step enables the growth of self-assembled and well-aligned nanostructures along the ripples, as we will show below, all over a macroscopic area, which is comparable to USD and Euro coins (shown in Fig. 1b).

Then, a selection of samples proceeded to step 4, which is the in situ capping by an ultra-thin (5 nm) amorphous SiNx layer, grown by RF magnetron sputtering from a Si3N4 ceramic target, in order to prevent oxidation and provide long-term stability to the nanostructures [START_REF] Rodriguez | Stabilizing the Plasmonic Response of Titanium Nitride Nanocrystals with a Silicon Oxynitride Shell: Implications for Refractory Optical Materials[END_REF]. SiNx was selected due to its dielectric character as well as the thermal stability of the conductive nitride/Si3N4 nanostructures [START_REF] Uglov | Features of microstructure of ZrN, Si3N4 and ZrN/SiNx nanoscale films irradiated by Xe ions[END_REF]. For selected samples, steps 3 and 4 were repeated in situ to form multi-stacks of TiN or ZrN nanostructures to enhance their far-field plasmonic features.

The formation of periodic ripple patterns on amorphous alumina surfaces subjected to 1 keV-Xe + ion bombardment at oblique incidence is well-documented and robustly reproducible [START_REF] Camelio | Anisotropic optical properties of silver nanoparticle arrays on rippled dielectric surfaces produced by low-energy ion erosion[END_REF][START_REF] Babonneau | Tunable plasmonic dichroism of Au nanoparticles self-aligned on rippled Al2O3 thin films[END_REF][START_REF] Babonneau | Formation of nanoripples on amorphous alumina thin films during low-energy ion-beam sputtering: experiments and simulations[END_REF]. The process results in near-parallel and slightly wavy ripples perpendicular to the ion-beam projection, as shown in the AFM images in the upper side of Fig. 2, all over the area of a macroscopic sample as shown in Fig. 1b. The AFM images can be quantified by considering the height variations along line scans in directions parallel or perpendicular to the ripples, as shown in Fig. S1 of on-line supplemental information. The height variations along the bare ripples are negligible, while the height variations across the ripples indicate a period  of 26.0±1.1 nm, and an average peak-to-valley height of 2.6±0.2 nm (note that the reported height and period values might be slightly affected by the radius of curvature of the tip, which is ~10 nm).

After the GLAD of TiN at 500 o C (Fig. 2, intermediate panels, and Fig. S1, green lines), the ripples retain their morphology and only some faint grainy structures appear at the highest resolution, which indicate the formation of continuous, yet rough, TiN NWs along the ripples. The difference of the determined period of the ripples after TiN deposition ( = 28.6±1.5 nm) from the bare ripples is within the experimental error, while the peak-to-valley height across the ripples increases after TiN deposition to 3.3±0.4 nm. Along the ripples, the root mean square roughness increases after TiN deposition, but no individual TiN nanoparticles can be identified.

The GLAD of ZrN results in a quite different morphology (Fig. 2, lower panels, and Fig. S1, magenta/purple lines). Again, the period of the nanostructures across the ripples ( = 29.6±2.2 nm) is not substantially different from that of the bare ripples.

However, the average peak-to-valley height becomes 4.5±0.7 nm, indicating significant growth of ZrN on the alumina ripples. More striking differences are observed along the ripples after ZrN deposition. Indeed, clear features, characteristic of granular ZrN nanostructures, are identified along the ripples; their period and average peak-to-valley height are 19.1±3.4 nm and 1.5±0.1 nm, respectively. Based on the observed morphologies and the quantitative data, it can be safely concluded that ZrN after GLAD forms highly oriented arrays of densely-packed nanoparticles resulting in granular-like NWs along the underlying ripples. The difference in the morphology of the produced TiN and ZrN nanostructures, i.e. quite smooth NWs vs arrays of granular NWs, respectively, may be assigned to a variety of reasons, such as differences in angular and energy distribution of sputtered atoms reaching the substrate at glancing angle, and the reduced mobility of Zr adatoms on the active growing surface [START_REF] Abadias | Stress Evolution in Magnetron Sputtered Ti-Zr-N and Ti-Ta-N Films Studied by In-Situ Wafer Curvature: Role of Energetic Particles[END_REF][START_REF] Mareus | Study of columnar growth, texture development and wettability of reactively sputter-deposited TiN, ZrN and HfN thin films at glancing angle incidence[END_REF]. The reported morphological features do not change after the deposition of the homogeneous 5 nm ultrathin SiNx capping layer over the TiN or ZrN nanostructures (Fig. 1a,step 4). When the GLAD of TiN or ZrN and the sputter deposition of the homogenous SiNx capping layer are repeated three times, in order to produce triple arrays of nanostructures and increase their linear as well as non-linear plasmonic response, the surface morphology of the top array of nanostructures is less organized and coherent as it is shown in the on-line supplemental information (Fig. S2). A possible explanation for this difference in morphology is most likely due to the roughness and morphology of the underlying layers of nanostructures that alter the shadowing and surface diffusion conditions during GLAD. Such loss of coherency sets the limit of repetitions in this process and defines the maximum interaction volume with the incident light; it also affects the linear and nonlinear optical properties of the TMN nanostructures, as it will be discussed later.

It is worth noting that the proposed process is generic and might be very efficient for the production of nanostructures of a wide variety of alternative non-metallic conductors, which are usually grown by sputter deposition, such as molybdenum nitride (MoN) [START_REF] Kassavetis | Near-zero negative real permittivity in far ultraviolet: Extending plasmonics and photonics with B1-MoNx[END_REF], tantalum nitride (TaN) [20, 22 60], niobium nitride (NbN) [START_REF] Ubl | Optical properties of niobium nitride plasmonic nanoantennas for the near-and mid-infrared spectral range[END_REF], In-doped tin oxide (ITO) [START_REF] Naik | Alternative plasmonic materials: Beyond gold and silver[END_REF][START_REF] Hillier | When Ellipsometry Works Best -A Case Study with Transparent Conductive Oxides[END_REF][START_REF] Kapetanovic | Tunable Infrared Plasmon Response of Lithographic Sn-doped Indium Oxide Nanostructures[END_REF], aluminum-or gallium-doped zinc oxide (AZO, GZO) [START_REF] Naik | Alternative plasmonic materials: Beyond gold and silver[END_REF][START_REF] Hillier | When Ellipsometry Works Best -A Case Study with Transparent Conductive Oxides[END_REF][START_REF] Sarma | Al-doped ZnO transparent conducting oxide with appealing electro-optical properties -Realization of indium free transparent conductors from sputtering targets with varying dopant concentrations[END_REF][START_REF] Zhu | Zn-aided defect control for ultrathin GZO films with high carrier concentration aiming at alternative plasmonic metamaterials[END_REF], niobium-doped titanium oxide (NTO) [START_REF] Ngo | Nanoantenna structure with mid-infrared plasmonic niobium-doped titanium oxide[END_REF], partly reduced molybdenum oxides MoOx (x<3) [START_REF] Liu | High electrochemical performance and phase evolution of magnetron sputtered MoO2 thin films with hierarchical structure for Li-ion battery electrodes[END_REF][START_REF] Cauduro | Tuning the optoelectronic properties of amorphous MoOx films by reactive sputtering[END_REF], and vanadium dioxide (VO2), which exhibits a metal-insulator transition [START_REF] Collado | Temperature dependence of electrical resistivity in oxidized vanadium films grown by the GLAD technique[END_REF][START_REF] Okimura | Preparation of VO2 films with metal-insulator transition on sapphire and silicon substrates by inductively coupled plasma-assisted sputtering[END_REF]. All these conductors have been proposed as alternative plasmonic materials, whose plasmonic activity spans the far ultraviolet to medium infrared range [START_REF] Patsalas | Interface Properties and Structural Evolution of Nanoscale TiN/Si and TiN/GaN Heterostructures[END_REF][START_REF] Kassavetis | Plasmonic spectral tunability of conductive ternary nitrides[END_REF][START_REF] Naik | Alternative plasmonic materials: Beyond gold and silver[END_REF][START_REF] Kassavetis | Near-zero negative real permittivity in far ultraviolet: Extending plasmonics and photonics with B1-MoNx[END_REF][START_REF] Ubl | Optical properties of niobium nitride plasmonic nanoantennas for the near-and mid-infrared spectral range[END_REF][START_REF] Cortie | The Quest for Zero Loss: Unconventional Materials for Plasmonics[END_REF][START_REF] Gutiérrez | Plasmonics beyond noble metals: Exploiting phase and compositional changes for manipulating plasmonic performance[END_REF][START_REF] Zhang | A metallic molybdenum dioxide with high stability for surface enhanced Raman spectroscopy[END_REF][START_REF] Etman | Molybdenum Oxide Nanosheets with Tunable Plasmonic Resonance: Aqueous Exfoliation Synthesis and Charge Storage Applications[END_REF][START_REF] Zhan | Phase-controlled synthesis of molybdenum oxide nanoparticles for surface enhanced Raman scattering and photothermal therapy[END_REF][START_REF] Osgouei | Active Tuning from Narrowband to Broadband Absorbers Using a Subwavelength VO2 Embedded Layer[END_REF][START_REF] Duan | Reconfigurable Multistate Optical Systems Enabled by VO2 Phase Transitions[END_REF][START_REF] Schrecongost | Rewritable Nanoplasmonics through Room-Temperature Phase Manipulations of Vanadium Dioxide[END_REF][START_REF] Butakov | Switchable Plasmonic-Dielectric Resonators with Metal-Insulator Transitions[END_REF].

b. Vibrational Properties and Photothermal Stability

In addition to the morphological features, it is important to clarify whether the formed TMN nanostructures adopt the B1 rocksalt crystal structure, which is the phase that exhibits strong plasmonic performance [START_REF] Patsalas | Conductive nitrides: Growth principles, optical and electronic properties, and their perspectives in photonics and plasmonics[END_REF][START_REF] Patsalas | Zirconium nitride: A viable candidate for photonics and plasmonics?[END_REF], or not. The Raman spectra of ZrN and TiN are characterized by distinct features [START_REF] Spengler | Raman scattering, superconductivity, and phonon density of states of stoichiometric and nonstoichiometric TiN[END_REF][START_REF] Patsalas | Zirconium nitride: A viable candidate for photonics and plasmonics?[END_REF][START_REF] Christensen | Phonon anomalies in transition-metal nitrides: ZrN[END_REF][START_REF] Gioti | Plasmonic and phononic properties of epitaxial conductive transition metal nitrides[END_REF][START_REF] Spengler | First and second order Raman scattering in transition metal compounds[END_REF][START_REF] Stoehr | Raman scattering from TiNx (0.67≤x≤1.00) single crystals grown on MgO(001)[END_REF][START_REF] Abdallah | Structural, mechanical, electrical and wetting properties of ZrNx films deposited by Ar/N2 vacuum arc discharge: Effect of nitrogen partial pressure[END_REF], which are associated with the first order transverse acoustic (TA) and longitudinal acoustic (LA) modes. These peaks are normally forbidden because of the Oh symmetry of the rocksalt structure, and they emerge only in the presence of defects that break the local symmetry. Therefore, the usefulness of the Raman spectra is twofold; it serves the non-destructive identification of the B1 rocksalt phases of ZrN and TiN, as well as the detection of point defects that break the symmetry of the cubic, centrosymmetric ZrN and TiN.

Figure 3 shows the polarized Raman spectra of the ZrN and TiN nanostructures (both single-and triple-arrays) with the electric field of the probe laser beam being linearly polarized across (denoted ⊥ for perpendicular) or along (denoted || for parallel) the ripples. For the Raman experiments, silicon replaced silica as the substrate for the process presented in Fig. 1, in order to eliminate the broad Raman signal from silica [START_REF] Zhang | A metallic molybdenum dioxide with high stability for surface enhanced Raman spectroscopy[END_REF] that overlaps with the spectra of TiN and ZrN. The corresponding Raman spectra from opaque epitaxial films from Ref. 81 are also shown for comparison. All the spectra exhibit a doublet at low frequencies (below 350 cm -1 ), which is assigned to the first order TA and LA modes. As mentioned previously, this doublet is exclusively due to the presence of point defects that break the Oh symmetry of the rocksalt (B1) structure. The Raman features at higher frequencies originate from the overlapping of the first order optical (O) and the second order acoustic modes (2A), according to Spengler et al. [START_REF] Spengler | Raman scattering, superconductivity, and phonon density of states of stoichiometric and nonstoichiometric TiN[END_REF].

The comparison of the Raman spectra of the nanostructures with those of the epitaxial films firmly identifies the B1 rocksalt crystal structure. This was further confirmed by Xray and electron diffraction studies (not reported here). The epitaxial films exhibit sharper features than the reported nanostructures due to the improved crystallinity. It is important to note that for all samples, there is no significant change of the broadening of the Raman features between the spectra of the two beam polarizations (along and across the ripples), indicating that there is no preferable axis of improved crystallinity despite of the linear ordering of the nanostructures along the ripples.

Spengler et al. also associated the intensity ratio of the first order acoustic modes and the broad envelope of O and 2A modes with the density of point defects that break locally the Oh symmetry. Indeed, it has been very recently demonstrated that this ratio is well-scaled with the formation energy of vacancies in all the conductive TMNs [START_REF] Gioti | Plasmonic and phononic properties of epitaxial conductive transition metal nitrides[END_REF]. The frequencies ωTA, ωLA, ωO&2A of the TA, LA modes and of the O&2A envelope, respectively, as well as the intensity ratio ITA/IO&2A of the TA peak over the O&2A envelope are summarized in Tables S1 andS2 of the on-line supplemental information for ZrN and TiN, respectively. Several aspects have been revealed from the quantitative results. Firstly, the frequencies of the TA and LA modes and the O&2A envelope of the epitaxial films are in remarkable agreement with previous studies on stoichiometric nitrides [START_REF] Spengler | Raman scattering, superconductivity, and phonon density of states of stoichiometric and nonstoichiometric TiN[END_REF][START_REF] Spengler | First and second order Raman scattering in transition metal compounds[END_REF][START_REF] Stoehr | Raman scattering from TiNx (0.67≤x≤1.00) single crystals grown on MgO(001)[END_REF]. Secondly, the TA and LA modes of both TiN and ZrN nanostructures are shifted compared to the corresponding epitaxial films; this is more pronounced for the TA mode that exhibits shifts higher than 17 cm -1 and 5 cm -1 for TiN and ZrN, respectively. Such shifts were assigned to deviations from stoichiometry both for TiN [START_REF] Spengler | Raman scattering, superconductivity, and phonon density of states of stoichiometric and nonstoichiometric TiN[END_REF][START_REF] Stoehr | Raman scattering from TiNx (0.67≤x≤1.00) single crystals grown on MgO(001)[END_REF] and ZrN [START_REF] Abdallah | Structural, mechanical, electrical and wetting properties of ZrNx films deposited by Ar/N2 vacuum arc discharge: Effect of nitrogen partial pressure[END_REF]. Therefore, the produced TMN nanostructures may not be stoichiometric. Importantly, for the SHG applications, the ITA/IO&2A ratio is increased by 66% and 18% in TiN and ZrN nanostructures, respectively, compared to the epitaxial samples. This ratio was associated with the point defects [START_REF] Gioti | Plasmonic and phononic properties of epitaxial conductive transition metal nitrides[END_REF] and the results show that the TiN nanostructures incorporate more defects than the ZrN nanostructures. Indeed, it was shown that at the very initial stages of growth, for thickness less than 6 nm, TiN is not stoichiometric and exhibits a high concentration of defects [START_REF] Patsalas | Interface Properties and Structural Evolution of Nanoscale TiN/Si and TiN/GaN Heterostructures[END_REF]. Last but not least, this ratio does not vary significantly along and across the ripples indicating that there is no preferred directionality for the accumulation of defects.

The Raman experiments may also be used to test the photothermal stability of the produced nanostructures. To check and demonstrate the photothermal stability, especially across the ripples where the size of the nanostructures is just few nanometers, Raman spectra were acquired from the single-array ZrN nanostructures with polarization across the ripples, varying the power of the probe laser beam. The relevant spectra for 1, 5, and 20 mW continuous wave laser probe are shown in the upper panel of Fig. 3 with different shades of grey; all the spectra are identical with no traces of bulk oxidation or formation of additional defects in the ZrN nanostructures. Given that the spot size of the probe laser beam is ~1 m on sample, the maximum power density used is 2.5 MW/cm 2 , which is one order of magnitude higher than the power density used for the sintering of colloidal Ag inks [START_REF] Balliu | Selective laser sintering of inkjet-printed silver nanoparticle inks on paper substrates to achieve highly conductive patterns[END_REF] and comparable with the pulsed laser power used for the melting of Ag layers [START_REF] Kalfagiannis | Selective modification of nanoparticle arrays by laser-induced self-assembly (MONA-LISA): putting control into bottom-up plasmonic nanostructuring[END_REF] and the diffusion of melted Ag out of yttria scaffolds [START_REF] Siozios | Sub-surface laser nanostructuring in stratified metal/dielectric media: a versatile platform towards flexible, durable and large scale plasmonic writing[END_REF]. Therefore, the ZrN nanostructures exhibit remarkable and extraordinary photothermal stability that easily outperforms plasmonic Ag nanostructures.

c. Linear and Non-Linear Optical Properties

The far-field optical properties of the formed TMN nanostructures are characterized by strong dichroism and plasmonic effects. This is clearly demonstrated in the polarized optical absorbance spectra of Fig. 4 (calculated from raw optical transmission spectra). The left and center panels show the optical absorbance spectra for two bare and two SiNx-capped TiN and ZrN arrays of nanostructures, respectively, with electric field across (⊥, solid lines) and along (||, dotted lines) the ripples. For both cases, the dichroism is not prominent for short wavelengths (shorter than 400 nm or 3.1 eV and 350 nm or 3.5 eV for TiN and ZrN, respectively); the increased absorbance in this range is due to the interband transitions of TiN and ZrN and it is consistent with the interband absorption reported for continuous opaque TiN and ZrN films [START_REF] Patsalas | Interface Properties and Structural Evolution of Nanoscale TiN/Si and TiN/GaN Heterostructures[END_REF][START_REF] Kassavetis | Plasmonic spectral tunability of conductive ternary nitrides[END_REF][START_REF] Patsalas | Conductive nitrides: Growth principles, optical and electronic properties, and their perspectives in photonics and plasmonics[END_REF][START_REF] Patsalas | Zirconium nitride: A viable candidate for photonics and plasmonics?[END_REF].

In the case of the bare TiN nanostructures, there is a strong dichroism mostly in the NIR range, with a prominent localized absorbance peak centered around 1200 nm (1.03 eV) for the electric field in the perpendicular direction, i.e. across the ripples and nanostructures. This peak is assigned to LSPR due to the infinitesimal size of the TiN nanostructures across the alumina ripples. The plasmonic hypothesis is confirmed by the comparison of the optical absorbance spectra of bare (Fig. 4, red solid line) and SiNxcapped nanostructures (Fig. 4, orange solid line). Τhe absorbance peak of the SiNxcapped nanostructures is substantially redshifted to 1860 nm (0.67 eV) due to the higher refractive index of the surrounding SiNx (n~2) compared to air (n~1), as indicated by the black arrows. This proves unequivocally the plasmonic nature of this optical feature. The LSPR of the bare TiN nanostructures lies into the NIR range and not in the visible as it would be anticipated for spherical nanoparticles [START_REF] Patsalas | Conductive nitrides: Growth principles, optical and electronic properties, and their perspectives in photonics and plasmonics[END_REF]. This can be explained by the high concentration of defects, as indicated by the Raman spectra, as well as the flatness of the TiN NWs and the short edge-to-edge distance between NWs. LSPR can be also substantially redshifted and weakened in ultrathin TiN due to stoichiometry variations [START_REF] Patsalas | Optical properties and plasmonic performance of titanium nitride[END_REF] and the incorporation of surface oxygen species [START_REF] Shah | Controlling the plasmonic properties of ultrathin TiN films at the atomic level[END_REF], which are both depleting the conduction electrons. In addition, it was previously shown that for ultra-thin TiN deposits (<6 nm) the nitride is inherently defective even if the deposition conditions are appropriate for the growth of stoichiometric TiN at higher thicknesses [START_REF] Patsalas | Interface Properties and Structural Evolution of Nanoscale TiN/Si and TiN/GaN Heterostructures[END_REF]. In the parallel direction of the electric field (along the ripples and nanostructures), no such peaks are observed in the experimental spectral window, and therefore the optical response of TiN nanostructures is more consistent with an infinite 1D conductor.

The case of the ZrN nanostructures is more complex and interesting. The dichroism and the plasmonic features are manifested not only in the NIR, but also extend into the visible range (Fig. 4, center panel). In the perpendicular direction of the electric field (across the ripples), the LSPR of bare ZrN nanostructures is vaguely manifested through a shoulder around 500 nm (2.48 eV) (Fig. 4, blue solid line), a value consistent with what would be expected for ZrN spherical nanoparticles [START_REF] Patsalas | Conductive nitrides: Growth principles, optical and electronic properties, and their perspectives in photonics and plasmonics[END_REF][START_REF] Patsalas | Zirconium nitride: A viable candidate for photonics and plasmonics?[END_REF]. This LSPR is redshifted, like in the case of TiN, to 580 nm (2.14 eV) for the SiNx-capped ZrN nanostructures (Fig. 4, cyan solid line). However, as opposed to the case of TiN NWs, for the parallel field (along the ripples) both bare and SiNx-capped ZrN nanostructures exhibit another plasmonic absorption band in the NIR around 1130 and 1200 nm (1.10-1.03 eV), respectively. In contrast to TiN, where dichroism originates from the existence or not of a plasmonic absorption band, in ZrN nanostructures the dichroism is due to the existence of two distinct plasmonic features at the two polarizations of the incident light.

While the LSPR in the visible range and for the perpendicular field is very well understood and resembles the optical response of the TiN nanostructures, the nature of the plasmonic feature in the NIR for the parallel field may originate from the granular structure of the ZrN nanostructures. Indeed, the particle-particle near field interaction of the ZrN dense granular nanostructure may result in the observed substantial redshift [START_REF] Hentschel | Plasmonic oligomers: The role of individual particles in collective behavior[END_REF][START_REF] Sweatlock | Highly confined electromagnetic fields in arrays of strongly coupled Ag nanoparticles[END_REF][START_REF] Camelio | Optimization of growth and ordering of Ag nanoparticle arrays on ripple patterned alumina surfaces for strong plasmonic coupling[END_REF]. Finally, the ZrN and TiN multi-stacks retain the plasmonic features of their capped single-array counterparts, albeit with stronger far-field optical absorption due to the cumulative interactions of each array of nanostructures with the incoming light (Fig. 4, right panel).

In order to shed some light on the plasmonic performance of the ZrN and TiN nanostructures interacting with the perpendicular electric field (⊥), Finite-Difference Time-Domain (FDTD) calculations were carried out. An in-house 3D FDTD simulator [START_REF] Lidorikis | Effective medium properties and photonic crystal superstructures of metallic nanoparticle arrays[END_REF][START_REF] Schedin | Surface-enhanced Raman spectroscopy of graphene[END_REF] was used in all calculations. Arrays of one-dimensional rods with infinite length and semi-elliptic cross section at a constant center-to-center distance of  = 30 nm (roughly the period of ripples on the substrate) were considered for the calculations, as shown in Fig. 5a. The calculations were carried out for a range of aspect ratios H/D (where H and D are the height and diameter of the rods, respectively) in adequacy with the AFM experimental observations and with the aim of reproducing the experimental position of the absorbance maxima. We used dielectric functions retrieved from ellipsometric measurements of continuous opaque ZrN and TiN films deposited at near-normal incidence in the same conditions as the TMN nanostructures. The incoming electric field was set perpendicular (⊥) to the infinite axes of the rods. The far-field optical absorbance (see on-line supplemental information, Fig. S3), as well as the distributions of the electric near-field enhancement (NFE) for the excitation wavelength (1030 nm) of the laser used for the SHG experiments, were calculated. The FDTD results unequivocally show that the far-field properties (Fig. S3) are substantially affected by the morphology of the nanostructures, as it is shown in the contour plots of the wavelength of the absorbance maximum for varying D and H values for ZrN (Fig. 5b) and TiN (Fig. 5c). In Fig. 5c (upper panel), the thick green line corresponds to the experimental position of the LSPR maximum for the bare TiN nanostructures (i.e., 1200 nm). Our calculations confirm that the LSPR redshift compared to the expected position for spherical TiN nanoparticles in air (520 nm [START_REF] Patsalas | Interface Properties and Structural Evolution of Nanoscale TiN/Si and TiN/GaN Heterostructures[END_REF]) originates from flat and coupled TiN NWs. Several (H,D) couples lead to consistency between the experimental and calculated spectral positions of LSPR maximum. Nevertheless, considering the nominal equivalent thickness of TiN (3.7 nm, Table 1), the best (H,D) couple that could explain the experimental position is D = 28 nm and H = 2.4 nm, corresponding to very flat (H/D < 0.1) and highly coupled (x =  -D = 2 nm < D/10) TiN nanostructures. The existence of nanogaps between plasmonic nanostructures associated with LSPR coupling were reported to result in strong redshift of the far-field absorbance and reflectance as well as exceptional NFE at the formed hot spots [START_REF] Herzog | Dark plasmons in hot spot generation and polarization in interelectrode nanoscale junctions[END_REF]. As shown in Fig. 5c (lower panel), the corresponding calculated positions of the LSPR for SiNx-capped TiN nanostructures are even more redshifted in the NIR in agreement with the experimental observations. In the case of ZrN nanostructures, considering a nominal equivalent thickness of 7.6 nm (Table 1), the best (H,D) couple that explains the experimental position of LSPR maximum for the SiNx-capped sample, which is clearly identified at 580 nm in Fig. 4, center panel (see thick green line in the contour plot of Fig. 5b, lower panel), is H = 17 nm and D = 18 nm. Therefore, for the ZrN nanostructures, the out-of-plane aspect ratio H/D is close to 1 (0.9) with an edge-to-edge distance x = 12 nm (higher than D/2). The corresponding calculated position of the LSPR maximum for the bare ZrN nanostructures is obtained at 515 nm (Fig. 5b, upper panel) in line with the experimental shoulder around 500 nm (Fig. 4, center panel). In that case, FDTD calculations show that the deviation from what is expected for spherical nanoparticles in air in terms of LSPR spectral position is less prominent. The distributions of NFE at 1030 nm (the excitation wavelength which was used for the SHG experiments) for simulated bare nanostructures with appropriate (H,D) values are shown in Fig. 6a,b. In both cases, the maximum NFE is located at the bases and edges of the nanostructures. The NFE in the case of 'isolated' ZrN nanostructures is moderate (~30) and it is remarkably enhanced for the case of the coupled TiN nanostructures (~100) due to the formation of hot spots.

By comparing the Raman with the optical absorbance spectra, we reveal an extraordinary feature of conductive albeit defective nitride nanostructures: TMNs exhibit a unique combination of broken centrosymmetry (due to defects and refractory character)

and plasmonic activity with great potential for SHG. This is in contrast to the traditional plasmonic metals (Au, Ag), which exhibit zero second-order susceptibility due to their centrosymmetric fcc crystal structure [START_REF] Matranga | Absolute intensity measurements of the optical second-harmonic response of metals from 0.9 to 2.5 eV[END_REF][START_REF] Kujala | Multipole interference in the second-harmonic optical radiation from gold nanoparticles[END_REF][START_REF] Obermeier | Modal Symmetry Controlled Second-Harmonic Generation by Propagating Plasmons[END_REF]. Continuous stoichiometric TiN films exhibit very weak SHG [START_REF] Capretti | Comparative Study of Second-Harmonic Generation from Epsilon-Near-Zero Indium Tin Oxide and Titanium Nitride Nanolayers Excited in the Near-Infrared Spectral Range[END_REF], possibly due to the low concentration of defects in such TiN structures. Although the crystal symmetry may be broken at metallic surfaces [START_REF] Matranga | Absolute intensity measurements of the optical second-harmonic response of metals from 0.9 to 2.5 eV[END_REF], for centrosymmetric nanostructures (as those presented in this work) the contribution of two opposing surface elements cancels out in the far-field in the dipole approximation [START_REF] Kujala | Multipole interference in the second-harmonic optical radiation from gold nanoparticles[END_REF][START_REF] Obermeier | Modal Symmetry Controlled Second-Harmonic Generation by Propagating Plasmons[END_REF]. Consequently, plasmonic nanostructures consisting of Au or Ag that are capable of SHG are based on metamaterials with non-centrosymmetric shapes [START_REF] Obermeier | Modal Symmetry Controlled Second-Harmonic Generation by Propagating Plasmons[END_REF][100][101][102]]. In the same context, TiN was implemented as a conventional metal component of noncentrosymmetric metamaterials, such as split-ring resonators [103] and chiral core-shells

[104] for SHG, regardless of its own intrinsic broken centrosymmetry. So far, there is no report on optical non-linearities of ZrN whatsoever.

Figure 7 shows the polar diagrams of polarized SHG (PSHG) with a pump beam of 1030 nm (1.204 eV) using the same gain for all cases; in this diagram zero degree is for the electric field of the pump beam being aligned along the ripple direction. An exceptionally interesting observation is that the SHG is emitted along (|| field) the ripples for ZrN and across (⊥ field) the ripples for TiN. This strikingly different behavior of the two sets of TMN nanostructures may be well explained if we consider their optical properties presented in Fig. 4. Indeed, ZrN exhibits a broad LSPR band around 1200 nm (1.03 eV) when the electric field of the light is parallel to the ripples, possibly due to the granular structure of the ZrN nanostructures that promotes the particle-particle near field interaction. On the other hand, TiN exhibits a similar LSPR band around 1200 nm (1.03 eV) when the electric field is perpendicular to the ripples due to the infinitesimal size of the TiN nanostructures across the alumina ripples. In both cases the 1030 nm probe beam lies within the bandwidth of the LSPR, while it lies far beyond the strong LSPR of ZrN across the ripples (< 600 nm). So, the TMN nanostructures act simultaneously as the probe (defects and broken centrosymmetry) and pump (LSPR field enhancement) medium for SHG. Note that this plasmon-pumped generation of second harmonic is contrastingly different to the work of Huang et al [105], where the SHG from the defective tantalum nitride (TaN) is pumped by the enhanced near field due to the LSPR in adjacent gold nanopillars, i.e. the pumping plasmon is not due to the nitride conductor itself.

The SHG is apparently stronger for ZrN, although TiN seems more defective both in the Raman (Fig. 3) and absorbance (Fig. 4) spectra. This might be attributed to the stronger LSPR in NIR for ZrN nanostructures, as well as the better overlap of ZrN LSPR with the probe beam. For optimal plasmon-pumped SHG in such defective TMN nanostructures there is a sensitive interplay between two competing mechanisms; a lot of defects increase the density of sites of broken symmetry, potentially enhancing the SHG, but at the same time they deteriorate the LSPR and the near-field enhancement, which is necessary for the SHG. In addition, the granular nature of the ZrN nanostructures may promote the creation of a high linear density of hot spots with great NFE along the ripples; this is supported by the redshifted LSPR along the ripples.

It is also interesting to follow the variations of directionality of PSHG polar diagrams; this is better illustrated in the cartesian SHG diagram presented in the on-line supplemental information (Fig. S4). The bare nanostructures, both ZrN and TiN, produce highly directional PSHG polar diagrams. This is well understood by considering that the non-linear dipoles, which define the directionality of SHG [106], are strictly directional due to the geometry of the pumping LSPR. The SiNx-capping does not alter substantially the directionality. Finally, the multi-stacks clearly exhibit a partial loss of directionality, possibly due to the scattering of the SHG from the incoherent top arrays of nanostructures (see Fig. S2). Such variations of directionality along or across the ripples, or their reduction due to loss of coherence of the nanostructures, provide exceptional versatility for optical devices and applications.

III. CONCLUSIONS

An etchless, cost-effective and novel bottom-up fabrication route to produce nanostructures of the refractory and alternative plasmonic materials TiN and ZrN was presented. It is based on glancing angle deposition by reactive magnetron sputtering on prepatterned (nanorippled) alumina substrates. The process is generic and could be applied to a wide range of alternative plasmonic materials that are traditionally grown by reactive magnetron sputtering, such as the entirety of TMNs and transparent conductive oxides.

The produced nanostructures are nearly one dimensional, and they are strongly dichroic due to the different LSPR features along or across their axis. TiN and ZrN are refractory and so the produced nanostructures exhibit exceptional photothermal stability as demonstrated by Raman spectroscopy. Raman spectra also revealed the existence of a higher density of point defects than that of the corresponding continuous epitaxial films.

Remarkably, these defects break the centrosymmetry of the cubic crystal structure of TiN and ZrN, while the plasmonic performance of the nitrides is retained. The combination of broken centrosymmetry with the plasmonic activity results in strong SHG. The directionality of the SHG depends on the type of plasmonic interaction (along the nanostructures for ZrN and across the nanostructures for TiN) and the coherence of the individual linear nanostructures. Their traits of strong dichroism and SHG make them suitable for laser technology as optical components such as polarizers and SHG emitters.

IV. METHODS

Experiments:

The surface morphology of the nanostructures was analyzed immediately after deposition by AFM using a multimode Digital Instrument microscope operating in tapping mode at ambient conditions. Raman spectroscopy measurements were conducted with a Horiba LabRAM HR instrument with a 515 nm excitation laser; the excitation laser was selected such that its energy is spectrally located beyond the reflectivity edge of TiN and ZrN and, therefore, to minimize the skin effect. A linearly polarized excitation beam was used to identify potential variations of the Raman spectra with beam polarization parallel (||) or perpendicular (⊥) to the TiN and ZrN nanostructures.

The far-field optical response of the self-assembled nanostructures was investigated by means of transmission spectroscopy using a Cary 5000 spectrophotometer from Agilent Technologies equipped with a rotating polarizer. Transmittance measurements were performed at normal incidence (beam diameter ∼3 mm) in the spectral range 250-2200 nm (4.96-0.56 eV) with a 1 nm step using parallel (||) or perpendicular (⊥) polarization with respect to the alumina ripple direction. The thick green lines correspond to the experimental position.

Fig. 1 :Fig. 2 :Fig. 3 :

 123 Fig. 1: (a) Schematic of the process used for the bottom-up fabrication of dichroic and

Fig. 4 : 29 Fig. 5 :

 4295 Fig. 4: Optical absorbance spectra of TiN (lower left panel) and ZrN (lower center panel)

Fig. 6 :

 6 Fig. 6: Near-field enhancement distributions around bare ZrN (a) and TiN (b)

Fig. 7 :

 7 Fig. 7: Polar diagrams of polarized SHG of the ZrN and TiN nanostructures probed by a 1030 nm laser beam; at 0 o the electric field polarization is parallel with the ripples of the

Table 1 :

 1 Conditions of the GLAD growth of TiN and ZrN

	TiN	ZrN

*The deposition time used for ZrN multistacks was shorter than that for single arrays of nanostructures in order to minimize the loss of organization of the upper layer due to the granular morphology of the underlayers.

ACKNOWLEDGMENTS P.P. acknowledges the University of Poitiers for granting him a visiting Professor position under which part of this work was performed. This work pertains to the French Government program "Investissements d'Avenir" (LABEX INTERACTIFS, reference ANR-11-LABX-0017-01, and EUR INTREE, reference ANR-18-EURE-0010).

Polarization-resolved SHG measurements were carried out using a custom-built laser scanning microscope as described in detail in Ref. [START_REF] Psilodimitrakopoulos | Ultrahigh-resolution nonlinear optical imaging of the armchair orientation in 2D transition metaldichalcogenides[END_REF], using the beam from a diodepumped Yb:KGW fs oscillator (1.2 W, 1030nm, 70-90 fs, 76 MHz, Pharos-SP, Light Conversion) and an inverted microscope (Axio Observer Ζ1, Carl Zeiss).

Computational Method:

The calculations of the optical response of the TiN and ZrN nanostructures were performed by time-integrating Maxwell's equations on a computational grid utilizing the Finite-Difference Time-Domain (FDTD) method, using an in-house built 3D simulator provided by E. Lidorikis [START_REF] Lidorikis | Effective medium properties and photonic crystal superstructures of metallic nanoparticle arrays[END_REF][START_REF] Schedin | Surface-enhanced Raman spectroscopy of graphene[END_REF]; key features of the results were confirmed by FDTD calculations using Lumerical TM .

In the computational process, the material dispersion is introduced in the form of polarization equations coupled and solved concurrently with Maxwell's equations. To accurately reproduce the optical constants of B1-TiN and B1-ZrN, within the FDTD framework, we performed a Drude-Lorentz fit [START_REF] Patsalas | Conductive nitrides: Growth principles, optical and electronic properties, and their perspectives in photonics and plasmonics[END_REF][START_REF] Patsalas | Zirconium nitride: A viable candidate for photonics and plasmonics?[END_REF] to the experimental dielectric functions of opaque stoichiometric ZrN and TiN films obtained in the same conditions and measured by spectroscopic ellipsometry.

ASSOCIATED CONTENT Supporting Information. The SI section includes:

Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.


Evguenia Malchukova

Bruno Boizot

Guillaume Petite

Dominique Ghaleb

Eugenia Malchukova

Effect of Sm-, Gd-codoping on structural modifications in aluminoborosilicate glasses under beta-irradiation

Keywords: codes : 61.16.Fh, 61.72.Hh, 61.80.Fe keywords : Sm and Gd ions, luminescence, glasses, EPR, Raman, irradiation

Two series of Sm-, Gd-codoped aluminoborosilicate glasses with different total rare earth content have been studied in order to examine the codoping effect on the structural modifications of β-irradiated glasses. The data obtained by Electron Paramagnetic Resonance spectroscopy indicated that relative amount of Gd 3+ ions located in network former position reveals non-linear dependence on Sm/Gd ratio. Besides, codoping leads to the evolution of the EPR signal attributed to defects created by irradiation: superhyperfine structure of boron oxygen hole centres EPR line becomes less noticeable and resolved with increase of Gd amount. This fact manifests that Gd 3+ ions are mainly diluted in vicinity of the boron network.

By Raman spectroscopy, we showed that the structural changes induced by the irradiation also reveal non-linear behaviour with Sm/Gd ratio. In fact, the shift of the Si-O-Si bending vibration modes has a clear minimum for the samples containing equal amount of Sm and Gd (50:50) in both series of the investigated glasses. In contrast, for single doped glass there is no influence of dopant's content on Si-O-Si shift (in case of Gd) or its diminution (in case of Sm) occurs which is explained by the reduction process influence. At the same time, no noticeable effect of codoping on Sm 3+ intensity as well as on Sm 2+ emission or on Sm reduction process was observed.

Introduction

The modern concept of radioactive waste management suggests the confinement of radioactive wastes into the compositions of matrices that possess a chemical, mechanical and radiation resistance. High-level nuclear wastes have been vitrified with the use of, for example, aluminophosphate glasses (in Russia) or borosilicate glasses (in France, England and United States). These nuclear waste glasses are often complex materials containing up to thirty different oxides. In order to predict the long term behaviour under irradiation of these complex glass materials, we have focused our work on simplified glass composition, which reproduces the structure and the behaviour of nuclear wastes glasses. It is known that structural evolution of the nuclear glasses under irradiation (α, β, γ) can modify the confinement properties [weber et al, 1997]. In order to understand structural changes induced by ionizing radiation, the effect of β-irradiation on the glass structure has been investigated in a first step on aluminoborosilicate glass compositions with compositions similar to the matrix of the complex nuclear glasses [1][2][3]. Then we have introduced in these simplified nuclear glass compositions different rare earth (RE) doping ions. Because of the similarity of chemical properties between rare earth (RE) elements and actinides (An), some RE ions are used as surrogates for the actinides during the development of actinide-containing radioactive waste glasses [4][5][6]. Moreover, a certain amount of RE ions also exists as fission products in some actinide-bearing radioactive wastes [7,8]. Our previous investigations on single REdoped aluminoborosilicate glasses showed that ionizing radiation leads to the RE reduction [2,3,9]. We analyse also that the structural changes at one integrated dose (defects production , glass densification and polymerization increase) are decreasing as a function of RE doping ion content. This influence of RE doping upon the glass structural changes under irradiation seems to be correlated with the relative stability of the RE different oxidation state For example, the structural changes in Gd-or Nd-doped glasses where the RE reduction processes are either negligible or absent at all are not limited by the RE doping content [2,3]. Thus, our recent studies have been directed on the investigation of interplay between structural evolution and reduction process occurring under irradiation in aluminoborosilicate glass and estimation of some general trend for behaviour under irradiation of single RE-doped glasses. But, taking into account that complex nuclear glasses contain mixture of RE ions as fission products and/or as non-radioactive actinides (An) surrogates, the simulation of RE mixed effect is of a great importance from both scientific and technological points.

Meanwhile, to the best of our knowledge, the effect of codoping and codopants concentration on the structural modifications in the glass matrices has not been investigated.

In this paper, initiated by previous results of the luminescence of Sm-doped and structural evolution of Gd-doped borosilicate glass under irradiation [2,9], the results of analysis of codoping effect on the changes in glass structure, RE environment and also on RE reduction process are presented. The choice of Sm 3+ and Gd 3+ codoped materials is based on the following criteria. The Gd 3+ ions are paramagnetic and, as our recent study has showed, can be easily observed by EPR even at room temperature [2]. As for samarium, both Sm 3+ and Sm 2+ ions can be detected by means of luminescence spectroscopy since they have a very high efficiency of the emission excited in visible (Sm 3+ ) and near IR (Sm 2+ ) region [9]. Therefore, codoping seems to be a good way for observation of the changes occurring both in glass structure and in RE ions charge state.

Experimental

Two glass series were fabricated with two total concentration of the Sm 2 O 3 and Gd 2 O 3 mixture -0.17 and 0.34 mol.% of the total RE oxide quantity. All glasses were prepared by mixing proper amounts of SiO 2 , Al 2 O 3 , H 3 BO 3 , Na 2 CO 3 , ZrO 2 (see table 1). The mixture was crushed to a fine powder and homogeneous mixing was ensured. The homogeneous powder was put in a Pt crucible and was gradually heated to 1500°C in an electric furnace during 14 h in air atmosphere. The glass melt was then poured on a copper plate. In order to favour homogenous distribution of rare earth ions into the glasses, another 2 hours melting at 1500 o C was effected and the samples were then grinded and mixed again into a homogeneous state.

After quenching, glass samples were annealed at 500 o C in order to decrease strain.

All samples were irradiated using 2.5 MeV electrons beam generated by a Van de Graaf accelerator. The temperature during irradiation was maintained around 50 o C by water cooling of the sample holder. Glass thickness was about 0.5 mm in order to reach a homogeneous irradiation in the glass volume. 4x10 9 Gy irradiation dose was reached on each sample with 14 μA electron beam.

Electron Paramagnetic Resonance (EPR) spectra were obtained using a X band (ν ≈ 9.420 GHz) EMX Bruker spectrometer at room temperature with a 100 kHz field modulation, 1 G amplitude modulation and 1 mW microwave power. All EPR spectra have been normalized to the samples weight of 100 mg. Error estimation has been done for all data obtained taking into account contribution of the random and systematic errors. Systematic error included uncertainty in the mass measuring. Mass measurement uncertainty was of 0.1 mg. In order to minimize the random error measurements of the spectra were repeated 10 times. All uncertainties have been considered and displayed in the table and on the data in the figures.

The microwave frequency has been determined using a Hewlett Packard 5352B frequency counter.

Luminescence and Raman spectra were collected after irradiation on a Labram HR microspectrometer using the 514.5 nm line of an Ar + laser. Experiments were carried out through x100 Olympus objective and a laser power about 80 mW on the sample was used to avoid significant heating of the samples. All Raman spectra of pristine and β-irradiated glass samples have been normalized to the intensity of the band near 460 cm -1 .

Results

EPR spectra

The EPR spectra for pristine and heavily irradiated codoped glass, illustrated in figure 1, show well-known features at g ~ 2.0; 2.8 and 6.0 of "U" spectrum [2,10], typical for Gd 3+ ion in vitreous materials as well as in disordered polycrystalline matrices. According to literature, the prominent EPR signal at g ~ 4.8 is also observed in disordered materials [2,11]. These two groups of features associated with a distinct type of crystal field exists to be known as corresponding to Gd 3 + ions located in the network modifier (Gd[n.m.] 3+ ) and network former (Gd[n.f.] 3+ ) positions, respectively [12] (figure 1). One can see that β-irradiation at doses higher than 10 9 Gy affects the content of Gd 3+ ions in network former sites and leads to the appearance of the additional EPR defects line around g ~2.0. We investigated correlation between the EPR lines attributed to the Gd[n.m.] 3+ and Gd[n.f.] 3+ sites before and after irradiation for both series Sm-,Gd-codoped aluminoborosilicate glasses (1SG and 2SG) in dependence on Sm/Gd molar ratio that are shown in figure 2a. The obtained ratio between Gd[n.f.] 3+ and Gd[n.m.] 3+ sites follows a non-linear behaviour with Sm/Gd ratio accompanied by abrupt increase of Gd[n.f.] 3+ sites with increase of Gd fraction in the total molar RE content. The tendency is more important for irradiated samples (figure 2a). By contrast, in single Gd-doped glasses, the relative proportion of Gd 3+ ions located in network former is decreasing with the gadolinium oxide concentration for both pristine and irradiated aluminoborosilicate glasses as it is shown in figure 2b. Sm/Gd Codoping therefore influences the speciation of Gd 3+ between a former and modifier network positions in both pristine and irradiated glasses.

Figure 3 presents the defects EPR spectra produced by the irradiation for 1SG and 2SG glass samples. We observed that superhyperfine structure of BOHC defects [13] appears with increase of the Sm/Gd ratio. This structure of defect line is noticeable at rather low content of Sm 2 O 3 in codoped samples for 1SG series (1SGb sample) and becomes significant for single Sm-doped glass (1SGs sample) (figure 3a). However, the ability to discern the structure of the defects diminishes with increase of the total RE content from 0.17(1SG) to 0.34 (2SG) mol.% and resolution can be distinguished only for the samples with Sm/Gd ratio above 3/1 (2SGc) (figure 3b). It is interesting to notice that amount of defects varies with codopant concentration as it shown in figure 4 for both glass series. The defect concentration follows a non-linear dependence on Sm-,Gd-codoping (figure 4) for the lowest irradiation dose which is not similar to the single-doped glass studied before. For all single-doped glasses recently investigated we obtained a decrease of the defect amount with dopant concentration [2,9]. Therefore, codoping can strongly affect also the defect creation processes at least for the lowest irradiation dose.

Raman spectra

Fig. 5 displays the Raman spectra of irradiated and non-irradiated Sm-,Gd-codoped aluminoborosilicate glasses. One can notice that the band attributed to the Si-O-Si bending vibration modes (at 460 cm -1 ) is shifted in the glasses irradiated with 10 9 Gy. The decrease of Q 2 species (Q n species correspond to SiO 4 units with n bridging oxygens) stretching band that appears around 1000 cm -1 in comparison with Q 3 ones (1100 cm-1 ) as it can be seen from the figure is also typical for these glasses both without dopants and RE-doped and has been observed previously [2,9,14,15]. But strong overlapping of the intense Sm luminescence arising from 1100 cm -1 with Raman peaks of Q 2 and Q 3 species prevents quantitative estimation of Q 3 /Q 2 ratio changes as well as observation of other modifications in Raman spectrum. It is possible however to measure the Si-O-Si bending vibration modes shift with the relative fraction of gadolinium oxide (460 cm -1 ) (Fig. 6a) which apparently revealed nonlinear character. On th contrary, the same dependence of Si-O-Si shift on RE concentration in single Sm-or Gd-doped aluminoborosilicate glasses showed completely different behaviour as it is shown in figure 6b.

Discussion

Recently it was shown that structural modifications of aluminoborosilicate glasses matrix doped with rare earth ions under the ionizing radiation are correlated with the content and the nature of the RE dopant [2,3,15]. It was established that the matrix changes under ionizing radiation like the decrease of average Si-O-Si angle, the glass polymerization increase and also point defect creation processes can be linked to the relative stability of the RE ion redox states. By contrary, for RE ions with negligible reduction processes under ionizing radiation like Gd and Nd [, ], the host glass matrix changes under irradiation doesn't depends on the RE doping content. Double-doped and single doped aluminoborosilicate glasses containing Sm and/or Gd show different behavior under irradiation, evidenced from both EPR and Raman spectroscopy. It could be suggested that mixing of these two dopants in investigated glasses should show monotonic dependence of structural changes and defect creation on RE content in the mixed series from Gd to Sm related with increase of Sm/Gd ratio. In fact as it can be seen from the figure 2a for each glass series (1SG and 2SG) the Gd[n.f.]/Gd[n.m.] follows a non-linear dependence on the Gd content. It is interesting to notice that without mixing the same dependence reveals clear monotonic decreasing character (figure 2 b). Apparently it should be explained by some RE codoping effect.

The EPR lineshape belonging to point defects is changed in both series of glasses with variations in the proportions of Sm, Gd up to single-doped samples. Indeed, we can observe on figures 3a and 3b a broadening of the BOHC superhyperfine structure with the increase of the relative proportion of Gd inside the glass. It is well known that dipole-dipole interaction results in the broadening and non-resolved EPR lines. Thus we can make an assumption that Gd ions are located in the vicinity to BOHC defects created by irradiation. Such assumption of specific distribution of Gd 3+ ions in borosilicate glasses has been already proposed by Hong Li et al from Raman spectroscopy measurements on borosilicate glasses doped with the highest Gd 2 O 3 content [16]. But, these BOHC EPR lines changes did not show a Sm/Gd codoping effect. By contrary, it should be noted that the defect concentration produced during irradiation depends on the Sm/Gd relative proportion and reveals clearlty a non-linear character. Moreover, one can see that this effect has a tendency to decrease with increase of the irradiation dose (Fig. 4). One explanation could be that the environment of both Gd and Sm ions (coordination number and redox state) changes also as a function of the dose and therefore could have an influence on the defects production.

N. Ollier et al has observed that defect concentration produced during irradiation in Na/Li and Na/K mixed alkali aluminoborosilicate glasses follows a non-linear behaviour [17][18]. It has be shown also that mixed-alkali effect (MAE) leads to the decrease of alkaline migration under ionizing radiation and therefore to a strong decrease of matrix structural changes under irradiation : namely densification and polymerization increase at one integrated dose. Unlike MAE, RE codoping does not prevent structural changes at integrated dose higher than 10 9 Gy as it can be seen by Raman spectroscopy (figure 5). The shift of Si-O-Si bending vibration modes around 480 cm -1 also revealed clearly a non-linear character with Sm/Gd ratio (figure 6a). On the contrary, for single RE doped irradiated glasses, this ionizing radiation effect is decreasing with the RE content with a slope directly correlated to RE ions reduction efficiency (figure 6b). For example, the increase of Gd amount inside the glass has no effect on the structural modifications observed by Raman spectroscopy (figure 6b). On the other hand, presence of strong reduction process like in case of Sm-doped glasses (Sm 3+ + β → Sm 2+ + defects) limits the structural modifications (figure 6b) and the defects produced at one integrated dose [15]. Therefore, impacts of RE codoping should reveal linear dependence of both defect amount and structural changes as a function of RE content moving along the series from single Gd to single Sm doping case. But, observed correlation between non-linear characters of both structural evolution and defect creation let us suppose that some additional processes exist connected with possible energy transfer between RE ions [20]. Moreover, taking into account that processes accompanying absorption of ionizing radiation by codoped glasses may be different [19], it is obvious that the explanation of RE codoped effect is not so unequivocal and request further investigation with use of luminescence measurements. Indeed, codoping effect of certain rare-earth elements with other ones is well-known and can lead either to enhancement or quenching of the intensity and the duration of RE luminescence [20,21].

Conclusion

Our results on structural evolution under β-irradiation of two glass series joins attest the presence of rare earth mixed effect in aluminoborosilicate glasses and its influence on the glass behaviour under irradiation. First, we showed a non-linear character in the speciation of the Gd 3+ ions residing in the network former sites with the Sm/Gd ratio. The Raman shift of Si-O-Si bending vibration modes induced by β-irradiation in Sm-,Gd-codoped glasses revealed the same non-linear character along with RE concentration as a function of the RE mixing. Finally, the point defect produced during ionizing radiation at one integrated dose follows also a non linear behavior with the relative proportion of Sm and Gd inside the glass. Therefore, it looks apparent that both the glass structure and the structural modifications in βirradiated glass are affected by the Sm/Gd-codoping. Besides, changes of EPR lineshape attributed to the BOHC defects with addition of Sm ions confirmed that Gd 3+ ions are preferably in a closed environment from boron ions.. The present work further enriches the efforts being made in the development of reliable model for interplay of reduction mechanisms and structural changes taking place under irradiation in rare earth doped aluminoborosilicate glasses. is for 0.17 and 0.34 mol.%, respectively. The systematic error was less than 0.005 mol. % of each component used for glass preparation.

Figure 1 .

 1 Figure 1. Evolution of EPR spectra of 0.34 mol.% Sm-, Gd-codoped aluminoborosilicate

Figure

  Figure 2a and 2b. Relative amount of Gd 3+ ions located in the network former position as a

Figure

  Figure3aand 3b. Changes in EPR spectra of (a) 0.17 mol.% (b) 0.34 mol.% Sm-, Gd-

Figure 4 .

 4 Figure 4. Defect concentration as a function of Sm content for 105 and 109 Gy -line is the

Figure 5 .

 5 Figure 5. Raman spectra of 0.34 mol.% Sm-,Gd-codoped aluminoborosilicate glasses for

Figure

  Figure 6a and 6b. ΔRaman shift of Si-O-Si bending vibration modes for (a) 0.17 and 0.34

  Figure 1

Table 1

 1

		1SGs	1SGa	1SGb	1SGc	2SGs	2SGa	2SGb	2SGc
		(1SGg)				(2SGg)
	x	1.00	0.25	0.50	0.75
	y					1.00	0.25	0.50	0.75
	SiO 2	59.77				59.67
	Al 2 O 3	3.84				3.83
	B 2 O 3	22.41				22.38
	Na 2 O	12.12				12.09
	ZrO 2	1.70				1.69
	Sm 2 O 3	0.17	0.045	0.087	0.131	0.34	0.084	0.174	0.263
	(Gd 2 O 3 )		(0.126)	(0.084)	(0.042)		(0.253)	(0.167)	(0.081)

Table 1 :

 1 Nominal compositions of 5 oxide aluminoborosilicate glasses (in mol.%) The first line indicates glass names beginning by 1SG for 0.17 mol. % of mixture Sm 2 O 3 +Gd 2 O 3 series and by 2SG for 0.34 mol. % of mixture of Sm 2 O 3 +Gd 2 O 3 series. Second and third lines correspond to molar ratio: x, y = Sm 2 O 3 /(Sm 2 O 3 +Gd 2 O 3 ) where Sm 2 O 3 +Gd 2 O 3

Acknowledgements

We are grateful to Thierry Pouthier and Stephane Esnouf for their contribution during external β irradiation experiments. We also thank Gilles Montagnac and Bruno Reynard (Laboratoire des Sciences de la Terre, ENS Lyon, France) for assistance in Raman experiments.


Yiqin Gao
email: yiqin.gao@ens-lyon.fr

Guillaume Pallez
email: guillaume.pallez@inria.fr

Yves Robert
email: yves.robert@ens-lyon.fr

Frédéric Vivien
email: frederic.vivien@inria.fr

Work-in-Progress: Evaluating Task Dropping Strategies for Overloaded Real-Time Systems

Keywords: time system, overloaded system, task dropping

This paper discusses evaluation criteria and scheduling strategies for the analysis of overloaded real-time systems. This work builds upon techniques from queueing theory and proposes a new approach for real-time systems.

I. INTRODUCTION

Problem statement: This work deals with overloaded real-time systems. Specifically, we consider a single periodic task to be processed by a server: task instance number i (or job i), i ≥ 1, is released at time r i = T × (i -1) and placed into the waiting queue; it must complete not later than time d i = r i +D, where T is the period and D the relative arbitrary deadline (we have D ≥ T ). More precisely, if job i is not completed by its deadline d i , it is considered as failed, and any time spent by the server to execute part of it has been wasted. We further assume that job execution times are stochastic and obey some probability distribution D.

The goal is to optimize some metric. There are many important criteria to guide the scheduling strategies, some user-oriented, and some platform-oriented. We review several of them in Section III. Our major contribution is to use dynamic strategies that take decisions on the fly, and that decide to execute a job, or not, based upon the current system state. These strategies are described in Section IV.

Motivation: Overloaded systems have become the norm in scientific computing. With the advent of parallel computing for many societal advances, traditional computing systems are under a much higher load than was typically the case a few years ago. High-Performance Computing (HPC) systems run with a utilization of almost 100% and a job waiting time of days [START_REF] Patel | Job characteristics on large-scale systems: long-term analysis, quantification, and implications[END_REF].

In order to cope with this increase in submissions, several approaches are considered. In Cloud computing, the typical approach is to have dynamic pricing, with prices increasing as the resources become more scarce [START_REF] Yeo | A taxonomy of market-based resource management systems for utility-driven cluster computing[END_REF]. In HPC, high waiting times are usually a deterrent with important negative consequences, such as limiting the number of very large jobs on HPC platforms [START_REF] Patel | Job characteristics on large-scale systems: long-term analysis, quantification, and implications[END_REF].

In this work, we focus on real-time systems but borrow the idea from HPC or Cloud computing that when not all jobs can be executed, the server can refuse (or drop) some jobs.

A key difference with many existing approaches is that we allow the server to take dynamic decisions, i.e., to use up-todate information about the system state to make the decision of accepting or dropping a job. Such information includes much more than the number of jobs in the waiting queue; e.g., when the server becomes ready, it can accept a job based upon the time spent since its release or upon the time left before its deadline, or both.

Contributions: We review several optimization criteria and discuss new parametrized algorithmic strategies which select which jobs should be dropped and when. We provide ways to compute the optimal value of the parameters for these strategies.

II. RELATED WORK

Real-time applications: Real-time jobs are released periodically and must complete successfully before a fixed timeinterval called the deadline. In the literature, real-time systems are classified as hard or soft [START_REF] Bernat | Weakly hard real-time systems[END_REF]. For hard real-time jobs, no deadline should be missed: it is mandatory that each job completes before its deadline. On the contrary, for soft realtime jobs, some deadlines can be missed. There are two scenarios then: either a job that missed its deadline must still be completed, or it can be ignored. The latter case is referred to as firm real-time jobs: firm jobs are allowed to miss their deadline, and there is no value to complete them after their deadline. Applications of firm real-time jobs include multimedia applications, satellite-based tracking, financial forecast systems, and robotic assembly lines [START_REF] Kopetz | Real-Time Systems: Design Principles for Distributed Embedded Applications[END_REF], [START_REF] Lin | Diverse soft real-time processing in an integrated system[END_REF].

In the classical setting with firm real-time applications, several periodic tasks are input to the system. Each periodic task is composed of instances (or jobs) that are released with a given period and deadline. The objective is to maximize the number of jobs that successfully complete before their deadline. There are many variations: for instance, some tasks may have two different types, skippable or not [START_REF] Marchand | Dynamic scheduling of periodic skippable tasks in an overloaded real-time system[END_REF], and only skippable jobs are allowed to miss their deadline. Our problem is similar to scheduling overloaded systems that allow skips [START_REF] Koren | Skip-over: algorithms and complexity for overloaded systems that allow skips[END_REF]. However, contrarily to some related work we do not impose that at least m of k consecutive jobs must meet their deadlines [START_REF] Hamdaoui | A dynamic priority assignment technique for streams with (m, k)-firm deadlines[END_REF], [START_REF] West | Analysis of a window-constrained scheduler for real-time and best-effort packet streams[END_REF] and we do not allow to modify a task period [START_REF] Buttazzo | Elastic task model for adaptive rate control[END_REF].

In this work, we revisit the problem with firm real-time applications in a new framework. We deal with a single periodic task and assume that all jobs are skippable, which simplifies the scheduling, but we do not assume that the set of jobs admit a finite worst-case execution time (WCET), which dramatically complicates it. We assume that the job execution times obey some known probability distribution D, and we provide experiments with a wide range of standard distributions. Dealing with such stochastic execution times introduces new challenges: If the support of the distribution D is unbounded, some jobs may execute for an arbitrarily long duration, thereby putting the following jobs at risk. However, if we decide to interrupt a job to launch a new one, the time already spent to execute it is lost, and there is no guarantee that the new job will complete faster than the interrupted one.

This job model assumes that some jobs may not be executed in the end. In fact, there are three cases: (i) some jobs are launched and reach completion before their deadline, meaning that they are successfully executed; (ii) some jobs are launched, but they are interrupted before completion, either because their deadline is missed or the scheduler has interrupted them, meaning that their execution has failed; and (iii) some jobs are not launched at all.

Queuing networks: In this work, we deal with the list of waiting jobs via a queue: while the machine is occupied, all submitted jobs are waiting for their turn (or waiting to be killed) in a queue. It strongly differs from what is done in queueing theory systems [START_REF] Harchol-Balter | Open problems in queueing theory inspired by datacenter computing[END_REF] where the typical constraint is to select in which order jobs should be executed to optimize an objective such as response time. Furthermore, job execution times obey the same probability distribution; to select which jobs we execute next we use a simple First-Come-First-Served strategy amongst the jobs that have not been dropped.

The closer to our work is the job dropping model [START_REF] Chydzinski | Queues with the dropping function and non-poisson arrivals[END_REF], where upon arrival of a job, the system selects, at admission, whether the job should be dropped (i.e., should be rejected). The decision to drop a job often depends on a function of queue parameters (number of jobs in the queue, average load of the queue): linear function (average queue size) [START_REF] Floyd | Random early detection gateways for congestion avoidance[END_REF] or other more complicated functions [START_REF] Feng | Congestion control scheme performance analysis based on nonlinear red[END_REF], [START_REF] Rosolen | A RED discard strategy for ATM networks and its performance evaluation with TCP/IP traffic[END_REF]. Then, all jobs from the queue are executed (for example following a FCFS strategy). Contrarily to the job dropping model, we propose to drop jobs dynamically.

Job pruning in heterogeneous computing systems: In a series of publications [START_REF] Denninnart | Efficient task pruning mechanism to improve robustness of heterogeneous computing systems[END_REF], [START_REF] Denninnart | Improving robustness of heterogeneous serverless computing systems via probabilistic task pruning[END_REF], [START_REF] Gentry | Robust dynamic resource allocation via probabilistic task pruning in heterogeneous computing systems[END_REF], [START_REF] Mokhtari | Autonomous task dropping mechanism to achieve robustness in heterogeneous computing systems[END_REF], [START_REF] Salehi | Stochastic-based robust dynamic resource allocation for independent tasks in a heterogeneous computing system[END_REF], job pruning techniques have been investigated. The authors consider an oversubscribed system to which jobs are submitted at random times. When a job is released, it is at first stored in a batch queue, and then can be allocated to the machine queue of a processor by the mapping process. At each mapping event (completion oor arrival of a job), the success probability of all jobs in the machine queue is recomputed, via a costly convolution over all possible durations of the jobs in the queue weighted by their respective probabilities. Jobs in the batch queue are also considered when there remains a processor with available positions after the previous computation. Jobs with low probability to meet their deadline are dropped from the machine queue and deferred in the batch queue. Deferring jobs means that their assignment to a processor is postponed. In contrast, a job dropped from the machine queue is definitely removed from the system. Contrarily to our problem, there are several job types, several processor types, and job arrival dates are random rather than periodic. This calls for a very costly solution where a whole convolution over a large time window must be recomputed at each mapping event. Some jobs can be dropped after some duration, a strategy also investigated in this work. The striking difference is that with a single job type and periodic releases, we are able to determine the optimal value of the key scheduling parameters once and for all and to apply them on the fly, thereby providing a schedule whose cost is constant and independent of the number of jobs.

III. OPTIMIZATION CRITERIA

There are two natural important optimization criteria:

Deadline miss ratio [START_REF] Canon | Scheduling independent stochastic tasks under deadline and budget constraints[END_REF], [START_REF] Díaz | Stochastic analysis of periodic real-time systems[END_REF]: This is a frequently used metric in soft real-time systems where one is only looking for probabilistic guarantee that the deadline miss ratio of a task is below a threshold. Note that this metric is directly related to the task execution rate metric, which corresponds to the expected number of jobs executed per time unit.

Utilization [START_REF] Patel | Job characteristics on large-scale systems: long-term analysis, quantification, and implications[END_REF]: This corresponds to the proportion of time spent doing computation that ends up in a success (i.e., the completion of a job before its deadline).

In addition, we study the following two criteria: Mean response time of successfully completed jobs [START_REF] Díaz | Stochastic analysis of periodic real-time systems[END_REF], [START_REF] Patel | Job characteristics on large-scale systems: long-term analysis, quantification, and implications[END_REF]: This corresponds to the delay between the release of the job and its completion. When deadlines are enforced, this criterion is less important but does provide qualitative information about the various solutions.

Mean rejection time: This new criterion is important for an overloaded system. Intuitively, it corresponds to the time it takes for the system to inform a user that its job will not be executed. Of course, this criterion must be coupled with another, otherwise the best strategy would be to reject all jobs at submission time.

IV. SCHEDULING STRATEGIES

In all strategies, we execute the jobs admitted in the waiting queue in an Earliest-Deadline-First (EDF) fashion. Because all jobs have the same relative deadlines, this is equivalent to a First-Come-First-Served (FCFS) strategy.

Admission policies: As discussed in Section II, existing strategies in queueing networks focus on an admission policy: at submission, the scheduler decides whether the job is admitted in the queue or whether it is rejected by the system. This has the advantage of providing almost minimal rejection time. Such dropping mechanisms usually consider the size of the queue as input [START_REF] Chydzinski | Queues with the dropping function and non-poisson arrivals[END_REF]. As baseline strategies, we admit the next job only if the queue contains strictly less than k jobs, where k = 1 or k = 2. However, when jobs have stochastic nature and are presented with deadlines, we expect such solutions to have sub-optimal performance.

Dynamic dropping mechanisms:

The main focus of this work is on strategies with no admission policy (all jobs go in the queue) but with dropping mechanisms. We dynamically drop jobs if conditions are not met. There are three natural approaches:

• S-MAX: Drop the job if it has not started s max units of time after its release; • L-MAX: Drop a running job after l max units of time;

• D-MAX: Drop an unfinished job (running or not) d max units of time after its release. Due to lack of space, we only study the performance of the S-MAX strategy in this paper. Our main result relies on a Markov model to characterize the best parameter values: Theorem 1. Given a periodic inter-arrival time of jobs whose execution time follow a discrete probability distribution and with identical relative deadlines, we can compute in polynomial time the optimal value for the parameter s max to maximize either the deadline miss ratio, or the utilization of the system. Proof Sketch. We define a quantum duration q, and discretize time into quanta: a period T has T q quanta, and the server has S = D-T q + 1 possible states when job i arrives: it can be available 0, 1, ..., D-T q quanta after the release of the job. Thus, given a parameter set (T , D and D), we construct a Markov chain with S states, and we compute its transition probability matrix. We show that this Markov chain is irreducible and aperiodic, and compute its limit stationary distribution. Using this limit distribution and the success probability of a job after any given quantum, we determine the asymptotic expected probability that a task is executed successfully. This calculation is done in O(S 3 ) steps. The best value s max is computed either by trying all values (adding a factor S to the complexity) or via a binary search (adding a factor log(S) to the complexity).

V. EVALUATION

Framework: In order to evaluate the strategies, we propose a simple framework: we use standard probability distribution functions (PDF) to generate job execution times, namely Uniform, Exponential, and LogNormal. All distributions have a mean execution time normalized to 1. Jobs are released periodically with a period τ ∈ [0.1, 2], and we have a relative deadline D = 4T . We measure the average performance via simulation over the execution of 1,000,000 jobs.

We compare six strategies: admission policy with k = 1 or k = 2 (QUEUEOFONE and QUEUEOFTWO), random admission (RANDOMADMISSION) where the admission ratio is equal to the period (hence, a randomly admitted load of 1), the strategy that admits all jobs and never interrupts any (NEVERKILL), and the two S-MAX strategies that admit all jobs and use the optimal value of s max either for the deadline miss ratio (BESTSMAXFORMISSRATIO) or for system utilization (BESTSMAXFORUTILIZATION). Rejection time

Period

Analysis of the results: We sum up some high-level observations from the results displayed on Figure 1: • The S-MAX dynamic dropping policy is extremely efficient for utilization under heavy load. This is not surprising as it was designed for this objective. However, S-MAX seems to be quite dominant for all metrics for the distributions under study. More analysis should be done to understand if this is a bias due to the selection of the distributions, and find the limits of its performance. • The only metric for which S-MAX does not dominate is rejection time. It still has good performance which is correlated to the high-utilization. To have a high utilization, one needs to have few running jobs dropped before they reach their deadline. Hence, the only jobs dropped are those that have not started (this is less true for the Exponential distribution). This balances the fact that the S-MAX policy does not automatically drop jobs at submission. More analysis should be done here to study if one can improve this. In particular, we observe that the RANDOMADMISSION policy has a better rejection time under heavy workloads. This is coherent and hints that we can probably do better by coupling those two algorithms.

• With an overloaded system and deadlines, an admission policy with k = 1 is usually more efficient than with k = 2, which is consistent with the fact that there is no job shortage. In accordance, the NEVERKILL algorithm (corresponding to the behavior of an unbounded queue, or k = ∞) has poor performance. About the overall performance:

• Deadlines have an important impact on utilization: when the system is overloaded, without deadlines one could expect an utilization of 1 since there is always work to be done. With the best heuristic, we get a utilization close to 1 only with a deadline greater than 2 (corresponding to T ≥ 0.5). • The high rejection time for underloaded systems correspond to jobs dropped exactly at their deadlines: by construction the deadline increases with the period. This is inevitable because we use unbounded distributions. However, looking at the deadline miss ratio, we see that there are very few jobs that are dropped.

VI. FUTURE WORK

This work has discussed admission and dynamic dropping policies to deal with overloaded service systems. As a future step, algorithmically we plan to study the importance of various parameters for dynamic dropping (l max , d max ). Then we believe that combining a posteriori mechanisms with a priori mechanisms will lead to significant improvements. Indeed, with deadlines, one of the main drawbacks of mechanisms that admit jobs depending on the size of the queue is the following: when a job arrives while another is running, the new job is queued or dropped if the queue is full. Because of deadlines, the longer the wait of a job in the queue, the less likely its success. An alternative approach would be: when a new job arrives and the queue is full, drop the oldest job from the queue and replace it by the newest job. Of course, this comes with a cost for user-oriented objectives.

Fig. 1 .

 1 Fig. 1. Performance of the different heuristics.

Acknowledgments: The work of Yiqin Gao was supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).

We thank the reviewers for their valuable comments.